Message Delivery Capacity in Delay-Constrained. Mobile Wireless Networks: Bounds and Realization. Gabriel Y. Keung, Bo Li, and Qian Zhang. The Hong ...
Message Delivery Capacity in Delay-Constrained Mobile Wireless Networks: Bounds and Realization Gabriel Y. Keung, Bo Li, and Qian Zhang The Hong Kong University of Science and Technology
Abstract—In this paper, we study the delay-constrained message delivery capacity formulation in mobile wireless networks. The message delivery capacity specifies the maximum percentage of messages that can be successfully delivered to base stations within a given time constraint. By taking full advantage of node mobility and rendezvous during mobile node encounter, messages can be delivered to a base station either directly or through relays by other nodes. We first identify a number of unique challenges involved in such systems including message relay and buffer replacement mechanisms and then derive the capacity bound under perfect message relay and buffer replacement mechanisms. Due to the unrealistic assumption for the foreknowledge of node moving trajectories, we proceed to propose a practical algorithm to approximate the maximal message delivery capacity. Finally, we evaluate the algorithm and examine the sensitivity with delay constraint, buffer size, message relay and replacement schemes. Keywords— Mobile Wireless Network, Relay Assisted Information Coverage, Message Delivery Capacity
I. I NTRODUCTION In this paper, we study the information collection problem in mobile wireless network with uncontrolled node mobility. One typical example is the message transmission between mobile devices with communication ability (e.g. Pocket PC) and base stations. Mobile nodes need to carry the information (messages) and deliver to stationary base station. In such applications, the completeness and timeliness of sensing information collection become two important criteria. Given the nature of uncontrolled mobility in nodes and the delay sensitiveness of information, it is important to investigate what is the maximum capacity for collecting message that has certain delay constraint in a mobile wireless network, especially for those nodes are moving outside the transmission range of base stations. Under this scenario, there are a number of key factors that influence the information collection performance: Delay Constraint: Due to the timeliness nature of the information in the application scenario, messages are required to be delivered to a base station(s) within a time constraint. Message Relaying: A message can be delivered to a base station in two ways: either (i) by direct delivery when a mobile node moves into the proximity of a base station; or (ii) a mobile node carries the message and forwards to one or more relaying mobile nodes onward to a base station. In the second case, a message relay mechanism becomes essential in achieving maximal message delivery performance. ∗ The research was support in part by grants from RGC under the contracts 615608, and 616207, by a grant from NSFC/RGC under the contract N HKUST603/07. This work was also supported in part by National Natural Science Foundation of China with grant no. as 60933012.
Message Replacement: Each mobile node is equipped with limited buffer space. A message replacement policy becomes necessary, which specifies which message to be replaced when a node buffer is fully occupied during relay. Based on the above discussions, we can define the delayconstrained message delivery capacity as the maximum percentage of individual message delivery from covered mobile nodes by base stations along a time period T with given the time-sensitive nature of the message τ and limited buffer space B for storing message. To expedite our discussion, we use message delivery capacity to represent the “delay-constrained message delivery capacity” in the rest of this paper. The objective is to maximize the message delivery capacity. We also are interested in investigating the design trade-offs involving relay and replacement strategies. Specifically, during an encounter of any two mobile nodes, a message relay scheme determines which message to be forwarded; while a message replacement scheme determines which message to be dropped if a node buffer is full when a new message is accommodated. In this paper, we formally describe the message delivery capacity in delay-constrained mobile wireless networks. We formulate this into an optimization problem with the objective of maximizing messages delivery capacity. We show that this formulation has a pseudo-polynomial time complexity with unrealistic assumption for the foreknowledge of node moving trajectories. We then present a practical message relay and replacement algorithm to achieve suboptimal solution. Finally, we examine the impact on the message delivery capacity from a number of relevant factors. The rest of this paper is organized as follows. Sec. II reviews the related works. Sec. III describes the message delivery capacity problem and its formulations. Sec. IV discusses the upper bound of message delivery capacity and proposes a message relay and replacement scheme to approximate it. Sec. V examines the message delivery capacity sensitivity under different network parameters and configurations, and finally Sec. VI concludes the paper. II. R ELATED W ORK Recently, information collection in mobile sensor networks has received increasing attentions. Several schemes have been introduced, which can be broadly divided into two categories: epidemical-based [1] and probabilistic delivery approaches [2]. Apart from the message delivery capacity, there were also relevant studies on the buffer management in Delay-Tolerant Network (DTN). Several policies were briefly discussed in the
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related work section in [3]. These include: Drop Last, Drop Front, Drop Oldest and Drop Youngest. A resource allocation protocol for intentional DTN (RAPID) was proposed in [4], which focused on scheduling messages under limited bandwidth and derived a suboptimal algorithm. In [5], an optimal buffer management policy based on global knowledge for DTN was presented. Our focus in this paper is completely different from all prior works. There are two distinctive features in our work, 1) the timeliness of message, i.e., the delay constraint, and 2) the relay-assisted message delivery mechanism. The challenge is to maximize the percentage of successful message delivery within a delay constraint by employing a proper relayassist scheme in mobile wireless networks. III. P ROBLEM D ESCRIPTION AND F ORMULATION We now describe the delay-constrained message delivery problem with relay assistance and the formulation. A. Problem Description and Assumptions In this paper, we consider a mobile wireless network consisting of M mobile nodes, which follow a given mobility model within a 2-D geographic area A to deliver some data messages. There exist N stationary base stations for collecting data messages. Let the coverage range of a base station to be R and the region within the disk of radius R centered at the base station to be coverage region. The base station is capable of collecting information once a node enters the coverage region of a base station, we say the node is covered. We quantify the collected information into data message with equal length, and each message has the same delay constraint τ . The buffer size B constrains the maximum number of individual messages that can be stored in a node. We assume the message arrival is in a Poisson process with an average arrival rate λ. We denote k(t) as the set of individual at a time instant t. And there messages T are totally |K| = t=0 k(t) number of individual messages existed within a time period T , where τ T and K is the set of messages. To facilitate our discussion, we assume there is enough time to forward messages when two nodes encounter and messages can be delivered deterministically. A node can transmit target messages to another node if they are both inside the disks of radius Rs centered at each other, and the transmission range is assumed to be constant over A. From the message delivery method discussed previously, it is possible that multiple copies of an individual message are held by different nodes in the system at any given time. If a mobile node is covered by a base station, the individual messages stored in the node buffer are considered to be delivered to the base station. Each individual message copy occupies buffer space before that copy is delivered to a base station or its time τ expires. After that, the delivered message and its replicas are no more useful to the application and should be dropped. A message j has its own life time period between [tj , tj +τ ] in different mobile nodes, where tj is the time since j was generated. In this work, we only assume the message will be dropped by its holder(s) if the time τ is elapsed. It can be
easily achieved by storing the time ti , which is the time since the message i was generated, into the header of message. B. Problem Formulation We now formally define the delay-constrained message delivery problem as follows: Definition 1: Let ci (t) be a probability that the message i is delivered to a base station within a particular time t, i.e., the probability that a mobile node carrying the message i, is within the coverage region of a base station within a time t; we refer this as the message delivery probability for i. Definition 2: We define the message delivery percentage C(τ ) as a random variable between 0 and 1 representing the percentage of total generated messages that can be successfully delivered subject to time and buffer constraints. The objective of maximizing message delivery capacity can be obtained by maximizing the expected value of the message delivery percentage with a given delay constraint: s.t.
max E[C(τ )], bi (t) ≤ B for 0 ≤ t ≤ T, ∀ i ∈ N,
(1)
where bi (t) is the state of buffer occupancy of the mobile node i at time t. Then the expected value of message delivery percentage can be obtained by: 1 · E[ [cj (τ )]]. (2) E[C(τ )] = |K| ∀j∈K
This indicates that the message delivery capacity depends on the maximization of the total message delivery probabilities, while the maximization of message delivery probabilities are determined by message relay and buffer replacement scheme. The functionality of the scheme is to make a series of decisions during a rendezvous of two mobile nodes. Specifically, there are several scenario that need to be handled: (i) during the rendezvous of two mobile nodes, a node needs to determine a set of “selected” messages to forward to the other node; (ii) a stored message could selectively replaced, if a node buffer is full when a new message is generated; (iii) a stored message could selectively replaced, if a node buffer is full when a replay occurs. We will investigate the bounds on the message delivery capacity and a practical algorithm as realization in the next section. We will also closely examine their effects and implications on the message delivery capacity. IV. M ESSAGE R ELAY AND R EPLACEMENT S CHEME Under the delay and buffer constrains, we are interested in finding out how the maximal message delivery capacity can be achieved, and more importantly how this can be realized by a concrete message relay and replacement algorithm. A. The Capacity Bound with Complete Information We now introduce the message relay and replacement scheme to achieve the maximal message delivery capacity. In this case, we basically assume the moving trajectories (paths) of all mobile nodes are known in advance, which means we can foresee the movement of nodes and know
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when they encounter each other. We denote the path, which consists rendezvous events with all corresponding mobile −−−→ nodes carrying the message k, as pathk . At the same time, we know which nodes carry which messages, and which messages can be relayed. It is impossible in the realistic world, but this scheme serves as an upper bound for the message delivery percentage under the given constraints (τ and B). In order to deliver a message k back to the base station(s) within τ , it needs to satisfy following requirements: (i) it requires a path from the source node of k to any base station within the constrained time τ . It is important to point out that multiple paths may be available for a particular message, as long as the path starts from the source node and ends at any base station; (ii) the message delivery requires buffer space for all corresponding mobile nodes along the path, so that the −−−→ buffer usage bj (t) ≤ B of mobile node j ∈ pathk at any time −−→ between tk ≤ t ≤ (tk + τ ), for ∀k ∈ K. We define bufk as the −−−→ buffer requirement of all mobile nodes along the path pathk . To calculate the maximal message delivery capacity, we need to examine different paths from the source node of a message to any base station subjected to the two requirements. A simple greedy algorithm cannot guarantee to provide the solution of the maximal capacity problem, since a change −−−→ of pathi may result a violation of buffer size by the second −−−→ requirement, and may require a further change of pathj , where i = j ∈ K. Thus, we propose a dynamic programming (DP) algorithm to achieve our goal. The inputs of the DP algorithm −−−→ are the different path options pathi and the corresponding −−→ buffer requirements bufi for any message i ∈ K, and the maximum number ofall possible paths for a particular message ti +τ I(j, k, t)|), where j, k ∈ M ∪ N and i is at most O(| t=t i I(j, k, t) is the rendezvous event at time instant t. We define the function C[j, k] as the maximal message delivery percentage with two indices j and k, where j is the −−−→ index of all possible pathi for ∀i ∈ K, and k is the index of all possible buffer configurations along the time period T . The recursive formulation of the DP algorithm is: −−→ 1 + C[i − 1, k − bufj ], C[i − 1, k] , (3) C[j, k] = max |K| −−→ for bufj does not excess the buffer size, and message i is not delivered yet. 0, for j = 0, k = 0 C[j, k] = C[j − 1, k], otherwise. The recursive formulation solve the subproblems by a bottomup fashion. The time complexity of the algorithm is related to the number of the recursive steps used to calculate the function −−−→ C[j, k], which is O(B · N · T · | ∀i∈K pathi |). Since the DP algorithm is a pseudo-polynomial time algorithm and it cannot be finished in polynomial time. Then a realistic scheme is needed to approximate the maximal message delivery capacity. B. An Approximated Scheme Due to the unrealistic assumption for the foreknowledge of node moving trajectories, we are now presenting a realistic
scheme to approximate the capacity. In this scheme, we assume the global knowledge of the network at any time instant is known, which includes the number of individual messages existed in the network, and the number of mobile nodes carrying a specified message. We remove the unrealistic assumption for the foreknowledge of node moving trajectories, in stead of that, we use probabilistic model to describe the node mobility in this case. We use the cumulative distributed function (CDF) Di,k (ti − t) as the probability along time t for a message i can be directly delivered by a mobile node k to any base station, where i has its own life time period between [ti , ti + τ ]. The probability for k containing i cannot be delivery is Di,k (ti − t) = 1 − Di,k (ti − t). The probability used above is related to the inter-contact time distribution between mobile nodes with different mobility model, which has been well discussed and studied in [7]. We admit that leveraging precise mobility model to improve the accuracy of the CDF is not the main goal of the work. The upper layer applications can leverage the existing work of machine learning on the node mobility model such as HMM [8] to improve the accuracy. Recall that, the message delivery probability ci (t − ti ) of a message i is the probability of i can be delivery to base stations within its current life time (t − ti ). Then the expectation value of message delivery at time instant t is: ⎫ ⎧ ⎬ ⎨ [Di,j (t − ti )] [Di,j (τ + ti − t)] , 1− ⎭ ⎩ ∀i∈k(t)
j∈mi (t)
j∈ni (t)
where k(t) is the set of individual messages existed in the network at time instant t, ti ≤ t ≤ ti + τ , ni (t) is the set of mobile nodes currently carrying the message copies of i, and mi (t) is the set of mobile nodes saw i before time t. With the calculation of message delivery probability, we can now introduce the realistic message relay scheme. And the decision should be formulated like following: ∂ argi max ∂|ni (t)| ⎡ ⎤ ⎣1 − [Di,j (t − ti )] [Di,j (τ + ti − t)]⎦ . (4) j∈mi (t)
j∈ni (t)
Above equation illustrates which message can maximize the increment of the expectation value of message delivery percentage, if the number of message copies increases. Since our goal is to maximize the message delivery percentage, as long as free buffer space is available from the receiver side during the rendezvous between two nodes, forwarding message i results increment of the delivery percentage. If the buffer space of a mobile node is full and a new (generated or forwarded) message is accommodated, the message replacement scheme is required for this case. The decision for replacement of new message k is following: argi max [Di,j (t − ti )] · [Di,j (τ + ti − t)] − j∈mi (t)
j∈ni (t)
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[Di,j (t − ti )] ·
j∈mi (t)
[Dk,j (t − tk )] ·
j∈mk (t)
[Di,j (τ + ti − t)] +
j∈ni (t)\y
[Dk,j (τ + tk − t)] −
j∈nk (t)
[Dk,j (t − tk )] ·
j∈mk (t)
[Dk,j (τ + tk − t)].
(5)
j∈nk (t)∪y
If a message i from the buffer of node y can maximize the change of expectation value of message delivery percentage, it will be dropped and replaced by message k. The equation demonstrates two important criteria for message relay and replacement scheme: the number of message copies and the remaining life time of individual messages. This provides some design insight for considering a distributed message relay and replacement scheme for this problem in future. C. Analysis of the Capacity Bound With the proposed message relay and replacement algorithm, we are going to derive the capacity bound under this algorithm. Recall that the message delivery percentage can be obtained by (2) and the message delivery probability ci (τ ) of the individual message i at the delay constraint τ is: ci (τ ) = 1 −
mi (τ )
j=0
ti ≤tstart ≤ti +τ j
[Di,j (tend − tstart )], j j
(6)
where mi (τ ) is the number of node saw i before delay constraint τ . And the first Cartesian product is counting the probabilities of any mobile nodes j carried i for a time period to tend start at tstart j j ; the second Cartesian product is counting for j carried i for multiple periods of time. We denote the set of rendezvous events I and each event I(i, j, t) represents the rendezvous between two mobile nodes or between a base station and a mobile node at the time instant t and i, j ∈ M∪N. Since the maximum number of individual messages could be carried by a mobile node is limited by B, the message deliv B ery capacity should be bounded by: |K| · ∀ i,j∈M I(i, j, t). It is a simple upper bound of the message delivery capacity, which is directly related with the buffer size B and number of rendezvous between a base station and a mobile node. Then the upper bound of message delivery capacity (UB) becomes: ⎛ ⎞ T − − − → 1 min ⎝B I(i, j, t) , (k(t) : ∀pathi )⎠ , (7) |K| ∀i,j∈M t=0 where the second component is the total number of messages, −−−→ which are generated along all possible paths (pathi ) to any base stations by relaxing the delay constraint (τ → T ). This shows the capacity bound is affected by buffer size, delay constraint and the mobility nature of mobile nodes. The equation also illustrates that under the buffer size limitation, the node mobility cannot be further exploited for improving the message delivery capacity by relaxing the delay constraint. We then would like to examine the message delivery capacity calculated by using the DP algorithm and the approximation with the realistic scheme. Fig. 1 demonstrates the
comparison between the upper bound of the message delivery capacity and the approximated scheme. The realistic scheme well approximate the two ends of the delay constraint (< 2500, and > 5000). Due to the lack of foreknowledge of node moving trajectories and the use of multiple message copies to increase the delivery probability, the approximated scheme shows the suboptimal results. The both schemes cannot further improve the message delivery percentage after delay constraint = 4500, which is due to the limitation of buffer size. In order to study the effects of two constraints: buffer size and delay constraint with the message delivery capacity problem. Fig. 2 and 3 show the three-dimension plot of buffer size, delay constraint and the message delivery capacity by using the approximated scheme under random walk and community mobility model. From the comparison between two mobility models, we find the similar sensitivity between message delivery percentage with delay constraint and buffer size. The results from both figures are consistent with our previous formulations and discussions in this section. V. R ESULTS AND A NALYSIS In this section, we use simulations to verify and examine the message delivery capacity with respect to the delay constraint, buffer size and message relay/replacement scheme. We develop a simulator that captures the essential aspects of the mobile wireless network described in Sec. III. The simulator provides the flexibility in selectively changing the configuration by setting various parameters including: (i) the area size (|A|); (ii) the number of mobile nodes (M ); (iii) the speed of mobile nodes (v); (iv) the coverage range of a base station (R); (v) the number (N ≥ 1) and positions of base stations; (vi) the length of the simulation time interval (T ); (vii) the delay constraint of message (τ ); (viii) buffer size of mobile nodes B; (ix) the mobility models of nodes, e.g., the random walk or the community mobility models [6]. Unless otherwise specified, we use the following default settings: we define 100 mobile nodes randomly distributed in an area of size 100 × 100 with the mobile node speed v = 1 per unit time and the coverage range R = 10 of a base station. In additional to the upper bound of message delivery capacity obtained by DP algorithm and the proposed realistic scheme, two simple message relay and replacement schemes are introduced to compare the message delivery percentage performance. The flooding scheme is an epidemical-based approach, which allows a mobile node to forward all new messages to another node during rendezvous between two mobile nodes. A node will randomly drop a message, if the buffer of the node is full when a new message is accommodated. The oldest drop first scheme uses the same message relay policy of the flooding scheme. For message replacement, it controls a node to drop the oldest message among its buffer, if its buffer is full when a new message is accommodated. Fig. 4 examines the message delivery percentage performance sensitivity with delay constraint, buffer size and different message relay and replacement schemes. From the results, we find that the proposed approximated scheme outperforms
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Fig. 1. The message delivery percentage comparison between the upper bound of the message delivery capacity and approximated scheme under random walk mobility model
Fig. 2. The message delivery percentage performance sensitivity with delay constraint and buffer size under random walk mobility model by using the approximated scheme
Fig. 3. The message delivery percentage performance sensitivity with delay constraint and buffer size under community mobility model by using the approximated scheme
Fig. 4. Message delivery percentage against different delay constraint requirements under random walk mobility model
Fig. 5. Buffer length requirement for 90% message delivery percentage compared with upper bound against delay constraint
Fig. 6. Number of message copies distribution after for different schemes
the oldest drop first scheme and flooding scheme. The results also shows the message delivery capacity is bounded by the buffer limitation B and the delay constraint τ . Fig. 5 shows buffer length requirement for 90% message delivery percentage compared with the upper bound against delay constraint, this also shows we can guarantee the message delivery message by using certain among of buffer length under different delay constraint requirements. Fig. 6 demonstrates the frequency count distributions of existed message copies are similar between different schemes. The proposed approximated scheme shows a “fatter” distribution compared to other schemes. From the results of fig. 4, it shows for larger number of copies the better message delivery performance. However, due to the buffer size limitation of mobile nodes, the number of message copies (message delivery performance) is bounded by this limitation. VI. C ONCLUSION In this paper, we present and study the message delivery capacity problem with three fundamental factors: buffer size, delay constraint and message relay/replacement scheme. We examine the relationship between the message delivery capacity and the factors by both formulation and simulation. In order to capture the message delivery capacity, the upper bound is studied with complete knowledge assumption. Due to the unrealistic assumption of the problem, we further present a realistic scheme to approximate the maximal message delivery capacity. From the approximation, we demonstrate that two
important criteria for message relay and replacement scheme are: the number of message copies and the life time of messages. There are several avenues for further study on this problem: (1) to additionally consider energy consumption constraint; (2) to study the trade-off between the energy consumption constraint and message delivery capacity; (3) to consider a practical distributed message relay and replacement scheme. R EFERENCES [1] T. Small and Z. J. Haas, “The shared wireless infostation model- a new ad hoc networking paradigm (or where is a whale, there is a way),” in Proc. of ACM International Symposium on Mobile Ad Hoc Networking and Computing (MOBIHOC), June, 2003. [2] B. P´ asztor, M. Musolesi, and C. Mascolo, “Opportunistic Mobile Sensor Data Collection with SCAR,” in Proc. of IEEE International Conference on Mobile Adhoc and Sensor Systems (MASS), October, 2007. [3] A. Lindgren and K.S. Phanse, “Evaluation of Queuing Policies and Forwarding Strategies for Routing in Intermittently Connected Networks,” in Proc. of the First International Conference on Communication System Software and Middleware (Comsware), January, 2006. [4] A. Balasubramanian, B. N. Levine, and A. Venkataramani, “DTN routing as a resource allocation problem,” in Proc. SIGCOMM, August, 2007. [5] A. Krifa, C. Baraka, and T. Spyropoulos, “Optimal Buffer management Policies for Delay Tolerant Networks”, in Proc. of SECON, June, 2008. [6] T. Spyropoulos, K. Psounis, and C. S. Raghavendra, “Performance Analysis of Mobility-assisted Routing,” in Proc. of Mobile ad hoc networking and computing (MobiHoc), May 2006. [7] T. Karagiannis, J. Y. L. Boudec, and M. Vojnovic, “Power Law and Exponential Decay of Inter Contact Times between Mobile Devices,” in Proc. of Mobile Computing and Networking, September, 2007. [8] J-M. Francois, G. Leduc, and S. Martin, Learning movement patterns in mobile networks: a generic method,” in Proc. of European Wireless, February, 2004.
