Randomized Energy Aware Routing Algorithms in Mobile ... - CiteSeerX

Randomized Energy Aware Routing Algorithms in Mobile Ad Hoc Networks Israat Tanzeena Haque

Chadi Assi

J. William Atwood

Computer Sci. and Soft. Eng. Concordia University Montreal, ´ Quebec, ´ Canada

Concordia Institute for Information Systems Eng. Montreal, ´ Quebec, ´ Canada

Computer Sci. and Soft. Eng. Concordia University Montreal, ´ Quebec, ´ Canada

it [email protected] [email protected]

[email protected]

ABSTRACT

1. INTRODUCTION

We consider the problem of energy aware localized routing in ad hoc networks. In localized routing algorithms, each node forwards a message based on the position information about itself, its neighbors and the destination. The objective of energy aware routing algorithms is to minimize the total power for routing a message from source to destination or to maximize the total number of routing tasks that a node can perform before its battery power depletes. In this paper we extend our previous work on randomized localized routing algorithms that achieve high packet delivery rates and show that they have good overall power consumption. We present two different variants of energy aware randomized routing, namely “greedy” and “compass”, and we study their performance using different cost metrics (e.g., forwarding power, remaining node energy, or a combination of both). We study their performance experimentally on different topologies and compare it with other existing algorithms. Our simulation results show that energy aware randomized algorithms achieve superior packet delivery rates and moderate energy consumption.

We consider the problem of energy aware routing, in which a message has to be sent efficiently from a source node to a destination node, in a sensor or an ad hoc network. A mobile ad hoc network (MANET) is a collection of autonomous mobile devices that can communicate with each other without having any fixed infrastructure. Each node in the network can have an omni-directional antenna and communicate using wireless broadcasts with all nodes within its transmission range. A multi-hop routing protocol is needed to enable communication between nodes that are not in transmission range of each other. However, the absence of infrastructure in MANETs, the dynamic topology of these networks, the autonomous heterogeneous nodes, and the resource constraints (battery power, bandwidth, computational power, etc.) all contribute to make the problem of routing a tremendous challenge. The constrained resources, especially the battery power, make the routing problem a very challenging issue for MANETs. Moreover, mobile devices in an ad hoc network need to forward packets for other nodes; this extra activity consumes a significant amount of the energy of mobile devices. Therefore, it is critical to design efficient routing algorithms with the objectives of (1) minimizing the overall energy usage in routing packets and (2) maximizing the packet delivery rate. Applications of minimum energy networks include soldiers deployed on a hostile terrain and multisensor networks, where sensors communicate with each other without having any central control or base station. Energy is consumed at two levels during routing, namely communication energy and the energy dissipated at the nodes. The communication energy or the energy needed per routing task can be optimized if nodes can adjust their transmission power to efficiently select the next hop along the route. This is equivalent to hop count if the transmission power is kept constant [13]. Routing algorithms that solely focus on the communication energy are not eventually a good choice for network lifetime. Some of the nodes (hot spots) in this approach will be chosen very often and this will ultimately drain out the battery power of these nodes quickly, therefore partitioning the network abruptly. Routing algorithms that minimize the energy required per routing task are called power/energy aware algorithms [13]; on the other hand, cost aware routing algorithms ensure optimal use of node’s battery power and hence prolong network’s life time [13]. In this paper we focus on the energy aware localized routing algorithms, where each node forwards the routing packets based only on the position or geographic coordinates of

Categories and Subject Descriptors C.2.1 [Computer-Communication Networks]: Network Architecture and Design—Wireless communication; C.2.2 [Computer-Communication Networks]: Network Protocols—Routing protocols

General Terms Performance, algorithms

Keywords wireless networks, mobile ad hoc and sensor networks, routing, position based routing, energy aware routing

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. MSWiM’05, October 10–13, 2005, Montreal, Quebec, Canada. Copyright 2005 ACM 1-59593-188-0/05/0010 ...$5.00.

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itself, its neighboring nodes, and the destination. We model a MANET by a unit disk graph, where two nodes are connected if and only if their Euclidean distance is at most the transmission range [1]. We classify the position based algorithms as deterministic and randomized. In the first category, the current node holding the packet selects the next node deterministically out of its neighbors, whereas in the second case the selection is random. Extensive work has been done on energy efficient routing, however in this paper we will mainly focus on position based routing. We will start first by presenting the performance analysis of some of the existing energy aware and non aware position based routing algorithms. Then, we will present our new randomized algorithms that control the communication power with high packet delivery rates. Essentially, to decide on the next node to which the packet should be forwarded, our algorithms pick one neighbor of the current node from above the line passing through the current node and the destination, and another neighbor below this line. Then the next node is chosen randomly from these two neighbors according to some probability distribution. The exact choice of the neighbors and the probability distribution determines the algorithm. In our simulation, we consider both the uniform and cluster distributions of nodes in a given area. Our simulation results show that on both data distributions our algorithms have a substantially higher delivery rate than the deterministic algorithms in unit disk graphs. The rest of the paper is organized as follows. The next section reviews some of the enery aware routing related to our work. Section 3 gives a brief description of the exsisting position based routing algorithms. Section 4 describes the system model and other relevant preliminaries. Randomized energy aware routing algorithms are presented in Section 5. The simulation environment is given in Section 6. In Section 7 we present the empirical results of our simulations and provide an interpretation of the behavior of the algorithms. We conclude with a discussion of the results and future directions of this research in Section 8.

2.

[13], is then subject to k−1

P (ni , ni+1 )

min ej , ej = ∀j

(1)

i=1

Minimum cost per packet: This metric tries to prolong the lifetime of the nodes and networks through the careful selection of next route node with plenty of energy. Let fi (xi ) be a function that denotes the cost or weight of node i, where xi represents the total energy that node i already expended. Hence, the total cost cj of sending a packet j from the node n1 to nk is sum of the cost of the entire route. The optimization of this metric is then subject to, k−1

min cj , cj = ∀j

fi (xi )

(2)

i=1

Another way of using the cost metric is normalizing fi to reflect remaining battery lifetime of a node. Then, fi (xi ) can be modified as fi (zi ) = 1/1 − g(zi ), where zi is the measured voltage and g(zi ) is the normalized (between 0 and 1) remaining lifetime of node i. Singh et al. consider a static random graph, where two nodes are connected with a fixed probability p, and use nonlocalized Dijkstra’s shortest path algorithm to evaluate the performance of their proposed metrics compared to the hop count metric. In [13], the authors proposed localized power, cost, and power*cost algorithms based on the observation that if additional intermediate nodes are placed at desired positions between the source and destination that are at distance d apart, then transmission power may be linearly dependent on d, rather than dα (α ≥ 2). The authors first defined two optimal algorithms in terms of communication energy and battery lifetime. The SP − power algorithm that minimizes the total communication energy of a packet is obtained by applying Dijkstra’s shortest path algorithm with the weight of the edge as adα + C, where a, α, and C are constants that depend on the radio models. On the other hand, the SP − cost algorithm, which maximizes the node’s lifetime, can be obtained by applying Dijkstra’s shortest path algorithm with an edge-weight of f (ni ) = 1/gi . Stojmenovic et al. use these optimal energy aware algorithms to compare the performance of their proposed algorithms. Let c, Ni , and d be the current node holding the packet, set of neighbors of c, and destination of the packet, respectively. The distance between c and one of its neighbors is x and the remaining distance to the destination is d−x, where d is the total distance between c and d. The localized loop free power and cost aware algorithms are next defined: In the power algorithm, c selects the next neighbor such that

OVERVIEW OF ENERGY AWARE ROUTING ALGORITHMS

In [12], the authors propose several power aware routing metrics to increase the lifetime of the nodes and the network. Conventional routing protocols in ad hoc networks use delay or hop count to calculate the path to the destination. This approach might accelerate the battery drainage of some specific nodes, which forward packets for many sourcedestination pairs. The effect would be early node failure and network partition. Following a longer path of a set of nodes with plenty of energy would be a better choice [12]. The energy-aware metrics proposed by Singh et al. are as follows. Minimum energy consumed per packet: This metric is used to minimize the total communication energy of a packet regardless of the available energy at the nodes. Assume a packet j traverses a set of nodes n1 , n2 , ........, nk , where n1 is the source and nk is the destination. Let P (ni , ni+1 ) be the power needed to forward j from the node ni to ni+1 . The total energy consumed by packet j to reach the destination is then the sum of the energy over the entire path. The optimization of this metric, which is called power metric in

min ej , ej = Ec→xi + Exi →d

(3)

Ec→xi = axα + C

(4)

∀j∈Ni

a(α − 1) 1/α a(α − 1) (1−α)/α ] + C(d−x)[ ] C C (5) Here, Ec→xi and Exi →d are the energy needed for direct communication. This process continues untill the packets reach the destination or the next node is one from which c received the packet, in which case algorithm fails to deliver the packet. In the cost algorithm, c selects the next neighbor

Exi →d = C(d−x)[

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and neighboring node and destination, respectively. Similarly, in projection power progress, the next node is the one that minimizes (dα cx + C)/cd.cx, where cd.cx is the dot product of two vectors.

xi such that min cj , cj = f (xi ) + t(d − x)/r,

∀j∈Ni

(6)

Here, r is the transmission radius of the nodes and t is a network dependent constant. For example, t = f (xi ) means the rest of the nodes have the same cost of the node xi , which is not realistic. Also, the other values of t, (f 0 (xi ) and 1/g 0 (xi )), where f 0 (xi ) and 1/g 0 (xi ) represent the average cost and remaining battery power of xi and its neighbors, respectively, and do not consider actual cost of the nodes. In our algorithms (presented in Section 4), nodes periodically broadcast their remaining battery power to their neighbors to make routing decision based on accurate energy level of nodes. Finally, the authors combine power and cost metrics in additive and multiplicative forms to obtain a single metric that considers both the communication energy and the battery power of the nodes. When multiplication of two metrics is used, the energy needed to forward a message from c to next node x is f (x)Ec→x . The additive form of the metrics is αEc→x + βf (x), where α and β may be fixed by the source node s as f 0 (s) and average of Es→Ni , respectively. The corresponding optimal energy consumption algorithms are called SP − power + cost and SP − power ∗ cost. The localized version of these algorithms select the next node that not only minimizes the communication energy but also selects the next node with plenty of remaining battery power. On the other hand, [3] proposed a distributed non-localized shortest cost routing algorithm for sensor networks, where the cost takes into account both the communication energy consumption and the residual energy level at nodes. Their objective is to maximize the network life by increasing the number of routing tasks, before the first node is dead (does not have enough energy to forward message). Their link cost can be formulated as follows. costij = (etij )x1 Er−x2 Eix3 + (erij )x1 Er−x2 Eix3 ,

3. POSITION BASED ROUTING ALGORITHMS In position based routing algorithms, each node makes a decision about which neighbor to forward the message to based solely on the location of itself, its neighboring nodes, and the destination. Although routing-table-based solutions merely keep the best neighbor information on a route toward the destination, the communication overhead for maintenance of routing tables due to node mobility and topology changes is quadratic in network size [6]. On the other hand, position-based algorithms do not require route establishment and maintenance, hence these algorithms may efficiently utilize the scarce memory resources and the relatively low computational power available to the wireless nodes. More importantly, given the numerous changes in topology expected in ad-hoc networks, no reconfiguration of the routing tables is needed and therefore we expect a significant reduction in the route maintenance overhead. Positionbased routing algorithms are classified in different ways in the literature, and we will describe some of those related to our work. In [6], the progress of a node x, with a given transmitting node c and the final destination node d, is defined as the projection of |cx| on the cd line. A neighbor of c is in the forward direction if the progress is positive; otherwise, it is said to be in the backward direction. The Most Forward within Radius (MFR) [14] routing algorithm maximizes the progress toward the destination by forwarding the packets to one of the neighboring nodes, whose projection onto the line between the current node and the destination is closest to the destination. However, it also consumes maximum power to cover maximum distance, which in turn increases collisions with other nodes. Hence, instead of forwarding packets to the farthest neighboring node, the Nearest with Forward Progress (NFP) [8] scheme sends the packet to the node closest to the sender. Recently in [13], Stojmenovic et al. have introduced another new routing method called Nearest Closer (NC), which is a variation of N F P method. In this method, c selects one of its closest neighbors that are closer to d than c. The transmission can thus be accomplished with minimum power; hence the interference with the other nodes is minimized, while the probability of a successful transmission is maximized. In Greedy routing [5, 6], a node forwards a packet to the neighbor that is closest to the destination. Compass or directional routing [9] moves the packet to a neighboring node such that the angle formed between the current node, the next node, and the destination is minimized. All of the above mentioned algorithms (except N C) choose the next node from among all the neighbors of the current node c, and fail to deliver the packet if the chosen next node is the one from which c receives the packet. Whereas, in N C, c considers only the neighbors closer to the destination than itself, and drops the packet if no such neighbor is available. Let us consider the example given in Figure 1 to illustrate the successful operation of each of the above mentioned algorithms, where the source and the destination are s and

(7)

Where Ei and Er are initial and remaining energy at nodes, respectively. x1 , x2 , and x3 are some positive weighting factors. For example, if x1 = x2 = x3 = 0, then the resulting path is the minimum hop path, however, if x2 = x3 = 0, then the path is the minimum energy consumption path. In [15] Xue et al. proposed a location aided or position based energy aware routing algorithm, called LAPAR, under node mobility and unidirectional links. In particular, each node constructs relay regions based on the position of the neighbors. The next node is then chosen such that its relay region covers the destination. In case of multiple neighboring nodes that covers the destination, the greedy approach is applied. This algorithm is loop free, but might fail if there is no neighbor whose relay region covers the destination. In this case LAPAR is combined with the perimeter routing. A planar subgraph of the original topology is required in perimeter routing, and the packet traverses those faces of the subgraph that are intersected by the source and destination nodes. Kuruvila et al. in [10] proposed another set of power and cost aware routing algorithms that guarantee progress In power progress algorithm the current node forwards the packet to one of its neighboring nodes that is closer to the destination than itself, such that it minimizes (dα cx +C)/dt − dr , where dcx , dt , and dr are the distances between the current and neighboring node, current and destination node,

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2. The transmission range of each mobile host is r, that is, two hosts can directly communicate with each other if their distance is at most r.

d1, respectively. In this example Greedy and M F R choose b, N F P and N C select a, whereas Compass routing picks c as the next node.

3. Each mobile host has an omni-directional antenna, which covers a circular area of radius r.

i

4. Communication links are bidirectional, that is, if a mobile host u is able to receive signals from a mobile host v, then v is also able to receive signals from u.

h e a

d2

f

Based on the above hypotheses, we can represent a MANET as a geometric undirected graph G = (S, E), where vertices represent mobile hosts and edges represent a link through which a pair of mobile hosts can communicate directly. The set of vertices S is thus a set of points in the Euclidean plane. Let d(u, v) be the distance between the points u and v in the plane. The set of edges E {{u, v} : u, v ∈ S, d(u, v) ≤ r} , that is, E contains all the pairs of mobile hosts at a distance of at most r [1]. The resulting graph UDG(S) is called a unit disk graph. For node u, we denote the set of its neighbors by N (u). Given a unit disk graph UDG(S) corresponding to a set of points S, and a pair (s, d) where s, d ∈ S, the problem of energy efficient position-based routing is to construct a path in UDG(S) from s to d, where in each step, the decision of which node to go to next is based only on the coordinates of the current node c, N (c), and d. At the same time, the energy consumption both at the nodes and for communication must be minimized to maximize the network lifetime. Here, s is termed the source and d the destination. Frequently, we will also refer to the line cd passing through c and d. An algorithm is deterministic if, when at c, the next node is chosen deterministically from N (c), and is randomized if the next step taken by a packet is chosen randomly from N (c). The routing algorithm may or may not succeed in finding a path from s to d. We use the following notion of a graph defeating an algorithm from [2]. A deterministic algorithm is defeated by a graph G = (S, E) if there is a pair (s, d) ∈ S such that a packet using the algorithm never reaches the destination d when beginning at the source s [2]. A randomized algorithm is defeated by a graph G = (S, E) if there is a source/destination pair (s, d) ∈ S such that a packet using the algorithm and originating at source s has probability 0 of reaching destination d in any finite number of steps. We are interested in the following performance measures for routing algorithms: the delivery rate, that is, the percentage of times that the algorithm succeeds and the power dilation, the average ratio of the total communication power consumption by the algorithm to the energy consumption of the shortest path in the graph.

b

s

g c

d1

Figure 1: A sample network topology to illustrate the operation of the algorithms. In most cases, M F R and Greedy require the same number of hops to reach the destination. However, Compass routing needs a few extra hops compared with the Greedy routing, while the delivery rate is similar. All these methods have high delivery rates for dense graphs, but low delivery rate for sparse graphs. However, the performance of M F R and Greedy routing come close to matching the path length given by the shortest path algorithm in case of successful delivery [6]. Hence, we might expect similar energy consumption for M F R, Greedy, and Compass algorithms. N C has fewer eligible next nodes compared to the other algorithms, which in turn reduces its delivery rate. However, when this algorithm succeeds, the probability of achieving the best power dilation is high. The energy consumption of N F P may have the power dilation in between N C and the other algorithms. Now we will present another example given in Figure 1, where the above mentioned algorithms fail to deliver the packet. In this example, the destination is d2. The next node is b in M F R, Greedy, and Compass algorithms. Then, b selects f which forwards the packet to the next node g. At this point, g forwards back the packet to f since its the only closest neighbor to the destination. According to the definition, all these algorithms will drop the packet at g. In the Greedy algorithm, if a node does not have any neighbors closer to the destination than itself, the packet gets stuck at the local maximum that reduces the delivery rates. N F P and N C also face a similar problem, when the packet arrives at f both the algorithms will drop the packet, where f does not have any neighbors closer to the destination than itself. All of these algorithms are routing loop free except Compass algorithm.

4.

SYSTEM MODEL AND PRELIMINARIES

5. RANDOMIZED ENERGY AWARE ROUTING ALGORITHMS

In our MANET model, which is adapted from [1], a set of mobile hosts are spread out in an environment that is modeled by the Euclidean plane: each mobile host with xand y-coordinates is represented by the point (x, y). All distances are Euclidean distances in the plane. We use the following hypotheses and notation:

The deterministic algorithms that follow a path, constructed based on a specific heuristic, might face the local maximum or routing loop, or lack of eligible neighbors during the packet forwarding process. Therefore, this may reduce their packet delivery rate, which is the primary objective of a routing algorithm. The deterministic face and GF G algorithms successfully overcome the above mentioned problems

1. Any mobile host knows the coordinates (x,y) of its position.

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and always guarantee packet delivery at the price of following a long path. Another simple and efficient way to avoid these problems might be using randomization when choosing a neighbor. A position-based routing algorithm is randomized if the next node is chosen randomly out of the neighbors of the current node. In [2], Bose and Morin proposed a randomized algorithm called Random Compass in the context of triangulations. In Random Compass the next node is chosen uniformly at random from the two nodes that satisfy the directional heuristic going in the clockwise and the anticlockwise directions. The algorithm has a higher delivery rate than the deterministic algorithms at the price of a longer path length. In [4], the authors proposed a new set of randomized algorithms, called AB algorithms, to increase the packet delivery rate with the control over the path length. In AB algorithms, the current node selects two neighboring nodes (candidate nodes) from above and below the cd line based on either the greedy or compass heuristic. The next node x is then chosen from these two nodes based on some probability distribution. For example, in uniform distribution, the next node is chosen uniformly at random out of the candidate nodes. Let na and nb be the candidate nodes from above and below the cd line, and disna d and disnb d be their distances to the destination d, respectively. Furthermore, let θna = na cd and θnb = nb cd be the angles formed by na and nb with c and d, respectively. In case of distance based biasing, the candidate nodes na and nb have weights of disnb d /(disna d + disnb d ) and disna d /(disna d + disnb d ), respectively. The next node is chosen such that the probability to pick the candidate node closest to the destination is high. However, in angle based biasing, the weights of na and nb are θnb /(θna + θnb ) and θna /(θna + θnb ), respectively. The next node is the one that minimizes the angle to the direction of the destination with high probability. The simulation results in [4] shows that AB algorithms not only have high packet delivery rates but also have good stretch factor. However, the above mentioned randomized algorithms do not consider the energy constraints while routing packets, and as a result might not be directly applicable in an energy constrained environment. In this paper we extend the above schemes and propose new energy aware randomized algorithms that enable higher packet delivery rates and efficient utilization of energy in the network. Below, we present variant of our algorithms: Let na and nb be the neighbor of c from above and below the cd line, respectively. Furthermore, Pca = dis2cna + C and Pcb = dis2cnb + C are the power needed to forward one bit information from c to na and nb , respectively. Here, discna and discnb are the Euclidean distance between c and na and nb . Finally, the cost at na and nb are Costna = 1/g(na ) and Costnb = 1/g(nb ), respectively. Our randomized algorithms can then be defined as follows:

such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that Costb ∗ disnb d is minimized among such neighbors. The next node x is chosen from na and nb with probability disnb d /(disna d + disnb d ) and disna d /(disna d + disnb d ), respectively. 3. Power*CostGreedy: Let na be the neighbor of c from above the cd line such that (Pca ∗ Costa )disna d is minimized among such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that (Pcb ∗ Costb )disnb d is minimized among such neighbors. The next node x is chosen from na and nb with probability disnb d /(disna d + disnb d ) and disna d /(disna d + disnb d ), respectively. 4. PowerCompass: Let na be the neighbor of c from above the cd line such that Pca ∗ θna is minimized among such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that Pcb ∗ θnb is minimized among such neighbors. The next node x is chosen from na and nb with probability θnb /(θna +θnb ) and θna /(θna + θnb ), respectively. 5. CostCompass: Let na be the neighbor of c from above the cd line such that Costa ∗ θna is minimized among such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that Costb ∗ θnb is minimized among such neighbors. The next node x is chosen from na and nb with probability θnb /(θna +θnb ) and θna /(θna + θnb ), respectively. 6. Power*CostCompass: Let na be the neighbor of c from above the cd line such that (Pca ∗ Costa )θna is minimized among such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that (Pcb ∗ Costb )θnb is minimized among such neighbors. The next node x is chosen from na and nb with probability θnb /(θna + θnb ) and θna /(θna + θnb ), respectively. This process continues until the packet reaches the destination or traverses a number of hops equal to three fourths of the total number of nodes in the network. In the later case, we drop the packet where randomized algorithms fail. Also, in cost aware routing, current node c needs to know the exact remaining power available at its neighbors. Hence, in our work, nodes periodically broadcast the battery power information to their neighbors. It is obvious that the smaller the broadcast period, the more accurate the battery power information. However, if the period is too small, it will increase the communication overhead. On the other hand, a larger period might lead the nodes to use inaccurate battery power level. Hence, we choose the broadcast period in between two extremes. When a node forwards 3 consecutive routing packets, it broadcast its remaining battery lifetime to the neighbors. Our algorithms first select two candidate nodes according to the definition to control both the energy requirement and distance or direction (ensure progress) to the destination. The difference between our algorithms and the AB algorithms is that in AB algorithms the communication and/or battery power is not taken into account to select the candidate neighbors. In its new version we expect a performance that is similar to AB algorithms in terms of the packet delivery rates; however, since we control the energy consumption both at communication and node levels, we expect a better

1. PowerGreedy: Let na be the neighbor of c from above the cd line such that Pca ∗ disna d is minimized among such neighbors. Similarly, let nb be the neighbor of c from below the cd line such that Pcb ∗ disnb d is minimized among such neighbors. The next node x is chosen from na and nb with probability disnb d /(disna d + disnb d ) and disna d /(disna d + disnb d ), respectively. 2. CostGreedy: Let na be the neighbor of c from above the cd line such that Costa ∗disna d is minimized among

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fail when the number of hops in the path computed so far exceeds three fourths of the number of nodes in the graph. To compute the average packet delivery rate, this process is repeated with 100 random graphs and the percentage of successful deliveries determined. Additionally, the average power dilation is computed. There are different radio models for energy aware routing. For example, in [11], Rodoplu et al. proposed a power consumption radio model based on the observation that direct transmission is more power consuming than relaying messages through the intermediate nodes. Also, transmission power is related to the path loss as 1/dn , where n is the path loss exponent. Finally, transmit and receive circuitries are subject to the energy consumption. Hence, the energy required to transmit a message from node A to node B at distance d is dn + c/t, where t and c are the required energy at the transmitter and receiver, respectively. In their simulation, they adopt the values of n, t, and c as 4, 10−7 mW, and 20 mW. The power needed to transmit one bit of information is then d4 +2∗108 . In [13] Stojmevovic et al. call this model as RM model. Another radio model is, called HCB model in [13], proposed by Heinzelman et al. in [7]. In their model, the energy needed to transmit one bit of information between two nodes is amp dn + 2Eelec , where n = 2 is the path loss exponent. The radio dissipates at transmitter and receiver circuitry is Eelec = 50nJ/bit and transmitter amplifier is amp = 100pJ/bit/m2 . We can further normalize the energy requirement by setting E = Eelec /amp . Hence, the final expression becomes (d2 + 1000) to transmit one bit of a message. We use HCB model to evaluate the performance of the algorithms. In addition to the uniform distribution, we also consider cluster distribution. In such distribution, 60 nodes are distributed randomly on the above mentioned two dimensional plane, and the remaining 27 nodes form three clusters A, B, and C. They are centered at (15, 15),(25, 25), and (75, 75) each with 9 nodes. Uniformly distributed nodes are used to ensure connectivity among clusters. The first two clusters are overlapped and C is disjoint. We randomly pick the source from A and the destination from C. We mainly consider 100 source-destination pairs from 100 differnt topologies and compute the average packet delivery rates and power dilation of all the algorithms.

overall energy consumption when routing packets between source and destination. Let us again consider the second example in Figure 1, where the deterministic algorithms fail to deliver the packet. The PowerGreedy will follow c − b − f , whereas PowerCompass will follow a − b − f with higher probability. At this point the algorithms may pick g and forward the packet to it. However, at some point h will be chosen as the next node, and the packet may reach the destination through the node i. Hence, it is clear that our randomized algorithms still offer higher packet delivery rates. In addition, we consider energy consumption of routing packets and nodes to pick the next node, this in turn ensures moderate power dilation. An immediate application of this class of algorithms is as follows: Mobile nodes may not always be distributed uniformly. For example, two sessions of a conference on wireless communications are going on in two different rooms of a building. People from these two rooms might need to communicate with each other through some of the intermediate mobile devices placed in between them. The resulting topology forms two distinct clusters of mobile devices. In this topology, only a few mobile devices might be placed between the clusters, we call this sparse part of the topology as a hole. Another such topology may appear in sensor networks. In sensor networks there are thousands of tiny sensors with constrained resources that are placed in an inaccessible environment. These sensors monitor the target area and send back the gathered information to one or more sinks (devices with more energy and computational power), which are accessed by the end users. Although the sinks are not usually energy constrained, direct communication between sinks and sensors might dissipate their energy abruptly. In [13] the authors mentioned that multihop routing might achieve better performance than direct communication to save the energy of sensors. However, in multihop routing, nodes close to the sink become hot spots and may lose their energy quickly because they are continuously forwarding messages for other sensors. This will create a hole around the sink. Hence, energy aware randomized algorithms are well suited for this particular situation.

6.

SIMULATION ENVIRONMENT

In the simulation experiments, a set S of n points (where n ∈ {75, 100, 125, 150}) is randomly generated on a square of 100m by 100m. For the transmission range of nodes, we use 15m (experiments showed that with lower transmission radii, the graph was too often disconnected, and with higher transmission radii, the generated graphs were so dense that the delivery rate of all algorithms approached 100%). Each node has initial energy level between 3M and 4M, where M = 106 , which is assigned randomly. After generating a fully connected U DG(S), a set of 100 source-destination pairs is randomly chosen. A fixed size data packet of length 16 bytes is used in addition to a 6 byte control packet that contains the ID and current battery level of nodes. This control packet is periodically broadcast by a node to its neighbors to advertise the current energy levels (nodes broadcast their energy level after forwarding every three consecutive routing packets). All the routing algorithms are then applied in turn on the chosen source-destination pairs. Clearly, an algorithm succeeds if a path to the destination is discovered. The deterministic algorithms are deemed to fail if they enter a loop, while the randomized algorithms are considered to

7. DISCUSSION OF RESULTS Detailed simulation results for all the routing algorithms are given in Tables 1 and 2 for the case when the transmission radius is 15m. In particular, we are interested in the performance of our proposed randomized routing algorithms compared to the previously published routing algorithms Greedy, Compass, PowerProgress, and ProjectionPowerProgress. The randomization helps us to avoid the local maximum or routing loop and offers high packet delivery rate. However, this also means that the extra paths found can be long, and this contributes to the higher power dilations of the randomized algorithms. Our simulation results give us the exact expected results. It is immediately evident from the results given in Tables 1 to 2 that all the deterministic algorithms have the worst delivery rates but the best power dilations. All the randomized algorithms improve on the delivery rates of the four deterministic algorithms. Among all the algorithms, PowerProgress has the best power dila-

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Algorithms n = 75 Greedy 61.17 Compass 63.08 PowerProgress 52.82 ProjecPProgress 57.98 PowerGreedy 76.59 CostGreedy 70.99 Power*CostGreedy 69.18 PowerCompass 79.19 CostCompass 78.46 Power*CostCompass 78.99

n = 100 72.33 73.24 64.21 69.79 86.06 85.51 82.93 88.46 88.11 87.32

n = 125 84.52 85.81 78.68 84.12 94.82 95.54 94.29 96.59 96.14 96.29

n = 150 92.44 93.97 88.67 91.53 98.81 99.08 99.04 99.13 99.05 99.17

rent node is more energy consuming than picking a node that minimizes the angle formed between that node, the current node, and the destination. This contributes higher power dilation in our randomized greedy based algorithm. Also, in these algorithms, the packets may deviate from the direction of the destination, and this may offer low packet delivery rates. We also examine the behavior of the variants of our algorithms. In these variants, c picks the candidate nodes just as the above mentioned algorithms, however, the next node is chosen out of two candidate nodes that minimize power, cost, or power*cost to reach the destination. These algorithms also have delivery rates similar to our above mentioned algorithms. However, they have very high (almost three times larger than the deterministic algorithms) power dilations. This might be happening due to the fact that their corresponding biasing does not provide any progress (either following the direction or reducing the distance to the destination). The three compass based algorithms have similar performance. This is expected because three of the algorithms use the same biasing method to reach the destination. Hence, they have similar delivery rates. The power dilation is also similar with PowerCompass having slightly better results compared to the other two algorithms. We also expect it since PowerCompass always tries to minimize the power requirement during routing while following the direction to the destination. However, the CostCompass algorithm might follow a slightly longer path than PowerCompass, which mainly considers remaining lifetime of a node to pick the candidate nodes. Finally, Power*CostCompass algorithm just combines both power and cost to pick next nodes and has performance close to both methods. The greedy based algorithms also show similar behavior with PowerGreedy having the highest packet delivery rate as well as lowest power dilation. The other two algorithms have similar performance.

Table 1: Average packet delivery rate in terms of percentages on U DG, for transmission radius r = 15m. Algorithms n = 75 Greedy 1.02 Compass 1.05 PowerProgress 1.01 ProjecPProgress 1.07 PowerGreedy 2.24 CostGreedy 2.38 Power*CostGreedy 2.46 PowerCompass 1.74 CostCompass 1.75 Power*CostCompass 1.75

n = 100 1.03 1.07 1.02 1.09 2.19 2.39 2.51 1.64 1.66 1.70

n = 125 1.03 1.08 1.02 1.10 1.97 2.08 2.27 1.54 1.55 1.58

n = 150 1.03 1.09 1.02 1.12 1.68 1.69 1.87 1.34 1.37 1.38

Table 2: Average power dilation on U DG, for transmission radius r = 15m.

tion and the worst packet delivery rate. In this algorithm, candidates nodes are closer to the destination than the current node. Hence, if there is no such candidate node to forward the packets, the algorithm fails, however, when it succeeds the resultant path minimizes the total energy consumption of the packet. The greedy algorithm has power dilation close to the PowerProgress algorithm with much better delivery rate. The other two deterministic algorithms also have similar performance. We can divide our randomized algorithms in two groups based on their biasing strategy, namely greedy and compass based algorithms. The compass based algorithms dominate the other group both in terms of packet delivery rate and power dilation. PowerCompass, CostCompass, and Power*CostCompass algorithms select candidate nodes that are close to the direction of the destination in addition to minimizing the power and/or cost. After that the next node is chosen with higher probability such that it again minimizes the angle formed between that node, the current node, and the destination. In greedy based algorithms, PowerGreedy, CostGreedy, and Power*CostGreedy algorithms select the candidate nodes in a slightly different way. The candidate nodes minimize both the distance to the destination and energy requirement. The next node is picked with higher probability such that it reduces the distance to the destination. Our simulation results show that the first group has better performance in terms of both packet delivery and power dilation compared to the other one. Following the direction helps the packet to reach the destination early. On the other hand, in greedy based algorithms, candidate nodes are chosen such that these nodes minimize the distance to the destination and the energy requirement. However, choosing farthest neighbor of the cur-

Algorithms Greedy Compass PowerProgress ProjecPProgress PowerGreedy CostGreedy Power*CostGreedy PowerCompass CostCompass Power*CostCompass

Delivery Rate 52.17 51.01 37.20 45.31 68.03 66.73 66.74 77.03 75.73 74.74

Power Dilation 1.04 1.10 1.02 1.19 2.31 2.78 2.86 1.95 2.09 2.10

Table 3: Average packet delivery rate and power dilation on U DG of 87 nodes under cluster distribution, for transmission radius r = 15m.

The results of our simulation on cluster distribution are given in Table 3. We investigate the performance of all the algorithms on cluster node distribution. The randomize algorithms still dominate the deterministic algorithms in terms of packet delivery rate though the performance on power dilation is just opposite. Furthermore, compass based algorithms again have better performance than greedy based algorithms.

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7.1 Effect of Node Density

10. REFERENCES

As the number of nodes grows, the delivery rate of all the algorithms increases. However, we notice the significant change in delivery rates in the case of deterministic algorithms that perform well in dense networks. In contrary, our randomized algorithms show the same behavior in sparse to dense networks. This gives another explanation of the applicability of our algorithms in a sparse network with the presence of local maximum. In terms of power dilation, however, the results are different for the deterministic and randomized algorithms. For the deterministic algorithms, the power dilations increase very slightly as the number of nodes increases. Whereas, our randomized algorithms show the opposite trend. The power dilations in the UDG decrease as the number of nodes increases. For larger values of n, the number of possible paths available to the randomized algorithms increase, so the power dilation can be expected to decrease. For instance, in a denser graph, the algorithm can recover from a bad path earlier, which leads to a lower power dilation for the algorithm. The deterministic algorithms on the other hand might need to traverse some extra length in dense networks, which leads to slightly worse dilations.

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8.

CONCLUSIONS

In this paper, we extended our previous work [4] and proposed a set of new randomized energy aware routing algorithms for mobile ad hoc and sensor networks. In our algorithms, the current node holding the packet always forwards the packet to the next node based on the position of itself, its neighbors, and the destination. The compass based algorithms are called PowerCompass, CostCompass, and Power*CostCompass algorithms. In particular, to determine the next node at any point, these algorithms pick one candidate above and one below the line between the current node to the destination, by using the heuristic that minimizes both the power and/or cost and the angle (formed between the current node, candidate node, and the destination). The next node is then chosen with higher probability to be closer to the direction of the destination. On the other hand, the greedy based algorithms PowerGreedy, CostGreedy, and Power*CostGreedy first minimize the energy requirement and the distance to the destination for the chosen candidate notes. Then, the next node is the one that is closest to the destination, which is picked with higher probability. Our simulation results demonstrate that our randomized energy aware algorithms yield a definite improvement over all deterministic algorithms studied in terms of the delivery rate. The best power dilations are achieved by the deterministic algorithms, however, by using weighted randomization based on the angles created by the candidate neighbors and the cd line, we can maintain the improved delivery rates while greatly reducing the power dilations of the randomized algorithms. Our algorithms retain their performance even under cluster node distribution while deterministic algorithms lose their performance.

9.

ACKNOWLEDGMENTS

The anonymous referee’s comments are gratefully acknowledged. This research is supported in part by the Natural Sciences and Engineering Research Council, Canada.

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