3.4.2 Reducing the HHR Scheme to a Store and Forward Queuing Network. ... 3.4.3 Optimal Allocation of Per-Hop Reliabilities for HHR Scheme .
ALGORITHMS FOR RESOURCE UTILIZATION IN SENSOR NETWORKS by BUDHADITYA DEB A Dissertation submitted to the Graduate School-New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Computer Science Written under the direction of Prof. Badri Nath and approved by ________________________ ________________________ ________________________ ________________________ New Brunswick, New Jersey [Month, Year]
Abstract ALGORITHMS FOR RESOURCE UTILIZATION IN SENSOR NETWORKS By Budhaditya Deb Dissertation Director: Prof. Badri Nath Wireless sensor networks are being envisioned for various civil, military and environmental monitoring applications. Since they are highly resource constrained, algorithms and protocols are designed to minimize resources. Minimizing the resources consumed for different protocols usually leads to degradation in the performance provided in terms of the application requirements. For mission critical applications, we sometimes need to provide guarantees even at the cost of extra overhead. Thus, efficient and intelligent use of resources rather than minimization of resource consumption should be the major focus in sensor network research. This is the primary theme of this thesis. We divide different protocols under two major categories, namely sensor data dissemination and network management and propose adaptive algorithms with the aim of maximizing resource utilization in sensor networks. For the data dissemination problem, we show that providing different levels of assurances (reliability, latency etc.) in data delivery based on the information content in the sensed data is critical for proper utilization of resources. For the network management problem we consider two important functions: network state retrieval and topology control. We show that retrieving the network state at different levels of resolution and adaptively controlling the topology based on the current network conditions, application characteristics leads to a more prudent approach while conserving resources. I.e. we shift the algorithm design paradigm from an unconstrained optimization problem (the fixed schemes which aggressively minimize resource consumption) to optimization under application constraints. We propose adaptive algorithms for both the data dissemination and the network management problems which exploit the fundamental tradeoffs between resource consumption and protocol performance. The algorithms are not only more efficient in the resources they consume, but more importantly are also aware of the requirements of the applications they are designed for.
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Acknowledgements
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Table of Contents
Abstract ........................................................................................................................................ 1 Chapter 1 Introduction ............................................................................................................ 1 1.1 Main Contributions.......................................................................................................................2 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5
1.2
Analytical Models for Sensor Networks .................................................................................................. 2 Information Aware Data Dissemination in Sensor Networks .................................................................. 2 Multi-Resolution State Retrieval.............................................................................................................. 3 Adaptive Topology Control for Network Power Conservation................................................................ 3 Minimum Virtual Dominating Sets (MVDS)........................................................................................... 4
Organization of the Thesis............................................................................................................5
Chapter 2 Background and Related Work ............................................................................ 6 2.1 Ad Hoc Sensor Networks .............................................................................................................6 2.2 Information Aware Data Dissemination .......................................................................................8 2.2.1 2.2.2 2.2.3
2.3
Connected Dominating Sets and its Applications.......................................................................16 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5
2.4
Topology of Wireless Networks using Unit Disc Graphs ...................................................................... 16 Definitions ............................................................................................................................................. 17 Applications of CDS in Sensor and Ad-Hoc Networks ......................................................................... 18 Algorithms for Computing CDS in Ad Hoc Networks .......................................................................... 20 CDS at Variable Cardinalities and Density Distributions ...................................................................... 21
Sensor Network Management and State Retrieval .....................................................................22 2.4.1 2.4.2 2.4.3 2.4.4
2.5
Information Assurance Metrics and Mechanisms .................................................................................... 8 Schemes for Reliable Information Forwarding ...................................................................................... 11 Related Work in Reliable Information Forwarding................................................................................ 13
Sensor Network States ........................................................................................................................... 23 Sensor Network Management Functions ............................................................................................... 24 Basic Topology Discovery Approaches................................................................................................. 24 Related work in Sensor Network State Retrieval ................................................................................... 27
Topology Control for Ad-Hoc and Sensor Networks .................................................................28 2.5.1 2.5.2 2.5.3
Topology Control using Transmission Power Control........................................................................... 30 Topology Control Using Node Scheduling ............................................................................................ 31 Computing Average Paths in Unit Disc Graphs..................................................................................... 32
Chapter 3 Information Aware Data Dissemination in Sensor Networks.......................... 34 3.1 Introduction ................................................................................................................................34 3.2 Network Model...........................................................................................................................37 3.3 Information Assurance with End-to-End Retransmissions (EER)..............................................39 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5
3.4
Information Assurance with Hop-by-Hop Retransmissions (HHR) ...........................................46 3.4.1 3.4.2 3.4.3 3.4.4 3.4.5 3.4.6
3.5
HHR under Memory Constraints ........................................................................................................... 47 Reducing the HHR Scheme to a Store and Forward Queuing Network................................................. 48 Optimal Allocation of Per-Hop Reliabilities for HHR Scheme ............................................................. 49 Distributed HHR Delivery ..................................................................................................................... 50 The Blocking Probabilities at each hop for the HHR Scheme ............................................................... 51 HHR Discussion .................................................................................................................................... 51
Information Assurance with Hop-By-Hop Broadcast (HHB).....................................................53 3.5.1 3.5.2 3.5.3 3.5.4
3.6
Overhead of EER Delivery .................................................................................................................... 39 EER scheme under Memory Constraints ............................................................................................... 40 Distributed EER Delivery ...................................................................................................................... 42 Buffer Allocation for Minimum Overhead ............................................................................................ 43 EER Discussion ..................................................................................................................................... 43
Operations while Forwarding a Packet using HHB ............................................................................... 55 Operations while Receiving a Packet in HHB ....................................................................................... 55 Overhead of HHB .................................................................................................................................. 55 Optimal Allocation of per hop reliabilities for HHB.............................................................................. 56
Comparing the EER, HHR and HHB Schemes ..........................................................................57
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3.6.1 3.6.2 3.6.3 3.6.4
3.7 3.8
Comparing the Overhead of HHR and EER Schemes ........................................................................... 58 Comparing the HHR and HHB Schemes ............................................................................................... 58 Comparing the Latency in Data Delivery for EER and HHR ................................................................ 59 Network Scalability and lifetime ........................................................................................................... 59
Summary ....................................................................................................................................61 Appendix ....................................................................................................................................62 3.8.1 3.8.2 3.8.3
Buffer Allocation for Minimum Overhead ............................................................................................ 62 Optimal Allocation of per-hop Reliabilities in the HHR Scheme .......................................................... 63 Latency in Data Delivery ....................................................................................................................... 65
Chapter 4 Minimum Virtual Dominating Sets (MVDS) ....................................................... 67 4.1 Introduction ................................................................................................................................67 4.1.1 4.1.2
4.2 4.3
Definitions ..................................................................................................................................69 Distributed algorithm for the Construction of MVCDS .............................................................70 4.3.1 4.3.2
4.4
Node Coloring Algorithm ...................................................................................................................... 72 Timers in Algorithm for MVCDS.......................................................................................................... 74
Analyses of VCDS Algorithm ....................................................................................................79 4.4.1 4.4.2 4.4.3 4.4.4
4.5
MVDS for Multi-Resolution State Retrieval.......................................................................................... 67 MVDS for Node Scheduling.................................................................................................................. 68
Worst Case performance bounds of VCDS............................................................................................ 79 Average Case Analysis .......................................................................................................................... 81 Request Propagation and Black Node Tree............................................................................................ 84 Adapting the algorithm to Handle Node Mobility ................................................................................. 85
Summary ....................................................................................................................................86
Chapter 5 Multi-Resolution Network State Retrieval using MVDS ................................... 87 5.1 Introduction ................................................................................................................................87 5.1.1 5.1.2
5.2
Sensor Topology Retrieval at Multiple Resolutions (STREAM) ...............................................91 5.2.1 5.2.2 5.2.3
5.3
Node Constrained Query:..................................................................................................................... 101 Edge Constrained Query: ..................................................................................................................... 102 Overhead Constrained Query............................................................................................................... 104
STREAM Under Arbitrary Network Conditions......................................................................105 5.5.1 5.5.2 5.5.3 5.5.4
5.6
Retrieved Topology Resolution ............................................................................................................. 94 Overhead for retrieving topology at different resolutions ...................................................................... 96
Mapping Topology Retrieval Queries to STREAM Parameters.................................................99 5.4.1 5.4.2 5.4.3
5.5
STREAM Request Phase ....................................................................................................................... 92 STREAM Response Phase..................................................................................................................... 93 Acknowledgement Timer....................................................................................................................... 93
Analysis of STREAM.................................................................................................................94 5.3.1 5.3.2
5.4
Multi-Resolution Topology Retrieval .................................................................................................... 87 The Network Model............................................................................................................................... 90
Non-Uniform Topology ....................................................................................................................... 105 Impact of Sleeping Nodes .................................................................................................................... 107 Handling Channel Errors ..................................................................................................................... 108 Handling Node Failures. ...................................................................................................................... 111
Summary ..................................................................................................................................113
Chapter 6 Topology Control Using Node Scheduling with MVDS ................................. 114 6.1 Introduction ..............................................................................................................................114 6.2 Topology Control Using Node Scheduling ..............................................................................114 6.3 Network Model and Assumptions ............................................................................................116 6.4 Impact of density on Shortest Path lengths...............................................................................117 6.5 Effects of Increased hop lengths on reliable delivery Overhead...............................................120 6.5.1 6.5.2
6.6
End-To-End Retransmission ................................................................................................................ 120 Hop-By-Hop Retransmission............................................................................................................... 122
Effect of Reduced network degree at low density ....................................................................122 6.6.1 6.6.2 6.6.3
Hop-by-Hop Broadcast Reliability (HHB)........................................................................................... 123 Expected number of Next hop Neighbors ............................................................................................ 123 Overhead of HHB ................................................................................................................................ 124
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6.7 6.8
Increased traffic load in memory constrained nodes ................................................................126 Algorithms for Node scheduling ..............................................................................................129 6.8.1 6.8.2
6.9
Constructing Poisson Distributed Active Node Set (Voronoi Aggregative Processes) ........................ 130 Scheduling with Virtual Connected Dominating Sets.......................................................................... 131
Distributed Computation of Density Conditions ......................................................................132 6.9.1 6.9.2
6.10 6.11
Handling non-uniform distribution of nodes........................................................................................ 133 Computing the local density distribution for node scheduling............................................................. 133
Simulation Results....................................................................................................................135 Summary ..................................................................................................................................137
Chapter 7 Future Research Directions.............................................................................. 138 7.1 Short Term Research Objectives ..............................................................................................138 7.2 Long Term Research Objectives ..............................................................................................139 7.2.1 7.2.2
Chapter 8
Managing Sensor Networks as a Single Entity .................................................................................... 139 Information Awareness in Sensor Networks........................................................................................ 140
Thesis Summary ................................................................................................ 142
References ............................................................................................................................... 143
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Chapter 1 Introduction Sensor networks are being envisioned for various civil, military, industrial and environmental monitoring applications. A collection of cheap and redundantly available sensor nodes with sensing, computing and wireless communication capabilities forming an adhoc network, has emerged as the prime candidate architecture for these applications. Another design philosophy is that sensor nodes should have low price and small form factor for ubiquitous operability and ad-hoc deployment of these networks. This limits the amount of resources such as battery and memory that can be put in sensor nodes. The different characteristics and constraints in wireless sensor networks pose many new challenges. As a researcher in sensor networks, it is essential to find out whether sensor networks are really so different that the vast experience from traditional networks cannot simply be reused An important contribution in my research has been to identify the unique characteristics and challenges, which makes sensor networks significantly different from traditional wireless networks. Based on these, we formulate three guidelines which form the recurring themes and motivations behind the research work. First, the proposed algorithms need to exploit the fundamental tradeoffs to maximize resource utilization. Second, the highly dynamic nature of network conditions and requirements makes it necessary for algorithms to be adaptive (or tunable) at a very fundamental level to work efficiently under different conditions. Third, the information centric nature of sensor networks dictates the design of network protocols. Sensor networks research has mainly looked at the problem of minimizing the resources consumed for different protocols primarily because the sensor nodes are resource constrained. However, reducing the resource consumption usually comes at the cost of degradation in services provided by the network. Algorithms need to gracefully tradeoff between resources consumed and levels of services provided. We divide these different algorithms under two categories namely data dissemination and network management. The thesis highlights the need for shifting the primary design goals for these algorithms
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from resource minimization to the intelligent utilization of resources and proposes adaptive algorithms for these purposes. For data dissemination, we exploit the tradeoff between reliability of data delivery (based on the information content) and overhead. For network state retrieval, we exploit the tradeoff between the resolution level of retrieved state information and the overhead. Finally for topology control, we exploit the tradeoff between the number of nodes kept active for routing and the overhead of reliable packet delivery (whose performance degrades with lesser active nodes).
1.1 1.1.1
Main Contributions Analytical Models for Sensor Networks
Design of future protocols for sensor networks require a good understanding of the network behavior with characteristics specific to sensor networks. (e.g. high network density, many-to-one traffic, high channel errors, low memory, limited energy) in nodes. Based on these factors we derive analytical models for the network graph (e.g. path lengths under different density conditions), traffic behavior (queuing theoretic models) etc. and answer fundamental questions on network lifetime, capacity and protocol design. These models form the basis for solving many different networking problems such as efficient resource allocation, network management, data dissemination and topology control in my thesis For example the queuing theoretic models are used in designing efficient data dissemination protocols and the path length behavior is used in the topology control problem. Accurate modeling and analysis of also revealed a highly non-intuitive nature of sensor networks and put forth many deficiencies in common perceptions and assumptions (when trivially extending concepts from traditional networks). Thus the models can serve as theoretical guidelines for future research in these areas.
1.1.2
Information Aware Data Dissemination in Sensor Networks
We identified that sensor networks need to provide different assurance levels to the sensed data based on their information content. Thus information assurance emerged as an essential network functionality to cater to the information centric nature of data and to exploit the fundamental tradeoffs between
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network resources expended and assurance levels provided. We identified reliability as the most important assurance metric and proposed different protocols for reliable information delivery based on End-to-End retransmissions, Hop-by-Hop Retransmissions (HHR) and Hop-by-hop Broadcast (HHB, a generalization of the HHR scheme which exploits the broadcast redundancy in wireless channel to reduce the overhead). Each of the protocols intelligently exploits some network functionality and provides novel solutions to the resource utilization problem in data dissemination. To compare and analyze these schemes, we derive queuing theoretic models. One of the most important results to emerge out of the analysis was that HHR based schemes may have higher overhead and lower network capacity and lifetime than EER schemes for characteristics specific to sensor network. This is in stark contrast to the notion of HHR being better suited for all wireless networks.
1.1.3
Multi-Resolution State Retrieval
Knowledge of the current network state is required for different network management functions. Due to large scale and density, the network can consume a lot of energy to retrieve these states. On the other hand, large scale also makes it feasible to infer statistically relevant state information from relatively lowresolution state data. The level of detail required is dependent on the network property being inferred or the network management operation. To this end, we describe a distributed parameterized algorithm for Sensor Topology Retrieval at Multiple Resolutions (STREAM), which makes a tradeoff between topology details and resources expended. The underlying idea in STREAM is to select a subset of nodes which forms a good representative spatial sample of the state according to the density distribution of the state under observation. The algorithm is also generalized to efficiently extract other network states such as energy dissipation and network lifetime by selecting different parameters for different queries. By retrieving the state at the resolution required and at proportionate cost, the algorithm achieves the prime objective of resource utilization for network state retrieval.
1.1.4
Adaptive Topology Control for Network Power Conservation
Topology control is an important mechanism in network management for resource conservation. In this thesis, we concentrate on the node-scheduling approach of topology control for reliable data
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dissemination in sensor networks. In dense ad hoc networks, node scheduling usually means maintaining the network at the lowest possible density while maintaining connectivity. Our analysis reveals that low network density adversely affects the overhead of packet transmissions due to significant increase in average path lengths, network congestion and buffer overflows, and reduction in path redundancy. When packets need to be reliably delivered the above increases the overhead by a large factor. Thus, aggressively minimizing the density is actually counter-productive. We develop probabilistic models to capture the relation between network density and network power consumption and derive optimal density conditions for minimum overhead. We also propose a distributed and message optimal algorithm which takes into account the application characteristics (e.g. traffic volume or whether data dissemination is reliable or not) and the current network conditions (e.g. channel errors, buffer capacity in nodes etc.) for scheduling nodes at the optimal density. For proper resource utilization, thus we need to maintain the network at different density distributions based on current conditions as opposed to the fixed scheme of maintaining only at the minimum density and thus again motivates the importance of intelligent resource utilization as the design paradigm in sensor networks.
1.1.5
Minimum Virtual Dominating Sets (MVDS)
We see that both the multi-resolution state retrieval and topology control problems require nodes to be selected at different density distributions based on the requirements or the network conditions. We proposed the concept of MVDS as a generalization of the normal dominating set concept capable of generating dominating sets at different density distributions. We also provide a distributed message optimal algorithm for creating MVDS with the best-known approximation bounds at constant message complexity. We extensively use the concept of MVDS for retrieving network state and for scheduling nodes and literally forms the backbone of the proposed algorithms. Moreover, since dominating sets are perhaps the most important graph theoretic structures used for different purposes in adhoc and sensor networks, the concept of MVDS and the proposed algorithm are important contributions for future research work in many different areas.
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1.2
Organization of the Thesis
We begin in Chapter 2 with a brief background and related research work on the topics covered in this thesis. In Chapter 3 we look at the problem of adaptive data dissemination in sensor networks. As described earlier, the novel concept of Minimum Virtual Dominating Sets is used in solving the network state retrieval and topology control problem. Hence we first introduce MVDS concept and our proposed algorithm to create MVDS in a sensor network in Chapter 4. We then look at multi-resolution network state retrieval problem in Chapter 5 and propose an algorithm called STREAM to retrieve network topology using the MVDS concept. Chapter 6 analyzes the node scheduling approach of topology control to minimize the power consumption for reliable packet delivery in sensor networks. Again we propose an algorithm based on the MVDS concept for adaptive node scheduling. In Chapter 7 we provide a brief outline on the future research direction and we conclude the thesis in Chapter 8.
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Chapter 2 Background and Related Work The thesis covers a broad range of issues in ad hoc and sensor networks. It is not possible to cover the entire spectrum of research work which is directly or indirectly related to the topics addressed in the thesis. In this chapter we briefly highlight the relevant topics and provide the background for the concepts, which are most closely associated with the particular problems being solved. We also provide references which deal with some of the topics in greater detail.
2.1
Ad Hoc Sensor Networks
Recent advances in MEMS technology have resulted in cheap and portable devices with formidable sensing, computing and wireless communication capabilities ([41], [43], [56], [104]). A network of these devices could be invaluable for automated information gathering and distributed micro-sensing in many civil, military and industrial applications. The use of wireless medium for communication provides a flexible means of deploying these nodes at some inhospitable terrain without any fixed infrastructure. Once deployed, they should require minimal external support for their functioning. Wireless sensor networks pose many new challenges because they are significantly different from traditional networks. We briefly outline some of the major differences: •
Resource Constraints: Sensor nodes have limited processing power, memory, communication capability and energy. Thus, the protocols operations have to be resource-efficient.
•
Modes of Operation: Sensor nodes operate in various modes to optimize their resource consumption. For example, nodes may periodically sleep.
•
Frequent failures: In a sensor network nodes may frequently fail. This is because sensor nodes may be deployed in hostile terrain or may die off due to energy dissipation. Thus protocols have to be designed keeping such failures as the norm rather than as exceptions.
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•
High Density and Scale: Sensor networks are envisioned to be dense networks comprised of thousands of tiny sensor nodes. Managing individual nodes in such networks is infeasible, inefficient and impractical. Thus protocols not only need to be scalable in terms of high density and scale but also need to have mechanisms to efficiently handle the large scale since the nodes are resource constrained.
•
Different Traffic Characteristics: Sensor networks would have many-to-one traffic characteristics with most packets being transmitted between the sensor nodes and a central monitoring node which collects data with virtually no inter node communication. This is significantly different from traditional networks where the communication is more evenly distributed.
•
Communication vs. Computation Trade-off: The energy consumed in performing computation at a sensor node is significantly lower than that in communication. Thus, energy constraints of a sensor node mandate the use of computation wherever it reduces communications. However, limited memory and processing power at the nodes, do not allow an arbitrarily large reduction in communication by performing computation.
•
Uncontrollable Environment: The performance of sensor networks depends on the surrounding environmental conditions. The network administrator has no control over the environment in which nodes are deployed. Thus, operating parameters have to be set dynamically to account for unforeseen circumstances.
•
Information centric network: Sensor networks are deployed to monitor environmental phenomenon, events etc. The sensed data would have different levels of criticality for different applications. Since sensor networks are resource constrained this fact could be used for resource utilization by conserving resources for unimportant data so that resources can be used for disseminating critical data with high levels of reliability and low delay. This concept is applicable in traditional networks but is more important for sensor networks because of their resource constraints.
•
Human Intervention: Conventional networks provide the capability to manually administer individual nodes (e.g., rebooting a malfunctioning node). Sensor networks may be deployed in inhospitable territories where no such intervention is possible and need to be managed remotely.
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•
Network is an entity: Traditional networks are comprised as a collection of network elements which collectively form a system to achieve their individual goals. A sensor network on the other hand behaves as a single entity with one collective goal. A fundamental problem in sensor network management would be to manage the network as a whole giving less importance to the individual node behavior. A global management scheme might also save resources since managing nodes on an individual basis will incur a huge overhead. The above characteristics in sensor networks makes them significantly different from traditional
networks. It not only introduces new challenges but also highlight that the typical networking problems need novel solutions under the unique characteristics of sensor networks. We next discuss the specific problems that were addressed in this thesis.
2.2
Information Aware Data Dissemination
Sensor networks are deployed to monitor the surroundings and keep the end-user informed about the events witnessed. Different types of events have different levels of importance for the user. Information Assurance is an ability to disseminate different information at different assurance levels to the end-user. The assurance level is determined by the criticality of the sensed phenomenon. Thus, information assurance capability allows a sensor network to deliver critical information with high assurance, potentially at a higher cost, while saving energy by delivering less important information at a lower assurance level. We look at this problem id detail in Chapter 3. In this section we give provide a brief survey of related work in this context.
2.2.1
Information Assurance Metrics and Mechanisms
We fist look at the various metrics for information assurance. Based on the criticality of a sensed event, a packet needs guarantees or assurances on the following: •
Reaching Probability: When a packet is sent from a source to sink separated by multiple hops, the probability of the packet reaching the sink successfully is low due to wireless link errors along the path. To increase the probability of packet delivery protocols typically retransmit lost packets. In the high speed networks, the reliability is usually provided through the use of end-to-end retransmissions
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using TCP. On the other hand for wireless networks, a hop-by-hop retransmission scheme is preferred. In sensor networks we are posed with two problems. First we need to design protocols which are adaptive in terms of the reliability i.e. have mechanisms to deliver packets at different levels of reliability. Second we have to reevaluate the typical retransmissions schemes in the context of sensor network. We do this in detail in Chapter 3. •
Latency: A high priority packet should reach the monitor with a lower latency than a low priority packet. This would mean that a low priority packet could take a longer route to make way for higher priority packets. The packets would thus be allowed to deviate from the shortest route path with the deviation bounded by the required latency in delivery. Latency would be dependent on channel error along with propagation delay. For example if a scheme provides reliable delivery through retransmissions and acknowledgments, expected number of retransmission would depend on the channel error rate and the expected latency would thus depend on number of retransmissions. Since channel errors could be high, the effect of retransmissions on the latency could be significant. This is different in data networks, which have low channel error rates, and latency typically depends on the propagation delay and queuing delay along the path.
•
Resolution of Delivered Data: Delivering data at various levels of granularity based on the information content can save valuable resources for the network. For example a critical event could be delivered to the monitoring node with high levels of details whereas a normal event could be sent at a low resolution to save resources for the network. Further we can define the resolution based on the dynamic nature of events. For example we can control the Temporal Resolution of data or the Spatial Resolution of data based characteristics of the sensed phenomenon. Next we look at how different levels of assurances can be guaranteed in sensor networks. We compare
against the well known schemes from traditional networks and discuss the issues when using them for sensor networks. The problem of information assurance is similar to priority based forwarding for service differentiation in traditional data networks. For normal data networks, the Integrated Services (IntServ, [86]) and Differentiated Services (DiffServ, [95]) models exist to provide service differentiation. The IntServ
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model gives special treatment to packets from a given flow. Each router stores per-flow states and maintains a token bucket for the flow to forward its packets. In the DiffServ model no guarantees are provided to flows. Instead, the edge routers mark a per-hop-behavior (PHB) in the packet header. The core routers use the PHB in their packet scheduling decision. Packets from different flows having same PHB are treated identically. Effectively, DiffServ provides service on a per-aggregate basis whereas IntServ provides service on a per-flow basis. We now discuss the implications of using the two concepts for Information assurance in sensor networks.
2.2.1.1
The IntServ Model for Information Assurance
For sensor networks, the concept of a flow can have two interpretations namely, Node-Centric and Data Centric. The data-centric interpretation would be to treat one type of data as one flow, e.g., information from all temperature sensors from a particular region is one flow. This kind of interpretation is used in protocols like directed diffusion ([26]), which use soft-state mechanism to periodically refresh interest in a particular flow. Each sensor node would cache the interests which are being routed through it. An interest would expire once the monitor node stops refreshing it. The node-centric interpretation would treat the stream of packets between a source and the monitor as a flow. Applying an IntServ-like model to sensor networks would require maintenance of per-flow states at each sensor node. Given the limited amount of memory at a sensor and the large number of sensors in a field, keeping states for node-centric flows is not feasible for sensor networks. With data-centric flows, the IntServ model might be feasible only if the number of interests in a sensor field is small. Hence, providing special service to packets from a particular flow is only feasible for data-centric flows with limited number of interests. In such a model, the monitor node could enclose the desired service for a particular interest in the interest refresh-message. For example, when issuing a refresh message for a particular interest i, the monitor node might enclose a desired level of service saying: “If a packet with priority p, is received for an interest i, forward it with probability f(p)”. Note that the update interval parameter specified in the interest refresh message of directed diffusion, provides a mechanism to have more frequent updates for more important interests, but it does not allow differentiation between packets of different importance belonging to the same interest.
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2.2.1.2
DiffServ Model
The DiffServ concept can be more easily adapted for sensor networks. The source knows the degree of importance of each packet it is sending which can translated into predefined priority levels. The other sensors would use the packet priority to decide the kind of service it should provide to the packet. This type of model is highly scalable because no matter what the number of sensors in a field, a forwarding node has just to contend with one packet at any given point of time.
2.2.2
Schemes for Reliable Information Forwarding
In this thesis we only consider the problem of delivering packets with reliability based on the information content. In this section we outline some basic schemes for reliable information delivery that can be used for sensor networks.
2.2.2.1
Classification Based on Redundancy Schemes
When packets are lost due to channel errors, extra redundancy can be introduced to increase the reliability of packet delivery. Different schemes for redundancy can be classified as follows: •
Packet Redundancy: In this approach for providing desired reliability, multiple copies of a packet may be transmitted from each node which receives a copy of the packet. By introducing such redundancy, the system can compensate for packet losses due to the local channel error. This approach incurs extra overhead in the number of packets. However it does not require retransmissions and acknowledgments due to which latency in packet delivery would be low. Hence this approach would be better if the latency in packet delivery is the more important factor rather than communication over-head.
•
Forward Error Correcting (FEC) Codes: FEC codes maybe used for correcting the corrupted packets. Effectiveness of FEC would depend on the degree of redundancy in codes. Thus we can increase reliability by introducing a stronger FEC for a high priority packet based on local channel error.
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2.2.2.2
Classification Based on Forwarding Schemes
Introducing extra redundancy can increase the reliability of packet delivery. When a packet has to travel multiple hops, different schemes could be used for forwarding the packets. •
Multi-packet Forwarding: In this approach, only the designated next hop node forwards a packet. The redundancy is introduced by forwarding the packet multiple times. Again a node may use the local channel error rate and the required reliability to calculate the number of copies of a packet that needs to be forwarded. Using totally local information may not allow a node to perform at a desired level. Moreover, since this approach uses single path only, a hot spot in the network would translate into extra load on a subset of nodes.
•
Multi-path Forwarding: This approach, takes advantage of wireless being a broadcast medium. All the neighbors of a forwarding node can listen to the transmissions. Although there would be a single next hop node based on the routing algorithm used, multiple nodes may actually forward a packet. A probability function based on the local channel error rate, desired reliability and node degree may be used by the receiving nodes to decide whether it should forward a packet. Thus in the lowest priority packets only a single node needs to forward whereas in the highest priority case the node forwarding may degenerate into a network flooding. To get multi-path forwarding approach to work, it is desirable to have a forwarding probability function which works independent of topology.
•
Probabilistic Flooding: Forwarding packets in unicast method leads to a single path forwarding with low reliability. Network wide flooding using packet broadcast leads to high probability in packet delivery but at a much higher cost. By choosing a probabilistic flooding scheme, (where packets are broadcasted with some probability) we can control the reliability rate of packet delivery along with the overhead.
2.2.2.3
Classification based on Retransmission Schemes
Instead of sending multiple copies of packets, the network can deliver a packet reliably by retransmissions of lost packets. In retransmission schemes, a destination node usually replies with an
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acknowledgement for a lost packet. The source node waits for the acknowledgement for a time period and retransmits a packet. Retransmissions with acknowledgements can significantly reduce the overhead for reliable packet delivery. We have analyzed three different schemes of retransmissions in Chapter 3. These are described in brief as follows: •
Reliability with End-To End Retransmissions (EER): In ad hoc networks source and sink may be multiple hops away from each other. In EER, the source waits for an acknowledgement from the destination and retransmits a packet. Here the source has to wait long enough for a packet to reach the destination across multiple hops and the acknowledgement to return back. Thus this method can have high delays. However the advantage is that intermediate nodes do not need to cache the packets.
•
Reliability with Hop-By Hop Retransmissions (HHR): In this method, a packet is reliably delivered on a hop by hop basis as it travels multiple hops from the sink. Thus at each hop, the packet is retransmitted until reliably delivered to the next hop. The next hop then starts transmitting the packet reliably to its next hop and so on. Using this method, the overhead can reduce as compared to the EER scheme. However this has more protocol complexity and packets need to be cached at each of the intermediate hops until they are successfully transmitted. In sensor networks with high memory constraints this aspect may significantly affect the choice of HHR over EER.
•
Reliability with Hop-By-Hop Broadcast (HHB): This is a generalization of the HHR concept which utilizes the wireless broadcast. When a packet is transmitted in the wireless medium, all neighbors can potentially receive the packet, automatically creating multiple copies from a single transmission. We show in Chapter 3, section 3.5 that this can be exploited to reduce the overhead of retransmission for reliable packet delivery.
2.2.3
Related Work in Reliable Information Forwarding
In traditional networks the protocols consider reliability as a binary metric: best effort and reliable. In best effort delivery no additional redundancy or retransmissions are introduced to increase the probability of packet delivery. On the other hand in reliable delivery, the number of retransmissions or redundancy is usually designed to ensure that a packet reaches the destination with probability of one.
13
Reliable delivery with such binary notion is a well-researched topic in both traditional networks and sensor networks. In [4], authors present a comparison of end-to-end and hop-by-hop schemes for reliable packet delivery. In [93] authors analyze different reliable transport protocols for ad-hoc networks and propose algorithms to find the minimum energy paths for reliable delivery. Reliable transport protocols for sensor networks (again with reliability as an absolute metric) have appeared in references [4], [28], [40] and [93]. Such a binary definition of reliability does not adaptively trade off reliability and overhead. To solve this problem we introduced the concept of assigning packet priorities based on the information content for sensor data in [20]. We consider the reliability of packet delivery as a metric for QoS in sensor networks and put forth the problem as a differentiated services paradigm for sensor networks. Similar service differentiation concept for sensor networks also appears in [80] which assumed events belonging to two classes as critical and non-critical and routed critical packets along less congested part of the network to provide lower delay guarantees. Reference [115] describes the same problem as eventto-sink reliability. The scheme described in [115] is based on ideas of feedback network to deliver at a required reliability similar to our scheme in [20]. We propose mechanisms for Reliable Information delivery using the EER, HHR and HHB schemes ([19]) and [21]) schemes. A hybrid approach has also been considered (RMST [40]) for sensor networks although it considers reliability as a binary metric and hence not useful in our study. As described in the previous section, probabilistic flooding can be a mechanism to achieve multi-path forwarding. The idea of probabilistic flooding has been extensively used in ad-hoc networks [105], [116], [118]. A notion of probabilistic reliability is also given in [58] to achieve reliable multi-cast in ad-hoc networks using gossip-based algorithms. Having proposed schemes for reliable information delivery in sensor networks, we turn our attention to analyzing these protocols and selecting the optimal scheme for sensor networks. The choice of the suitable scheme for reliable delivery depends upon the network parameters and application requirements. EER schemes are generally preferred for the high speed Internet with highly reliable links to maximize throughput and maintain end-to-end semantics [4]. On the other hand, for wireless multi hop networks with high channel error rates, HHR schemes are preferred. Thus for sensor networks, which are viewed as
14
a special case of multi-hop wireless networks the HHR schemes are believed to be better suited for reliable delivery ([40]). However under severe memory constraints and many-to-one traffic characteristics (as will be frequently encountered in sensor networks) the network behaves in a completely different manner. We see in Chapter 3 that in fact EER schemes might be better suited in terms of overhead, capacity and lifetime of the network. This is because end-to-end schemes in general have lesser protocol complexity and timeout functions and require lesser buffer space at intermediate hops to store undelivered packets. HHR schemes on the other hand have higher memory requirements since packets need to be buffered at each hop. The many-to-one traffic has been considered as the normal traffic behavior in sensor networks ([34], [35], [115]). Reference [34] studies the many-to-one capacity in sensor networks. References [35] and [115] on the other hand look at reliable data delivery for many-to-one traffic. Many-to-one traffic behavior leads to non-uniform traffic density based on the distance of a node from the single sink (or the source) with the nodes in immediate neighborhood of the sink becoming the bottleneck in the network since they have to support the largest traffic volume. However in contrast to [34] we show that bottlenecks are not necessarily formed at the first hops (they may be pushed back to the middle) when the nodes have high memory constraints. Finally we refer to papers [38] and [83] to derive the queuing model for our system. An important contribution in the analysis in Chapter 3 is the derivation of the queuing network under many-to-one traffic behavior and data requiring multiple levels of reliability. Previous work has only considered a uniform traffic behavior and a binary definition of reliability and do not consider the many-to-one traffic model in the network. Different levels of reliability can also be provided using adaptive forwarding error control schemes ([8], [31], [1], [8], [37]). However these require complicated dedicated hardware or computation which might not be suitable for current configuration of sensor nodes. Reference [34] exploits a similar tradeoff between reliability and efficiency in source coding. Reliability in sensor networks has also been looked at in the context of fault tolerance in routing paths between source and sink. In sensor networks, nodes may frequently fail leading to reduced reliability of
15
Communication Range
Figure 2.1 Topology of Wireless networks modeled using unit disc graphs paths from a source and sink pair. In [32], multiple paths from source to sink are used in diffusion routing ([26]) framework to quickly recover from path failure. The multiple paths provided by such protocols could be used for sending the multiple copies of each packet. However it incurs extra overhead of multiple path formation and maintenance of path state in each node and is not adaptive to channel errors. Similarly for mobile ad-hoc networks different multi-path extensions to well known routing algorithms have been proposed [9], [74], [75]. However, the main purpose of these multi-path routing schemes is to increase the robustness, by quickly recovering from broken paths. They are also not adaptive to local channel errors.
2.3
Connected Dominating Sets and its Applications
Connected dominating sets and its variants are perhaps the most important graph theoretic structs that are used in ad hoc and sensor networks and many algorithms and protocols use connected dominating sets for efficient functioning. In this section, we give an introduction to this concept, discuss briefly the applications and provide a brief literature survey of algorithms for creating connected dominating sets in ad hoc networks.
2.3.1
Topology of Wireless Networks using Unit Disc Graphs
We first look at how the network topology is represented for wireless sensor networks. A network is usually represented as a graph consisting of nodes (the network elements) and edges (the links connecting the network elements). In wireless network with no physical links, the network graph is usually modeled using disc graphs [14]. In this section we discuss briefly the modeling of wireless networks based on disc graphs.
16
A disc graph is a special case of geometric graphs whose edges are determined by the positions and the communication ranges of the nodes. Here, given the position of nodes, an edge between two nodes exist if they are within a communicating distance from each other (given by the communication range). This is illustrated in Figure 2.1. When this distance the normalized communication range (i.e. equal to one) the graph is called a unit disc graph. The communication range by definition is the distance till which a node can receive messages from other nodes using the wireless medium. Since wireless signal is assumed to propagate symmetrically along all directions, the range is assumed to be circular around the center of the transmitting node. Thus given a set of nodes with their wireless communication range and the position of nodes, a wireless network can be depicted by considering the edges formed using the definition presented above. If {V(x,y)} is the set of nodes then the communication range determines the set of edges and together form the unit disc graph G{V(x,y), E(r) }.
2.3.2
Definitions
Consider an ad-hoc wireless network represented by a unit disc graph G{V(x,y), E(r) } with V the set of vertices and E(r) the set of edges. •
Minimum Dominating Set (MDS): A subset of vertices V’Õ V such that each of the vertices in set V-V’ (the rest of the nodes) are adjacent to some node in V’, forms the dominating set on the graph. A Minimum Dominating Set (MDS) is the minimum cardinality set which is dominating. Finding a minimum-sized dominating set or MDS is NP-Hard [76].
•
Minimum Connected Dominating Set (MCDS): A connected subset of nodes which is also a dominating set on the graph. Thus the CDS is the set which is connected among itself and covers all other nodes in the network. A Minimum Connected Dominating Set (MCDS) is the minimum cardinality set which is dominating. MCDS is also NP-Hard [76].
•
Weakly Connected Dominated Set (WCDS): A dominating set V’ such that it induces a connected subgraph of G. In other words, the subgraph weakly induced by V’ is the graph induced by the vertex
17
d
d
f e
a
f e
a
b
b
g c
g
c
Figure 2.2 A Dominating Set
Figure 2.3 A Connected Dominating Set
set containing V’ and its neighbors. Given a connected graph G, all of the dominating sets of G are weakly connected. Computing a minimum WCDS is NP-Hard [76]. •
Maximal Independent Set: A subset of V’ such that no two vertices within the set are adjacent in V’. Maximal Independent Set (MIS), is an independent set such that adding any vertex not in the set breaks the independence property of the set. The subset V’ is called the dominating set, and the subset V-V’ is called the dominated set. When the
network topology is wireless, the DS and the CDS are computed on the corresponding unit disc graph for the network. Finding the MDS and the MCDS for a unit disc graph is also NP-Hard ([14], [44], [77]). Figure 2.2 illustrates the DS (the nodes b and f). Note that nodes b, f cover all other nodes. However the nodes b and f are not connected in the subgraph formed by the DS and its neighbors. Figure 2.3 illustrates the CDS (nodes b, e, f) on the same simple wireless network. In this case we note that the subgraph formed with nodes b, e, f is connected.
2.3.3
Applications of CDS in Sensor and Ad-Hoc Networks
Connected Dominating Sets are one of the most important graph theoretic structures used in Ad Hoc and Sensor Networks. Reference [51] provides a detailed survey on CDS applications in Ad Hoc and Sensor Networks. In this section we briefly discuss some of the applications that are relevant in this thesis. •
Routing using Virtual Backbones: Wireless ad-hoc networks do not have a physical backbone like traditional wired/wireless IP networks for routing purposes. Connected Dominating sets can be
18
constructed over the network forming a virtual backbone. Since a CDS forms a connected set of nodes and covers every node in the network, packets can be routed on top of such a virtual backbone. Routing over CDS is usually referred to as Spine Routing or Backbone routing. It is extremely useful in Ad-Hoc networks since it reduces the routing update overhead, as the cardinality of the CDS is usually much lower than the actual network graph. References [7], [16], [73], [85], [94], [97] and [98]are some well known algorithms using this concept. •
Spatial Reuse for Quality of Service: Creating a CDS structure in the network can lead to higher spatial reuse of the wireless channel. A dominating node with its set of dominated nodes can form clusters and use orthogonal spread spectrum codes in the neighborhood as described in [87]. This can lead to higher throughput and lesser collisions for transmitted packets.
•
Efficient Broadcast and Mutlicast: In wireless network, when messages are broadcasted or multicasted, nodes often hear the same message multiple times. This problem also known as the broadcast storm problem [105] can be greatly reduced by using a CDS for broadcasting messages ([46], [45]).
•
Network State Retrieval: Ad hoc and sensor networks require the current network state for network management purposes. For a large-scale network, retrieving network state can consume a lot of resources for the network. Using a dominating set, we can select a subset of nodes in the network to retrieve the partial state of the network. Since the dominating set covers all other nodes (i.e., all dominated nodes are neighbors of this set) approximate cached state about the neighbors could also be retrieved. This problem has been studied for the topology retrieval problem in [72]. We propose a Multi-Resolution state retrieval algorithm based on concepts of dominating sets in Chapter 5.
•
Node Scheduling for power conservation: Ad hoc networks are infrastructureless and run on battery-powered nodes. Thus it may be difficult and expensive to replace or recharge nodes drained of energy. Minimizing the network power consumption is not only important for maximizing the network lifetime but also for improving the maintainability of the network. CDS structure provides an efficient means of conserving power since a network can be maintained by keeping the minimal set of nodes required for connectivity. By switching off redundant nodes (the non-CDS nodes are
19
not required for forwarding packets) valuable network power can be saved. This is usually referred to as node scheduling ([13], [36], [63]). We look at the node scheduling problem in Chapter 6.
2.3.4
Algorithms for Computing CDS in Ad Hoc Networks
As noted earlier, the problem of finding the MCDS is NP-Hard due to which we have to use approximation algorithm. The quality of the approximate CDS formed is usually determined by the approximation ratio. The usability of different algorithms for CDS construction is determined by the message complexity and whether the algorithms are distributed or not. Finding the MCDS for general graphs was first shown to be NP-Hard by reduction from the set cover problem [76]. The MCDS for unit disc graphs is also NP-Hard ([14], [44], [77]). However it was shown in [112] that the approximation ratio of CDS for unit disc graphs can be constant up to any factor based on the time complexity of the algorithm. In [97] and [98] two centralized algorithms appeared for finding connected dominating sets for unit disc graphs with an approximation ratio of 2(1 + H(D)), where H is the harmonic function and D is the maximum degree. Many algorithms appearing later in Ad hoc network research ([84], [112], [110], [111]) are based on the algorithms described in [97] and [98]. For ad-hoc and sensor networks, it is important that algorithms for CDS construction be distributed due to the lack of any centralized management authority and global knowledge of the network state. In [15] and [16] the fully distributed implementations of algorithms proposed in [97] and [98] appeared for the first time. Since then a lot of other distributed algorithms for constructing connected dominating sets have emerged ([50], [62] [68], [84], [89], [108]). In Chapter 4, we improve upon these approaches by proposing a distributed algorithm for constructing approximate MCDS. To the best of our knowledge, our algorithm for node scheduling has the best known approximation ratio at linear message complexity (8opt-2, n). Table 1 compares the approximation ratios of different distributed algorithms for CDS construction with our algorithm described in Chapter 4.
20
[15],[89],[108]
[62]
[50]
[84]
[68]
Our Algorithm
Approx. Factor
Q (log n)
n
n/2, n
§8
192opt-48
§8
Msg. Complexity
O(n2)
Q (m)
O(n2)
O(nlogn)
O(n)
O(n)
Time Complexity
O(n2)
O(d3)
W(n)
O(n)
O(n)
O(n2)
Table 1 Comparing the Approximation Bounds of Different Algorithms for Finding Connected Dominating Sets. m=the number of edges and d= highest degree The constant approximation bound algorithms usually are based on a two-phase construction: in the first phase an MIS is created and in the second phase the MIS is connected using a second set of nodes to form a CDS. Our algorithm is also based on the MIS construction along with the algorithms described in [68] and [84]. However we improve the approximation factor over these algorithms while keeping the message complexity linear.
2.3.5
CDS at Variable Cardinalities and Density Distributions
Research in ad hoc networks has mostly looked at algorithms for minimum connected dominating sets. In many situations however, a minimum connected set of nodes is not the ideal structure required for various applications. In our thesis we encounter two cases where connected sets are required at different cardinalities and density distributions. In Chapter 5 to retrieve the sensor network state we need connected sets at various cardinalities. In Chapter 6 we need to schedule nodes at different density distributions using CDS at multiple cardinalities for minimum power consumption. This is because the cardinality and the density distribution of the selected node set plays a crucial role in the both the resolution of the retrieved state and the incurred power consumption in the network. To get CDS at multiple density distributions, we propose a novel concept know called Minimum Virtual Connected Dominating Sets (MVCDS) in Chapter 4. MVCDS is a generalization of the normal MCDS concept and is shown to be very effective in selecting dominating sets at various density distributions. It uses a parameter called as virtual radius which acts as a control knob for density.
21
A parameterized approximation algorithm to create an approximate MCDS appears in [39]. However the parameter is used to control the optimization bound and is not useful for controlling the network density. A concept similar to the virtual radius used in MVCDS is used in [69] to form clusters given a specified radius of coverage by cluster heads. The algorithm has an approximation bound of O(n1/2) with message complexity of O(n2) for a two dimensional network. However, both [39] and [69] do not provide any density distribution results which might have proved useful while applying the methods for node scheduling and multi-resolution state retrieval. Finally, the concept of MVCDS is comparable to the concept of Weakly Connected Dominating Sets (WCDS) [61]. WCDS is a relaxed formulation of the MCDS problem creating a larger cardinality CDS than a normal MCDS. Compared to all the above, our algorithm based on the concept of MVCDS is ideal for selecting multi-cardinality connected node sets since it has the best approximation bounds along with the ability to accurately control the network density.
2.4
Sensor Network Management and State Retrieval
In section 2.1 we discussed the fundamental differences between traditional network and sensor networks. These differences motivate researchers to look at the problem of Sensor Network Management from a completely different perspective. In this thesis we consider two important functions in sensor networks management: network state retrieval and topology control. Before we go into these specific aspects, we discuss some general issues related to sensor network management and provide a brief overview of related work with respect to network state retrieval. The background on topology control is discussed in the next section. Although management of traditional networks is a well-researched topic, research in Sensor Network Management is currently in its nascent stage. The network management problem is one of defining a set of metrics and functions by which to keep the network at accepted levels of performance. Since there are very few currently deployed and operated sensor networks, the requirements and issues involved are at most times, ill defined. Authors have tried to address the problem by defining some generalized guidelines, models and functions. The first such effort appears in our paper on topology discovery (an essential part of sensor state retrieval) in [17]. It describes a simple topology discovery algorithm along
22
with an architecture for sensor network management. In the book wireless sensor networks, [27] similar concepts are covered on network management issues. Reference [78] discusses efficient monitoring and management of sensor networks states. In [60] authors describe a comprehensive architecture for sensor network management called as the MANNA architecture. Next, we briefly discuss some of the issues related to network management that were raised in [17].
2.4.1
Sensor Network States
One of the primary challenges in managing sensor networks would be to view and infer the current state of sensor networks. As discussed in the previous section, the state retrieval problem is unique since we have to view the network as a single entity. We need to not only look at statistical inference techniques to derive large-scale behavior (which is used even for traditional networks), but also infer the complex collective behavior from interaction of nodes at a local scale. We now list some of the required models for sensors sensor network management that were described in [17] •
Network Topology: This describes the current connectivity/reachability map of the network and could assist routing operations and in future deployment of nodes.
•
Energy Map: This gives the energy levels of the nodes at different parts of the network. The spatial and temporal energy gradient of the network nodes may also be modeled. Coupled with network topology, this could be used to identify “weak areas” of the network.
•
Usage Pattern: This describes the network activity in terms of periods of activity for nodes, amount of data transmitted per unit time and tracking of hot spots in the network.
•
Cost Model: This represents the network in terms of equipment cost, energy cost, and human cost for maintaining the network at desired performance level.
•
Statistical Models: In large scale and dense sensor networks it is not practical to retrieve the exact state of the network. Moreover we are more interested in the global behavior of the network as a whole rather than have information about individual nodes. Thus, statistical models and metrics for evaluating different network states could prove to be much more suitable for sensor networks.
23
2.4.2
Sensor Network Management Functions
Based on the current network state, different network management functions need to be carried out to maintain the network at the expected performance levels. We list some of these functions. •
Deployment of sensors: Typically sensors would be deployed at random with no prior knowledge of the terrain. Future deployment of sensors would depend upon the present state of the network.
•
Setting Network Operating Parameters: This involves setting up of routing tables, node duty cycles, timeout values of various events, position estimation etc. These differ from normal monitoring operations because many protocols may require accurate information of the above for their optimal performance.
•
Monitor Network States using Network Models: Take periodic measurements to obtain various states like network connectivity, energy map etc.
•
Network Maintenance: By monitoring the network, regions of low network performance could be traced with reasons for such performance could be identified. Corrective measure like deployment of new sensors or directing network traffic around those regions could be useful. This is Reactive Network Maintenance.
•
Predict Future Network States: From periodic measurement of network states it could determine the dynamic behavior of the network and predict future state. This could be useful for predicting network failures and preventive action could be taken. This is Proactive Network Maintenance.
•
Design of Sensor Networks: The models on Cost Factor and Usage Patterns could be used for design of sensor network architectures.
2.4.3
Basic Topology Discovery Approaches
In Chapter 5 we look at the problem of retrieving the state of a sensor network to infer its properties and performance. We specifically select the topology of the network and propose an algorithm to retrieve the sensor topology at multiple resolutions. Next we briefly discuss some trivial algorithms for topology retrieval.
24
Figure 2.4 Example illustrating difference between topology discovery approaches •
A monitoring node requiring the topology of the network initiates a "topology discovery request".
•
This request diverges throughout the network reaching all active nodes.
•
A response action is set up which converges back to the initiating node with the topology information. We assume that the request divergence is through controlled flooding so that each node forwards
exactly one "topology discovery request". Note that each node should send out at least one packet for other nodes to know its existence. This also ensures that all nodes receive a packet if they are connected. However various methods may be employed for the response action. The different mechanisms described, differ only in this aspect and affect the overall efficiency of the process. For illustrating the response action of these methods we consider the network in Figure 2.4 with node a as the initiating node. The topology discovery request reaches node b from a and nodes c and d from node b. Requests are forwarded only once.
2.4.3.1
Direct Response
In the first approach when a node receives a topology discovery request it forwards this message and sends back a response to the node from which it received the request. The response action for the nodes in Figure 2.4 is: •
Node b replies back to node a.
•
Node c replies to node b; node b forwards the reply to node a.
25
•
Node d replies to node b; node b forwards the reply to node a.
•
Node a gets the complete topology We note that even though parent nodes can hear the children while they forward a request (for example
a can know about b when it forwards), this is not useful as its neighborhood information is incomplete. Hence an exclusive response packet is needed for sending the neighborhood information.
2.4.3.2
Aggregated Response
We next solve the problem of extra communication in the previous section by aggregating the responses. Here, all nodes wait for the children nodes to respond before sending their own responses. After forwarding a topology discovery request, a node gets to know its neighborhood list and children nodes by listening to the communication channel. Thus the number of communications is reduced. The response action for the nodes in Figure 2.4 is: •
Node c and d forward request; node b listens to these and deduces them to be its children.
•
Node c replies back to node b; Node d replies back to node b.
•
Node b aggregates information from c, d and itself; node b forwards the reply to node a.
•
Node a gets the complete topology
2.4.3.3
Clustered Response:
In this approach the network is first divided into set of clusters. Each cluster is represented by one cluster head with every non-cluster-head node being in range of at least one cluster head. The response action is generated only by the cluster heads, which send information about nodes in its neighborhood. Similar to aggregated response, cluster heads can aggregate information from other cluster heads before sending response. The response action for the nodes in Figure 2.4 is: •
Assume that node b is a cluster head and nodes c and d are part of its cluster.
•
Node c and d do not reply.
26
•
Only node b replies to node a.
•
Node a does not get link c↔ d. We note that whereas direct and aggregated response methods give complete network topology,
clustered response provides only a partial topology. However we can get more details if a larger number of nodes decide to respond back i.e., by controlling the cardinality of the responding set we could extract different levels of topology details. This forms the underlying theme behind the multi-resolution topology retrieval algorithm, STREAM described in Chapter 5.
2.4.4
Related work in Sensor Network State Retrieval
Researchers have proposed different mechanisms for topology discovery of data networks in [6], [23], [113], [81] primarily using probing techniques. In [3] given a set of network endpoints, end-to-end Bayesian probing is used to infer properties of IP networks using correlations. Probing techniques are not very efficient since they consume a lot of resources. Further, many nodes in sensor networks could be sleeping with the network being partially disconnected. A common topology discovery technique is based on aggregating routing table information from routers. Using routing tables is not feasible in sensor networks because traditional routing tables may not be available if data centric model of routing is used in [26]. Moreover, in ad-hoc deployments, routing tables are often inaccurate or incomplete. In [88] authors propose a mobile agent based framework to distribute topology information. Here, the optimal number of agents required is half the number of nodes. The overhead with this approach would not scale for sensor networks where the number of nodes is large. Reducing the agents would increase the expected time that any agent takes to reach the initiating node. References [13], [36], [63], [107] and [117] describe routing in ad hoc networks using minimal connected dominating sets. Using each of the above methods, a simple topology discovery algorithm can be designed to query the dominating nodes, which provide their neighborhood lists. However, our algorithm differs significantly from the above in its multi-resolution nature. Hence, the topology returned
27
is not limited to the minimal backbone. In fact, the topology that each of the above can provide is only a special case of lowest resolution topology recovered by our algorithm. In [32] a conceptual framework called DIMENSIONS was introduced for multi-resolution data-access in sensor networks. STREAM provides a framework to spatially sample a sensor network to select a subset of representative nodes for the network at a given resolution. The algorithm proposed in [32] focus compression of data at each node using spatial and temporal correlations and maybe used in conjunction with STREAM after using STREAM to first select the subset. STREAM provides the network topology at different resolutions where the main problem is to represent the network graph correctly (i.e. for inferring topological properties within specified error bounds) using as few edges as possible. A centralized scheme for representing topology with sparse graphs is presented in [12]. Similarly, graph compression schemes for representing dense graphs in an efficient manner (e.g. as described in [5]) could be used for retrieving multi-resolution topology. Designing distributed algorithms for topology retrieval based on the above ideas is part of future work. Apart from network topology, research has also looked at retrieving the current residual energy of sensor networks ([65], [64], [66]). STREAM could be used as the underlying scheme for selecting a representative subset to retrieve the energy state while using the schemes described in [65], [64] and [66] for aggregating the information.
2.5
Topology Control for Ad-Hoc and Sensor Networks
In Ad Hoc Sensor Networks, minimizing the network power consumption is not only important for maximizing the network lifetime but also for improving the maintainability of the network. In this context topology control is one of the most important mechanisms used for reducing network power consumption. Recall that the node positions and the communication range of the wireless channel determines the wireless network topology. Topology control is the mechanism by which the naturally formed wireless topology is changed to suit the network conditions and application requirements. We broadly divide the different topology schemes as follows:
28
•
Transmission Power Control: A large communication range leads to higher transmission power consumption and also increases receive power consumption since wireless is a broadcast medium. A smaller transmission radius on the other hand can lead to a disconnected network or an increase in the number of hops between nodes. The optimization problem is to find the communication radius at which the power consumed is minimum. This problem has been studied in [42], [70], [91]. [93], [99].
•
Node Scheduling: Due to high density many nodes are redundant for routing purposes. By using a scheduling algorithm to turn nodes on and off, significant power can be saved. The optimization problem here is to find the set of nodes to be kept active to minimize power consumption. The problem has been studied in ([13], [15], [36], [63], [107], and [117]).
•
Channel Assignment: The wireless channel can be divided into multiple channels using different schemes such as Time Division Multiple Access (TDMA), Frequency Division Multiple Access (FDMA) and Spread Spectrum Code Division Multiple Access (CDMA). The channel assignment for different nodes determines the connectivity graph for the network. The channel assignment problem tries to assign non overlapping (or orthogonal) channels among neighbor nodes to minimize interference and collisions among simultaneous transmission of packets. Usually this falls under the category of optimal node coloring problem where nodes are colored such that no two neighbor nodes have the same color as described in [106].
Another approach is using dominating sets with
orthogonal spread spectrum codes in the neighborhood of each cluster as described in [87]. •
Changing Node Position:
By changing the position of nodes, we can change the network
connectivity to suit the application requirements. For example if the connectivity for a node is really weak, node can change locations to go closer to another node. Reference [2] proposes a scheme for deforming the network by changing the node positions to bound the delay and throughput of the network based on traffic characteristics. Among the above schemes, Transmission Power Control and Node Scheduling are the most important approaches for topology control in ad hoc and sensor networks in context of minimizing energy consumption. In this thesis we only concentrate on the node scheduling approach (Chapter 6). Here we give a brief overview of both the approaches.
29
2.5.1
Topology Control using Transmission Power Control
As described in the previous section in transmit power control algorithms try to find the transmission range which leads to a topology with minimum overhead. In wireless transmissions, the maximum power consumptions occurs when transmitting packets. Thus minimizing the transmission range can reduce the power consumption for data delivery. Moreover, with a smaller transmission range, the number of collisions also reduce since there is lesser overlap among the transmission ranges of neighborhood nodes. However transmission range needs to be maintained such that network remains connected. This idea was the motivation behind the work in [82] where the critical transmission range for connectivity was derived using probabilistic methods. Having a transmission range a little larger than the critical connectivity leads to a network with a high probability of connectedness while minimizing the power consumption from packet transmissions. However, decreasing the transmission range can lead to increase in overhead since it can lead to larger number of hops between nodes. This effect (increase in the average path lengths with reduced transmission range in a multi hop wireless network) was first studied in [47] and [70]. Based on their models, the optimal transmission range was derived. Similarly PAMAS, [103] proposed the idea that a node may choose a smaller number of hops using a longer range so as to reduce the overall power consumption. Research then turned the attention to reliable packet delivery where packets need to be retransmitted when lost in the noisy wireless channel. To minimize overhead for reliable data delivery, the problem formulation needs to include the overhead of retransmission as well while choosing the transmission range. This problem was studied in [93], and a mechanism was proposed to route along minimum energy paths using different communication ranges and to find the optimal paths for different reliability schemes. Finally the optimal transmission power when packets are delivered reliably using TCP over wireless links was studied in [48] and [71].
30
2.5.2
Topology Control Using Node Scheduling
In this thesis, we analyze the node-scheduling scheme for topology control in the context of reliable packet delivery in Chapter 6. Although a lot of work exists in this direction for transmission power control, this is the first time the problem has been looked in detail for the node scheduling approach. In node scheduling, instead of reducing the transmission power, we could actually shut down the radio circuits for many nodes which are redundant for routing purposes. This actually leads to reduction in power dissipated from idle listening which is also a significant proportion of the energy drain in ad hoc networks. Based on this intuition, node scheduling algorithms try to find the minimum set of nodes which need to be kept active for routing purposes. This is usually done with an approximation of Minimum Connected Dominating Set (MCDS). Routing over MCDS is usually referred to as Spine Routing or Backbone routing and [7], [16], [73], [85], [94], [97] are some well known algorithms using this concept. CDS reduces the routing overhead and minimizes the idle power dissipation in nodes. Switching off nodes can lead to a few nodes (the CDS) taking the load of the entire network and can get drained of energy. To solve this problem, reference [63] looks at load sharing and energy dissipation in nodes to reduce unequal depletion of energy in cluster heads of the CDS. Reference [52] looks at scheduling nodes in such a manner so as to balance the load of communication and sensing among different sensor nodes. Switching off nodes can also result in degradation of the network quality (in terms of capacity, delay, overhead, coverage etc.). Reference [36] tries to conserve the network coverage in sensor networks while minimizing the number of active nodes. Reference [53] uses a clustering scheme for switching off nodes while maintaining sensor coverage. Reference [13] proposes a localized algorithm called SPAN for node scheduling. Using simulations they show that the node-scheduled topology does not suffer too much in terms of latency and capacity of the network while reducing redundant power consumption.
31
2.5.3
Computing Average Paths in Unit Disc Graphs
One of the consequences of reducing the density by switching off nodes in node scheduling is the increase in path lengths. The accurate theoretical model of the path length distribution as a function of the network density is very important for analyzing the node scheduling approach. We derive the path length distribution as a function of network density in section 6.4. Since the approach in our analysis is similar to ones in [47] and [70] (but is more accurate under different network density conditions) we outline the derivation of average path lengths in [47] and [70] The node distribution in the network is assumed to be Poisson with intensity λ . In [47] and [70], it is assumed that when a packet travels from the source to the sink, it moves along the straight line joining the nodes. At each hop, the packet is forwarded to the node which is closest to the sink. The expected forward progress at each hop is used to compute the average number of hops Since the nodes are Poisson distributed, the expected progress, z in one hop by a packet for a given density is given by: 1 − λr 2 cos −1 (t )−t 2 z = r 1 + e −λπr − ∫ e −1
1−t 2
dt
The expected number of hops between any two nodes is approximated by dividing the distance between the two nodes by the expected progress z. To compute the average hops between all pairs of nodes, a second approximation is taken by considering the expected length between any two randomly chosen nodes in a circle of radius R (given by D (average) ) and dividing it by the progress z. Thus the average number of hops is given by the following equations. D ( average) =
H=
128 R 45π
128R 1 −δr 2 cos−1 (t )−t 2 45πr 1 + e −λπr − ∫ e −1
1−t 2
dt
As can be seen from the above, the analysis for average path lengths in [47] and [70] is rough approximation and fails when the network density is low. We improve upon the analysis with more accurate modeling of the paths in section 6.4. In our analysis we consider a radial distance of progress
32
towards the sink (which means all forms of trajectories between source and sink are considered in the probability distribution) whereas in [47] and [70] the progress is assumed along a straight line joining the source and sink. Also we consider the actual probability distribution of distances in bounded regions instead of the expected distance. Thus we get a more accurate estimate of the average hops.
33
Chapter 3 Information Aware Data Dissemination in Sensor Networks
3.1
Introduction
Sensor networks are being deployed to monitor physical environments to responding to situations such as forest fires, chemical leaks in industrial plants, and enemy infiltration in hostile terrains by disseminating the sensed data to the end user. In general, the sensed data would have different levels of importance and would require different levels of guarantees in delivery. For example, the information of a potential chemical leak is more important than knowing that everything is fine (which might be the norm) and should have higher reliability and lower delay in delivery. Moreover, there could be intermediate levels of leaks for which the messages could tolerate different levels of reliability and delay. We refer to any such guarantees in sensor data delivery as Assurance Levels where the assurance level for any data packet is determined by the information content that it carries. Information Assurance is the ability to disseminate different information at different assurance levels to the end-user. In this chapter, we look at the problem of reliable information dissemination in sensor networks, i.e., consider reliability in data delivery as the metric for Assurance levels. The existing solutions treat the process of sending information to the end-user in two extreme ways: unreliable transmission or reliable transmission. The example of monitoring chemical leak clearly illustrates that disseminated data has different levels of importance for the user. Thus, reliability should have multiple levels of reliability (and not be a boolean task) in the context of sensor networks. Moreover having multiple reliability levels can ensure proper resource utilization by exploiting tradeoffs in resources consumed and reliability achieved in data dissemination. In this chapter, we propose different schemes for reliable information delivery and then using queuing theoretic modeling (of conditions and characteristics specific to sensor networks) we
34
comparatively analyze these protocols to find the most suitable algorithm for reliable information delivery. Reliable data delivery protocols typically use either End-to-End (EER) or Hop-by-Hop Retransmissions (HHR). We propose extensions of the basic EER and HHR schemes to deliver data at various levels of reliability. We then propose a Hop-by-Hop Broadcast (HHB) scheme which exploits the broadcast capability of the wireless medium to further reduce the overhead of reliable data dissemination. Having described the different methods for reliable information delivery, we try to understand which schemes are suitable for sensor networks. The choice of the suitable scheme usually depends upon the network parameters and application requirements. EER schemes are generally preferred for the high speed Internet with highly reliable links to maximize throughput and maintain end-to-end semantics [4]. On the other hand, for wireless multi hop networks with high channel error rates, HHR schemes are preferred. Thus for sensor networks, which are viewed as a special case of multi-hop wireless networks the HHR schemes are believed to be better suited for reliable delivery ([28], [40]). However sensor networks are significantly different than normal wired and wireless data networks as was discussed in the previous chapter. The following characteristics become critical while analyzing the reliable data delivery protocols: •
Severe resource constraints: Energy, Memory, Computation
•
Higher link error rates: Sensor networks may be deployed in uncontrollable hostile terrain
•
Many-to-one traffic characteristics: In sensor networks communication is between sensor nodes and a central collection node (monitoring node) with limited inter-node traffic. In our analyses we try to answer the question that under the aforementioned conditions, whether the
existing knowledge about reliable delivery schemes is adequate. More specifically we want to qualitatively analyze how the different schemes behave in terms of delay, throughput, overhead and network lifetime under these conditions and how the behavior can impact the design of reliable data dissemination scheme in sensor networks.
35
Hth
2nd 1st
z
Monitorin
Figure 3.1: The network model used in this paper. The monitoring node is at the center and each hop is of width z. We consider a many-to-one network as (shown in Figure 3.1) with a monitoring node at the center of the network collecting data from sensor nodes multiple hops away from the monitoring node. We derive queuing theoretic models for the EER and HHR schemes with reliable information delivery (multiple levels of reliability classes for packets) and many-to-one traffic. The queuing network and its characteristics derived is an important and non-trivial contribution in this thesis. Based on the queuing theoretic models, we evaluate how the EER, HHR and HHB schemes perform in relation to the overhead, capacity and lifetime of sensor networks. The behavior of the queuing network that emerges out of the analysis is significantly different from that commonly assumed in literature. The most important (and in fact surprising) effect of the queuing behavior is that EER schemes may actually have lower overhead than HHR schemes for many-to-one traffic, high channel errors and low buffer capacities which would be the norm in sensor networks. Moreover the queuing behavior has a non-trivial effect on the network capacity and lifetime, and how bottlenecks are formed for the EER and HHR schemes. Although similar comparative studies exist for traditional wired and wireless data networks [4] the results are not extendible to sensor networks. To the best of our knowledge this is the first work of its kind. The non-intuitive nature of the results itself is an important contribution in the understanding the behavior of sensor networks. For sensor networks it is more important to minimize the energy consumption than maximize throughput. Such a requirement may again alter resource allocation decisions. For example we prove that a cumulative buffer to support all classes of flows is more efficient in terms of energy consumption than
36
dividing up the buffer for different classes of flows (which may be required for delay guarantees). We also propose efficient and distributed EER and HHR schemes for practical implementation. The results are not only important for designing reliable information delivery schemes but also useful for efficient network management and resource allocation in sensor networks. For example in a many-toone traffic scenario, the bottleneck nodes (with highest incoming traffic and highest energy consumption) are often trivially assumed to be the first hop nodes [35]. However the analysis shows this to be far from the reality for both the EER and HHR cases. Thus we need to allocate resources differently based on the actual behavior of the queuing network.
3.2
Network Model
We assume a circular field of diameter D with N uniformly distributed set of nodes. All nodes report their sensed data to a monitoring node present at the center of the circular area. Each node has a communication range of r meters. After detecting an event, a node reports it to the monitoring node. An important consequence of the many-to one traffic scenario is that nodes closer to the monitoring node have to support a higher traffic. This is because nodes have to forward packets for other sensors in outer layers (see Figure 3.1). To account for the traffic generated due to forwarding, we need to compute number of nodes at the hth layer. Let z be the width of the ring consisting of nodes equidistance in terms of hops from the sink. The number of nodes at the hth hop is given by:
(
)
n(h) = ( zh) 2 − (z (h − 1) ) δπ 2
( 3.1 )
h = 1,K, H
where, δ = 4 N / πD 2 = Average density
The maximum number of hops for the network is given by H = D / 2 z . The actual width z is dependent on the communication range and the network density. However we will see that it is not required in modeling our system. Next we define terms related to multiple reliability levels for generated events. We define L reliability levels as follows:
{R1 ,K, RL },
( 3.2 )
0 ≤ R j ≤ 1 , R1 < R2 < K < R L
37
λe = Total Rate of event occurrence per node =Rate of packet generation per node.
λej = Rate of events belonging reliability class j per node Thus the number of events in reliability class j, generated at the hth hop is given by:
(
)
N e (h, R j ) = ( zh) 2 − (z (h − 1) ) δπλej 2
( 3.3 )
For generating some of the results in this paper, we assume L=10 and a uniform distribution of events in for different classes i.e., λej = λe / L . For the reliabilities, we use R j = j / L . Thus we have ten different reliability levels {0.1,…,1}. The following definitions are used throughout the analysis: M = Total buffer capacity at each node p = packet loss probability due to channel errors bh = blocking probability due to finite buffer space. fh = packet loss rate at hth hop= f h = bh + p − pbh
( 3.4 )
ttrans =fixed packet transmission rate at each link We use a transmission rate of ttrans=10-3s. The link propagation delay is negligible for simplicity of the derivations. We also assume that packet transmissions do not suffer losses due to interference (i.e. they have TDMA type of MAC scheduling). Each node has a limited buffer capacity for supporting network traffic. However, the locally generated events at each node are buffered in a separate cell with infinite capacity i.e., local packets are never lost due to buffer restrictions. Only packets coming from outer layers may be dropped due to limited buffer space available. Now if the required reliability for a packet is Ri and the probability that a packet successfully reaches the 0
monitoring node h hops away is P ( success ) = Π (1 − f i ) ≤ Ri , then we have to provide extra redundancy i = h −1
to deliver packets at the desired reliability. We will evaluate different methods for delivering packets at the desired reliability and also analyze the expected latency, overhead and transport capacity for the different schemes.
38
3.3
Information Assurance with End-to-End Retransmissions (EER)
In End-to-end reliability schemes, a lost packet is retransmitted from the source back to the sink after a timeout for an acknowledgement has expired. In the classical definition of reliable transfer of packets, the maximum number of retransmission attempts is usually kept very large so that the packet is ultimately delivered with reliability 1. When delivering packets with multiple levels of reliability, we need to find the maximum number of retransmission attempts for reliability Rj. By adjusting the retransmission attempts according to the criticality of information in the packets, the network saves resources while delivering low priority packets. Consider a source which is h hops away from the sink with a packet belonging to the jth class. The probability that a packet arrives at the sink h hops away from the source is
0
∏ (1 − f ) . Let NEER (h, R j) be i
i = h −1
the number of retransmissions required to deliver it with reliability Rj. Then the probability that all the
NEER copies were lost is given by: 0 1 − Rj = 1 − ∏ (1 − fi ) i =h−1
N EER
Thus the number of retransmission required for reliably delivering a packet with probability Rj is given by:
N EER (h, R j ) =
3.3.1
ln(1 − R j )
( 3.5 )
ln1 − Π (1 − fi ) i = h −1 0
Overhead of EER Delivery
Next we evaluate the overhead. A packet is retransmitted if it gets corrupted or lost, or the acknowledgement for a correctly received transmission gets lost and the timeout for retransmission expires. The expected number of transmissions out of the maximum NEER retransmissions is given by:
Nˆ EER (h, R j ) =
N EER ( h , R j )
∑ k =0
k
h 1 − (1 − p ) ∏ (1 − f i ) = i =h 1
0 h 1 − 1 − (1 − p ) ∏ (1 − f i ) i = h −1 0
(1 − p )h ∏ (1 − f i ) i = h −1
39
N EER
λh+1
Nh
λh
λh-1
Nh
p bh-1
λ0
λ
λ
λ
…
λ1
N1
p bh-2 Acknowledge
N0 p b0
Figure 3.2: The Queuing network for EER. Packets are dropped due to channel error p and buffer blocking probability bh
M λh+1 λe
p bh Locally
Figure 3.3: The Queue at each node. At each hop packets are dropped due to channel error p and buffer blocking probability bh The probability that a copy of a packet is forwarded at the hth hop is a geometric distribution of the packet success rate at each hop. Thus the expected overhead incurred for each of the above copies as it goes to the destination and the acknowledgement returns to the source is given by: Oh =
∑ ∏ (1 − f ) + ∏ (1 − f )∑ (1 − p ) 0
0
i
i = h −1 k = h −1
k
h
h
i = h −1
( 3.6 )
k
k =1
The first summation denotes the data overhead and the second summation denotes the acknowledgement overhead if a data packet successfully reached the destination across h hops. The total overhead is then given by: O EER = Oh Nˆ EER ( h, R j )
3.3.2
( 3.7 )
EER scheme under Memory Constraints
To compute the actual overheads we need to compute the blocking probabilities fi at each hop. The queuing network under the EER scheme is depicted in Figure 3.2 and Figure 3.3. We assume a fixed buffer size of M, and packets are serviced in FCFS order.
40
Consider the packets generated at the hth hop belonging to reliability class j as shown in Figure 3.2. Let th λh,p j be the outgoing rate of events in a node at the h hop belonging to class j. Then total events at the (h-
1)th hop node is the sum of the incoming events from hth hop and the locally generated events. The packets from the outer layers maybe lost due to channel errors and buffer overflow. Since there are a larger number of nodes in the hth hop than the h-1th hop, this load has to be shared between the nodes in the (h-1th) hop. Thus we have the following recurrence relation for the outgoing rates.
λhp, j = λ pj +
(
λhp+1, j (1 − f h ) (z(h + 1))2 − ( zh) 2
((zh)
2
− ( z(h − 1))
2
)
)
( 3.8 )
To solve for the blocking probabilities at each hop we start with the sink or the monitoring node. We call this the 0th hop where all the events are collected (the monitoring node with infinite buffer capacity). Since the flows are conserved in this closed loop network and since packets belonging to the reliability class j have an associated probability of delivery Rj, the rate for each class that enters the monitoring node at the 0th hop is λ0p, j = R j λej N . The incoming rate at the monitoring node is due to the cumulative outgoing rates from the nodes at the first hop, which is given by:
R j λej N = λ1p, j (1 − p)πz 2δ ⇒ λ1p, j =
R j λej N (1 − p)πz 2δ
Since there are H hops comprised of rings of width z, and there are N nodes in the network the density is given by δ = N / π (zH )2 . Hence we have the incoming rate at the first hop as: L
λ1p = ∑ j =1
R j λej N
=
(1 − p) z δ 2
N (1 − p)πz 2 δ
L
∑R λ j =1
j
=
e j
H2 L R j λej (1 − p ) ∑ j =1
( 3.9 )
Next we rewrite the recurrence relation in equation 8 for the cumulative rates comprised of all the reliability classes: λ hp =
(λ (1 − f ))((z ( h + 1) ) p h +1
h
(( zh )
⇒ λ hp+1 (1 − p ) =
2
(λ
2
− ( zh ) 2
− ( z ( h − 1) )
p h
−λ
e
)(h
2
2
)
− ( h − 1)
(1 − b h )((h + 1)
2
−h
2
)+ λ 2
)
e
( 3.10 )
)
41
L
where λhp = ∑ λhp, j j =1
The source is assumed to have infinite capacity for its own packets. Hence the blocking probability for incoming packets at each hop is given by: bh = Pr[L = M ] =
(1 − λ (1 − p )t )(λ (1 − p)t ) 1 − (λ (1 − p )t ) p h +1
M +1
p h +1
trans
trans
( 3.11 )
M +1
p h +1
trans
Now the blocking probabilities for each hop can be computed in a recursive manner by solving equation 5.10 and 5.11 starting with the value for the first hop given by equation 5.9.
3.3.3
Distributed EER Delivery
In the derivations described above, we see that a source node needs to have the knowledge of the blocking probabilities at each hop to the sink. Keeping the entire network state at each node is not desirable for sensor networks. We now show that reliabilities can be provided in a distributed manner (without the knowledge of the blocking probabilities at each hop) albeit a larger overhead. The scheme is based on the simple observation that if a source forwards a packet with probability Rj but keeps infinite retransmissions attempts for any packet, the packet would be delivered to the sink with the given reliability. The end-to-end reliability would be same as before i.e., Rj but there would be a difference in the overhead. A packet is retransmitted when it is lost or the acknowledgement is lost. The probability of retransmission is: 0 h P[retransmis sions = n] = 1 − (1 − p ) ∏ (1 − f i , j ) i = h +1
n
Since the maximum number of retransmission attempts is infinite, the expected number of retransmissions is given by: E[retransmis sions ] = 1 / (1 − p )
h
∏ (1 − f ) 0
i, j
i = h +1
Again for each of these retransmissions the overhead is: O1 =
∑ ∏ (1 − f ) + ∏ (1 − f )∑ (1 − p ) i
0
i = h +1 k = h +1
0
k, j
h
i, j
i = h +1
k
k =1
42
Since the packet is put in the outgoing queue for delivery with probability Rj according to its class, the total overhead is:
O EER (distributed ) =
h 0 i 0 k R j ∑ ∏ (1 − f k ) + ∏ (1 − f i )∑ (1 − p ) 1 1 i = h − k = 1 1 k h = − i = h −
( 3.12 )
0
(1 − p )h ∏ (1 − f i ) i = h −1
Comparing the overhead of the centralized and the distributed schemes given by equation 5.7 and 5.12 respectively, we see that: 0 R j = 1 − 1 − ∏ (1 − f i , j ) i =h+1
N EER
0 h ≥ 1 − 1 − (1 − p ) ∏ (1 − f i , j ) i = h +1
N EER
( 3.13 )
Thus OEER (distributed) is always larger than OEER from the inequality (5.13). However the advantage of using this scheme is that we do not need to compute the retransmission numbers for each class, or keep track of the blocking probabilities and channel error rates for the entire network.
3.3.4
Buffer Allocation for Minimum Overhead
Finally we consider the buffer allocation required to minimize overhead. In traditional data networks, different buffer sizes are allocated for different service classes mainly to provide delay guarantees to different flows. For sensor networks, it is more important to minimize energy consumption. We show that having multiple buffers has no advantages in terms of overhead of transmissions as given by the following theorem: THEOREM 5.1. A single buffer supporting all the flows in different classes always has lesser overhead
than the case where the buffer is divided into smaller parts to support each reliability class. PROOF. See appendix.
3.3.5
EER Discussion
We first look at the behavior of the network with infinite buffer capacity in the nodes. Figure 3.4 shows the incoming rates at each hop in this network for different error rates. We expect the nodes closer to the monitoring node to have larger incoming rates since they have to support the flows from the outer layers. However it is interesting to see that the incoming rates do not have such a monotonic behavior based on
43
Traffic Rates: Infinite Buffer
p=0.1 p=0.2 p=0.25
600
Blocking Probability
500 400
Rate
EER: Effect of Channel Errors
1
300 200 100
M=1 M=2
0.9
M=5
0.8
M=10
0.7 0.6 0.5 0.4 0.3 0.2 0.1
0 1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0 0
Hops
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Channel Error
Figure 3.4: Incoming Packets rates for EER with
Figure 3.5: EER blocking probability, for a 2-hop
infinite buffer capacity
n\w p=0.1,λe=10,H=2, ttrans=10-4
EER: Blocking Probability
Overhead of EER
100 90
M=1
0.08
80
M=2
0.07
70
M=10
Overhead
Blocking Probability
0.1 0.09
0.06 0.05
p=0.01
0.04 0.03 0.02
60 50 40
p=0.05
30
p=0.1
20 10
p=0.15
0
0.01
1
0 1
3
5
7
Hops
9
11
13
15
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
Hops
Figure 3.6: Effect of channel error on blocking
Figure 3.7: The effect of buffer size on the overhead for EER. p=0.1, λe=1.0, H=2, ttrans=10-3
e
probability. λ =10, H=20,M=5
the hop distance from the sink. This can be attributed to the nature of EER scheme which generates the extra packets from the source for reliable delivery based on the number of hops it is away from the source. Thus at large hop distances, nodes have to generate a larger number of packets even for moderate channel error rates whereas nodes closer to the sink have to support a larger rate from outer layer. These two factors have opposite monotonicities (i.e., one increases with hops and the other one decreases) due to which we see the interesting curve in the figure. Under buffer constraints the high incoming rate would increase the buffer blocking probability which would in turn increase the rate and could potentially lead to a buffer overflow problem. Thus the graph highlights that we would expect a more complex behavior of the network with buffer constraints. We first consider a trivial network of only two hops with buffer restrictions. Figure 3.5 shows the blocking probabilities at the only intermediate hop which is the first hop for different channel error rates. We see that the blocking probability shows a threshold phenomenon i.e., after a certain increase in channel error rate, it suddenly goes to 1 and has buffer overflow. Although this presents the results for a trivial network, it gives us a preliminary insight into the behavior with large number of hops.
44
Rate=0.1
Buffer Size vs Network size
0.8
Rate=0.5
50
0.6
40
0.5
Overhead
45 35 Hops
error=0.01
0.7
Rate=1
30 25
EER: Centralized vs Distributed
error=0.05 error=0.1 error=0.15
0.4 0.3 0.2
20
0.1
15
0
10 0
5
10
15
20
25
0.1
30
0.2
Buffer Size
0.3
0.4
0.5 0.6 Reliability
0.7
0.8
0.9
Figure 3.8: EER: Network Size that can be
Figure 3.9: Localized scheme for EER. λe=1.0,
supported. p=0.1, H=20
H=20,M=5, ttrans=.0.001
We now consider a network with 20 hops. Figure 3.6 shows the blocking probabilities at different hops for the network for different channel error rates. We see that blocking probabilities are high at hops very near or very far away from the monitoring node and relatively small in the middle. The curves depict the essential difference between the unconstrained and constrained buffer cases. In the constrained buffer case, incoming rate and the blocking probability are inter-related such that each increases the other and hence we get the non-trivial curve in the figure. Figure 3.7 shows the overhead to reliably transmit a packet from different hops using EER. From equation 6 we see that the overhead is dependent on the product of the node blocking probabilities (due to channel error and buffer overflow). Since the product is monotonically decreasing with respect to hop lengths irrespective of the actual values of the hop errors (since blocking probability is less than 1), the overhead is also monotonically increasing as depicted in Figure 3.7. The interdependence of blocking probability and incoming rates can actually lead to buffer overflows at hops far away the sink. Figure 3.8 shows the buffer space required to support a network of a given size (in terms of number of hops) for different packet rates. We see that for each of the rates the network becomes unstable beyond a certain point and increasing the buffer capacity does not really have any impact stability of EER. This is because the incoming rates at any node increases exponentially with hops and beyond a certain point the arrival rate at each node becomes higher than the transmission rate creating buffer overflows. We now look at the differences in overhead between the centralized scheme and the distributed EER scheme for providing different reliabilities. Figure 3.9 shows the overhead difference for the two cases.
45
Rj1/h
Rj1/h
…
Rj1/h
Rj1/h
R Figure 3.10. Illustration of the Hop-by-Hop retransmission scheme for providing a desired reliability. The end-to-end Reliability is divided into different per hop Reliability. The Differences are normalized with respect to the required reliability. The trends show that at low and high required reliabilities, the differences between the two reduce. This is because at low reliabilities, the overhead itself is very low, whereas when the required reliability is very high, the distributed scheme converges towards the centralized scheme as was expected. We also see that at low channel errors, the two schemes perform similar which again can be attributed to equation 13, which has equality when channel error rate is zero.
3.4
Information Assurance with Hop-by-Hop Retransmissions (HHR)
Reliability can also be provided with retransmissions at each hop. Here an intermediate node forwards the packet reliably to the next hop to provide the end-to-end reliability. For multi hop networks, with high channel errors, such a Hop-by-Hop scheme is usually more suitable in terms of overhead and latency. In HHR schemes, a packet is buffered until it is reliably forwarded to the next hop (usually referred to as a store and forward approach). This causes extra buffer occupancy leading to higher blocking probability. With small buffer capacity in sensor networks, this might even increase the packet drop probability such that the overhead is adversely affected. In this section we analyze the behavior of HHR schemes to compare the performance with EER for sensor networks. We start by giving a brief overview of how multiple levels of reliability is provided using the HHR scheme. Consider a path from source to sink consisting of h hops as given in the Figure 3.10. Let Rih, j be the required reliability for the jth class at the ith hop for a source which is h hops away from the sink. Then h
if
∏R i =1
h i, j
=R j , we get an end-to-end reliability of Rj which is the desired reliability. A Trivial way to
provide the reliabilities is depicted in Figure 3.10 (at each hop provide R1/h reliability so that the end-toend reliability is R). For required reliability
R ih, j ,
the maximum number of retransmissions is given by:
46
Node at hth hop
Packets from outer layers
Acknowledgement
1-p
1-p bh
p
λh+1,j
p
bhNode at h-1th hop
Rj Locally generated packets
Acknowledgement
Figure 3.11: The HHR queuing model for a node at the hth hop using the localized HHR scheme
Rih, j = 1 − f i NHRR ⇒ N HHR (i, j ) =
(
ln 1 − Rih, j
)
ln( f i )
The expected number of retransmissions is given by: Nˆ HHR =
N HHR
∑ i =1
fi
i −1
=
N Rih, j 1 − f i HHR = 1− f i 1− f i
( 3.14 )
Thus the total overhead incurred over h hops, when trying to provide a reliability Ri,j at each hop is: h i −1 R h R h i, j k , j OHHR = ∑ ∏ i =1 k = 0 1 − f i
( 3.15 )
Next we derive the queuing network model for the HHR scheme.
3.4.1
HHR under Memory Constraints
The HHR scheme is similar to a store and forward scheme where an intermediate node buffers an incoming packet until it is reliably forwarded to the next hop. However the differences lie in that any packet has an associated reliability Rj based on its class due to which the computations would be different. Figure 3.11 illustrates the queuing network for the HHR scheme. Let, kj = total number of packets in the jth class on the transmissions queue for the output lines nj = number of packets in the jth class waiting acknowledgement from the next hop L
s = ∑ k j + n j , the total number of occupied buffers j =1
λj=Incoming packet rate in the jth reliability class
47
We assume that there are no processing delays in the node such that packets are instantaneously queued for transmissions. The stationary distribution of the buffer occupancy, assuming each of the reliability classes has associated reliability of 1 (and only differ in their incoming rates) has been shown to have a product form in [83] and is given by: L α nj L k Ph [k1, K, kL , n1, K, nL ] = Ph (0)∏ ρh, j j ∏ h, j , s ≤ M j =1 j =1 nj!
( 3.16 )
Where Ph(0) is the normalizing coefficient given by: L ρ 1 h, j =∑ Ph (0) j =1
M +1
S M (λ h , j Thhold / ρ h, j ) ,j
φ h , j ( ρ h , j − 1)
+ S M (λ h Th
L
L
L
i =1
j =1 j ≠i
j =1
hold
( 3.17 )
) /ηh xj j =0 j! M
λh = ∑λh, j , φh, j = ∏(1− ρ h, j / ρ h,i ) , ηh = ∏(1− ρ h, j ) , S M (x) = ∑
ρ h, j
= λ h, j (1 − p )t trans ,
α h, j = λ T
Hold , h, j h, j
Th
hold
L
= ∑ λh , j Thhold ,j j =1
th ThHold , j = Holding time for a packet in the j reliability class.
3.4.2
Reducing the HHR Scheme to a Store and Forward Queuing Network
Supporting information assurance leads to a more complex queuing network due to different number of retransmissions for different reliability classes. The most important aspect in the queuing network with multiple reliability classes is that packets of each class are buffered for different time periods based on their reliability class. In HHR, a packet is emptied from the buffer either when an acknowledgement arrives or when the number of retransmissions attempts, N HHR (i, j ) expires. The expected number of retransmissions actually depends not only on N HHR (i, j ) but also the blocking probability at the next hop. Theorem 5.2 shows that the complex network model of HHR with information assurance can be reduced to a classical store and forward model given by equation 16 and 17 with a simple transformation.
48
Rj
1
…
1
1
Rj Figure 3.12: Illustrates the optimal assignment of reliabilities for the HHR scheme.
THEOREM 5.2. If λej is the packet generation rate for the jth class, and
Rj the required reliability, the
queuing network for multiple reliability classes can be modeled using equation 5.16 by substituting
λej → λ ′j ,h = R j λej . PROOF: See appendix.
3.4.3
Optimal Allocation of Per-Hop Reliabilities for HHR Scheme
Next we look at the per-hop reliabilities Rhk, j which are required to solve the blocking probabilities at each hop. This is because, the incoming rate for packets in each class, λ = λe h Rk would be dependent h, j j ∏ i, j i=k −1
on the allocation sequence. This in turn would affect the blocking probability at each hop and the overhead for reliable delivery. The allocation of Rhk, j which minimizes the overhead of delivering a packet reliability Rj is given by Theorem 3 and 4. THEOREM 5.3: The allocation of per-hop reliabilities, which minimizes the overhead, is given by:
Rhh, j = R j , Rkh, j = 1, h − 1 < k ≤ 1, , H < h ≤ 1 =Rj. PROOF: See appendix. THEOREM 5.4: The allocation sequence given by theorem 5.3 minimizes the blocking probabilities at
each hop and also minimizes the overhead of HHR scheme for any number of classes and sources and also for non-uniform channel error rates and buffer availability in nodes. PROOF: See appendix.
49
Figure 3.12 illustrates the per-hop optimal allocation sequence for HHR from Theorem 3. We see that that λh, j = λej R j and Rhk, j = 1 for packets coming from the outer layers belonging to the jth class. Thus we h
have λh, j = λej ∏ Rik, j = λej R j which can be used to find the buffer blocking probability (see appendix). i = k −1
In the optimal per-hop reliability allocation sequence, although from second hop onwards from the source, we actually allocate a per-hop reliability of 1 (which means that the overhead for any packet will be the highest), the actual total overhead is minimized. The intuition behind this is that by assigning Rhh, j = R j , we are actually minimizing the source rate which leads to minimization of the rates at any
subsequent hop. Moreover the allocation sequence does not depend on the actual channel error rate at each link and the buffer capacity at each intermediate node and hence is completely general.
3.4.4
Distributed HHR Delivery
Based on the analysis in the previous section and the theorem 5.3 and 5.4 for optimal allocation of reliabilities we design a simple localized scheme for HHR. This is described as follows •
Each locally generated packet of the jth reliability class is forwarded with probability Rj.
•
Any packet is which enters the buffer is assigned an infinite maximum retransmission attempts. This is exactly similar to the distributed EER scheme. To understand why the above scheme works we
see that the stationary distribution for packets in different classes with different maximum retransmission numbers (hence different holding times) is same with an input rate Rjλj (but with infinite retransmissions). From theorem 5.2 if locally generated packets enter the queue with rate Rjλj and are assigned infinite maximum retransmissions irrespective of their classes, we would get the same queuing network as with input rate of λj but NEER(j) retransmission attempts. Here each node does not need to know the blocking probabilities for the next hop to compute the maximum retransmission number. This makes the forwarding completely localized. There will be a small difference in the overhead just like in the EER case, but since these are restricted only to single hops, the differences in this case would be negligible. Moreover the scheme automatically accomplishes the optimal per-hop reliability allocation as given by theorem 5.4 and will deliver packets with the minimum overhead as well.
50
3.4.5
The Blocking Probabilities at each hop for the HHR Scheme
The blocking probabilities for the distributed scheme are derived next. In the distributed HHR scheme, the incoming packets from outer layers are not treated differently according to their classes because the per-hop reliability at each intermediate hop is 1, except at the first hop, where it is Rj. Hence the blocking probability has to be calculated only for the cumulative incoming rate from outer layers. The blocking probability at the hth hop can be calculated with L=1 in equation 5.16 as follows: bh =
ρ h M S M (λinhThhold / ρ h ) P1 (0) φh
ρ 1 = h Ph (0)
M +1
( 3.18 )
S M (λinh Thhold / ρ h ) S M (λ h Th + φ h ( ρ h − 1) ηh M
φ h = 1, η h = (1 − ρ h ) , S M ( x) = ∑ x j j! ,
hold
)
ρ h = λinh t trans
j =0
The total incoming rate at the hth layer is due to the packets originating at the outer layers. Since in HHR, any flow entering a node is conserved on a per hop basis, we have:
(H λ = in h
2
− h2
(h
)∑ λ L
j =1
2
in h, j
− (h − 1)
2
Rj
( 3.19 )
)
We again solve the blocking probabilities in a recursive manner starting with the first hop. The overhead for the HHR case is given by equation 5.15.
3.4.6
HHR Discussion
For the analyzing the HHR queuing network, we assume TAck =0.02s, and Ttimeout =2TAck =0.04s. The blocking probability for the HHR scheme at different hops is given Figure 3.13 for different traffic rates. We see that the blocking probability curves are completely opposite of the EER blocking probabilities. I.e., the blocking probabilities are high in the intermediate hops and low at hops near to the sink or far away from the sink. This is highly surprising and non-intuitive since from equation 5.19, the incoming rates at nodes increase monotonically as we get near the sink and we expect the blocking probabilities to increase as well.
51
0.9
Rate=1.0
0.8
Rate=1.5
0.7
Rate=2.0 Rate=2.5
0.6
Rate=3.0
0.5 0.4 0.3 0.2 0.1 0 1
3
5
7
9
11
13
15
17
HHR Overhead
p=0.2, M=1
200 180 160 140 120 100 80 60 40 20 0
p=0.1, M=5 p=0.2, M=5
Overhead
Blocking Probability
p=0.1, M=1
Blocking Probability with Traffic Rate
1
1
19
Hops
3
5
7
9 Hops
11
13
15
17
19
Figure 3.13: HHR: The node blocking probability
Figure 3.14: HHR: Overhead Curves N=2500,
N=2500, p=0.3, M=5, hops =20
p=0.3, M=5, hops =20
In HHR the blocking probability at any hop is dependent on two parameters: the incoming rate and the blocking probability of the next hop. Since the 0th hop (monitoring node) has infinite buffers, the blocking probability at the first hop is only due to the incoming rates. However from second hop onwards, the blocking probability is dependent on the incoming rate (decreasing with hops) and the blocking probability at the previous hop. When buffer capacity is small, initially the effect of reduced incoming rates with increased hops is overshadowed by the effect of the blocking probability in the previous hop. This results in the bottleneck hop (hop with the highest blocking probability) being shifted away from the first hop. This is an important result in this paper. Later we will see that this adversely affects the overhead, network capacity and lifetime such that EER schemes may become more favorable for sensor networks. Next we look at the overhead of HHR for different channel error rates and buffer capacities in Figure 3.14. In general the HHR overhead is expected to be linear since it is a sum of a function of the blocking probabilities as opposed to the function of the product in the EER case. However, we see that the overhead is linear only with a large buffer capacity whereas for low buffer capacity and high channel errors the overhead increases rapid at initial hops and then flattens at the end. This can be attributed to the nature of blocking probability curves due to which per hop overhead at the intermediate hops is significantly high resulting in the non-linear curve of the overhead. Finally we evaluate the distributed HHR scheme having optimal allocation of reliabilities. We compare against a trivial scheme where each of the Rhi,j=Rj1/h. The overhead curves are shown in Figure 3.15. We
52
Trivial HHR vs Optimal HHR 16
Overhead
14 12 10 8 HHR: Trivial
6
HHR: Optimal
4 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Reliability
Figure 3.15: HHR: Optimal allocation of reliabilities in HHR p=0.2, λe=1, H=10, M=5 see that at high reliabilities, the trivial scheme converges towards the optimal scheme since Rj1/h converges to 1 as Rj goes to 1. However at low reliabilities, the optimal scheme has significant savings over the trivial scheme. Thus using the optimal allocation of per hop reliabilities, energy can be saved for delivering data having low criticality.
3.5
Information Assurance with Hop-By-Hop Broadcast (HHB)
In this section we show that the use of one-hop broadcast capability of wireless medium can achieve significant reduction in overhead while providing the desired reliability. Consider a source and sink node separated by h hops with a required reliability r. For ease of exposition, let us assume that the channel error is a constant p at each hop. Using the hop-by-hop methods, we require r1/h reliability at each hop for which we require NHHR copies as given for the HHR case. Our aim is to reduce the number of copies required to less than NHHR while still attaining the desired reliability of r1/h at any hop. To attain this objective, we utilize the broadcast property of the wireless medium. Since sensor networks typically have high density, there would be multiple nodes which are h-1 hops away from sink. Thus, for a packet to reach the sink with reliability r, it is sufficient to have any one of these h-1 hop nodes receive the packet with reliability r1/h. Consider the example shown in Figure 3.16. In HHR, the source sends a packet (or any copy of the packet) to one of its three next-hop neighbors a, b or c. However, if the source uses one-hop broadcast instead of unicast, then one copy of the packet acts as three copies, one sent to each of source’s next-hop neighbors. Similarly, one packet sent by c acts like four retransmissions (because it has four next-hop neighbors d, e, f and g).
53
a Source
d
b
Sink …
f
c h
e
h-2 hf
h-1
g …
Figure 3.16: Illustration of the basic strategy used in hop-by-hop broadcast There are two key issues to be addressed in order to use hop-by-hop broadcast: 1.
Determining the number of copies, NHHB, of a packet required to be sent at each hop. Thus if ki is the i
number of next hop neighbors, and Rh , j is the required reliability for the jth class at the ith hop, then R hi , j = 1 − p N HHB ( i ) k i N HHB (i ) =
2.
(
log 1 − R hi , j k i log p
)= N
( 3.20 ) HHR
ki
≤ N HHR
Ensuring that at most one of the next-hop neighbor forwards the multiple copies of the packet. Since, more than one next-hop neighbors can receive a packet correctly, it is a waste of resources if all of them send NHHB copies of a packet to the next-hop. For example, in Figure 3.16, if nodes d, e and f receive a packet correctly from c, then only one of them (node e in the figure) should forward it to nodes at h-3 hops from sink.
To strictly enforce the rule that exactly one node forward a packet at each hop, we would have to incur extra overhead in form of some control packets. Instead we use a probabilistic method so that we have only one node forwarding a packet at each hop, in the expected. Thus, each next-hop node that receives the packet, forwards the packet with a probability Pforward which is given by: Pforward =
1 k 1 − p N HHB
(
( 3.21 )
)
Thus, to use HHB to provide desired reliability, at each hop one node broadcasts NHHB packets. This ensures that the packet reaches one of the next-hop nodes with a probability r1/h. Each next-hop node that receives the packet correctly, forwards (broadcasts NHHB copies) the packet with a probability Pforward. Although each receiving node probabilistically forwarding the packet does not guarantee that at least one node would forward it, the probabilistic guarantee in packet delivery is still maintained.
54
3.5.1
Operations while Forwarding a Packet using HHB
When a source detects any sensed event, first it computes the criticality of the detected event and maps it to the required reliability R j
.
The source computes the number of packets required to provide this reliability, given by NHHB. It then generates a packet with the detected information and the following additional packet fields: •
R: Required packet reliability (set to Rhi , j )
•
h: Hops from the source to sink
•
i: The hop distance of the sender to the sink
•
ki : Number of next-hop neighbors at the Ith hop from the sink.
These values are used by the receiving next hop nodes to decide the forwarding probability.
3.5.2
Operations while Receiving a Packet in HHB
When a node (other than the sink) receives a packet, it first checks if it is one hop nearer to the sink node than the sender, (i.e., it is h -1 hops away from the sink, it decides to process the packet). Otherwise the packet is dropped. It then decides to forward the packet with probability Pforward using equation 5.21 and the number of next hop neighbors ki and required reliability Rhi , j . If the node decides to forward the packet, it sends an acknowledgment as a confirmation of forwarding, back to the source. Thus, the acknowledgment serves the purpose of letting the source know that some node which is willing to forward the packet has already received the packet and the source should stop further retransmissions. The packet is retransmitted with a gap of Tdelay between each retransmission. Tdelay needs to be large enough so that each packet has sufficient time to get through i.e., including the time required for acknowledgements to come back. Thus, Tdelay=TRTT where TRTT is the one-hop round trip time.
3.5.3
Overhead of HHB
Now we compute the overhead of HHB and HHBA. We define the following variables used in this section:
55
pi = average channel error at ith hop ki= expected number of next hop neighbors at ith hop Rj = end-to-end required reliability
Rhi , j = per-hop reliability From equation 5.20, we know the maximum number of retransmission for the HHB scheme at each hop. The acknowledgment packets are likely to be received correctly because of their small size and hence we do not consider them in the computation of the overhead. Thus, the sender transmits the ith copy of the packet only if all of the previous i-1 copies were incorrectly received at all the next-hop neighbors. In this case, the expected number of copies of the packet transmitted at the ith hop, is given by: N HHB ( i )
∑
OHHB (i ) =
i =1
pi
ki ( i −1)
kN
=
1 − pi i HHB k 1 − pi i
(i )
Substituting the following for Ni :
N HHB (i ) =
O HHB (i ) =
(
log 1 − R hi . j k i log p i
) = log (1 − R ) i h. j
pi
ki
Rhi , j 1 − ei
( 3.22 )
ki
Thus, the total end-to-end overhead using HHB is given by: i t R i −1 h h ∏ h. j OHHB (total ) = ∑ OHHB (i )∏ Rht . j = ∑ t =1 ki i =1 t =1 i =1 1 − pi
3.5.4
( 3.23 )
Optimal Allocation of per hop reliabilities for HHB
Now we look at the optimal allocation of per-hop reliabilities for the acknowledgement-based method. Again, we neglect the overhead of the acknowledgement packets. Acknowledgement packets have negligible overhead as we saw in the previous section in results for overhead incurred HHR and HHRA, when the required reliability, channel error or number of hops is large. Thus, allocating reliabilities to each hop in order to minimize OHHBA reduces to the following optimization problem:
56
i t ∏ Rh. j h min O HHB = ∑ t =1 k i i =1 1 − p i h
s.t
∏R
i h. j
= Rj ,
( 3.24 )
R j ≤ Rht . j ≤ 1
i =1
In this formulation, the value of OHHB is used from equation 5.23. We use the following simple technique to solve the above problem. Let, ri = Rhi . j and α = i OHHB
1 . Then the overhead O HHB is given by: 1 − piki
= α 1 r1 + α 2 r1 r2 + L + α h r1 r2 L rh
( 3.25 )
≥ α 1 r1 r2 L rh + α 2 r1 r2 L rh
O * HHB
= R j (α 1 + α 2 + L + α h )
The optimal allocation sequence is now given by theorem 5.5. THEOREM 5.5: The overhead is minimized only when r1=Rj and all other rj are 1, irrespective of the
values of pi and ki. The optimal distribution of reliability, r*, is hence given by: r1* = r
( 3.26 )
ri* = 1 ∀i ≠ 1
PROOF: We see that the the minimum value of OHHB (denoted by O*HHB) occurs when the coefficient of
each αi in each of the terms in the summation is Rj . Thus above solution for ri is the optimal solution of the problem. We also see that the above is a feasible solution to the constraints of the problem and hence it is the optimal solution. The solution is exactly similar to the HHR case as given in theorems 5.3.
3.6
Comparing the EER, HHR and HHB Schemes
In this section we compare the three schemes. The results discussed in this section attempt to illustrate the significant effects due to each of the governing parameters and the results which are interesting from the perspective of future protocol design covering the effect of all the parameters. The parameter values are specified in the figure captions.
57
OverHead EERvsHHR
EER: p=0.1
EER: M=2
70
HHR: p=0.15
Overhead: EERvsHHR
HHR: M=1
80
EER: p=0.15
100
HHR: M=2
60
80
Overhead
Overhead
EER: M=1
HHR: p=0.1
120
60 40
50 40 30 20
20
10 0
0 1
3
5
7
9
11
13
15
17
19
1
3
Hops
5
7
9
11
13
15
17
19
Hops
Figure 3.17: Overhead of HHR vs EER, , λe=0.4,
Figure 3.18: Overhead of HHR vs EER, p=0.15,
H=20, ttrans=.0.001
λe=0.55, H=20, ttrans=.0.001
3.6.1
Comparing the Overhead of HHR and EER Schemes
We compare the overhead of EER and HHR schemes for a network of size 20 hops. The overhead curves are shown in Figure 3.17and Figure 3.18 for different channel error rates and buffer sizes respectively. The figure plots the overhead to deliver a packet reliably from each hop. The overhead curves highlight the conditions at which EER has lower overhead than the HHR case. For traditional data networks, in general the HHR overhead to send a packet from a large hop distance and non-negligible channel error rates is always lesser than the EER overhead. However for sensor networks with many-to-one traffic, and low buffer capacity, the queuing network has a significantly different behavior. With a high buffer capacity the HHR overhead is more similar to the expected behavior of being lower than the EER overhead as seen in Figure 3.18. The results discussed above shows that for sensor networks, a scheme based completely on either the EER or the HHR scheme is not efficient in terms of overhead. Since at hops near to the monitoring node, the EER scheme performs better than the HHR, a reliable delivery scheme should stop buffering packets as it nears the monitoring node. The design of such a protocol which progressively goes from an HHR based scheme at large hop distances towards an EER scheme is part of future work.
3.6.2
Comparing the HHR and HHB Schemes
Figure 3.19 shows the effect of increasing hop distance on the overhead of various schemes. The channel error was set to 0.5 and the desired reliability to 0.7. We see that the overhead for HHR scheme
58
HHR: p=0.01 EER: p=0.01
HHB
35
0.3
HHR
30
EER: p=0.1 HHR: f=0.1
0.25
25
0.2
20
Delay
Packet Overhead
Packet Delivery Delay
Comparison of Schemes
40
15
0.15
10
0.1
5
0.05
0 2
3
4
5
6
7
8
9
10
11
12
13
14
0
15
1
Hops
3
5
7
9
11
13
15
17
19
21
23
25
27
29
Hops
Figure 3.19. Comparison of HHR with HHB: for
Figure 3.20: End-to-End Delay, λe=1, H=30,M=10
Rj=0.7 and p=0.5
ttrans=.0.001
increases rapidly as the number of hops increases. The HHB scheme has a lesser increase in overhead since it uses the broadcast redundancy in the wireless channel. Thus under the stringent conditions of high error, high desired reliability and large number of hops, HHB is the suitable choice for reliable information delivery.
3.6.3
Comparing the Latency in Data Delivery for EER and HHR
An important aspect in sensor networks is the timely delivery of critical data. Although in this paper we have not discussed the aspect of guaranteeing latency of data delivery, it would be interesting to study the packet delays for ultimate delivery using the HHR and EER schemes. Appendix 3.8.3 derives the latency in packet delivery for the two schemes. Figure 3.20 plots the packet delays from each hop for the EER and HHR cases for different channel error rates. Not surprisingly, we see a very similar trend in the delays as compared to the previous graph on overhead. At low channel error rates and low hop lengths, EER scheme has a lower delay since it would have lower buffering delays and end-to-end acknowledgement delays. However at higher channel errors the delay increases very rapidly.
3.6.4
Network Scalability and lifetime
In this section we compare the performance of EER and HHR schemes related to network scalability and lifetime and the design decision which result from using these schemes. Let us first look at the energy consumption incurred by nodes at different hops for the EER and HHR schemes. The energy consumption of packet transmissions by nodes at any hop is proportional to the
59
Outging Rates, HHRvs EER
700 600
EER: Lifetime
HHR: M=1, f=0.15
0.018
HHR: Lifetime
EER: M=2, f=0.25
0.016
EER: Connected Time
HHR: M=2, f=0.25
0.014 Lifetime
500
# of Packets
Lifetime
0.02
EER: M=1, f=0.15
400 300
HHR: Connected Time
0.012 0.01 0.008 0.006
200
0.004
100
0.002 0
0 1
3
5
7
9
11
Hops
13
15
17
1
19
3
5
7
9
11
13
15
17
19
Hops
Figure 3.21: Outgoing rates at different hops,
Figure 3.22: The comparison of network
p=0.15, lambda=0.25, M=1
lifetime and the actual cut-off time for EER and HHR
outgoing rates at each hop. Figure 3.21 shows the outgoing rates for the EER and the HHR cases. For the EER case we see that the rates are higher at the outer hops whereas for the HHR case the rates are higher in the intermediate hops. Thus in EER outermost nodes would be depleted of energy the earliest whereas in HHR the intermediate nodes die of before other nodes. Since the inner layers form the cut set for the outer layers in our network, when inner layers die of due to energy depletion, the outer layers are also cut off. Figure 3.22 plots the actual lifetime of node depletion and the cut-off time of nodes at different hops. For the HHR scheme we see that actual time of node death due to energy depletion is very small at hops near the sink and high at the larger hops. The effect is completely opposite in case of EER. Although the total energy depletion of the network may actually be larger for the EER case, in HHR a large fraction of nodes get cut off very early even though they have energy left due to the low lifetime in inner hops. The outer layers may also be cut off due to buffer overflows at inner layers. Consider Figure 3.23 and Figure 3.24 which depict the blocking probabilities for the HHR and EER schemes. We again see a similar trend i.e., with EER buffer overflows happen at out layers (blocking probability =1.0) whereas with HHR the intermediate layers have buffer overflows. Thus using the HHR scheme may cut off the outer layers although they have favorable buffer blocking probabilities. The above results highlight the non-trivial nature of the queuing network for sensor networks. We clearly show that the network bottleneck in terms of capacity and lifetime may not be the first hop nodes as assumed in papers such as [35]. Such trivial notions of capacity can also lead to inefficient resource
60
HHR: Blocking Probability
M=1 0.4
M=3
0.9
M=5
0.8
M=10
0.35
M=10 M=100
0.7 0.6 0.5 0.4 0.3 0.2
0.25 0.2 0.15 0.1 0.05
0.1 0
M=5
0.3
Blocking Probability
blocking Prob
EER: Buffer Overflow for Diff. Buffer Size
M=2
1
0
1 1
3
5
7
Hops
9
11
13
15
17
2
3
4
5
6
19
7
8
9
10
11
12
13
14
15
16
17
18
19
Hops
Figure 3.23: Buffer overflow for HHR N=2500, e
Figure 3.24: The buffer overflow for EER: N=2500, p=0.3, λe=2.5, hops =20
p=0.3, λ =2.5, hops =20
allocation in sensor networks. For example using the HHR scheme we should deploy nodes with higher resources (energy/memory) at the intermediate nodes instead of at the first hop nodes as frequently proposed in literature.
3.7
Summary
In this chapter we look at the problem of resource utilization while disseminating sensed data to the end user in a sensor network. We argue that sensed data would have different levels of importance and need mechanisms to be delivered with different levels of assurances based on the information content in the data. Such information assurance is critical for resource utilization since the network usually incurs overhead proportionate to the level of assurance provided in data delivery. We consider the problem of reliable information delivery. We provide extensions to known reliable dissemination schemes (EER, HHR) and propose a novel Hop-by-Hop Broadcast (HHB) scheme for supporting multiple levels of reliability. To derive the most efficient algorithms for reliable information delivery, we theoretically model the unique characteristics of sensor networks and compare the performance of these schemes. HHR schemes are generally preferred in multi-hop wireless environment due to the presence of high channel errors. However due to the many-to-one traffic characteristics and severe memory constraints in sensor nodes a purely HHR based scheme not only incurs more overhead than EER schemes but also leads to degradation of network capacity and lifetime. Thus the main contribution in this chapter is the accurate reevaluation and comparative analysis of EER and HHR schemes in the context of sensor networks. The analytical models reveal interesting and non-trivial
61
behavior of the network. The non-intuitive nature of the results makes them an important contribution for understanding the behavior of sensor networks and can serve as theoretical guidelines for other network functions such as resource allocation, node deployment strategies.
3.8
Appendix
3.8.1
Buffer Allocation for Minimum Overhead
THEOREM 5.1. A single buffer supporting all the flows in different classes always has lesser overhead
than the case where the buffer is divided into smaller parts to support each reliability class. PROOF: We prove the theorem only for a single hop network having one source and sink and with two
reliability classes with required reliability R1 and R2. The proof can be extended for any arbitrary network. The blocking probability with buffer size Mi is given by bi = (1 − λt trans )(λt trans )M
j
+1
1 − (λt trans )
M j +1
and the
overhead is given by Oi = Ri /(1 − bi ) for each class. Let α i = λi t trans and α 1 + α 2 ≤ 1 (i.e., the network transmission rate is more than the packet arrival rate). The overhead for the multiple and the single buffer cases are as follows:
(
R1 1 − α 1
Omultiple = =
(1 − 2α
Osin gle =
1 − α1
M 1 +1
1
+ α1
)
− (1 − α 1 )α 1
(R1 + R2 )(1 − α 1 M 1 +1
M 1 +1
M1 + 2
M 1 +1
M 1 +1
−α2
)(1 − 2α
+
M 2 +1
M 2 +1 2
(
(
R2 1 − α 2 1−α2
M 2 +1
)
+α2
M 2 +2
)
)
− (1 − α 2 )α 2
R 2α 1
+
M 2 +1
M 1 +1
(1 − 2α
(1 + α α 1
M 1 +1 1
M 2 +1 2
+ α1
)
( R1 + R2 ) 1 − (α 1 + α 2 ) M +1 M +2 1 − 2(α1 + α 2 ) + (α 1 + α 2 ) M +1
( 3.27 )
M 2 +1
) + R α (1 + α α ) )(1 − 2α + α ) M 2 +1
1
2
M1 + 2
M 2 +1
2
M 1 +1
2 1 M 2 +2
2
( 3.28 )
Now for any α 1 + α 2 ≤ 1 and M 1 + M 2 = M , we have:
(1−2α
M1 +1 1
(1−α
M1+1 1 M1+2 1
+α
−α2
M2 +1
)(1−2α
2
)
M2 +1
(1−(α +α ) ) M+1
+α2
M2 +2
)
≥
1 2 M+1
1− 2(α1 +α2 )
+ (α1 +α2 )
M+2
.
Hence Omultiple ≥ Osin gle i.e., the overhead for the single buffer case is always lesser than the multiple buffer case which proves the theorem. Although we have shown the result for only two flows and two sets of buffers, this can be generalized for a set of L buffers for L classes and is in fact also valid for the HHR scheme.
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We note here that a single buffer scheme although minimizes the overhead, while guaranteeing the reliability, does not however give any assurances in delay incurred for different classes of flows. However guaranteeing delays using sophisticated scheduling algorithms on multiple buffers is a wellstudied problem in QoS and is not interesting or different from the perspective of sensor networks.
3.8.2
Optimal Allocation of per-hop Reliabilities in the HHR Scheme
THEOREM 5.2.The number of retransmissions follows a geometric distribution but is limited up to N HHR (i, j ) . The acknowledgement to a packet arrives with probability Rih, j , the associated reliability for
the jth class (due to the limit on the number of retransmissions). If TTimeout is the mean timeout interval for a packet retransmission, TAck is the mean time for acknowledgement and bh-1 is the blocking probability for the next hop nodes, the mean holding time for the jth class is: = ThHold ,j
Th
hold
(1 − (1 − p )(1 − bh−1 ))Rhk, j T Timeout (1 − p)(1 − bh −1 )
+ Rhk, j T Ack
L L (1 − (1 − p )(1 − bh −1 ))T Timeout = ∑ λ h , j Thhold = ∑ λ h , j Rhk, j + T Ack ,j p b ( 1 )( 1 ) − − j =1 j =1 h −1
( 3.29 )
where we assume T Timeout = 2T Ack Now the joint distribution of the ktotal and ntotal can be calculated by summing over all ktotal and ntotal from equation 5.16 and using Thold from equation 5.27 and is given by: L ρ j ,h P[k total = k , n total = n] = Ph (0) ∑ j =1 φ h, j
(λ h Thhold ) n , k+n≤ M n!
The probability that s buffers are full is: L
P[k total + ntotal = s ] = Ph (0)∑ j =1
ρ jj, k S M (λ h Thhold / ρ h , j ) φ h, j
Finally the blocking probability bh that the buffers are full is then given by: M
L
m−0
j =1
bh = ∑ P[ktotal + ntotal = M ] = P(0)∑
ρ j ,h M S M (λhThhold / ρ h, j ) φh, j
63
( 3.30 )
In equation 5.27 and 5.28 we clearly see that by using the transformation λej → λ ′j ,h = R j λej , the distributions reduce to the case of a classical store and forward network, with infinite number of retransmissions, (reliability of delivery equal to 1) but with an input rate of λ ′j . THEOREM 5.3: The allocation of per-hop reliabilities, which minimizes the overhead, is given by: Rhh, j = R j , Rkh, j = 1, h − 1 < k ≤ 1, , H < h ≤ 1 =Rj.
PROOF: We have to solve the following constrained optimization problem to find an allocation
sequence which minimizes the overhead. min
i 1 R kh, j O HHR = ∑∏ , i = h k = h (1 − f i )
h
s.t
∏R
h k, j
( 3.31 )
= Rj
k =1
We assume there is only one class of flow in the network. Let, α i = 1 / (1 − f i ) , a function of the per hop node blocking probabilities. The overhead OHHR per packet is given by: = α h R j ,h + α h −1 R j ,h R j ,h −1 + L + α 1 R j ,h R j ,h −1 L R j ,1
OHHR
OHHR is minimized by the allocation sequence based on the following observations: Observation B1: Any allocation of per hop reliabilities satisfy the condition R j ≤ R j ,k ≤ 1 since h
∏R k =1
j ,k
= R j and R ≤ 1 j
Observation B2: The blocking probability bi is monotonic with respect to the input rate λj,i and hence
αi,j is monotonically increasing with input rate. Let λ j,i be the input rate of packets in the jth class at the ith hop. Since the reliability at each hop is Rkh, j , i +1
the input rate is given by λ j ,i = λ j ∏ Rkh, j . We see that α i is dependent on the input rate at each hop k =h
i +1
λ j ,i = λ j ∏ Rkh, j . Thus in the overhead equation, each term in the summation has the product of per hop k =h
reliabilities and α i which is a function of the product of reliabilities up to that hop. Since α i is
64
monotonic in the input rate from observation 4, the overhead would be minimized for the per-hop allocation which minimizes the product of reliability terms at each hop. For the optimal allocation sequence as proposed in the theorem we have O HHR = α h ( R j ) R j + α h − 1 ( R j ⋅ 1 ) R j ⋅ 1 + L + α 1 ( R j ⋅ 1 K ⋅ 1 ) R j ⋅ 1 K ⋅ 1 = α h (R j)R j + α h(R j)R j + L + α h (R j)R j
( 3.32 )
We see that such an assignment of per hop reliabilities minimizes all the product of reliability terms given the reliability constraint. This follows from observation B1 since we have the minimum value of Rj,i = Rj. This completes the proof. THEOREM 5.4: The allocation sequence given by theorem 5.3 minimizes the blocking probabilities at
each hop and also minimizes the overhead of HHR scheme for any number of classes and sources and also for non-uniform channel error rates and buffer availability in nodes. PROOF: We have already proved the special case in theorem 5.3 which assumed a single source and a
single class reliability class. We see from the previous proof that the allocation sequence minimizes the incoming rate of packets in each hop. When there are multiple classes and sources (as in the network model in this paper), the cumulative rates at each hop (the sum of rates over all reliability classes) are also minimized using the allocation sequence. This is true even with different channel error rates at each hop. Now since the blocking probability at each hop is monotonic with respect to the cumulative incoming rate of packets (from observation B2), the blocking probabilities are also minimized. Hence the overhead for the general case is also minimized.
3.8.3
Latency in Data Delivery
The end-to-end delay for the EER case is derived as follows. Since the source node has unlimited buffer capacity, the expected length of this queuing network is given by:
(λhp+1 (1 − p )t trans ) λ hp+1 (1 − p )t trans − ( M + 1) M +1 p (1 − λh+1 (1 − p )t trans ) 1 − (λ hp+1 (1 − p )t trans ) M +1
E[ LEER ]= h
From Little’s theorem, we have the expected waiting time for a packet in node given by:
65
( 3.33 )
E[WhEER ] =
E[ LEER ] h λh +1 (1 − p )
Since each packet is retransmitted Nˆ EER (h, j ) times for the given reliability, the expected end-to-end overhead for a packet starting at the hth hop is given by:
(
)
h
( 3.34 )
Ack E[WhEER ] = Nˆ EER (h) E[WhEER ] + TEER + ∑ E[WhEER ] j =1
where, Nˆ EER (h) =
1 L ˆ ∑ N EER (h, j ) L j =1
Ack We assume TEER = 2hT Ack the acknowledgement timeout for a node h hops away because the
acknowledgement packet has to travel back h hops. Next we derive the latency for the HHR scheme. Since the HHR scheme behaves as a single flow with cumulative rate λh = ∑ λh , j R j , the waiting time for packets at each hop is given by: ( 3.35 )
E[WhHHR ] = E[k ] / λh + E[l ] / λh L
λ j R j (1 − (1 − p)(1 − bh−1 ) )(1 − bh )T timeout
j =1
(1 − p)(1 − bh−1 )
E[l ] = ∑
( 3.36 )
ρS (λ T hold ) ρ M S M (λhT hold ) 1 ρ M λhT hold S M −1 (λhT hold / ρ M ) M + + E[ k ] = P (0) M h − 1− ρ 1− ρ 1 − ρ 1− ρ
( 3.37 )
The total end-to-end delay for the HHR scheme can be computed by summing the delays at each hop given by the above equations.
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Chapter 4 Minimum Virtual Dominating Sets (MVDS) 4.1
Introduction
Efficient retrieval of sensor network state and network topology control are two of the most important functions in sensor network management. Accurate knowledge of current network state is required for inferring about the performance of the network and to maintain the network at desired levels of performance. Topology control on the other hand is essential for reducing the energy consumption of the network. In this chapter we introduce the concept of Minimum Virtual Dominating Sets (MVDS) which is used in solving the two network management problems in Chapter 5 and Chapter 6 for maximizing the resource utilization in sensor networks. MVDS is a simple generalization of the normal dominating set concept (introduced in section 2.3) which could be used for selecting random sample of nodes with different properties. We also propose a distributed and parameterized algorithm for creating the MVDS in a sensor network. The proposed algorithm is message optimal and has the best known approximation bound. Before we go into the details of MVDS, we first discuss how the two problems stated above lead to the development of the MVDS concept.
4.1.1
MVDS for Multi-Resolution State Retrieval
Knowledge of the current network state is required for different network management functions. Due to large scale and density, the network can consume a lot of energy to retrieve these states. On the other hand, large scale also makes it feasible to infer statistically relevant state information from relatively lowresolution state data. The level of detail required is dependent on the network property being inferred or the network management operation. To retrieve data at different resolutions, we have to select a sample subset of nodes to reply back. However the sampling mechanism may need to adhere to the following constrains based on the requirements of the application.
67
•
Good Statistical Representation: Based on the characteristics and the required resolution of the
state being retrieved, the sample set would vary. The selection should find a good representative sample set based on the requirements. •
Adapt to different density distributions: Sample selection mechanism needs to adapt to the local
non-linearities in distributions. •
Dominating Set: Even though we want random subset of nodes to reply back, it would be good if the
subset of nodes covered all other nodes (or more specifically are dominating). This would ensure that data about all nodes are taken into consideration when the subset is chosen to reply back. We will show in this chapter that the MVDS concept can be used to select dominating set of nodes at different density conditions. The conceptual framework would be used to retrieve the network state at different resolution incurring proportionate overhead and is extremely powerful for maximizing the resource utilization for the network state retrieval problem.
4.1.2
MVDS for Node Scheduling
Large scale and dense sensor networks have many nodes which are redundant for routing purposes. A lot of resources can be conserved if these nodes are kept inactive when they are not used for forwarding packets. Switching off redundant nodes is usually referred to as the node scheduling approach of topology control. Research in this direction has mostly looked at selecting a minimal set of nodes to be kept active at the lowest possible density while maintaining connectivity. However our analysis reveals that low network density adversely affects the overhead of packet transmissions due to significant increase in average path lengths, network congestion and buffer overflows and reduction in path redundancy. Moreover, when packets need to be reliably delivered in mission critical applications, the penalty for low density is even more significant. Based on the network conditions, traffic characteristics etc., the network needs to be maintained at different densities for maximizing the utilization of resources and minimizing the overhead. Again we need a subset of nodes to be selected to be kept active under different constraints as follows:
68
•
Follow Required Density Distribution: Based on the network conditions (channel errors, traffic
rate etc..) the required density distribution of active nodes for minimum power consumption would vary. The node selection mechanism need to adapt to the different requirements. •
Adapt to non-linearities in network conditions: The computation of ideal density conditions would
assume certain network characteristics. The selection mechanisms need to adapt to local aberrations in these assumptions (for example high traffic condition in certain regions). •
Connected Dominating Set: The selected set of nodes need to be a connected dominating set since
they would be used for forwarding packets from all other nodes. Again we use the concept of MVDS to create connected dominating sets at required density conditions. MVDS proves to be a powerful concept for resource utilization in the context of topology control since we find the optimal density for node scheduling under different constraints and is a significant shift from the previous research which has only looked at fixed schemes which aggressively minimize the density. The rest of this chapter is organized as follows. We introduce the concept of MVDS and its associated definitions in section 4.2. We provide a distributed algorithm for creating MVDS in section 4.3. We provide analytical results related the algorithm in section 4.4. We conclude in section 4.5.
4.2
Definitions
Consider a graph G (V, E(r)) where an edge exists between nodes vi and vj if their distance is at most r. For wireless networks, having communication range as r defines the network graph. When the transmission range is unit radius they are more generally called as unit disc graphs [14]. We will consider different subsets of this underlying graph and which would be used to create dominating sets at different density distributions and cardinalities based on the following definitions: •
Virtual Edge Set (E(rv)): The subset of E(r) such that each edge in the set has endpoints at most
distance rv apart. In this case, rv is called the Virtual Range. •
Virtual Graph G(V, E(rv)): The sub-graph of G which has only edges from E(rv ).
•
Minimal Virtual Dominating Set MVDS(r): A minimal dominating set on G (V, E(rv)).
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Communication Range
Virtual
Figure 4.1 MVDS at virtual range equal to the
Figure 4.2 MVDS at a virtual range smaller than
communication range. The two black circles show
the communication range. The three black circles
the dominating nodes.
show the new set of dominating nodes on the virtual graph.
•
Minimal Virtual Connected Dominating Set MVCDS(r): A minimal connected set of dominating
nodes on G (V, E(rv)). Figure 4.1 and Figure 4.2 illustrate the use of virtual range for selecting dominating sets at different cardinalities (the virtual dominating sets). In the first figure, we have the virtual range equal to the communication range (r=rv) and the MVDS is given by the two black circles. In the second figure, the virtual range is much smaller than the communication range and many of the edges of the original graph are missing (shown by dotted lines) in the virtual graph. The MVDS is given by the three black circles in this case. We see that a decrease in virtual range rv causes a decrease in the number of virtual edges and hence an increase in the cardinality of MVDS(rv). Thus MVDS is a relaxed optimization problem of the MCDS version similar to Weakly Connected Dominating Sets (WCDS) [61]. However WCDS does not allow a variable cardinality definition, which makes the MVDS definition much more useful. We note here that the virtual range is not the reduced transmission range but simply a definition on the graph, which allows for variable cardinality of dominating sets. The transmission range still remains r although the dominating set is created with range rv. Intuitively, the virtual radius only acts as a knob to control the cardinality of the dominating set formed.
4.3
Distributed algorithm for the Construction of MVCDS
In this section we describe our algorithm to construct the virtual dominating set. Any node randomly decides to initiate the process (the initiator node). Nodes are selected in layers i.e. first the nodes one hop
70
RECEIVE_REQUEST_PACKET( recvColor, distance, rv ) 1. 2.
if ( (recvColor==BLACK) & ( selfColor==WHITE) ) if (distance < rv )
3.
selfColor = GRAY
4.
FORWARD_REQUEST_PACKET(R_F_T(distance))
5.
if (distance> rv)
6.
selfColor= DARK-GRAY
7.
B_F_T(distance)
8.
FORWARD_REQUEST_PACKET(R_F_T(distance))
9.
if ((recvColor==BLACK) & ( selfColor==DARK-GRAY ) & ( distance< rv ))
10.
selfColor=GRAY
11.
cancel (B_F_T)
12.
if (a request packet has not been Forwarded )
13. 14.
FORWARD_REQUEST_PACKET (R_F_T(distance)) if ( (recvColor==GRAY) OR ( recvColor==DARK-GRAY ) )
15.
if (selfColor == WHITE)
16.
selfcolor = DARK-GRAY
17.
B_F_T(distance)
18.
FORWARD_REQUEST_PACKET(R_F_T(distance))
Figure 4.3:The Coloring Algorithm
away from the initiator are selected and then nodes two hops away and so on. We will see that this layered scheme is required for good approximation bounds for the VCDS. However it also creates a uniformly distributed set of active nodes. This is useful since it prevents any bottlenecks from forming (if the structure and the routing paths are too skewed) on the node-scheduled graph. We also note that the initiator node may be dependent on the application. For example if we are selecting the VCDS for collecting network state, the initiator node is usually the monitoring node. In case of other applications such as node scheduling, the initiator node could be any random node.
71
Dark-gray nodes 2 hops Dark-gray nodes 1 hop
initiator
Figure 4.4 Illustration of the VCDS algorithm. The larger and the smaller circles depict the communication and the virtual range. The coloring takes place in layers (using timer mechanism). The dark gray nodes one hop away 1st change color and then forward a packet. Only after the 1st hop nodes change color, the 2nd hop nodes receive a packet and change color.
4.3.1
Node Coloring Algorithm
To find the VCDS we use four colors. The initiator node sends out a node selection request packet. As the algorithm propagates, different nodes are colored according to their definitions given below: •
White: Yet unmarked node
•
Black: A node in the VCDS which would be used for state retrieval and control. After becoming
black, a node discards all other request packets. •
Gray: A node which is virtually dominated by at least one black node, i.e., it is inside the virtual
range r of the black node. After becoming red, a node discards all other request packets. The node is said to be attached-to the corresponding black node.
72
•
Dark-Gray: A node which receives a packet from a red or blue node or a node which is within
communication range of a black node but outside its virtual range. It waits for a time period for some other node in its virtual range to become black. Otherwise it itself becomes a black node. Initially all nodes are white. The initiating node is colored Black and starts the process by broadcasting a node selection request packet with its node id and color. The basic idea in the node coloring is that all nodes which are within the virtual radius rv of a black node are the dominated nodes and colored gray and all other nodes are colored dark-gray (and thus need to be either covered by some other black node or they themselves become a black node). As the request propagates, each node is colored black, gray or dark-gray according to the coloring algorithm. Figure 4.3 describes the coloring algorithm based on the action taken by a node on receiving a node coloring request packet. The algorithm is also illustrated in Figure 4.4 and explained as follows: •
Lines 1-13 describe the operations when a node receives packet from a black node. Nodes which are within the virtual range of this black node change their colors to gray. All other nodes which receive the packet and are currently colored white change their color to dark gray. Thus all white and darkgray nodes within virtual range of a black node become gray. Each node after changing their color to gray or dark gray forwards the request (broadcasts) after R_F_T(distance) time delay. Note a node may change its color after it has already forwarded a packet (changing from dark gray to gray). In that case it does not forward again. All dark-gray nodes start a timer to become black with a Black Node Formation delay function B_F_T(distance).
•
Lines 14-18 describe operations when a node receives a packet from a gray or dark-gray node. In this case only when a white node receives a packet it becomes dark-gray. It then starts a timer B_F_T(distance), to become black.
•
Now all dark-gray nodes are the candidates to become black nodes. Thus all dark gray nodes have a B_F_T associated with them. However, if any dark-gray node receives a packet from a black node in its virtual range, it cancels its B_F_T and becomes gray. Once nodes are gray or black, they ignore other packets.
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Figure 4.5: Topology of network with (1000 nodes in a 400x400 m2 field, communication range 40m) retrieved at multiple resolutions. In all cases topology discovery is initiated at the center node. The nodes on the tree form the responding MVIDS. The effectiveness of STREAM in selecting the MVIDS is clearly depicted by the uniform distribution of the MVIDS across the sensor field. •
When a node becomes black, it selects the node from which it initially got the request packet as the intermediate node to a parent black node. The parent black node would be the black node at the previous hop to which this intermediate node would be connected. When the algorithm terminates, each node in the network are dominating nodes (the black nodes),
dominated nodes (the gray nodes) and the connecting set of intermediate nodes. The set of black nodes and the intermediate nodes form a Virtual Connected Dominating Set (VCDS) whose cardinality can be controlled using the virtual range as the input parameter to the algorithm. Figure 4.5 shows the VCDS structure of black node set with intermediate nodes, for three different virtual radii.
4.3.2
Timers in Algorithm for MVCDS
In the algorithm described in Figure 4.3, we introduced the two delay functions associated with packet forwarding (request forwarding timer, R_F_T) and node color change (black node formation timer, B_F_T). The timer mechanisms are intended to achieve the following: •
Maximal number of nodes become blue (using R_F_T delay).
•
R_F_T tries to reduce the collisions between forwarded packets
•
R_F_T is designed such that a node h hops away from the monitoring node forwards before a node h+1 hops away thus preserves a layered coloring sequence.
74
Extra Nodes Covered by b b
Nodes Covered by c
a c
Figure 4.6: Illustration for the delay heuristic.
•
The better candidates among the candidate blue nodes become black with a lower delay. (using B_F_T delay).
•
The timers work without any global time synchronization.
Next we derive the timer function and discuss how they achieve the above objectives.
4.3.2.1
Request Forwarding Timer (R_F_T )
Each white node (node which has not received any node coloring request packet) forwards in again after a time delay. This time delay is computed using the R_F_T (distance) function where the parameter distance is the distance between the receiving node and the sending node (We assume that distance between nodes can be approximated by using GPS, signal strengths, etc). The R_F_T (distance) delay function is given by the following set of equations. Tdelay ( d i ) =
Epoch (i ) di
(4.1)
Epoch(i ) = 2πr 2 nTg (t p + c )
(4.2)
R _ F _ T (d i ) = Tdelay (d i ) + Epoch(i − 1) − Tdelay (d i )
(4.3)
Where, Tg = the granularity of the timer. tp = time to broadcast a packet based on the packet size. c = constant time delay added as a guard band
75
n = number of neighbors in the neighborhood. di = distance between sending node at the i-1th hop and the receiving node at the ith hop. Epoch(i) =The time period within which the packet will be forwarded by the node at the ith hop. Tdelay(i)=The forwarding delay from a node at the ith hop. The timer function is designed with the following properties in mind: first it ensures that coloring takes place in layers (i.e. nodes i hops from the initiating node forward before nodes i+1 hops) and second it reduces the probability of simultaneous forwarding of packets by different nodes (and thus reduces collisions). The first property of the timer is very important for proper ordering of the coloring algorithm in the network and is described in detail as follows: Based on the timer equation we describe the properties associated with packet forwarding and the rationale for using them. Property 1: When a node broadcasts a request packet to its neighbors, the nodes at next hop forward
the request packet in the reverse order of their distance from the sending node This property can be seen from equation 3.1 where the Tdelay(i) timer for any node is inversely proportional to the distance, from the sending node. The rationale for selecting such a delay function is explained as follows: Ideally a minimum number of nodes should be to selected based on the application requirement (in this case a connected dominating set of the virtual graph). Moreover we want black nodes to cover the maximum uncovered nodes as the algorithm progresses (also means minimum overlap between black node covers). Note that the dark gray nodes are the candidates to become black in the next hop. Thus maximum number of dark-gray nodes should be formed (and thus will start their B_F_T timers, which are described in the next section). The timer delay tries to achieve this in a greedy manner without any global knowledge of the node neighborhoods sets as is illustrated in Figure 4.6. Consider node a which forwards a packet to its neighborhood consisting of node b and c. Since b is farther away than c, using Tdelay, node b forwards
76
before node c, as it is expected to reach a larger set of uncovered nodes. Hence we have a larger candidate set to become black. Property 2: Choosing the timer delay from equation 3.1 and time period for packet forwarding from
equation 3.2 minimizes the collisions among forwarded packets. Although the distances di of each forwarding node from the sending node are unique with high probability, due to the discrete nature of the system clocks, the delay function may not have unique mapping and can lead to simultaneous packet forwarding in the neighborhood leading to collisions. Let us consider an interval dx such that Tdelay ( x +
dx d ) = Tdelay ( x − x ) 2 2
(4.4)
I.e. due to discrete nature of timers, all timer values in the interval dx are mapped to the same value. To have unique values of timers, we require that expected number of nodes in an interval dx should be less than one. The interval dx corresponds to a ring of width dx in the neighborhood of the node. Thus the expected number of nodes in a ring of width dx at distance x is 2PIxdxn. Since the outermost ring at distance R would have the maximum number of nodes, by computing dx such that it contains one node in the expected, we ensure that any ring of width dx inside would have lesser than one nodes i.e., dx =
1 2πRn
(4.5)
Approximately, the number of rings of width dx that are present in communication range r is = 2PIr2n. I.e. Epoch should be chosen such that due to discrete Tg, we have 2PIR2n discrete levels each separated by packet transmission times. Hence:
Epoch(i ) = 2πr 2 nTg (t p + c )
( 4.6 )
Where c is a constant time delay that may be added as a guard band and the scaling of the delay function by the timer granularity ensures unique timer delays for all nodes within the one hop neighborhood.
77
Property 3: R_F_T (distance) ensures that node at ith hop always forwards the request packet before the
(i+1)th hop Lets assume that each node decides a time period EpochStart, to EpochEnd within which they should forward the request. EpochStart is the time when it first received a request packet. Let us consider two nodes a and b which are i and i+1 hops away respectively from the initiating node. Suppose the node a decides to forward with delay Tdelay(a) in its Epoch, then node b should forward after the EpochEnd(a) of previous hop node. To ensure this, we need to add the value of (EpochEnd(a) - EpochStart(a) - Tdelay(a)) to the Tdelay(b) of the next hop. This is passed inside the request packet forwarded by each node. We note here that here we don’t need the actual time of epoch EpochStart, and EpochEnd but only the time difference EpochEnd - EpochStart. The above scheme would ensure that a node at ith hop would always forward before (i+1)th hop. Epoch in R_F_T(distance) given by equation 26 also reduces the collisions between forwarded packets.
4.3.2.2
Black Node Formation Timer (B_F_T)
Recall that all dark gray nodes are uncovered nodes in the algorithm and hence are candidates to become black unless they are covered by some other node which becomes black in their virtual range. Thus when a white node changes its color to dark gray, it starts a timer to change its color to black. If it receives any packet from a black node in its virtual range within this time frame, it cancels its own B_F_T and changes its color to gray. . The B_F_T function is given by: ( 4.7 )
B _ F _ T ( d i ) = c 2 2r − d i
Where c2 is a constant calculated similar to the Epoch in the R_F_T case which ensures the proper ordering of the coloring sequence. We see in the equation that a node closest to the 2r distance (2 hop distance) from the sending node becomes black first. The rationale for using this function is similar to the forwarding request function i.e. the better candidates among the candidate dark gray nodes become black with a lower delay. The better candidates are the ones which are expected to cover the maximum uncovered nodes and have minimum overlap with a black node in the previous hop. We see that in Figure 4.6, a node at 2r distance from the
78
previous black node usually achieves this objective. Hence B_F_T delay is a heuristic to approximate the greedy selection of black nodes.
4.4
Analyses of VCDS Algorithm
Based on the algorithm described we highlight the following properties of the VCDS. These are essential to derive the performance bounds for the algorithm Property 4: No two black nodes are virtual-neighbors of each other i.e. the set of black nodes actually
form a Maximal Independent Set or more specifically a Maximal Virtual Independent Set (MVIS) since it is formed on the virtual graph. Property 5: The Virtual Connected Dominating Set (VCDS) consists of two sets of nodes, the Maximal
Virtual Independent Set (black nodes) and the intermediate or connector node set. Each black node is connected to at most one black node through one connector node. Property 6: Packets are forwarded in layers according to their hop distances from the root. Thus a node
h hops away from the root node forwards before a node h+1 hops away. This also means that nodes are colored in layers and know the hop lengths from the root, as well as the parent node to the root since it can receive the first packet only from the previous hop nodes. This layered forwarding can be achieved by choosing a forwarding delays as described in the previous section. Property 7: Each node sends at most two messages, one when color changes and the other when an
intermediate node has to notify another black node about its children.
4.4.1
Worst Case performance bounds of VCDS
Although the main aim of using VCDS instead of CDS is to have variable cardinality dominating sets, we note that the algorithm described can be used to create a normal CDS as well (by using virtual radius equal to communication radius). In this section we show that the algorithm described is an important contribution from the perspective of creating a normal CDS with constant approximation bounds and
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optimal message complexity. The results are also valid if the virtual graph is also a connected sub-graph of the original connected graph. If the virtual graph is connected, let opt=|MVCDS| be the cardinality of the minimum virtual connected dominating set. THEOREM 3.1.Any MVIS on G(V, E(rv)) is at most 4opt+1. PROOF: The case for any MIS on a unit disk graph was proved in [84]. The case for virtual graph is not
different since the MIS is created on a disk graph of virtual radius and hence the result holds for the virtual graphs as well.
THEOREM 3.2. If the virtual graph is connected, the size of the Virtual connected dominating set
(VCDS) is within a constant approximation bound given by VCDS ≤ 8opt − 3 . PROOF: Let |MVIS| be the cardinality of the maximal independent set of black nodes. The black node tree
would consist of at most |MVIS|-1 intermediate nodes. Hence the cardinality of VCDS is at most 2|MVIS|-1.From lemma 2, the approximation bound for the VCDS is given by VCDS ≤ 8opt − 3 .
THEOREM 3.3. The VCDS algorithm has a message complexity of O(n) and a time complexity of
O(n2). PROOF: From the algorithm description we see that each node whether a black node or a gray node sends
at most two messages. Thus the message complexity is O(n). For the time complexity, we note that each node needs to have a unique time delay to change color since otherwise two or more nodes can become black at the same time. Hence each node needs to select a time delay from O(n) time period so that n nodes have n unique delays. Since the maximum number of hops is n-1, the worst case time complexity is O(n2). However the time complexity can be reduced to O(nd) by considering the degree of each node where d is the maximum degree of the graph.
Let us compare our bounds with the two known algorithms for CDS with constant approximation bounds. In [68], the CDS construction has approximation bound of 192opt+48 with message and time
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complexity O(n). In [84], the CDS construction has approximation bound similar to ours (8opt) but has a message complexity O(nlog(n)) and time complexity of O(n). Thus our algorithm has the message complexity of O(n) but the worst time complexity O(n2). It also has the best known approximation ratio with optimal message complexity of n. Both the CDS approximation algorithms described in [84] and in this paper rely on a spanning tree structure to attain the optimality bound. In [84], the spanning tree construction itself takes O(nLog(n)) steps. In our case the spanning tree is implicitly created while the algorithm progresses. This is achieved with only n messages using unique delays for each node and layered forwarding based on hop lengths from the root resulting in the worst time complexity of the three algorithms but the best message complexity. We note however that the spanning tree construction in O(n) steps is only possible in broadcast networks where only one message is required to communicate with all neighbors.
4.4.2
Average Case Analysis
In this section we analyze the expected cardinality and the node distribution properties of the VCDS formed using probabilistic methods. We first look at the cardinality of the black node set and then see how many intermediate nodes are added to form the VCDS. The black node set in the VCDS is a maximal virtual independent set. Thus a node in the graph can be black only if all its neighbors are not black. Consider different runs of the algorithm with different root nodes. Suppose a particular node i, becomes black, bi times out of the k different runs of the algorithm. Then we define, pi = probability of node i to become black for a randomly selected initiating node = bi / k for large k. We assume that pi is dependent only on the local degree of node (di) and uncorrelated with the rest of the network. The independence assumption is not valid for a particular run of the algorithm but for a large-scale network and large number of runs. Thus the expected number of times each node becomes black across all runs of the algorithm is the expected number of times all its neighbors become gray. Then for each node the following set of equations are satisfied: d1
p i = ∏ (1 − p j )
( 4.8 )
where p j = j th neighbor of i.
j =1
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Multiplying the set of equations for all i, each 1-pj appears di times in the set of equations. Thus we get: N
N
∏ p = ∏ (1 − p ) i =1
i
j =1
( 4.9 )
dj
j
Let E[ pi ] = pˆ . For large N and large number of runs k, pi for each node tends to the expected value pˆ . N
From law of large numbers, as ∏ p i → pˆ N , we get: i =1
d Nd pˆ N = (1 − pˆ )∑j j = (1 − pˆ )
( 4.10 )
d ⇒ pˆ = (1 − pˆ ) 2
d = Nrv / R 2 − 1 , the expected virtual degree.
For sufficiently large d the above can be approximated using: pˆ MVIS =
ln(d ) − ln(ln(d )) d
for
( 4.11 )
d ≥1
The expected cardinality of VCDS is thus O (N ln(d ) / d ) , i.e., at large scales, it actually behaves like a random d-regular graph which also has a dominating set cardinality of O (N ln(d ) / d ) ([79]). Equation 3.11 is valid only when the expected number of nodes within a virtual region is greater than 1. As the virtual range reduces, this value can become less than 1. This means that each node has less than one node in its virtual range on an average. Under such conditions we should consider the binomial distribution of the number of virtual edges to find the expected number of black nodes. Let E = Expected virtual degree of a node N −1 N − 1 πr 2 1 − E = ∑ x x A x =1
x
πr 2 A
N − x −1
( 4.12 )
The total number of virtual edges in the network is half the sum of the expected degree of all nodes, i.e., 0.5NE. Since the number of virtual edges is less than the number of nodes (recall d1
for
d ≤1
( 4.14 )
Finally we account for the extra intermediate nodes which connect the chosen set of nodes to form the connected tree. We see that any node is dominating with probability p (given by equation 3.11) only if it falls outside the virtual range of another black node. Using area arguments, the probability that an intermediate node is also dominating is:
(
2
)
( 4.15 )
P = p MVIS r 2 − rv / r 2
Hence the cardinality of MVCDS is given by:
(
(
2
) )
( 4.16 )
VCDS = Np MVIS 2 − p MVIS r 2 − rv / r 2
The expected cardinality of VCDS is thus O (N ln(d ) / d ) , i.e., at large scales, it actually behaves like a random d-regular graph which also has a dominating set cardinality of O (N ln(d ) / d ) [79]. Figure 4.7 plots the average number of black nodes formed in simulations compared to the number of black nodes from the analytical expectation. The analytical expectation is very close to the simulation results. In Figure 4.8 and Figure 4.9 the performance of the algorithm is compared against centralized greedy algorithm (Cormen, Leiserson, Rivest (1990)) to find the black nodes. The nodes for this simulation are uniformly spread in 200m x 200m field and the virtual range is equal to the
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50
60
STREAM
STREAM
50
Number of Black Nodes
Number of Black Nodes
70
Centralized
40 30 20 10 0
40
Centralized
30
20
10
0
20
25
30
35
200
40
Communication Range
400
600
800
1000
Number of Nodes
Figure 4.8: Number of Black nodes vs
Figure 4.9: Number of Black nodes vs. total
communication range for STREAM and
number of nodes for STREAM and Centralized
centralized greedy algorithm. The size of sensor
greedy algorithm. The communication range is
field is 200x200m2 and the field has 1000 nodes.
30m and sensor field dimensions are is 200x200m2
communication range. In Figure 4.8 and Figure 4.9 the impact of communication range and increasing the number of nodes in the field is shown for 1000 nodes in the field. In both cases, STREAM performs almost as well as the centralized solution which has global knowledge. This result shows that the heuristics used in STREAM timers perform well.
4.4.3
Request Propagation and Black Node Tree
The black node set along with the set of intermediate nodes form the virtual connected dominating set. The set also forms a tree structure with the initiating node as the rooted node. In this section we discuss some of the properties of this black node tree which is implicitly formed during the algorithm. THEOREM 3.4. Each node receives the first request packet in minimum number of hops. PROOF: Consider a node which is h hops away from the initiating node. From property 3 due to the
R_F_T mechanism, a node which receives a packet that has traveled h-1 hops, always forwards before a node which receives a packet which has traveled h hops. Thus the first packet the node receives always travels the shortest number of hops from the initiating node. THEOREM 3.5. If the network has symmetric links, the black node tree is optimal in the number of
hops. As a consequence every black node and intermediate node is optimal number of hops from the sink in the retrieved network graph. PROOF: The next hop to parent node for any node is the one from which it received the request packet
first. Since the first packet is received in the optimal number of hops (proposition 1), by sending the
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topology response packet to the next hop to parent node, the initiating node can be reached in optimal number of hops. Since the aggregation tree is included in the retrieved topology, by default, every node in the black node tree is optimal number of hops away from the monitoring node in the retrieved network graph.
4.4.4
Adapting the algorithm to Handle Node Mobility
Next we briefly describe how the VCDS structure is maintained in the presence of node mobility. There are three kinds of nodes in the VCDS, the dominating black nodes, the connector or intermediate nodes and the dominated gray nodes. We consider the mobility of each of these cases separately. Movement of gray nodes
When a dominated gray node n moves to a new location, i.e., moves out of virtual range of its dominator black node, it becomes a dark gray node and sends out a message to query for any black node in the new neighborhood. If a black node in its virtual neighborhood replies back, it becomes a gray node. Otherwise it changes itself to a black node. Next it sends out a 3-virtual-hop broadcast message to connect itself to its neighboring black nodes. This is required since although the original structure has at least one black node within two hops, once the new black node is added to the structure, that condition would not hold. The intermediate nodes for each black node in its three-virtual-hop-neighborhood are also added. Movement of Intermediate nodes
An intermediate node is the one which connects a child node to its parent. Between the two pairs if an intermediate node leaves, only the children black nodes associated with it have to react. When the dominator node learns that the intermediate node has left, it sends a three virtual-hop broadcast message to connect itself. It always finds at least one black node if it is still connected to the network and adds the intermediate nodes associated with the connections. Movement of Black Nodes
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When a black node moves to a new location it first sends a broadcast to find if any of its new neighbors is black. If finds a black neighbor, it changes its color to gray. If it does not find any black neighbor it remains black but acts as a new black node and performs operation described in first case of node movement. Reconstruction of Black Node Tree
Finally the structure can be recomputed by running the complete algorithm again. The initial algorithm based on MIS and the black node tree can be started fresh from a random node after some time interval. Since the message complexity of the entire reconstruction is O(n), it is not a bad idea to reconstruct the tree after some sufficient time interval since our algorithm has very good approximation bound which we will show in the next section. When the VCDS is restructured due to node mobility, our algorithm would behave similar to one described in [68] with similar approximation bounds.
4.5
Summary
In this chapter, we introduced the concept of minimum virtual dominating sets and discussed why it is required for resource utilization in the network state retrieval and topology control applications. We provided a distributed, parameterized algorithm for creating connected dominating sets at varying density distributions. The algorithm has currently the best known approximation bounds at linear message complexity. We also provide analytical results on the average case performance of the algorithm and show that the parameter virtual range can be computed analytically and used in practical networks to create the VCDS at varying cardinalities. In the next two chapters we will extend this concept and use the proposed algorithm for retrieving network state at multiple resolutions and scheduling nodes for topology control at varying densities.
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Chapter 5 Multi-Resolution Network State Retrieval using MVDS 5.1
Introduction
Large-scale sensor networks need energy-efficient mechanisms to extract the current network state for various aspects of sensor network management. Some network properties can be inferred from a relatively low-resolution representation of network state. Different resolutions suffice for different management applications to perform at a desired level. In these cases, it is an overkill to retrieve the entire state of a large-scale network particularly because sensor nodes are energy constrained. Retrieving the network state according to the requirements of the applications can lead to good utilization of resources. In this chapter we specifically look at the network topology and propose an algorithm for Sensor topology Retrieval at Multiple Resolutions (STREAM) at proportionate cost. STREAM is a parameterized algorithm whose parameters could be generalized to handle other network state retrieval queries such as residual energy. We also describe various classes of state retrieval queries suitable for sensor networks. The parameters for the queries are derived to minimize the overhead under the constraints of the required resolution.
5.1.1
Multi-Resolution Topology Retrieval
Network Topology is an important attribute of the network state as it aids in network management and performance analysis. Accurate knowledge of network topology is a prerequisite to many critical network management tasks, including proactive and reactive resource management, server siting, event correlation, root cause analysis, growth characteristics and even for use in simulation for networking research. In sensor networks ([59]), where redundant, cheap nodes operating on battery, are used to form a largescale dense network, energy-efficiency is the key criterion guiding the design of network protocols. Accordingly any topology discovery algorithm should also be energy-efficient. We show that different
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levels of partial topology suffice for determining different network properties. In fact, this observation opens up a new avenue for energy-efficient topology discovery where we can save a large amount of energy by aiming to retrieve only the partial topology without paying much in terms of inference of network characteristics. To further motivate our problem we consider the following observations about large-scale dense networks: •
Sensor networks typically would have a high density, since sensing range is much smaller than communication range (E.g. in heat sensors and motion sensors). A network barely sufficient to provide sensing coverage could be sufficiently dense for providing communication coverage. We found that many topological properties in dense networks can be inferred accurately using a lowresolution representation of topology. For example, if the network administrator needs to test the robustness of the network by verifying that each node in the network has at least three disjoint paths (to a monitoring node), using STREAM she only needs to recover 17% of the edges (in a network of 1000 nodes with average degree 25). This results in reduction of 83% in energy consumed with respect to retrieving the complete network graph.
•
Different topology resolutions are required for different applications to perform at a desired level. For example, estimating the overhead incurred in disseminating data to the monitoring node requires single source shortest path lengths (SSP). On the other hand, network-wide distribution of source coding symbols requires all pairs shortest paths lengths (ASP) ([10]). For SSP, on retrieving only 10% of the network edges (for a 1000 node network with average degree of 25) the average single source shortest path length increased by only 6% of the hops from the optimal. Similar bound (6% increase in hop length) for ASP required 25% of the network edges.
•
Many sensor network specific properties such as field exposure and coverage ([100], [101]) and connectivity [22] also converge quickly towards the real values of the network even for at fairly low granularity of location information. Thus we could save resources by retrieving the topology just above the critical value.
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•
In some situations, network administrators may not be willing to spend more than a given amount of energy to retrieve certain network state. Given the energy budget we should be able to retrieve the maximum amount of information about the network. Note that the energy budget allotted to infer different network properties would also depend on the importance of the network property evaluated and how much the administrator is willing to tolerate in terms of the property degradation. The above observations clearly highlight the need to have an ability to retrieve topology at multiple
resolutions. To this end, we describe a distributed parameterized algorithm for Sensor Topology Retrieval at Multiple Resolutions (STREAM) that trades between topology details and resources expended in large scale, dense networks. STREAM provides the flexibility to control the topology resolution and the overhead incurred in the process. The concept of Minimum Virtual Dominating Sets forms the basis for extracting the topology at the required resolution. For topology discovery we derive rules for optimal parameter selection under different constraints. We describe various types of topology discovery queries relevant for sensor networks and applications that can use these queries and use the rules to map the queries to the parameter values in STREAM. Moreover, the algorithm is generalized to extract other network states such as energy map, and takes application specific parameters (e.g. based on the specific types of state queries) for adaptive spatial sampling of network states. STREAM utilizes the broadcast property of wireless medium called the Wireless Multicast Advantage ([54], [55] A node can detect the presence of its neighbors by eavesdropping on the communication channel. Thus by selecting a subset of nodes, approximate topology can be created by merging their neighborhood lists. The resolution of the topology depends on the cardinality and structure of the chosen set of nodes. For example, to construct a minimal backbone tree of the network, we only need to merge the neighborhood lists of the minimum connected dominating set of the network graph. STREAM runs in two stages. First a monitoring node, which requires the topology, sends a topology discovery request to all the nodes in the network by controlled flooding. The request contains two parameters called virtual range and resolution factor. These parameters are used to select a minimal set of nodes required to retrieve topology at a desired resolution. The selection of the appropriate set makes
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use of the VCDS concept and creates a black node tree (as described in chapter Chapter 4 for the VCDS) rooted at the monitoring node. In the second phase, the black nodes reply back to the request with a subset of its neighborhood list, determined by the resolution factor. Each black node aggregates the topology data received from its children black nodes and sends it to its parent in the tree. Finally at the monitoring node the aggregated topology at the required resolution is available at the end of this stage. The following factors make STREAM extremely suitable for sensor networks: •
The MVCDS is created using message complexity of N (number of nodes in the network), and compares well to a centralized scheme. Use of only one packet per node makes it highly energy efficient. The above is made possible by using local timer mechanisms (which do not require any time synchronization)
•
The cardinality of the MVCDS is dependent only on the network field dimensions, the communication radius, and required resolution and is almost constant with respect to density of the network. Hence the message complexity does not increase with increase in density of the network.
•
The entire message complexity to retrieve topology is proportionate to the resolution retrieved. If x% of the topology (see definitions in section 2) is retrieved, the message complexity is N(1 + x + c), (0 ≤ c ≤ x ≤ 1) .
5.1.2
The Network Model
The sensor network topology is a connected disk graph with a fixed circular communication range R (the radius of the disk). We assume that there are no node failures and channel errors, and nodes can collect their neighborhood lists by eavesdropping on the communication channel1. For complete evaluation of STREAM, we relax some of these assumptions in section 5. We assume that the network has a central monitoring node, which keeps track of the current network state and issues management requests such as state retrieval to the network. Based on the application
1
Note that each node must send at least one packet for other nodes to know its existence. A topology discovery request from
each node ensures: 1) All nodes receive a packet; 2) Nodes have complete neighborhood lists
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requirements, the monitoring node decides an appropriate resolution for the state (in this case, the topology). In this work, we define the resolution of the topology as: −
Edge Resolution: Ratio of the number of retrieved edges and the actual number of edges in the
network. −
Node Resolution: Ratio of the number of retrieved nodes and the actual number of nodes in the
network. To evaluate the performance of STREAM we describe two trivial approaches for multi resolution topology discovery for comparison. •
Direct Response: When a node receives a topology discovery request, it forwards this message and
immediately sends back a response with its neighborhood list along the reverse path. In the probabilistic variation, nodes decide to reply back with some probability p and report all their edges. •
Aggregated Response: When a node receives a packet, it forwards the request immediately but waits
to aggregate the information from its children before sending its own response. In the probabilistic variation, all nodes respond but report each of their edges with probability p. The probabilistic variations can be used to retrieve topology at multiple resolutions. In the first case, since nodes respond back randomly, aggregation is not guaranteed. In the second case, since all nodes are reporting back, the number of responses is same as its non-adaptive counterpart, albeit with a reduced per- response cost. In the probabilistic variations, no guarantees can be provided that each node will be covered.
5.2
Sensor Topology Retrieval at Multiple Resolutions (STREAM)
STREAM selects a subset of nodes to reply to the topology discovery query with their neighborhood information. The cardinality of this subset determines the resolution of retrieved topology. The conceptual framework, which allows STREAM to control the cardinality, is a Minimum Virtual connected Dominating Set. STREAM creates a VCDS(rv) based on the virtual range rv. Since a decrease in rv causes a decrease in the number of virtual edges, the cardinality of VCDS(rv) increases as rv decreases. STREAM selects the
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nodes in VCDS(rv) to respond to topology discovery queries with rv providing a resolution control parameter. STREAM consists of two phases. In the first phase the subset of nodes are selected. In the second phase, the nodes respond back to the monitoring node with their neighborhood lists. We describe the two phases next.
5.2.1
STREAM Request Phase
The request propagation in STREAM uses the VCDS algorithm to select a subset of nodes to respond back with the topology discovery request from the monitoring node. The algorithm takes three user-specified parameters that control the topology resolution from the minimal backbone tree to the complete network graph. The parameters are defined as follows: −
Virtual Range (rv ∈[0,R]): The virtual range controls the cardinality of the VCDS.
−
Resolution Factor (f∈[0,1]): Each member of the VCDS reports this fraction of edges
originating from it. For example, if f=0.4, then a node should return 40% of its neighborhood list. −
Query Type (Q): This defines the set of queries, which STREAM can support. The query type
maps to specific filters and aggregating functions to support more sophisticated queries. Thus, each query is of the form: STREAM (rv, f, Q). Here only the virtual range rv is used to select the set of nodes to respond back. The parameter virtual range is computed based on the required resolution of the request, the query type and other factors (these will be discussed in detail later). Recall that coloring algorithm for the construction of the VCDS. The algorithm starts at the monitoring node (which needs the topology at some resolution) and is colored black. At the end of this phase, there are three sets of nodes, the black nodes (dominating nodes), the gray nodes (the dominated nodes) and the connector/intermediate nodes. The set of black and the connector nodes form a Black node tree (or the VCDS), rooted at the monitoring node. In the coloring algorithm, each node sends out at least one packet. Since wireless is a broadcast medium, all nodes can have complete neighborhood lists, just by eavesdropping on the communication channel. At the end of the first phase, thus each node can select a subset of edges from their complete neighborhood lists to respond back to the topology discovery query.
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5.2.2
STREAM Response Phase
During the first phase, the monitoring node becomes the root of the black node tree where the parent black nodes are at most two hops away from their children black nodes. Each node has the following information at the end of this period: −
The parent black node, which is the last black node from which the topology discovery was forwarded.
−
The intermediate connector node to which it should forward packets in order to reach the parent black node.
−
Complete neighborhood information by eavesdropping on the communication channel. Using the above information, the response action is described as follows:
•
When a node becomes black, it sets up an acknowledgement timer (described later) to reply to the discovery request. Each black node waits for this time period during which it receives responses from its children black nodes.
•
It aggregates all topology information from its children black nodes and randomly adds a fraction f of edges from its own neighborhood list.
•
When its acknowledgement timer expires, it forwards the aggregated neighborhood list to the intermediate connector node to its parent black node.
•
In the general framework for information retrieval, the parameter query-type is used to filter or aggregate the information. However for topology discovery we just merge the neighborhood lists.
5.2.3
Acknowledgement Timer
For the algorithm to work properly, timeouts of acknowledgements should be properly set. The acknowledgement timer of a black node should always expire before its parent black node so that each black node forwards only after receiving responses from its children. For this we set a timeout value inversely proportional to the number of hops a black node is away from the monitoring node (the number of hops is obtained from the discovery request packet). We need an upper bound on the number of hops
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between extreme nodes. If the extent of deployment region and communication range of nodes is known initially, the maximum number of hops can be easily calculated.
5.3
Analysis of STREAM
In this section we state some of the analytical results related to STREAM. Specifically we want to analytically find the retrieved resolution based on the STREAM parameters and it tradeoff with the incurred overhead. All the results assume that the node density is uniform and that the network is connected. We verify our analytical results with simulations, for which we modified the NS-2 simulator to incorporate details of STREAM. We use the following terminology in this section: N=number of nodes A= area of sensor field r= communication radius rv =virtual radius f= resolution factor 2 dr = average virtual degree for virtual radius rv = Nπrv − 1 . We omit the subscript if clear from the A
context. D = average node degree = dR.
5.3.1
Retrieved Topology Resolution
STREAM takes two parameters, which control the resolution of the returned topology. Thus for a given virtual range(r) and resolution-factor(f), the returned topology resolution is computed as follows: Recall from chapter 3 that the fraction p of black nodes formed for a virtual range r is given as:
p ( rv ) =
ln( d ( rv )) − ln(ln( d ( rv ))) d ( rv ) E 1− 4
for
d ( rv ) > 1
for
d ( rv ) ≤ 1
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( 5.1 )
Figure 5.1: Expected Resolution of topology as a
Figure 5.2: Expected Byte overhead of STREAM for
function of parameters Virtual Range and Resolution
different values of parameters (, N=1000, R=40m,
Factor. Each contour on the base shows the range of
400x400m). Constant packet overhead C=5 bytes,
parameter values for topology returned at a particular
Overhead per edge information Cf = 2 bytes. The
resolution.
contours show the equal overhead points.
Each of these black nodes formed, returns a fraction f of its edges. Also recall that two or more black nodes may be formed in the same communication range if virtual range is smaller than communication range. Since we assume edges to be symmetric, we should account for the edges which are reported by both its endpoints. Thus both nodes associated with any edge have probability p(rv) of becoming black. When a black node reports any edge with probability f (the resolution factor), the actual probability of that edge being reported is:
(
)
X ( p ( rv ), f ) = p ( rv ) 2 f 2 + 2 f (1 − f ) + 2 p ( rv ) f (1 − p ( rv ) ) = 2 p ( rv ) f − f p ( rv ) 2
( 5.2 )
2
e( rv, f) = number of edges reported back = X(p(rv), f)ND
( 5.3 )
Figure 5.1 shows the plot of expected topology resolution as a function of STREAM parameters virtual range(r) and resolution-factor (f). The contours at the base are the iso-resolution curves obtained by intersection of a resolution plane with the curve. Each contour shows the set of (r,f) pairs which return the same resolution. We show in section 5.4, how these can be used for selecting the parameter values for specific queries.
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5.3.2
Overhead for retrieving topology at different resolutions
In this section we derive the overhead for the retrieving topology at various resolution. We consider the packet overhead and the overhead of multi-resolution topology data carried in the packets. First we consider the packet overhead of the algorithm. For the initial phase of the algorithm, since all nodes forward the topology discovery request at most once, the overhead is N. During the acknowledgement phase, a subset of nodes are involved namely the black nodes and the intermediate nodes between two black nodes. Hence the total number of packets transmitted (Op) in the acknowledgement phase is equal to the sum of number of black nodes and the number of intermediate gray nodes. Since each black node except the monitoring node has one intermediate node, the total expected overhead is given by: 2 r 2 − rv O P ≤ N + Np 2 − p r2
( 5.4 )
Next, the overhead incurred by STREAM is analyzed in terms of total bytes transmitted. Let, Cp = constant header overhead per packet. Since the packet overhead is given by Op, the overhead due to the constant packet overheads is simply CpOp. To compute the overhead of edge information, we need to estimate the number of hops the retrieved edge information travels, to reach the monitoring node. For simplicity of exposition we assume that the sensor field is circular with radius RA and the monitoring node is at the center of the deployed region. From reference [114] we know that for sufficiently dense graphs, the maximum number of hops H, any node is away from the monitoring node is given by: H = ceil (R A / R )
( 5.5 )
w.l.o.g. let RA=rH We can divide the circular region into H concentric rings of width r. The nodes in the ith ring are i hops away from the monitoring node. Thus all edges which are reported by nodes in the ith ring, travel i hops to
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the monitoring node (This is enforced by the aggregation tree properties). Let us now find the expected number of edges reported in the ith hop. Expected number of edges in the ith ring =
ND π (ri) 2 − π (r (i − 1) ) 2 πR 2
Expected number of edges reported in the ith ring Ei ≤ X ( p, f )
2
ND r 2 (2i − 1) 2 R2
( 5.6 )
( 5.7 )
Since each edge can be reported at most twice through independent paths, the total overhead due to the edges reported back is given by: H
O E ≤ C e 2∑ iE i = X ( p, f )C e ND i =1
R2 R A2
H
∑ i(2i − 1) i =1
2
R 1 1 H ( H + 1)(2 H + 1) − H ( H + 1) 2 R A2 6 = C e NDcX ( p, f )
= X ( p, f )C e Nd
( 5.8 )
Where Ce = Overhead per edge information sent Hence the total overhead OT of the process is give by: R 2 − r 2 OT ≤ O P + O E = C p N + Np 2 − p R 2 + C e NDcX ( p, f ) = C1 1 +
( 5.9 ) R 2 − r 2 + C 2 X ( p, f ) p 2 − p R 2
Where, C1, C2 includes the constant terms which are not dependent on the STREAM parameters. We state two important observations from the derivations above. These observations would later be used for optimal parameter selection for topology retrieval. Observation 1: For a given resolution, the overhead is a function of only the p, and independent of f.
We see that the second term in the overhead in equation is constant for a given topology resolution (X(p,f) is the required resolution) and the first part does not have the parameter f. Observation 2: Given a resolution, the overhead is monotonically increasing and convex with respect
to p. The second term in overhead is constant for a given resolution. Hence we have to show that the first term is monotonically increasing. I.e.
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Probabilistic Direct
70
Probabilistic Aggregated
120
60
Direct
100
50
Aggregated
Percentage
Percentage Gain
80
40 30 20
Retrieved Edge/ Actual Edges Overhead / Maximum Overhead
80 60 40 20
10
0 40
0 5
7.5
10
15
20
35
55
80
100
35
30
25
20
15
10
5
0
Virtual Range
Topology Resolution in % of Edges
Figure 5.3: Relative Gain in Overhead of STREAM
Figure 5.4:. Graph shows the relative overhead (with
over Aggregated, Direct and their probabilistic
respect to maximum overhead) incurred in retrieving
variations.
topologies at different resolutions. (N=1000 nodes, r= 40m, and field size 400x400 m2)
2 r 2 − rv Np 2 − p r2
2 r 2 − rv Np 2 − p r2
is monotonically increasing in p.
2 r = 2 Np − Np 2 1 − v = 2 Np (1 − p ) + Ap 2 d r 2 r2
( 5.10 )
Again the first term is monotonically increasing. We approximate the second term as follows:
(ln(d ) − ln(ln(d ) ) ln(d ) − ln(ln(d ) p2d ≈ d= d d 2
2
( 5.11 )
The above is monotonically decreasing for d (i.e. increasing w.r.t p). Hence the overhead is monotonically increasing w.r.t p. Next we consider the plots for the overhead functions. Figure 5.2 plots the STREAM overhead against different virtual-ranges and resolution-factors. Figure 5.3 and Figure 5.4 show the actual overhead incurred by STREAM. In evaluating the byte overhead, we assume a constant packet header of 5 bytes and an additional 2 bytes of information per edge reported in the packet. Total bytes transmitted during the entire operation characterizes overhead. Figure 5.3 shows the percentage extra overhead incurred by direct response and aggregated response and their probabilistic variations over STREAM to retrieve equal resolution topologies. The probabilistic variations perform better than their non-adaptive counterparts, but are significantly worse than STREAM.
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Moreover, STREAM guarantees that each node would be reported whereas the probabilistic variations cannot provide such a guarantee. Figure 5.4 shows the relative overhead for different virtual ranges. Observe that the relative overhead curve (as a percentage of the overhead incurred for retrieving the complete topology) closely follows the resolution. The relative difference at low resolutions is due to the constant overhead of the coloring phase of the algorithm.
5.4
Mapping Topology Retrieval Queries to STREAM Parameters
In this section we describe various types of queries that would be relevant in the context of sensor networks. We also provide the rules for selecting appropriate parameter values to be used in STREAM for these different queries. While this is not meant to be an exhaustive list of queries, it would encompass a broad range of applications. STREAM algorithm can be invoked using three parameters: virtual range, resolution factor and query type and take the form of STREAM (rv, f, Q). Before we go onto the different classes of Topology retrieval queries, we look at the relationship between parameter values, the retrieved resolution and the overhead. We state the following two theorems which provide rules for optimal parameter selection. THEOREM 4.1. To retrieve topology at a required resolution Tres, the minimum overhead is achieved by
selecting parameter r and f as follows: • If Tres < X(p(r) ,1), put rv=r, and find f such that X(p(r) , f)= Tres • If Tres ≥ X(p(r) ,1), then put f=1, and find rv such that X(p,f)= Tres. Thus the general rule is to first keep rv=r and increase f up to 1 and then decrease rv until we get the required resolution. PROOF: The proof of theorem 4.1 is the solution to the following non-linear optimization problem: Min(OT ( p, f )) s.t. X T ( p, f ) = Tres 0 ≤ f ≤ 1,
( 5.12 )
p( R) ≤ p ≤ 1
99
We see here that constraints of the algorithm are actually hidden in the above formulation. The above formulation ensures that at least a spanning tree of the network would be formed (since min(p)=P(r) ). From equation the overhead equation, we see that the above optimization can be reduced to the following: Min[F ( p )] 2 r 2 − rv F ( p ) = p 2 − p 2 r
where
( 5.13 )
s.t. X T ( p, f ) = Tres 0 ≤ f ≤ 1,
( 5.14 )
p(r ) ≤ p(rv ) ≤ 1
The monotonicity of F(p) makes the solution even simpler. We have to find minimum p, for which X(p,f)=Tres. Since X(p,f) is also monotonic, first put p=p(R) (the minimum value of p), and find f such that X(p(r), f)=Tres. If X(p(r), 1) < Tres i.e. maximum value of X such that p=p(r), is less than the required resolution, find p(rv), by solving X(p(rv), 1)=Tres for p(rv). Since the overhead function is convex, the solution is also a global maxima. THEOREM 4.2. Given an overhead OM, to retrieve topology at the maximum possible resolution Tres,
select parameter r and f as follows: • Put r=R, and find f such that OT(p(R) , f)= OM • If OT(p(R) , f)< OM, then put f=1, and find p(r) such that OT(p(R) , 1)= OM. Thus the general rule is to first keep r=R and increase f up to 1 and then decrease r to get higher resolutions until the overhead constraint is met.. PROOF: The proof of theorem 4.2 is the solution to the following non-linear optimization problem: Max( X ( p, f )) s.t. OT ( p, f ) = OM 0 ≤ f ≤ 1,
( 5.15 )
p( R) ≤ p ≤ 1
We write the Lagrangian for the above formulation: max L( p, f , µ ) = X ( p, f ) + µ [OT ] = X ( p, f )(1 + µ ) + µF ( p ) + µC1 ≡ Max[F ( p ) + µ ′X ( p, f ) + µ ′C1 ]
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( 5.16 )
Since the functions X(p,f) and F(p), are convex and monotonic, we have reduced the optimization problem in theorem 4.1 to the dual problem. From Lagrangian duality theorem for convex functions, if (po, fo) is the solution to primal problem (minimize F(p)), then there exists µ0 ≤ 0 such that (p0, f0 , µ0) solves the dual problem. Hence we get the necessary rules for selecting parameters. The intuition behind the above rules is that they are maximizing the amount of information per packet so that the effect of constant overhead is minimized. Note that, the retrieved topology at a desired resolution might be required to possess certain properties. The trade-off here is between the overhead and maintenance of graph properties. For example, to test whether each node has two edge-disjoint paths to the sink, the retrieved topology should have minimum node degree of two. However, exploring these tradeoffs is orthogonal to the focus of this paper and is part of our future work. In this work, we try to minimize the overhead or maximize the retrieved resolution for which theorem 4.1 and theorem 4.2 are used to select the parameter values. We use simulations (averaged over 20 runs with randomly chosen initiating nodes) to show the effectiveness of STREAM in dealing with different types of queries.
5.4.1
Node Constrained Query:
The node-constrained query is of the form: “Return a representation of the network with x% of the nodes”. Such a query helps in determining the approximate density distribution of the network while not requiring all the nodes to be returned. The query can also be used for other node centric queries such as the residual energy levels for nodes and by selecting a subset of the nodes we can create samples of the network residual energy at different granularities. For this query, we construct the minimal backbone topology by setting the parameter virtual range (rv) equal to communication range r and resolution-factor (f) equal to the desired fraction x. From theorem 4.1 we know that such a selection of parameter values will give the minimum overhead for the query. The following lists various forms of the node-constrained query: −
STREAM (r, x, Node-Constrained-Query).
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Returned Node Resolution
TABLE I.
Required Node Resolution
Average
Maximum
Minimum
0.8 0.6 0.4 0.2
0.813 0.621 0.439 0.238
0.832 0.646 0.456 0.269
0.791 0.605 0.416 0.213
Performance of Node-Constrained Query on a 1000 node topology in a 400x400m2, R =40m.
−
STREAM (r, 0, Node)
−
STREAM (rv, 0, Exposure)
−
STREAM (r, f, Residual Energy) When the parameter Node-Constrained-Query is passed, each black node finds its neighbor nodes,
which have not been reported by its children black nodes. Out of these unreported neighbors, it picks x% of nodes and aggregates with the children’s information before sending it upstream. The second query is used to find the virtual backbone of a network. In the query-type Exposure-Query, each node just returns its location. Finally in the residual energy query, the nodes return their residual energies and x% of the residual energies of the neighbors. The number of nodes responding is controlled by virtual range r. Table 1 gives the performance of node-constrained query using STREAM for the different required resolutions in the simulation setup described earlier.
5.4.2
Edge Constrained Query:
The form of an edge-constrained query is: “Return x% of the edges in the network”. This type of query is useful in determining the connectivity properties of the network. We evaluate properties such as shortest paths lengths and number of node disjoint paths in the network at different topology resolutions. To extract topology at the required resolution with minimum overhead, we use the solution in theorem 4.1 to find parameters rv and f. The resolution-factor is set to 1. Note that setting resolution-factor to 1 ensures that at least one edge pertaining to each node is reported back. Let, E= expected total number of edges in the topology. E = ND 2
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Since every black node reports all edges originating from it, the total number of edges reported back only depends on the number of black nodes chosen to reply back. To get the required fraction of edges (x) the following condition must be satisfied. 2 xE Bd = xE => B = D 2
This gives the virtual radius as follows: rv =
A Nπ
ln(x ) , if 1 + ln(1 − x )
ln(x ) >1 ln(1 − x )
( 5.17 )
Otherwise rv is given as: rv =
4 A(1 − x) Nπ
( 5.18 )
The topology discovery query takes the following form: −
STREAM(r, 1, Edge-Constrained-Query). The edge-constrained query is central to the multi-resolution topology extraction problem we are trying
to solve. The overhead equation for this type of illustrates the exact nature of trade-off between the resolution of topology retrieved and communication overhead expended. The edge-constrained query also illustrates the convergence properties of the multi-resolution network graph. Let us compute the average hop deviation for shortest paths from monitoring node as estimated from the partial retrieved graph as compared to the actual. From proposition 2 in section 4.4 we know that every black node and intermediate node is optimal number of hops away from the sink in the retrieved graph. All other nodes are at most two hops away in the retrieved graph since they are always neighbor of some black node. Hence the average deviation is bounded by the number of gray nodes which are not intermediate nodes. For required edge resolution x, this is given by: 2 r 2 − rv N gray = N − Nx 2 − x r2
2 r 2 − rv H deviation (x ) ≤ 21 − x 2 − x r2
( 5.19 )
( 5.20 )
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Returned Edge Resolution
Required Edge Resolution
Virtual Radius
Average
Maximum
Minimum
0.8 0.6 0.4 0.2
6.38 9.03 11.93 20.45
0.710 0.556 0.424 0.179
0.715 0.561 0.427 0.188
0.706 0.546 0.421 0.175
TABLE II. Performance
of Edge Constrained Query on a 1000 node topology in a 400x400m2, R =40m.
Thus if we retrieve x% of the edges using edge-constrained query the average deviation is bounded by equation 4.21. Thus STREAM can give guarantees on hop deviation and 100% node coverage which is not possible with probabilistic variations. Moreover the overhead incurred is lesser which makes STREAM suited for sensor networks. Table 2 shows the mapping of required edge-resolution to the value of parameter rv for the simulation setup described at the beginning of the section 5.4. Using the above queries, we see that STREAM is able to retrieve edges at resolution close to the desired values. Figure 5.4 showed that multi-resolution topology is retrieved at proportionate cost. Figure 5.5 plots the effect of edge resolution on the length of single source shortest path lengths (rooted at initiating node) and all-pairs shortest path lengths. The graph shows the relative deviation in the path lengths as the topology resolution increases. We see that even for low resolutions, paths do not deviate significantly from the optimal. Figure 5.6 plots the number of nodes and edge disjoint paths for different resolutions. We note that the savings in topology discovery increases as the density of the graph increases, i.e., lesser percentage of edges is required as density increases for similar deviation from optimal behavior. Also as density increases lesser percentage of nodes become black due to which the relative message complexity decreases. Using STREAM we can attain significant savings by discovering topology at the required resolution in dense sensor networks.
5.4.3
Overhead Constrained Query
This query has the form: “Return the best resolution topology using a given Overhead budget”. The total overhead in topology discovery is due to request forwarding (network-wide flooding) and the
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"All Pairs Shortest Paths" 100 90
Lost % of Edges
1200
Deviation: Shortest Paths
1000 # of nodes
Percentage
80 70 60 50 40
800 600 2 Paths
400
3 Paths
30
4 Paths
200
20 10
0
0 40
35
30
25
20
15
10
5
0
0
5
10
15
20
25
30
35
40
virtual range
Virtual range
Figure 5.5: Effect of Edge Resolution: Deviation
Figure 5.6: Effect of Edge Resolution on number
from All-pairs and single source shortest path
of Node Disjoint Paths to Sink. (N=103, r=40,
lengths, (N=1000, R=40, field size 400x400 m2)
400x400 m2)
discovery acknowledgement. To support this query, we use solutions from theorem 4.2 to select the optimal parameter values. In general, the analytical computation of parameters by this method is not required if we assume uniform random graphs. Setting resolution-factor to 1 and then selecting the minimum possible virtual range for the given overhead would retrieve the maximum resolution. However, other considerations may come into account when selecting the parameter values. For example we might want to further constrain the query so as to maintain error bounds on a particular estimate of graph property. The parameter selection in such a case would involve considering the analytical bounds derived on graph properties as derived earlier.
5.5
STREAM Under Arbitrary Network Conditions
In this section we analyze the performance of STREAM under arbitrary network conditions involving non-uniform topologies, sleeping nodes, channel error rates and node failures. We also propose methods which make STREAM adaptive to these varying network conditions.
5.5.1
Non-Uniform Topology
We had assumed a uniform node distribution for the network topology. Although this was useful to analyze the behavior of STREAM, practical sensor networks may not be uniform. Moreover, we may not have any information about the underlying distribution. For STREAM to be useful in practice, it should be able to function properly even without knowledge of node distribution.
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Consider the network to have any arbitrary distribution where a node i has some degree di whose distribution is unknown. For a uniformly distributed network we could select a constant virtual range across the network. When the distribution is non-uniform, each node has to compute its own virtual range based on the local network density and node degree. Consider the circular range of the node i with di neighbors. Let pi be the probability of nodes inside this region to become black. Expected number of black nodes in the region = pi di. Assuming that node distribution is uniform inside the communication range2, the number of edges reported for this region should be a fraction, x of the total expected edges in this region. Hence, we have to choose a virtual range so that pi=x where x is the required edge resolution.. Let the virtual range for node i be ri. ri =
r2 di
ln(x ) 1 + (1 − x ) ln
( 5.21 )
And for ln( p ) < 1 , we have ln(1 − p )
ri =
4r 2 (1 − x) di
(5.22 )
Thus every node computes its own virtual radius according to its local information. The coloring scheme and the algorithm remain the same while using these local values of virtual range. We evaluate this scheme using two different non-uniform topologies. In the first one we consider a square field of dimension 400x400 m2 and divide it up in four quadrants of size 200x200 m2. We put 100, 200, 300 and 400 nodes in the first, second, third and the fourth quadrant respectively resulting in different densities in each quadrant. In the second case we generate a gaussian distribution of 1000 nodes around the center of a 400x400 m2 field and variance 75m.
2
While the node distribution across the network is non-uniform, the distribution inside a node’s communication range can be
considered to be uniform when the range is much smaller than the network dimensions.
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Required Edge Resolution
Edge Resolution on Topology 1 assuming uniform
Returned edge resolution with local scheme
Edge Resolution on Topology 2 assuming uniform
Returned edge resolution with local scheme
0.8 0.6
0.66 0.497
0.706 0.56
0.503 0.353
0.71 0.56
0.4
0.38
0.411
0.23
0.43
0.2
0.160
0.184
0.087
0.208
TABLE III.
Performance of Edge Constrained Query on non uniform topologies of 1000 nodes in a 400x400m2, R =40m.
Table 3 shows the results of performing edge-constrained queries of the two non-uniform topologies. In the first topology, we divide up the network field in 4 quadrants with 100, 200, 300 and 500 nodes respectively in the four quadrants. In the second topology, we generate a gaussian distributed topology with the mean at the center of the field and standard deviation of 75m. We see that by assuming the topology to be uniform and computing a single global value of virtual range results in large deviations in the returned edge resolution compared to the required edge resolution. By using local virtual ranges the returned resolution is very near to the required resolutions and performs as good as with the uniform topology case.
5.5.2
Impact of Sleeping Nodes
One of the distinguishing features of sensor networks is that nodes would sleep periodically. In this section we analyze the impact of sleeping nodes on the returned topology resolution. We compute the fraction of sleeping nodes that would be reported by the responding active nodes. We assume that active nodes cache data about their neighbors and respond with information about these sleeping neighbors. Also we assume that the network of active nodes is connected. Let, x=fraction of nodes sleeping, ⇒(1-x) N = N’ is the number of active nodes. p =probability of node to become black ps=probability of a sleeping node being reported D D i D −i bx=expected black neighbors of each node = p∑ i(1 − x ) (x ) i i =1
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( 5.23 )
120
Number of Black nodes for Different Channel Errors 1200
100 Number of Black nodes.
% of Nodes Retrieved
1000
80 60 Range 40 m
40
Range 30m Range 20m
20
Error =0 Error= 0.1
800
Error=0.3
600 400 200 0
0 0
10
20
30
40
50
60
70
80
40
90
35
30
% of Nodes Sleeping
25
20
15
10
5
0
Virtual Radius
Figure 5.7: Effect of sleeping nodes on the Retrieved Topology.
Figure 5.8: The number of black nodes formed for different
The network consists of 1000 node in a field of 400x400m2.
channel error rates. 1000 node deplyed in 400x400 m2 field with
Simulations are carried out for three different communication
R=40m.
ranges with varying % of nodes sleeping.
Each active node is always reported because the network of active nodes is connected. The probability of a sleeping node being reported is the probability of that any one of its bx black neighbors report it. Since black nodes report a fraction f of their edges, the probability is given by: 1 − (1 − f ) x px = bx f b
bx ≥ 1 bx < 1
( 5.24 )
We tested the performance of STREAM with varying percentage of sleeping nodes. The simulations are set up for rv=r and f=1. The results in Figure 5.7 show that STREAM is able to retrieve information about nearly all the nodes if the number of active neighbors of each node is around 8. The results also imply that if the active node set is connected, then we get information about almost all nodes.
5.5.3
Handling Channel Errors
Sensor networks are envisioned to be deployed in uncontrolled environment with significant channel errors and frequent node failures. To make our proposed algorithm useful in practical scenarios, we have to ensure that it is fault tolerant. We first discuss the effect of channel errors and node failures during the request propagation phase. Channel errors affect the request propagation phase in the following ways:
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Average Deviation in Hops from the Optimal for Different Channel Errors
Number of Black Nodes for different Channel Error Rates.
400
Wthout correction for channel error
350
With correction for channel error
2
300
1.8
Without correction for Channel Error
1.6
With correction for channel error
1.4
250
Number of Hops
Number of Black Nodes
450
200 150 100
Hop Deviation for Black Nodes
1.2 1 0.8 0.6 0.4
50
0.2 0 0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
0
90
0
Channel error in %
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
Channel Error
Figure 5.9: The number of black nodes formed for
Figure 5.10: The average hop deviation of the
different channel error rates. 1000 node deplyed in
received request packets for different channel error
2
rates. 1000 node deplyed in 400x400 m2 field with
400x400 m field with R=40m, and r=40.
R=40m, and r=40. •
Increase in number of black nodes: Since packets may be lost when a black node forwards a request, a node may become black even if it is inside the virtual range. The effect is shown in Figure 5.8 and Figure 5.9.
•
Proposition 1 and 2 are no longer valid. This means that request divergence may not have a wave like propagation. Figure 5.10 illustrates this effect.
•
Timers may not work since nodes do not receive a topology discovery request packet in optimal hops.
•
All nodes may not receive a request packet. However for dense networks, this effect is negligible A node becomes black if no other node in its virtual range is black or if packets from one or more black
nodes in its virtual range are lost due to channel errors. Thus we need to incorporate the channel errors to get the analytical model for number of black nodes. Thus a node becomes black not only when it does not have any black virtual neighbors but also when a packet from a black neighbor is lost due to channel errors. Thus the modified equation to find the probability to become black is given by: d
p = ∑ e i p i (1 − p )
( 5.25 )
n
i =0
One may choose a value of virtual radius according to equation 4.26 to get the required resolution. We may also select timers such that it incorporates the expected deviation related to the error rate. However
109
computing the values may be too complex for sensor nodes. We choose a simpler method (albeit at a higher cost) by which the number of black nodes and hop deviation is reduced significantly. We see in Figure 5.8 and Figure 5.9.that increase in number black nodes and hop deviation is negligible for channel error rates lesser than 10%. This means that at such channel error rates, we could use the formulation where we had assumed zero channel errors. Let 0 < c < 0.1 be some channel error rate for which the effect is negligible. Suppose we send multiple copies of the request packet per node, neighbor nodes would receive the packet with higher probability because of this redundancy, thus reducing the apparent channel error rate. In particular to make the apparent channel error close to c, the number of copies m required is:
m=
ln (c ) ln(e )
( 5.26 )
However if each node sends m request packets the overhead would increase significantly. Instead if only a black node sends m copies, overhead is significantly lowered since black nodes are much lesser in number. We can use the scheme of only black nodes forwarding multiple packets if it is able to reduce the effect of channel errors significantly. We assume that nodes have knowledge of their local channel error rates. We use c=0.05 to compute the value of m. We use the scheme where only black nodes forward m copies. In Figure 5.9 and Figure 5.10 we show the effect of this scheme on the number of black nodes and hop deviation for different channel error rates. We see that the number of black nodes and average hop deviation is stabilized even for high channel error rates. By stabilizing the number of black nodes and average hop deviation for any channel error rate, the analysis in section 5.3 remains a good approximation. Hence the parameters for edge and node constrained queries can still be mapped as described in 5.4. The bounds on the properties of retrieved topology are also maintained with high probability. Since m increases sub-linearly with increase in e the overhead increases sub-linearly if the number of black nodes is small compared to the total number of nodes in the network.
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5.5.4
Handling Node Failures.
Sensor networks would also experience frequent node failures due to the hostile terrains they may be deployed or may die off due to energy depletion from their batteries. The effect of node failures during the request propagation phase is equivalent to the effect of sleeping nodes since nodes do not participate in the topology discovery if they are dead or sleeping. However a node may die in between the topology request phase and the acknowledgement phase. During the request propagation phase, black nodes send back their aggregated topology information upstream. Due to channel errors topology information is lost if an acknowledgement packet is lost. Similarly if any node in the black node tree dies, the information from its sub-tree would not reach the sink. A passive acknowledgement scheme from ([30]) is used to account for channel errors. We utilize the high density of sensor networks to account for node failures. These schemes are described as below: After a black node sends its aggregated edge information to the next hop to its parent black node it eavesdrops on the communication channel to see if the next hop node has forwarded its packet. If it does not hear within a certain time period it forwards the packet again. Since the next hop may be dead, the number of retransmissions cannot be allowed to go on indefinitely. We have to limit the number of retransmissions. From equation 4.27, we know that if a packet has been retransmitted m times, then it reaches the next hop with probability 1-c. If the black node does not receive the passive acknowledgement after m tries we may infer with high confidence that the next hop node is dead. After m retransmissions, a black node selects another node from its neighborhood with a gradient towards the sink (if the black node is h hops away from the sink, it selects a node which is h-1 hops away from the sink). Since the network density is high there are multiple such nodes with high probability. Each node just has to cache the information which it gathers during the request propagation phase. Let k be the number of neighbors of a node with a gradient towards the sink. If the node failure rate is f and local channel error rate is e, the probability that a packet is forwarded successfully to a node next hop is given by:
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Virtual range
Expected resolution to be retrieved
40 30 20 10 5
0.07 0.15 0.219 0.62 0.81
TABLE IV.
Retrieved Edge Resolution, e=0.15, f=0.05 0.16 0.262 0.304 0.706 0.859
# of nodes Retrieved e=0.15, f=0.05 982.46 980..5 995.57 999.95 999.71
Retrieved Edge Resolution, e=0.3, f=0.1 0.182 0.28 0.326 0.731 0.91
# of nodes Retrieved e=0.3, f=0.01 978.952 963.047 994.5 999.578 999.6
Performance of Edge Constrained Query with different channel error rates and node failures, 1000 nodes in a 400x400m2, R =40m.
P( success) = 1 − (1 − (1 − e )(1 − f ))
mk
( 5.27 )
Using the above method, information from active black nodes can be sent to the sink with high probability. For example with e=0.3, f=0.1, m=3 and k=2, gives P(success)= 0.999622. Thus we see that even if a node retransmits a packet at most three times, and has only two next hop nodes, then even for a channel error of 30% packets from downstream are lost with very low probability. However the neighborhood information of a dead black node is lost. To compensate for this loss, when an intermediate node infers after m transmissions that its black parent is dead, it adds its own neighborhood list to a packet when forwarding it upstream. Thus we are able to retrieve information about nodes which are in range of both the black node and the intermediate node. The only problem with this method is that there is no way to compensate for the death of a node in the last hop. We have to also take care of the timers for nodes which forward when some node in the black node tree dies. After receiving the topology request each node starts an acknowledgement timer in case it is required. With this method the layered aggregation semantics remain valid. This completes the description of our algorithm. Table 4 gives the performance of the edge-constrained query for different channel error and node failure rate. In the first case we look at mild conditions of 15% channel error and 5% node failure. In the second case we look at a harsher condition of 30% channel error and 10% node failure rate. The node failure rate is a relatively high since it is unlikely that 10% of the nodes would die within a time period of topology discovery. We see that retrieved edge resolution is always higher than the expected resolution although the average number of nodes, which are included in the retrieved topology, is not 1000. This is because to compensate for high failure rate, other intermediate nodes insert their own neighborhood lists.
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We also see that at low resolution there is a large difference in the expected and retrieved topology which reduce at higher resolutions. This is because at higher resolutions percentage of gray intermediate nodes is very low and the extra edges added by them correctly compensates for the missing information due to dead black nodes. This result is also favorable for sensor networks since we would ideally like to have better control on retrieved as resolution increases because retrieved resolution directly affects the overhead incurred in the process.
5.6
Summary
In this chapter, we look at resource utilization in the context of network state retrieval. We concentrate on the efficient retrieval of sensor network topology and propose an algorithm for sensor topology extraction at multiple resolutions (STREAM). STREAM may also be used as a general-purpose multi resolution information retrieval algorithm. For topology discovery the node selection process finds a suitable MVDS. In general, the node selection could depend on any characteristic of the information sought. For example, if the desired information has spatial gradient and neighborhood correlations, then such information can determine the virtual-range. The parameters resolution factor query-type and determines the filtering and compression of neighborhood done at the sampled node set. The resolution-factor determines the number of neighbors about which a reporting node responds. Each node can collect information about its neighboring nodes by eavesdropping. The responding node can choose a fraction f of these neighbors and apply the filter specified by the query-type parameter. Even, in case of topology discovery a more sophisticated aggregation scheme as proposed in [12] could be used. A different type of aggregation could be used for energy information as proposed in [64]. Finally use of schemes like smart messages as proposed in [24] which would carry specific code along with the query to support infrequent queries could save valuable memory resources in sensors. STREAM opens up a wide design space for multi-resolution information retrieval. This would require knowledge of the properties of queries and the corresponding aggregation/filter functions to compress that property. We intend to explore this design space for different types of information in future.
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Chapter 6 Topology Control Using Node Scheduling with MVDS 6.1
Introduction
In infrastructureless ad hoc networks with battery-powered nodes, it may be difficult and expensive to replace or recharge nodes drained of energy. Minimizing the network power consumption is not only important for maximizing the network lifetime but also for improving the maintainability of the network. Based on the current network conditions, a good topology control protocol can save resources while maintaining the network at the levels of performance required for the application the network is deployed. In this context topology control is one of the most important network management mechanisms used for resource utilization in sensor networks. Topology control can be achieved either by transmit power control or node scheduling. In transmit power control the optimization problem is to find the transmission power (while maintaining connectivity) at which the power consumed is minimum. In node scheduling, nodes redundant for routing are switched on and off using a scheduling algorithm. In this chapter we focus on node scheduling to optimize the power consumed for Reliable Packet Delivery in ad hoc networks. We derive theoretical models for the behavior of the network when nodes are switched off and derive the conditions for minimum power consumption. We also propose a distributed algorithm for scheduling nodes at density conditions which adapt to the changing network conditions and application characteristics. The node scheduling problem is solved under the constraints of application characteristics thus leading to a prudent approach for conserving resources in the network.
6.2
Topology Control Using Node Scheduling
Most node scheduling algorithms find a minimum set of nodes required to maintain connectivity, usually with approximations to Minimum Connected Dominating Set (MCDS). MCDS minimizes the idle
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power dissipation and the power consumption from unnecessary reception of packets in nodes redundant for routing purposes. However switching off nodes can change the properties of the network graph and may actually increase the power consumption. The intuitive idea that minimizing the network density may not achieve minimum power consumption forms the main motivation behind the analyses in this chapter. We specifically highlight the adverse effects of keeping low network density on the overhead of reliable packet delivery and derive density conditions for scheduling nodes to minimize power consumption. Low network density may affect overhead of reliable packet delivery in three ways. First, at low density, the average hop-length between nodes increases and may increase the power consumption. Second, many protocols such as the hop-by-hop broadcast (HHB) reliability protocol (chapter 3.5), rely on high network degree (not available at low density) to reduce the overhead. Third, at low network density, the few active nodes become congestion points. If these nodes are memory constrained, this causes buffer overflow and hence increases overhead for reliable packet delivery, even leading to network instability. Thus network density plays a key role in the overhead of reliable delivery and to minimize power consumption we need to find the optimal density based on current network conditions. Further, given that we can find the optimal density we need a practical, distributed and adaptive algorithm which can achieve the required density. In this chapter, we show how to find the optimal density and design a practical node-scheduling algorithm based on virtual dominating sets that achieves the desired density. We first analytically derive a relationship between the density of active nodes and the distribution of shortest path lengths (section 6.4). The analysis shows that the expression for path lengths used in prior work ([48], [92]) is not an accurate approximation at low densities. Next, we analyze the power consumption of reliable delivery and highlight the performance of the HHB protocol at different densities in section 4.1. We analyze the effect of constrained buffers in section 6.7. The results show that operating the network at low density is counter productive. Finally we propose a practical node scheduling algorithm based on the theoretical models. The proposed algorithm can achieve optimal density conditions in a distributed manner while being adaptive to different network conditions and parameters
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(section 5). Thus, the results in this chapter provide design guidelines for both theoretical analysis and practical implementation on topology control using node scheduling.
6.3
Network Model and Assumptions
We first describe the model and assumptions used in analysis of node scheduling. The nodes in the network are assumed to be distributed in a circular field of radius R according to a Poisson process. The methodology described can be easily extended for other network shapes and density distributions. The communication range of each node is fixed at r (no power control). We assume that the fraction of nodes kept active using some node-scheduling algorithm (possibly a connected dominating set) is also Poisson distributed (this assumption is relaxed). For a particular realization of the network, we assume that N nodes are uniformly distributed in the field and a fraction X of the N nodes form the active set. Thus we have: Avg. density, δ = NX / πR 2 , Avg. degree= d = δπr 2 − 1
( 6.1 )
The MAC layer is assumed to be based on TDMA scheduling [67] such that collisions and interferences do not occur. Channel error is a network-wide constant (f) and occur independently at each link. Generalization to non-uniform packet error rates (e.g. based on SNR) is part of future work. We use a simple traffic model where nodes generate a constant number of packets per unit time (λp) and send it randomly to some other node possibly multiple hops away from the source is. We assume a geographic routing protocol for packet forwarding. The energy consumption in node is divided in to four categories: ET Joules= Energy consumed per Packet Transmission ER Joules= Energy consumed per Packet Reception EI J /second = PI watts =Power consumption in Idle Mode ES J/second = PS watts =Power consumption in Sleep mode Etotal J/second = Ptotal watts = total power consumption
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Active nodes consume PI at all times even when receiving or sending packets (this corresponds to the power to keep the node circuitry running). Thus, the total power dissipation normalized with respect to the number of nodes in the network is: PTotal = λ P (ET OTrans + E R ORe c ) + PI X + PS (1 − X )
( 6.2 )
OTrans=Total number of packets transmitted. ORec=Total number of packets received. OTrans denotes the expected number of packets transmitted over multiple hops including overhead due to retransmissions between a source and sink. ORec denotes the overhead when nodes in the neighborhood of the transmitting node receive all these packets.
6.4
Impact of density on Shortest Path lengths
In this section we study the relationship between the density of nodes active for routing and the average hop length between nodes. The results in this section highlight some of the inadequacies in papers such as [48] and [92] which approximate the hop lengths as y / r for transmission radius r and distance between nodes y. The analysis for average path lengths is similar to ones in [47] and [70] with some important differences. The derivations consider all forms of trajectories between source and sink in the probability distribution whereas in [47] and [70] the progress is assumed along a straight line joining the source and sink. We also take into account the unequal distribution of the distances between randomly chosen points in a circle (and hence unequal distribution of the hop lengths) for minimizing the power. To derive the analytical model for the average number of hops in the network as a function of density for a pair of nodes separated by distance y we assume a geographic forwarding scheme with no holes in the network. Thus a packet progresses to the node closest to the sink. I.e., all next hop neighbors lie in the intersection of two circles of radii r (communication range) centered at source and y (the distance between source and sink) centered at the sink, as shown in Figure 1. Let x = radial distance of the forwarded node next hop and sink z =radial progress from the source towards the sink =y -x The probability that the next hop node is at a distance larger than z (i.e., the CDF) can be shown to be:
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source
r z
a x y
sink
Region of next hop neighbors Figure 6.1 Computing the expected number of paths assuming a geographic forwarding scheme. r 1 − exp − ∫ 2a ( y − t )δdt t=z FZ ( z ) = P ( Z ≤ z ) = FZ ( 0 )
Where, a =
(
cos −1 x 2 − y 2 − r 2 2 yx
( 6.3 )
)
r FZ 0) = 1 − exp − ∫ 2a ( y − t )δdt t =0
( 6.4 )
Fz(0) is the normalizing factor obtained by putting FZ (0) = 1 ( to account for the assumption that there are no holes in the network and there is at least one node closer to the sink). The PDF is simply given by:
f Z ( z) =
d FZ ( z ) = dz
r
(2δa( y − z ) ) exp ∫ 2a( y − t )δdt t = z FZ (0)
( 6.5 )
Next we define h (y, z) = 1/(y-z), the hops per meter progressed at distance y. To compute the number of hops from source to sink we integrate the expected hops per meter over the distance y from source to sink. 0
h( y ) = E ( h( y, z )) =
z =h
f Z ( z) dzdh (h − z ) h= y z =h−r
( 6.6 )
∫ ∫
Finally to find the average all pair shortest paths, let fY (y) be the PDF of the distance between two randomly chosen Poisson distributed points inside the circular field. The expected number of hops is: hˆ = E[h( y )] =
s =2 R
∫
s =0
0 z =h f Z ( z) dzdh f Y ( s)ds ∫ ∫ h = y z = h − r (h − z )
( 6.7 )
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Analytical Expectation
R
v
R
34
R [v]
Actual topology
32
average hop distance
R [v]
v
30
Expected Hops as a function of Density
28 26 24 22 20 18 16 14
Figure 6.2 The bounded region R shifted by
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Density of Active Nodes
vector v. The density of the parametric is proportional to the intersection area of the R
Figure 6.3 Comparison of the equation 3 with actual hops between a pair of nodes. R=600m, r=40m,
and R[v].
N=10000. We still need to compute the function fY(y) (look ahead to equation 6) in order to complete the analytical model. To derive this, consider a parametric formulation of a d dimensional bounded region R and any point defined by the vector v[x1,…,xd]. The following theorem is used to compute the density distribution of distances between randomly chosen pair of points inside a bounded region. THEOREM 5.1. The density distribution of a pair of points with magnitude and direction given by a
vector v in a bounded region R is proportional to the area formed by the intersection of R and a copy of R shifted by the vector v, (R[v]). Proof: We prove the above using area arguments (refer to ). For pair of points in a bounded region, R is
the possible starting points and R[ v] the set of end points for the any pair of points with direction and magnitude given by vector v. The density of such pair of points among all possible pair of points in the bounded region is proportional to the intersection area of the two regions. The concept is illustrated in Figure 6.2 for two different types bounded regions. Although the theorem 5.1 is conceptually simple, it gives us an elegant way of handling any bounded region of different arbitrary shapes. Applying this principle to unit disks, we see that the intersection area is symmetric and equal along any direction of vector v. The overlap area of circles shifted by distance y is: 1
4
∫
y/2
4 − y2 y − 1 − x 2 dx = 2 tan −1 4 − y2 2 y
( 6.8 )
Now to find the density distribution of a pair of points with magnitude |v|=y, we use the parametric form of the vector v=[ysin(t), ycos(t)] and integrate along the entire circular region. In the parametric
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form for the distances, y is the first quadrant of the circle ( π / 2 ) and the number of vectors with magnitude y is proportional to y. The integral of y from 0 to 2 is simply ( π / 2 ). Hence the density distributions (PDF and CDF) is given by: f Y ( y) =
4− y2 y 4 tan −1 y π y
FY ( y ) = ∫ f Y ( y )dy =
− y 4− y2
(
4 − y2 − y − y2
0
2π
( 6.9 )
) + 2 sin π
−1
y 2 2 −1 + y tan 2 π
( 4 − y / y) 2
( 6.10 )
We use fY(y) given by equation 6 in equation 5 to compute the expected number of hops as function of density. For infinite network density, the above equation would tend to the trivial approximation of hop lengths using y / r . However for networks with finite densities, the above formulation accurately estimates the path lengths. Figure 6.3, shows the number of hops between a pair of nodes for different densities from a randomly generated topology and compares with the analytical computation using equation 3. We see that the analytical model closely resembles the actual hops for the generated topology; thus, verifying our claim. An important observation is that the number of hops at low densities is large and has significant cost on the network overhead. In this case of the source destinations pair, with only 10% of the nodes active, the hop lengths double and increase by 30% with around 30% of the nodes active. Impact of this significant increase in hop length cannot be neglected especially for reliable transmissions.
6.5
Effects of Increased hop lengths on reliable delivery Overhead
To reliably transmit a packet between nodes, protocols employ end-to-end retransmissions (typically at the transport layer) or hop-by-hop retransmissions (typically at the link layer). We compute the impact of density on the overhead for both the cases.
6.5.1
End-To-End Retransmission
Since channel errors occur independently at each link, the number of transmissions for successful delivery follows a geometric distribution. Since the retransmissions stop only when packet travels h hops
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Overhead (HRR): Impact of Channel errors
Overhead (EER): impact of channel errors 150
450 400
300
140
Error=0.1
130
Error=0.3
120
Packet overhead
Overhead
350
Error=0.05
250 200 150 100 50
Error=0.1 Error=0.3 Error=0.5
110 100 90 80 70 60
0
50 0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.1
Fraction of active nodes
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Density of active nodes
Figure 6.4 The total expected power consumption 3
Figure 6.5 The total expected power consumption
using EER. N= 10 , R=400m, r=40m, ET = 1 units,
using N= 103, R=400m, r=40m, ET = 1 units, ER =
ER = 0.1, PI = 0.01, PS = 0.005, λp=1. Large dots
0.1, PI = 0.01, PS = 0.005, λp=1 Large dots indicate
indicate the minimum overhead point
the minimum overhead point
to the destination and the acknowledgement travels back h hops, the expected number of transmissions is given by: T=
1 (1 − f )2h
( 6.11 )
The expected number of hops each copy of the packet travels from the source is also geometrically distributed. Thus the expected number of times each copy of the transmitted packet is forwarded is: 2 h −1
1 − (1 − f ) f
2h
Oi (h) = ∑ (1 − f ) = j
j =0
( 6.12 )
Total expected number of packets generated to transmit a packet reliably over h hops is then: T
OEER ( h) = ∑ Oi ( h) = i =1
1 − (1 − f ) 2h f (1 − f )
2h
( 6.13 )
The expected number of packets generated is,
OEER =
2R z
1 − (1 − f )
∑ f (1 − f ) h =1
2h
2h
( 6.14 )
f H ( h)
The total power consumption is then given by: PEER = λP (ET OEER + ER dOEER ) + PI (1 − X ) + PS X
( 6.15 )
Figure 6.5 shows the overhead computed for different densities and different channel error rates using end-to-end retransmission schemes.
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6.5.2
Hop-By-Hop Retransmission
Reliable delivery of packets may also be provided using link-layer hop-by-hop retransmissions. Here, the expected number of transmissions per hop is given by: OT =
1
( 6.16 )
(1 − f )2
The probability of an acknowledgement packet being generated for any transmission is 1-f. Hence the expected number of packets (the data packet and its acknowledgement) generated per packet at each hop is 2-f. The overhead for h hops is then given by: h h 2− f 2− f + PI (1 − X ) + PS X + ER d ∑ PHHR (h) = λ P ET ∑ 2 2 i =1 (1 − f ) i =1 (1 − f ) 2− f (E + E d ) + PI (1 − X ) + PS X = λP h (1 − f )2 T R
( 6.17 )
The expected overhead is given by: E[PHHR (h)] = λ P = λP
2− f
(ET + E R d )E[h] + PI (1 − X ) + PS X
2− f
(ET + E R d )hˆ + PI (1 − X ) + PS X
(1 − f )2 (1 − f )2
( 6.18 )
Figure 6.4 shows the expected overhead for different densities of active nodes using hop-by-hop retransmission schemes.
6.6
Effect of Reduced network degree at low density
Many protocols in ad hoc networks exploit the broadcast nature of the wireless channel and high network degree to reduce their overhead (e.g., [49], [11]for routing and [54] for creating multicast trees). The Hop-by-Hop Broadcast (HHB) reduces the number of retransmissions for reliable delivery by exploiting the high network degree at high density. Low network density leads to a low network degree and may affect the performance of these protocols. This is analyzed in section 4.1. Instead we concentrate on a concept known as Reliability with Hop-by-Hop Broadcast (HHB) which is affected more significantly due to node scheduling. HHB significantly reduces the overhead of reliable transmissions over EER and HHR schemes by exploiting the wireless broadcast in a dense networks. In
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HHB, higher the local density, the lower is the retransmission overhead. Thus it is very interesting in context of this paper since, it shows that density not only affects the number of hops between nodes but also the per hop retransmission overhead.
6.6.1
Hop-by-Hop Broadcast Reliability (HHB)
We now briefly describe the HHB protocol and analyze the effect of density on its overhead. Consider a source and sink node separated by h hops. When networks have high density, there are multiple next-hop nodes as was shown in Figure 3.16. Since wireless is a broadcast medium, all the next-hop-nodes in the communication range can potentially receive a forwarded packet. This packet needs to reach at least one of these nodes so that it is forwarded with a gradient towards the sink. The probability of a packet being successfully transmitted to at least one of these nodes is higher than the probability of it being forwarded to one particular node. The success probability of a packet getting forwarded is thus: p= probability to reach next hop= 1 - f k(h) ≥ 1 - f k(h) = number of next-hop nodes at the hth hop. HHB exploits this inherent redundancy in the wireless medium and the independent nature of signal corruption at the receivers to reduce the per-hop retransmission overhead. Higher the network density, higher would be the number of next hop neighbors and a packet would have a higher probability to be forwarded to the next hop. This leads to lesser number of retransmissions. Intuitively HHB protocol is exactly same as a normal HHR protocol but with lesser number of retransmission at each hop. The idea is analogous to diversity decoding with multiple wireless antennas to increase transmission reliability with the different nodes acting as multiple antennas for the same signal. Next we compute the overhead for the HHB protocol. For this we first need to compute the expected number of next hop neighbors k(h), as a function of network density.
6.6.2
Expected number of Next hop Neighbors
Consider a node which is h hops away from the sink as shown in Figure 6.6. Since we assume a geographic routing scheme, the intersection area of circles centered at the source (radius r) and the sink (radius x) represents the area for all h-1 hop nodes. Now y is equal to (h-1)z (where z is the expected
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y
b
a
sink
source x A1
A2
Figure 6.6 Computation of expected number of next hop neighbors progress E[fZ(z)] per hop computed from equation 3) since we are considering the region of h-1 hop neighbors. Also the source is h hops away from the sink. Thus x (as shown in figure) is given by: x=z
(h
2
+ (h − 1) 2 2
)
The area of the shaded region is A=A1 + A2, where, sin(2b) sin(2a) 2 A1 = r 2 a − , A2 = y b − 2 2
a = cos −1
b = cos −1
(r
(y
2
2
+ x2 − y2 2rx
+ x2 − r 2 2 xy
( 6.19 )
)
( 6.20 )
)
( 6.21 )
The expected number of next-hop nodes at the hth hop is: ( 6.22 )
k (h) = δ ( A1 + A2 )
We verify the analytical expectation of k(i) using simulations for 20,000 nodes, randomly deployed in a 1000m2 field for various communication ranges. The comparison of the simulation results with the analytical model is shown in Figure 6.7. The results suggests that equation 5.22 is a good approximation of k(h) and may be used effectively to compute the overhead of HHB.
6.6.3
Overhead of HHB
Next we use the analytical expression for the expected number of next hop neighbors to compute the overhead of the HHB protocol. Consider the ith hop between a source and sink separated by h hops. A
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Expected number of next hop neighbors
Overhead (HHB): Impact of Channel Errors
100
12
95 90
10
Overhead
85
#nodes
8
Range=25, Simulation
6
Range=30, Analytical
9
Error =0.3 Error =0.5
50 0.1
0 5
Error =0.1
55
Hops
1
70
60
Range=30, Simulation
2
75
65
Range=25, Analytical 4
80
13
17
21
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Density of Active nodes
25
Figure 6.7 Comparison of equation 13 with
Figure 6.8 The expected power consumption using
simulation results on a real topology. R=600.
HHB. N=104, R=600m,r=40m, ET=1J ER=0.1J, PI =
N=20000
0.01J/s, PS=0.005J/s, λp=1
packet is successfully forwarded to the next hop with probability (1 - f k(i) ) and the acknowledgement is received from the next hop with probability (1 - f ) . The number of retransmissions follows a geometric probability distribution of (1 - f k(i) )(1 - f ) . Hence the expected number of retransmissions is 1 / (1 - f k(i) )(1 - f ) . The probability of an acknowledgement packet being generated for any transmission is (1 - f k(i) ) . Hence the expected number of transmissions per packet is (2 - f k(i) ). The total number of packet transmissions for a pair of points separated by distance y (and hence number of hops is h(y)) is: TTrans ( h( y )) =
h ( y )
(2 − f ) )(1 − f )
∑ (1 − f i =1
k (i )
k (i )
The expected number of packet transmissions for all nodes in the network is given by equation 10 (using the equation 4, 5). TTrans = E [TTrans ( h)] =
(
)
h ( y ) 2 − f k (i ) ∑ ∫y =0 i =1 1 − f k (i ) (1 − f ) f Y ( y) 2R
(
)
( 6.23 )
We have OTrans = TTrans , ORe c = TTrans d where d is the average network degree. The expected power consumption is given by:
E[ PHHB ] = λ P E[OHHB (h)](ET + E R d ) + PI X + PS (1 − X )
( 6.24 )
For simple unicast HHR scheme, we can put fk(i)=1 in equation 10 to model that packet is forwarded to only one node at each hop.
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lambdaP=0.1 lambdaP=0.3
100 90
lambdaP=0.5
80
lambdaP=1.0
Overhead(HHB): Impact of varying Trafffic
HHR 110 100
70
90
60 Overhead
Overhead
Comparison: HHR vs HHB
HHB
50 40 30
80 70 60
20 10
50
0 0.1
0.2
0.3
0.4 0.5 0.6 Density of VCDS
0.7
0.8
0.9
0.1
1
0.3
0.5
0.7
0.9
Density of Active nodes
Figure 6.9 The actual overhead using HHB.
Figure 6.10 Power consumption for the HHR and
R=600m, r=30m, f=0.3, ET = 1 units, ER = 0.1
HHB. N=104,R=600m, f=.3, r=40m, ET=1,
units, PI = 0.01 units/s, PS = 0.005 units/s. Large
ER=0.1,PI = 0.01, PS=10-3 J/s, λp=0.1
dots indicate the minimum overhead point Figure 6.8 plots the expected power consumption of the HHB method using equation 11. Figure 6.10 compares this with a hop-by-hop unicast reliability (HHR) protocol. In the graphs the optimal density point is marked with a large dot. We see that at very low and very high density the consumption is high and the optimal density point lies somewhere in the middle. The HHR and HHB plots have essentially the same curves but the HHB method has significantly lesser power consumption. We also see that increase in channel error does not have a significant impact on the overhead in the HHB case. When channel errors are high, the optimal density point for HHB lies at a much higher density than for the HHR case. This is because in HHB a high density not only lowers number of hops but also reduces the per hop retransmission overhead by exploiting the higher network degree.
6.7
Increased traffic load in memory constrained nodes
When very few nodes are active for routing purposes, the traffic load in these active nodes increase since a fewer number of nodes have to share entire the traffic load. With limited memory capacity, it becomes difficult to handle the increased load at low density. The overhead of reliable delivery is severely affected since buffer overflow causes extra packet losses. The effect of memory constraints on the overhead is analyzed in section 4.2. We now consider the effect memory constraints on the overhead of reliable delivery at different densities. The intuitive idea in the analysis is that with lesser active nodes, the traffic rates at these active
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nodes increase leading to buffer overflow and subsequent packet drops. This would increase the required number of retransmissions for reliable delivery. To find the overhead of reliable delivery in this case, we have to find the actual packet drop probability which now depends on the channel error rate, the buffer size at each node and the packet arrival rate. In a hop-by-hop reliability scheme, a node buffer gets occupied not only with new incoming packets but also with previously buffered packets which have not been reliably transmitted to the next hop. The holding time for such packets in the buffer actually depends on the rate at which packets are lost at the next hop nodes. A higher blocking probability at the next hop node thus increases the buffer occupancy at the forwarding node. Thus we need to find the buffer blocking probability at balance conditions for the network. Let us first look at how packet arrival rates at each node increase with reduction in network density. We assume that the distribution of packet destinations is uniform and each active node equally participates in the routing. We neglect edge effects. In the HHB scheme, each forwarding node is a source for a packet on a hop-by-hop basis irrespective of the original source of the packet. When a packet is generated (or forwarded), it is intended for one of the neighbors. Thus the average traffic volume at each node can be considered on a local basis. Based on the current network degree (from equation 1), the packet arrival rates at each node increases due to the decrease in network degree as follows. λinp = λ p
d actual
( 6.25 )
d node − scheduled
To solve the expected blocking probability at balance conditions we model the hop-by-hop retransmission scheme as a stable (arrival
