Capacity Deficit in Mobile Wireless Ad Hoc Networks Due to Geographic Routing Overheads

Transcription

1 Capacity Deficit in Mobile Wireless Ad Hoc Networks Due to Geographic Routing Overheads Nabhendra Bisnik, Alhussein Abouzeid Abstract Mobility of nodes may cause routing protocols to incur large overheads in order to maintain reliable paths between source destination pairs. These overheads diminish the capacity available for transmitting useful data over a mobile wireless ad hoc network. Understanding the lower bounds of the protocol overheads incurred by routing protocols in a mobile ad hoc network is important for developing efficient routing protocols, and for understanding the actual capacity available for network users. In this paper we use an information-theoretic approach for characterizing the lower bounds of protocol overhead associated with geographic routing. We formulate the minimum overhead problem as a rate distortion problem. The formulation may be applied to any traffic arrival process, mobility model and location service schemes. We evaluate a lower bound on the minimum protocol overhead, in terms of mobility pattern and packet/session arrival process, that is required for routing packets with desired level of reliability. We evaluate expressions for the lower bound for various packet arrival processes. We then characterize the deficit in transport capacity caused by the protocol overheads. We also characterize the critical network size, above which the capacity available to network users diminishes to zero because of the protocol overheads. I. INTRODUCTION Efficient routing of data packets is one of the fundamental challenges associated with mobile wireless ad hoc networks. Unlike wired communication networks, wireless ad hoc networks are highly dynamic in nature, which increases the complexity of finding and maintaining optimal routes. Due to fading and multipath effects the quality of paths changes over time [8], [35], [9]. The nodes in ad hoc networks may leave or join the network at random leading to frequent changes in possible routes to a destination. The routing problem in ad hoc networks in further exacerbated by node mobility [6], [5], which is an essential element of untethered networks. Due to node mobility the paths and neighborhoods of a node may change in an unpredictable manner. A routing protocol in ad hoc network is not only required to discover paths to the destination but also make the routing resilient against the above mentioned fluctuations in the state of an ad hoc network. Since the ad hoc networks lack any kind of infrastructure or centralized control, the routing decisions have to be made by the nodes in a completely distributed manner. The primary goal of a routing protocol is to gather state information from the network and make an informed forwarding decision in order to enable data packets to be received by the intended destination. The state information may comprise of node locations, link states, velocity and direction of node motion, queue lengths etc. The objective of the forwarding decision may be optimize delay, throughput, network lifetime or load balance while making sure that the packets eventually reach their destinations. There is a trade-off between the overhead incurred while collecting state information and the performance of a routing protocol - more state information that a routing protocol gathers, better forwarding decisions it may take. If infinite bandwidth is available and nodes communicate at arbitrarily high bit rate, it is possible to maintain accurate global information at all the nodes. Optimal forwarding decisions may then be made using routing algorithms, such as modified Dijkstra s algorithm, with link costs that reflect the quality of link [7] or any other metric of interest. However capacity of ad hoc networks is limited, thus routing protocols may have only limited or stale information which may lead to poor performance. For maintaining performance of a routing protocol above a certain threshold, the overhead incurred would increase with the rate at which the state information of the network changes. For a highly dynamic network, this overhead may consume a sizeable fraction of the network capacity leaving very little or no residual capacity for transmitting useful. We refer to the decrease in capacity available for transmitting data packets as capacity deficit caused by routing overheads In this paper we characterize the minimum routing overhead incurred in order to route packets with desired level of reliability. We investigate the dependence of the overhead on node mobility and rate at which the data packets are generated in the network. The family of routing protocols considered in this paper is geographical routing [8], [46], where each node maintains its location information at one or more location servers and this location information is used to route packets to the nodes. The node mobility model considered in this paper is the Brownian motion model [74]. Geographic routing protocol overheads may be divided into two categories i) Location update overhead: The overhead incurred by a node in order to maintain its location information at the location servers. ii) Beacon overhead: The overhead incurred by a node in order to allow neighbors to maintain consistent neighborhood tables and neighbor locations. The former overhead is critical for packets to be routed to their destination. If the accurate location information of a destination is not available at its location server then packets destined to the destination may never be delivered. The later overhead is critical for the source and intermediate nodes to make local forwarding decisions. If consistent neighborhood information is not maintained then an intermediate node may try to forward packet to a node that is no longer within its communication range. Thus in order to characterize the minimum routing overhead associated with geographical routing we investigate the following questions: i) What is the minimum rate at which a node must transmit in order to ensure that the location

2 information available at the location servers at the time the location server is queried) is within a distance ɛ of the accurate location? ii) What is the minimum rate at which a node must transmit beacons such that whenever a node needs to forwards a packet, the forwarding node knows with probability at least δ if the node belongs to its neighborhood or not? We use an information theoretic approach for answering the above questions. More precisely, both the problems are formulated as rate-distortion problems [9], [8]. For the former the measure of distortion is the squared error in the location information stored at location servers squared error distortion measure) while in the later problem the distortion measure is the probability that a perceived neighbor is not a actual neighbor of the node Hamming distortion measure). It should be noted that for reliable routing it is not required that accurate location information or consistent neighborhood information is maintained at all time instances. For reliably forwarding a packet, the source or intermediate nodes require consistent neighborhood information only when they have a packet to forward. Similarly the location servers are required to have accurate location information of the nodes they serve only when they are queried by source nodes for the location information. Thus in addition to the degree of mobility and parameters ɛ and δ, the minimum routing overhead also depends upon the packet arrival process. We compare the minimum geographic routing overhead with the transport capacity of multihop wireless networks evaluated in [3]. It is observed that when the node mobility is high and the average packet inter-arrival time is sufficiently small, complete transport capacity of an ad hoc network may be consumed by the routing overheads only. We derive an upper bound on the critical network size above which all the transport capacity of the network would be consumed by the routing overheads and no useful communication would be possible. In this paper we only consider the scenario where the routing protocol initiates the forwarding process as soon as a packet arrives at the source. Thus the capacity improvement due to node mobility achieved at cost of delay associated with waiting for the destination to move to a nearby location) pointed out in [3], [8] is not applicable to our work. A. Main Contributions The main contributions of this paper may be summarized in the following manner: ) We present a novel information-theoretic formulation for evaluating the minimum routing overhead incurred by the geographic routing. The formulation is very general, that is, it may be applied to any node distribution and may be extended to any location service scheme and mobility model. ) For the Brownian mobility model and various packet inter-arrival time distributions, we evaluate the lower bounds for the minimum rate at which a node must transmit its location information and beacons such that the packets are routed with desired level of reliability. Combining both overheads we find the lower bound on the capacity deficit caused by geographic routing overheads in wireless ad hoc networks. 3) For a given packet arrival process, standard deviation of Brownian motion and reliability parameters ɛ, δ), we evaluate the upper bound on the number of nodes the ad hoc network can support such that the complete transport capacity of the network is not used up by routing overhead. 4) We characterize the effective transport capacity of an ad hoc network after taking into account the minimum routing overheads that must be incurred for reliable routing. B. Paper Outline The rest of the paper is organized in the following manner. We review the related work in Section II. The network model is described in Section III. The lower bound of overhead associated with maintaining location information is evaluated in IV. The lower bound of minimum beacon rate required for maintaining consistent neighborhood information is evaluated in Section V. The capacity deficit caused by the routing overheads, the upper bound on the maximum number of nodes that may be deployed and the effective transport capacity is characterized in Section VI. The strengths and shortcomings of the approach used in this paper to characterize the routing overhead are discussed in Section VII. We conclude the paper with a discussion of future research direction in Section VIII. II. RELATED WORK In this paper we deal with many issues related to wireless ad hoc networks, namely routing protocols and their overheads, capacity of wireless networks and using information-theoretic approach to study communication networks. In this section we review previous works that have studied the above-mentioned issues. A. Routing Protocols and Their Overheads A lot of routing algorithms have been proposed for wireless ad hoc networks during the last two decades [], [36], [44], [75]. The routing algorithms may be broadly divided into topology based and position based or geographic routing protocols [57]. The topology based routing protocols make routing decisions based on the links that exist in the network. The topology based routing protocols may be further divided into the following three categories []: i) Proactive routing protocols - Information for routing packets between any node pairs is maintained within the network irrespective of whether the nodes are communicating or not e.g. [5], [6], [9], [37], [47], [6], [66], [67], [69]). ii) Reactive routing protocols - Information for routing packets between a source-destination is gathered only when source has packets to send to the destination e.g. [], [], [39], [4], [48], [65], [7], [7], [87]). iii) Hybrid routing protocols - Routing information for a subset of node pairs is maintained in proactive manner while routing information for other pairs is gathered in a reactive manner e.g. [4], [34], [6], [7], [9]). Since reactive routing maintains routes for nodes pairs that have an active session, the overhead of reactive routing protocols is smaller than that

3 3 of the proactive routing protocols when only few node pairs have active sessions at any given time. However since the source has to gather routing information when a new session arrives, the reactive routing protocols may lead to higher delays. The hybrid routing protocols try to achieve the best of both worlds. Since the routing information for a subset of nodes pairs is maintained in a proactive manner, source nodes may gather routing information for a new session with smaller delays. Thus on an average hybrid routing algorithms incur less overhead and delay than proactive and reactive routing protocols respectively. However in worst case the routing overhead incurred by all topology based routing protocols is similar. In contrast to topology based routing protocols, the geographic routing protocols use position of the destination nodes in order to make routing decisions [57]. Geographic routing requires nodes to know their locations. This may be accomplished using GPS [45] or other mechanisms [4]. Also geographic routing requires a distributed location service which allows source nodes to discover the location of the destination nodes []. All geographic routing protocols function in the following manner. When a new session/packet arrives at a source node, it queries the location service in order to discover the current position of the destination. The position of the destination is added to packet headers and the source and intermediate nodes forward the packet to a neighbor that is closer to the destination than them. If no such neighbor exists then the intermediate node resorts to a recovery mechanism in order to discover alternate paths to the destination. Due to the availability of additional information in the form of geographical locations of the nodes, the geographic routing protocols is free from the overhead caused link failure and hence has higher scalability than topology based routing protocols. The overhead of geographic routing largely depends on the overhead incurred by the underlying location service and the overhead required to maintain consistent neighborhood information. The location services may be categorized into flooding based and rendezvous based services []. The flooding based location services may be proactive or reactive. In proactive flooding based location services each node periodically floods its location throughout the network [8]. On the other hand, in reactive flooding based location services a source floods the location query for the destination when a new session/packet arrives and the source does not have fresh information about the location of the destination node [48]. In rendezvous based location services, each node maintains its location information at a one or more than one nodes using periodic updates. In contrast to flooding based location services the location information is never propagated throughout the network. The rendezvous based location services may be quorum-based or hashing-based schemes. In quorum based schemes, a node maintains its location at a subset of nodes. A source node queries another subset of nodes whenever it needs to know the location of the destination. The quorum rules decides which subset is chosen for maintaining location information and which subsets are queried by the source nodes. Partial overlap of the two subsets is necessary for proper functioning of quorum-based services. Examples of quorum-based services are [33], [85]. Hashing-based services use a well known hashing function in order to determine its location servers. When a node needs to discover the location of a destination, it uses the hash function in order to find out which nodes have the location information of the destination. The hash function may either map a node to another node s IDs or to a geographical locations, which may then be used to choose location servers. The hashing-based services may be flat [3], [83], [9] or hierarchical [54], [96]. A performance comparison of location services for geographic routing is presented in []. The effect of location errors and uncertainty on the performance of geographic routing protocols has been considered in [79], [8], [86]. The effect of error in location information on the packet delivery ratio and the power consumed at the nodes in an ad hoc networks in studied in [79]. It is observed that % error in node s location information may cause substantial loss in performance. An improvement, comprising of maintaining two hop neighborhood information, is proposed which makes geographic routing tolerant to 4% error in location. Similar observations regarding loss in performance due to location errors induced by increased mobility and beacon interval are made in [8]. The authors propose neighbor and destination location prediction schemes in order to mitigate the performance loss. The effect of uncertainty in location information on delay of geographic routing is studied in [86]. The uncertainty may be due to random GPS errors, coarse location information or availability of location information at only subset of node while other nodes randomly forward the packet. It is observed that the delay performance of geographic routing in presence of uncertain location information is within a factor of the delay performance of geographic routing using perfect routing information. The overhead incurred by routing protocols and their scalability properties has been studied in [3], [], [38], [4], [64], [76], [9], [93], [97], [98], [99], [], []. The initial studies [3], [], [4], [64] are mainly simulation based. The routing overhead incurred by DSDV [69], TORA [65], DSR [4] and AODV [7] is studied in [3] using detailed packet level simulations. It is observed that none of the routing protocol is the best across all scenarios. Instead each protocol performs well in some scenario and bad in others. It is found that TORA incurs the highest routing overhead and scales worst with the number of sources. Although DSR and AODV perform better than DSDV and TORA under moderate to high mobility, the routing overhead increases steadily with mobility. The performance of TORA is compared with that of ideal link state ILS) routing, a modification of OSPF [6] for mobile wireless networks, in [64]. It is observed that as the network becomes more dynamic the performance of TORA surpasses that of ILS in terms of both delay and routing overhead incurred. A exhaustive performance comparison of two most popular ad hoc routing protocols, DSR and AODV, is performed in []. It is observed that in most scenarios AODV performs well under low mobility while DSR performs well when mobility is high. Similar observations regarding performance of AODV, DSR and DSDV are made in [4].

4 4 Although simulation-based studies provide useful information about the performance of routing protocols, the observations may not be generalized to all scenarios. Therefore it is important to have analytical results for the performance of routing protocols in order to be able to develop a deep and general understanding of the trade-offs involved. In [38], [9], the authors present an analytical framework for characterizing the routing overhead for ad hoc routing protocols. Asymptotic results are provided for the overhead of proactive and reactive protocols in terms of network and routing protocol parameters such as packet arrival rate, hello packet transmission rate, hello packet size, size of route request packet, topology broadcast rate etc. The scalability properties of various routing protocols is studied in [76]. The authors find answers to the question that which routing protocol scales best with networks size, traffic intensity and mobility. The scalability of a routing protocol with respect to a parameter is defined as the maximum exponent of the parameter appearing in the asymptotic expression of overhead for that protocol. It is found that flooding and ZRP [34] scale best with mobility, link state [6] and ZRP scale best with traffic load while HSLS [77] is found to scale best with respect to the network size. The impact of traffic pattern on scalability of reactive routing protocols in ad hoc networks with unreliable but stationary nodes is studied in [], []. It is found that the reactive routing protocols may scale infinitely i.e. the routing overhead does not tend to infinity) with respect to network size for some traffic patterns. The analysis is extended to and similar results are obtained for cluster-based routing algorithms in [93]. In [97], [98], [99], the authors use an information-theoretic approach to characterize the minimum routing overhead and memory requirements of a topology-based proactive) twolevel hierarchical routing protocols for ad hoc networks. The entropy rate of topology change and the entropy of ad hoc network topologies is used to find the above mentioned bounds. Our work is along the same line. Instead of characterizing the overhead incurred by an implementation of a particular routing protocol, we present an information-theoretic analysis for finding the minimum cost, in terms of routing overhead, that must be paid in order to ensure that packets are routed with certain level of reliably using any implementation of geographic routing protocol. Similar to the impact of Shannon s work [8] on improving data communication and compression techniques, such bounds on performance of routing protocols may be helpful in better understanding and improvements of routing techniques for mobile ad hoc networks. B. Capacity of wireless ad hoc networks The characterization of the capacity of mobile wireless ad hoc network using a comprehensive model that takes into account mobility, delay, topology, energy, multiuser issues and characteristics of wireless channel is still a challenge. However in past few years a significant progress has been made in this direction. In their seminal work, Gupta and Kumar [3] used the concept transport capacity in order to characterize the throughput capacity of wireless ad hoc networks. The transport capacity of a network in bit-meters per second) is defined as the supremum over the set of feasible rate vectors of the distance weighted sum of rates [95]. It is shown in [3] for uniformly distributed stationary nodes, the transport capacity of wireless networks is Θ W ) An bit-meters per second, where W is the bandwidth of wireless channel and A is the area over which the network is deployed. This implies that W transport capacity per node is Θ A n ). Also it is shown that for random node-destination pairs the per-node throughput W scales as Θ n log n ). Both these results indicate that as the number of nodes deployed in an ad hoc network tends to infinity, the per-node throughput capacity of the network tends to. In other words, the wireless ad hoc networks at least with stationary nodes) do not scale well with network size. Following Gupta and Kumar s work, the capacity of wireless ad hoc networks has been investigated for various scenarios. A simulation based study of the capacity of multihop wireless ad hoc networks with IEEE 8. MAC has been done in [5]. It is found that with IEEE 8. MAC, the per-node ) capacity of multihop ad hoc network scales as O n more precisely as.47 n ). In [88] it is shown that under general model of fading channel ) the per-node throughput capacity scales as Θ. Similar bound for capacity is found when the nlog n) 3 nodes are mobile. However if On d ) delay is tolerable then n the per-node capacity is shown to scale as Θ d log n) ). 5 The capacity of arbitrary wireless networks with probabilistic receptions is evaluated in [58]. The capacity is expressed in terms of the packet reception probability density function and the distance between the source destination pairs. The authors then evaluate closed form expression of the capacity in regular networks in [59]. For Manhattan grid network the capacity is found to scale as O ) N, where N is the size of the Manhattan grid. The delay-throughput trade-off for the Gupta-Kumar model is investigated in [4] and it is found that D = Θ nt ), where D and T denote delay and throughput and delay respectively. A queuing theoretic approach for characterizing end-to-end delay and capacity of random access MAC based ad hoc networks is used in [] and []. It is shown that in presence with the collision avoidance mechanism, the maximum achievable per-node throughput scales as o nlogn ). The upper bound is not achievable because some of the network capacity is wasted due to the collision avoidance mechanism of the MAC protocol. The effect of TCP regulated traffic on the capacity of ad hoc networks is studied in [4] and [7]. In [7], it) is observed that RT T p the throughput of TCP scales as Θ, where RT T is the round trip time and p is the packet loss probability. Both RT T and p decrease with increase in the communication range transmission power) and thus TCP throughput also increases. However the increase in communication radius also leads to increase in MAC level interference and subsequent loss in capacity. Thus there exists an optimal communication radius that maximizes the capacity of ad hoc network with TCP regulated traffic. Simulations are used to verify the existence of such an optimal communication radius. Similar optimal communication radius is shown to exist in [4] even when

5 5 mobility is taken into account. The impact of multiple channels, multiple interfaces and directional antennas on the capacity of ad hoc networks is studied in [49], [53], [68]. OPNET simulations are used in [53] to characterize the guaranteed capacity for linear and grid network topologies with single and multiple channels. When RTS/CTS mechanism of MAC are enabled, it is found that for single channel case the per-node throughput scales as.67 n.744 in grid network. The use of multiple channels is observed to increase by per-node throughput by approximately 9.3 times. The impact of ratio of number of available channels to number of interfaces available at a node on transport capacity of ad hoc networks is studied in [49]. It is shown that if the ratio is below a certain threshold Olog n) for random networks and On) for arbitrary networks) then the transport capacity is worse than that of a network with single channel and interface. However if the ratio is above the threshold then capacity is improved. In [68] the capacity problem is formulated as network flows problem [3] and min-cut/max-flow theorem [6] is used to evaluate maximum stable throughput for wireless networks with omni-directional antennas, simple directional antennas and complex directional antennas. It is found that using arbitrarily complex signal processing techniques along with directional antennas, the throughput capacity is scaled by at most Θ log n) ). Although use of direction antennas and signal processing leads to some improvement in the throughput capacity of ad hoc networks, the network still cannot scale with number of nodes deployed. The above mentioned results regarding the capacity of multihop ad hoc networks are discouraging since they all indicate that ad hoc networks cannot scale with network size. However a few encouraging results have been found recently which indicate that it may be possible to construct scalable ad hoc networks. The transport capacity of wireless networks with fading channels is studied in [95]. It is shown that even when the channel state information is available the transport capacity grows as On). However if the network assumptions are relaxed to include independence and non-zeroness condition on the fading channel, the transport capacity of Ωn) may be achieved even without knowledge of channel state information. The transport capacity of ad hoc networks in terms of geographical locations of the nodes is evaluate in [43]. The transport capacity is shown to be c n, where c is a constant independent of n. These results are encouraging as they indicate that under certain conditions it is possible to obtain optimal capacity scaling of Θn) in wireless ad hoc networks. In [3], it is shown that if arbitrarily large delays are tolerable then the throughput capacity of mobile ad hoc network scales as O), that is the capacity is independent of the network size. This result is very important as it provides some optimism for designing scalable ad hoc network. The mobility of nodes, which was previously considered a nuisance in ad hoc networks, can be exploited in order to increase capacity. However the increased capacity is at the cost of delay which depends on the node mobility. This has triggered a large number of investigations into the delay-throughput trade-offs in mobile ad hoc networks. In [4], it is shown that for a Brownian n ) motion based mobility model the delay scales as Θ vn) in order to achieve O) throughput. Here vn) is the velocity of the nodes. The authors later extend their analysis to a more realistic random walk model in [3]. It is shown that in order to achieve any capacity improvement the delay scales as Θ n log n). In [56], it is shown that even if the mobility is restricted to one dimension the delay still scales as Θ n log n). Similar delay-capacity trade-offs are studied in [55], [8]. In [55], it is shown that in order to achieve throughput of Θ), an expected delay cost of Ω log n σ ) has to be paid, where σ is the standard deviation of the Brownian motion performed by the nodes. Also it is shown that any scheme that leads to a lower ) expected delay degrades the per-node throughput to O n The problem of finding the critical delay below which no capacity improvement is possible is also studied in [8]. In order to allow mobility models to be more realistic, the authors introduce hybrid mobility models. In this paper we are not concerned with capacity improvement achieved by increasing delay, since in most practical scenarios such a high delay may not be tolerable. Instead we are primarily concerned with the scenario where the routing protocol tries to send out a packet as soon as possible. In this case, the instantaneous transport capacity of a mobile ad hoc network is same as that of a static network. Thus we use Gupta-Kumar s result on transport capacity in order to characterize the capacity deficit, critical node density and resultant transport capacity. C. Information theory and communication networks So far information theory has not significantly influenced the design and understanding of communication network protocols. This unconsummated union is highlighted in [5] and several topics that may benefit from an information-theoretic insight are reviewed. One of the earliest attempts in using information theory to enhance the understanding of communication networks was made in [7]. In [7], Gallager used information-theoretic approach in order to characterize the lower bound on the amount of protocol information required to keep track of the sender, receiver and timing of messages in a communication network. Gallager analyzed a simple network consisting of two node with multiple synchronous sources and receivers. He also analyzed the effect of delay on the protocol information, i.e. how does delaying the transmission of message effect the amount of protocol information. It is found that although introduction of message delay decreases the protocol information, small average message length and high message arrival rate may lead to prohibitively high protocol information overhead. A information-theoretic investigation of transport capacity of wireless ad hoc networks is done in [94]. This study differs from previous studies in that the asymptotic bounds of transport capacity are found by taking position of nodes into account and without considering the interfering transmissions as noise. The distinction between interference and noise is particularly important. Although for the current technologies the interference is a nuisance that leads to packet collisions, use

6 6 of cooperative transmission and relaying [5], [63], [78] may allow the interference to be exploited for useful purposes. It is observed that for realistic propagation models the transport capacity scales as On). This result implies that no routing or node cooperation scheme can yield a transport capacity greater than cn, for some n. In this paper we are concerned with using information theory as a tool to analyze routing overhead in a mobile wireless network. The major portion of the overhead is incurred because topology of a mobile networks changes over time. Thus the routing overhead is directly related to entropy of node locations and therefore the entropy of network topology. A few papers have used information theory to understand the effects of node mobility on wireless networks. An analytical framework, based on entropy of node location, for characterizing delay and overhead associated with paging and routing a call to a mobile station in a cellular environment is provided in [73]. The complexity of tracking a mobile user in a cellular environment is studied using a information-theoretic approach and a position update and paging scheme is proposed in [9], []. An entropy based modeling framework for evaluating and supporting route stability in mobile ad hoc networks is proposed in [5]. In [89], the authors propose the entropy of link change as the metric for mobility models against which performance of wireless network protocols must be evaluated. III. NETWORK MODEL The ad hoc network consists of n nodes. The nodes perform Brownian motion with variance σ. In a multi-dimensional space, say with dimension k, the Brownian motion may be decomposed as independent Brownian motion in each dimension with variance σ /k. Let X i t) = {X i t), X i t),... X ik t)} denote the location of destination node i in a k-dimensional space at time t, where X ij t) denotes the coordinate of node i corresponding to the j th dimension. Let ˆXi t) = { ˆX i t), ˆX i t),... ˆX ik t)} denote the location information of destination i available at its location server at time t. Initially the location information of each destination is accurately known at its location server, i.e. X i ) = ˆX i ). In the following two sections, we first analyze overheads for k =. We then extend the results for k =. Note that k = and k = are cases of the most practical importance. However using similar approach the -dimensional results may be extended to any arbitrary k. The j th packet destined to destination i arrives at a node source of the j th packet) in the network at time T j. For all j >, S j = T j T j the packet inter-arrival time) is independently and identically distributed according to an arbitrary distribution with pdf f S t). In this study we first present results for an arbitrary f S t) and then investigate the special cases of deterministic, uniform and exponential interarrival time distributions. Similarly let τ i k) denote the time at which k th packet to be forwarded by node i arrives. The forwarded packets include both the packets generated by node i and the packets for which the node acts as an intermediate relaying node. The inter-arrival time of the forwarded packets τ i k + ) τ i k), k > ) is identically and independently distributed according to pdf denoted by f τ t). The communication radius of each node is r meters. Rendezvous based location service is used for maintaining locations of destinations at a subset of nodes acting as location servers. When a new packet arrives at a source node, it queries the location server of the packet destination for the location of the destination. The packet is routed to the destination according to greedy geographic forwarding using location of the destination returned by its location server. It is assumed the position of a destination does not change significantly while the location server is being queried by the source and the packet is being forwarded through the network. In other words the time scale of forwarding a packet is much smaller than that required for a significant change in position. Also the source node is always able to communicate with the location server and vice versa. IV. OVERHEAD ASSOCIATED WITH LOCATION INFORMATION UPDATES In this section we find a lower bound of the minimum rate at which each destination node must update its location servers such that the average squared error of the position available at the location server at the time when the location servers are queried is less than ɛ. We first introduce the notations and the rate distortion formulation for the minimum update rate problem. We then present the analysis for the one dimensional cases and extend it to the case of two dimensional network. A. Notation and rate-distortion formulation Definition : D i t) is the squared-error in the location information of destination i available at its location server at time t i.e. D i t) = X i t) ˆX i t) ) where X i t) ˆX i t) = l=k l= X il t) ˆX il t)). Definition : Xi N = {X i T ), X i T ),..., X i T N )} is the vector of locations of destination i at time instances T j, j N. Similarly ˆX i N = { ˆX i T ), ˆX i T ),..., ˆX i T N )} is the vector of location information at the location server of destination i at time instances T j, j N. Definition 3: Xi N and ˆX i N are defined as sets of all possible vectors Xi N and ˆX i N respectively. Definition 4: P N [x N i ; ˆxN i ] denote the probability that Xi N = x N i and ˆX i N = ˆx N i, where xn i ˆX i N and ˆx N i ˆX i N. Definition 5: D in is defined as D in = N N E[D i T j )] ) j= where E[D i T j )] is given by E[D i T j )] = x N i X i N ˆx N i ˆX i N P N [x N i ; ˆx N i ]D i T j ) 3) Definition 6: P N ɛ ) is defined as the family of probability distribution functions P N [x N i ; ˆxN i ] for which D in ɛ. Now we will present a rate-distortion theory based formulation to find the minimum number of bits required to represent the location information such that D in ɛ.

7 7 Definition 7: R N ɛ ) is defined as the minimum rate in terms of bits per packet) at which a destination must transmit its location information to its location server such that D in ɛ. According to [9], R N ɛ ) is given by R N ɛ ) = min P N P N ɛ ) N I P N Xi N ; ˆX i N ) 4) where I PN Xi N ; Xˆ i N ) is the mutual information between X i N and ˆX i N. The minimum rate at which a destination must update its location information such that a large fraction of packets are delivered, represented by Rɛ ), is given by Rɛ ) = lim N min R N ɛ ) 5) In the next two sections we characterize the lower bound of Rɛ ) for one and two-dimensional networks. B. Rate-Distortion Analysis for -Dimensional Network Lemma : The mutual information between Xi N satisfies the following relationship and ˆX N i inf I PN Xi N ; ˆX i N ) NR ɛ ) 6) P N P N Proof: This proof is very similar to the proof of [7, Theorem 3]. Using the standard definition of mutual information we get I PN X N i ; ˆX N i ) = HX N i ) HX N i ˆX N i ) 7) Now consider HX N i ˆX N i ) HXi N ˆX i N ) = HX i T ) ˆX i N ) N HX i T j ) X i T ), X i T ),..., X i T j ), ˆX i N ) j= Since conditioning cannot increase the entropy, we have From the above Lemma and definition of rate distortion HXi N ˆX i N ) HX it ) ˆX N i N ) + X function 4) and 5) it follows that HX i T j ) X i T j ) ˆX i T j )) 8) j= Rɛ ) R ɛ ) 9) 9) follows from 8) since HX i T j ) X i T j ) ˆX i T j )) is already conditioned on X i T j ), subtracting it from X i T j ) is similar to translating the random variable by a scalar. Define a new random variable, Y j j N ), such that Y j = ˆX i T j ) X i T j ) ) Since Y j depends only on ˆX i T j ) and X i T j ), we have HX i T j ) X i T j ) X i T j ), ˆX i T j )) = HX i T j ) X i T j ) Y j, X i T j ), ˆX i T j )) ) HX i T j ) X i T j ) Y j ) ) Equation ) follows from ) because conditioning does not increase entropy. Thus we get the following upper bound for HX N i ˆX N i ) HXi N ˆX i N ) HX i T ) ˆX i T )) + N HX i T j ) X i T j ) Y j ). 3) j= Now consider HXi N ), since X it j ) X i T j ) are independent of each other, we have HX N i ) = HX i T )) + N HX i T j ) X i T j )) j= Combining 7), 3) and 4), we get 4) I PN Xi N ; ˆX i N ) = IX i T ); ˆX i T )) + N IX i T j ) X i T j ); Y j ) 5) j= Notice that, the squared error in the location information may also be written as D i T j ) = Y j X i T j ) X i T j )) = X i T j ) ˆX i T j ) 6) Let d j = E[D i T j )]. Since X i T ) has the same distribution as X i T j ) X i T j ), and 6) is satisfied, therefore we have IIX i T j ) X i T j ); Y j ) R d j ) j 7) Define d i E[ X i T ) ˆX i T ) ], then by substituting 7) in 5) and using the convexity of the rate distortion function R, we get I PN Xi N ; ˆX N i N ) R d j ) NR N d j 8) N j= j= Now since P N P N, /N) N j= d j = D in ɛ, therefore from 8) we have I PN X N i ; ˆX N i ) NR ɛ ). = HX i T ) ˆX N i N ) + X HX i T j ) X i T j ) X i T j ), ˆX i T j )) The 9) following theorem provides a lower bound of the minimum rate at which a destination must update its location j= information. Theorem : In order to ensure high delivery ratio, the lower bound on the location update rate in bits per packet) is given by Rɛ ) hx i T )) log πeɛ ) where hx i T )) is the differential entropy of the location of destination i at the time when the first packet destined to it arrives in the network. Proof: From 9) we know that the minimum update rate is bounded by R ɛ ), which in turn is defined as R ɛ ) = inf I P X i T ); ˆX i T )) P P

8 8 Now consider I P X i T ); ˆX i T )), Here ) follows from ) since conditioning does not increase entropy. 3) follows from ) since for a fixed variance, normal distribution has the highest differential entropy. 4) follows from 3) since for P P E[X i T ) ˆX i T )) ] ɛ. Thus R ɛ ) hx i T )) log πeɛ ) and ) follows directly from it. Theorem implies that the minimum update rate largely depends on hx i T ), which in turn depends on two factors: i) the mobility pattern of the destination node and ii) the packet inter-arrival time process. Let f X x) denote the pdf of X i T ) without loss of generality, X i T ) = ). For Brownian motion with variance σ and packet inter-arrival time distribution f S t), f X x) is given by f X x) = and Rɛ ) is given by where τ= x e σ τ f S τ)dτ 5) πσ τ Rɛ ) hf X x)) log πeɛ ) 6) hf X x)) = x= f X x) log f X x)) dx 7) The lower bound of the minimum overhead incurred by location update information in terms of bits/second,denoted by Uɛ ), is given by Uɛ ) E[S] hf X x)) log πeɛ )) 8) In the remaining part of this subsection we evaluate f X x) and Rɛ ) for deterministic, uniform and exponential interarrival time distributions. ) Deterministic inter-arrival distribution: We first consider the case of deterministic arrival case because it is easy to obtain closed form results for this case and therefore we can focus on gaining insights without bothering much about complicated analysis. Suppose that packet destined to destination i arrives in the network after every T seconds, that is f S t) = δt T ) For such an arrival process, f X x) is given by f X x) = πσ T e x σ T and hf X x)) is given by I P X i T ); ˆX i T )) = hx i T )) hx i T ) ˆX i T )) = hx i T )) hx i T ) ˆX i T ) ˆX hf X x)) = i T )) ) log πeσ T ) hx i T )) hx i T ) ˆX Therefore the minimum update rate in bits per packet for i T )) ) hx i T )) hx i N, E[X i T ) ˆX deterministic packet inter-arrival time is given by i T )) ]))3) hx i T )) log πeɛ ) 4) Rɛ ) σ ) log T ɛ bits/packet 9) For the deterministic inter-arrival time the lower bound is similar to the minimum number of bits required to represent a Gaussian random variable with variance σ T subject to the constraint that the expected squared-error is less than ɛ. This is because the chance in position of the node between during a packet arrival interval is indeed a Gaussian random variable with variance σ T. The equation 9) bounds the minimum number of bits that a destination node must send to the location server for each packet destined to it that arrives in the network. For the deterministic arrival, the the packet arrival rate equals /T packets per second, thus minimum update rate in bits/second, represented by Uɛ ), is given by Uɛ ) T log σ ) T bits/second 3) Notice that Rɛ ) increases with both σ and T. This is because larger σ and T would imply larger uncertainty in the change in position of the destination during the packet arrival duration and hence more bits are required to represent the destination s position for each new packet. Uɛ ) also increases with σ for the same reason. However, Uɛ ) decreases with increase in T. This is because the increase in update rate due to higher uncertainty associated with high T is over-compensated by the fact that larger T implies that updates have to be made less often. For deterministic inter-arrival time it is also quite easy to construct a strategy to update the location server that achieves the bound in 3). The following strategy achieves the bound: Each destination updates its location at time kt k =,,... ) seconds by encoding N, σ T ) Gaussian random variable corresponding to the change of position since last update such that the squared error distortion is less than ɛ. However in a real-life scenario the arrival times of packets destined to a particular destination is not known a priori, although we might have some estimate of the arrival rate. It is expected that such uncertainty in the packet arrival rate would further increase the update rate. ) Uniform inter-arrival time distribution: Among all continuous time distributions with a given finite base and mean the uniform distribution has the highest entropy. Thus uniform distribution maximizes the uncertainty of packet arrival instances and thus would lead to maximum position update rate among all distributions with the same finite base and mean. Consider inter-arrival time to be uniformly distributed between [, T ] such that the inter-arrival time distribution is given by { f S t) = T, t T 3), otherwise ɛ

9 9 According to 5), f X x) is given by T f X x) = T x = πσ T e σ T + x σ T erf τ= x e σ τ dτ πσ τ x σ T ) x σ T 3) Let h U denote differential entropy of f X x), then update rate per packet is given by Rɛ ) h U log πeɛ ) bits/packet 33) and update rate in bits per second is given by Uɛ ) h U T log πeɛ )) bits/second 34) 3) Exponential inter-arrival time distribution: Among all the continuous time distribution with base [, ) and a given mean, the exponential distribution has the highest entropy. Also exponential distribution is widely used to model external inter-arrival time of packets to a given destination. This acts as a motivation for characterizing lower bound for position update when the packet inter-arrival time is exponentially distributed. We consider an exponential distribution with mean /α, given by f S t) = αe αt According to 5), f X x) is then given by f X x) = = x e σ τ αe ατ dτ πσ τ α x e σ ατ τ dτ 35) πσ τ Unfortunately the closed-form expression for the above integral cannot be evaluated for the above integral. So we will use numerical methods to calculate its value. Let h E be the differential entropy of the distribution f X x) given in 35). Then the lower bounds on update rate per packet and update rate per second are given by Rɛ ) h E log πeɛ ) bits/packet 36) Uɛ ) α h E log πeɛ )) bits/second 37) C. Update rate analysis for -dimensional network The update rate analysis for the two-dimensional case is based on the analysis for the one-dimensional case. The approach used in this analysis may also be extended to the three and other higher dimension case. Brownian motion in two dimension space may be decomposed into two independent one dimensional Brownian motions along x and y coordinates each with a variance σ /. Thus if X i t) = {X i t), X i t)} be the coordinates of destination i at time t then the distribution of X i t) is independent of the distribution of X i t). Recall the rate distortion function is Rɛ ) = lim inf N P N P N ɛ ) N I P N X N i ; ˆX N i ) Now consider I PN Xi N ; ˆX i N ). For the two dimensional networks, this may be written as I PN X N i ; ˆX N i ) = I PN X N i, X N i ; ˆX N i ) = I PN X N i ; ˆX N i ) + I PN X N i ; ˆX N i X N i ) = I PN X N i ; ˆX N i ) + I PN X N i ; ˆX N i ) = I PN X N i ; ˆX N i ) + I PN X N i ; ˆX N i ˆX N i ) + I PN X N i ; ˆX N i ) + I PN X N i ; ˆX N i ˆX N i ) = I PN X N i ; ˆX N i ) + I PN X N i ; ˆX N i ) where X N i = {X i T ), X i T ),..., X i T N )}, X N i = {X i T ), X i T ),..., X i T N )}, ˆXN i = { ˆX i T ), ˆX i T ),..., ˆX i T N )} and ˆXN i = { ˆX i T ), ˆX i T ),..., ˆX i T N )}. We know that D in ɛ implies N N j= N E [X i T j ) ˆX i T j )) + X i T j ) ˆX i T j )) ] N j= [ E X i T j ) ˆX i T j )) ] + N N j= ɛ [ E X i T j ) ˆX i T j )) ] ɛ Thus the rate distortion function Rɛ ) may be written as where Rɛ ) = min k r R) k ) + R ) ɛ k ) 38) R ) ɛ ) = lim R ) ɛ ) = lim inf N P N P N k ) inf N P N P N ɛ k ) N I P N X N i ; ˆX N i ) 39) N I P N X N i ; ˆX N i ) 4) From Lemma and Theorem, it follows that R ) ɛ ) hx i T )) log πee [X i T ) ˆX i T )) ]) 4) R ) ɛ ) hx i T )) log πee [X i T ) ˆX i T )) ]) 4) For a large fraction of packets to be routed to the destination E [X i T ) ˆX ] i T )) + E [X i T ) ˆX ] i T )) ɛ. Thus we have following upper bound for Rɛ ) hx i T )) + hx i T )) max k [,ɛ ] log πek) + log πeɛ k) ) ) 43) The RHS of 43) is minimized for k = ɛ /. Thus we have the following Theorem. Theorem : In order to ensure that the error in location

10 information used for forwarding packets is less than ɛ, the lower bound on the location update rate in bits per packet) for a two-dimensional network is given by Rɛ ) hx i T )) + hx i T )) log πeɛ ) bits/packet 44) Again here hx i T )) and hx i T )) depend on the mobility pattern and distribution of packet inter-arrival time. We now present the lower bounds of the update rate for various inter-arrival time distributions. ) Deterministic packet arrival: For the deterministic packet arrival process, where packets arrive at t = kt, k =,,..., the probability distribution functions of X i T ) and X i T ) are given by f X x) = f X x) = x e σ T πσ T and hx i T )) and hx i T )) is given by hx i T )) = hx i T )) = log πeσ T ) Thus the lower bound on location update rate in bits/packet and bits/second is given by σ Rɛ ) T ) log bits/packet 45) ɛ Uɛ ) T log σ T ɛ ) bits/second 46) The behavior of Rɛ ) and Uɛ ) is similar to that observed in the one-dimensional case. In fact the minimum rate is simply the double of what was observed in the one dimensional case. This is not surprising since the change in x-coordinate of the destination node is independent of the change in the y- coordinate of the destination. The change has variance σ / rather than σ as in the case of one-dimensional motion thus one might expect the minimum rate in -D to be less than times the minimum rate in -D case. However this decrease in entropy due to decreased variance is compensated by the fact that allowable squared error in both the coordinates is also decreased, leading to a factor of. ) Uniform distribution of packet inter-arrival time: For the uniform packet arrival process described in 3), the probability distribution functions of X i T ) and X i T ) are given by 4 x f X x) = f X x) = πσ T e σ T + x σ T erf ) x x σ T σ T 47) Let h U denote the differential entropy of X i T ) and X i T ) i.e. h U hf X x)) = hf X x)), then the lower bound on update rate is given by Rɛ ) h U log πeɛ ) bits/packet 48) Uɛ ) T hu log πeɛ )) bits/second 49) 3) Exponential distribution of packet inter-arrival time: For the exponential packet arrival process, the probability distribution functions of X i T ) and X i T ) are given by α x f X x) = f X x) = πσ τ e σ +ατ τ dτ 5) Let h E denote the differential entropy of X i T ) and X i T ) i.e. h E hf X x)) = hf X x)), then the lower bound on update rate is given by Rɛ ) h E log πeɛ ) bits/packet 5) Uɛ ) α h E log πeɛ )) bits/second 5) 4) Comparison of minimum update rates for various arrival processes: Figure shows how the minimum update rate varies with the variance of the Brownian motion while Figure shows how the minimum update rate varies with the the average update inter-arrival time. The uniform packet interarrival time distribution requires highest update rate in order to have large delivery ratio followed by exponential distribution. As expected the deterministic packet arrival requires least update rate, which is equal to value of rate distortion function of Gaussian distribution, with squared error distortion measure, at epsilon. The behavior of update rate bits/second) in Figures b) and b) is particularly interesting. These figures show that the update overhead in bits/seconds may be arbitrarily large if σ is large and E[S] is small. Thus if the motion of destination has a high variance or the packets destined to the destination arrive at a high rate then the location update overhead for reliable routing may become prohibitively high. V. OVERHEAD ASSOCIATED WITH BEACON TRANSMISSION A beacon is a broadcast packet transmitted periodically by each node to advertise its presence. Beacon are necessary so that nodes maintain consistent neighborhood information. In the case of geographic routing, apart from the header information such as node id and timestamp, the beacon packet also consists of location service information of the nodes. This is because in geographical routing each node not only needs to know which nodes belong to its neighborhood but it also needs to know their locations so that it may make correct forwarding decision. Thus apart from the location information, another component of the routing overheads is the beacon overhead. In this section we evaluate a lower bound on the minimum rate at which the nodes need to transmit beacons so that neighbors maintain a consistent neighborhood information. We then bound overhead in terms of bits per second. We first consider Brownian motion in one dimension and later extend the results to two dimension case. A. Notation and minimum beacon rate formulation All the notations defined in subsections III and IV-A are used throughout this section. In this subsection we define some additional notations for the analysis. Definition 8: N i t) is the set of nodes that belong to the neighborhood of node i. That is N i t) = {j : X i t) X j t) r, j n, j i} 53)