Link State Routing Overhead in Mobile Ad Hoc Networks: A Rate-Distortion Formulation

Transcription

1 Link State Routing Overhead in Mobile Ad Hoc Netorks: A Rate-Distortion Formulation Di Wang and Alhussein A. Abouzeid Department of Electrical, Computer and Systems Engineering Rensselaer Polytechnic Institute Abstract In this paper e use an information-theoretic formulation for characterizing the minimum routing overhead of link state routing in a mobile ad hoc netork. We formulate the minimum overhead problem as a rate-distortion problem. We evaluate loer bounds on the minimum overhead incurred by maintaining link state information hen link state routing protocols are designed ith guaranteed delivery ratio for data packets. We also characterize the deficit caused by the routing overhead on the overall transport capacity of a mobile netork. Further e derive a threshold value for the delivery error ratio, and conclude that no link state routing protocol can achieve a delivery error ratio smaller than this threshold. I. INTRODUCTION Routing in mobile ad hoc netorks faces strong challenges due to dynamically changing netork structure and limited communication resources. We vie that the primary objective of a routing protocol is to maintain state information such that packets can be forarded ithin an acceptable quality metric. The state information may be comprised of link states, node locations, velocity of nodes, queue lengths, etc. The quality metric could be minimum delay, maximum throughput, maximum lifetime or merely best effort delivery of packets. The design of routing protocols can be vieed as choosing a subset of classes of state information about the netork, and then specify hen and ho to collect and disseminate the chosen state information. The process of collecting state information incurs routing overhead. A number of routing protocols have been designed, and they differ from each other by selecting different subsets of classes of the state information. It may be difficult to make a comprehensive comparison among all the routing protocols, since there are many types of evaluation criteria to be used and many netork scenarios to be tested. The premise of this ork as ell as related prior ork [] and [] is that, instead of trying to find the best available protocol under certain cases, e could characterize the routing overheads incurred by a fe key classes of routing protocols. The ork in [] addressed the overhead for the class of topology-based hierarchical routing protocols, hile [] considers geographic routing protocols. This paper evaluates the overhead for the class of link state routing protocols under a flat non-hierarchical netork structure in hich nodes have identical protocol responsibilities. We present an information-theoretic frameork for characterizing the minimum overhead incurred by link state routing protocols. The key departure point of this frameork is to treat the link state as a random variable/process that exhibits random changes. The minimum routing overhead is the minimum amount of state information rate needed to be received by the nodes, such that the current state of the netork can be identified ithin a certain distortion bound. This necessitates the use of information theory and rate-distortion theory as a tool for developing loer bounds on the routing overhead. In this paper e study the class of proactive link state routing protocols, in hich each node collects the link state information of the netork from other nodes such that it can compute paths and thus make routing decisions immediately upon the arrival of data packets. In our analysis, the actual link status of a link and the perceived link status by an arbitrary node say, node k are both treated as random processes. To types of errors associated ith these to processes may occur at node k: the actual link status of a link is don but is perceived as up by k; the actual link status is up but is perceived as don by k. The link state routing overhead is the bit rate at hich a node must receive link state information such that the probabilities that the to types of errors may occur are bounded. We formulate the problems of finding the minimum value of these bit rates as rate-distortion problems see e.g. [3] for an introduction to rate-distortion theory. Using the rate-distortion formulation, e present loer bounds on minimum link state routing overhead for fixed distortions δ and δ for the to types of errors. We further sho that there is a connection beteen the probabilities that the to types of errors may occur, and the data packet delivery ratio η in the netork layer. This allos us to derive a loer bound on the minimum overhead for link state routing protocols ith guaranteed delivery ratio η. This result is then connected ith the results on the transport capacity of multi-hop ireless netorks evaluated in [4] in order to characterize the effective capacity available for users. It is observed that a delivery error ratio η cannot be achieved by any link state routing protocols if it is smaller than some threshold value η. In this paper e exclusively consider the scenario here the routing protocol initiates forarding process as soon as a packet arrives at the source. Thus the capacity improvement due to node mobility achieved at the cost of delay associated ith aiting for the destination to move to a nearby location pointed out in [5], [6] is not applicable to our ork. The class of link state routing protocols is dominant in ired netorks [7]. Hoever, hen migrating from the ired to ireless, conventional link state routing protocols may incur

2 large amount of overhead and thus face scalability challenges [8]. To improve the scaling properties of link state routing, a number of modified protocols have been proposed [9], [], [], [], [3], [4]. Specifically, fish eye routing described in [9], [] and [] introduces the notion of multilevel fisheye scope to reduce routing overhead through reducing the routing packet sizes and update frequency. Optimized link state routing protocols OLSR [] produce less control overhead as it forces the propagation of link state updates only at multipoint relay nodes [3]. Similarly, topology broadcast based on reverse path forarding TBRPF protocols [4] reduce link state update forarding at leaf nodes of each source tree and disseminates differential updates. To the best of our knoledge, there is still no theoretical ork reported on finding the minimum link state routing overhead. The objective of this ork is to fill in this gap. The relation beteen this paper and the existing related ork for modeling and analysis of protocol overhead can be summarized as follos. One of the earliest and pioneering orks that use information theory to understand protocol overhead in communication netorks can be found in [5]. Gallager [5] used an information-theoretic method to characterize the minimum amount of protocol information required to keep track of the sender, receiver and timing of messages for a simple stationary netork model. It is pointed out that if the packet lengths are small compared to the random interarrival times, then the protocol overhead required per packet can be prohibitively high. Several relatively recent papers that use information theory to understand the effects of mobility on ireless netorks can be found in [7], [8], [] and []. In [7], the authors propose the entropy of link change as the metric for mobility models against hich performance of ireless netork protocols must be evaluated. Authors of [8] use rate-distortion theory to investigate the optimal timing for updating a certain link state metric e.g. bandidth such that the update rate is minimized ithin a given cost constraint. One of our prior orks [] uses an information-theoretic approach to characterize the minimum routing overhead and memory requirements of hierarchical routing protocols for ad hoc netorks. Our ork shares a similar flavor ith another one of our prior ork [], in hich the family of geographic routing protocols are studied and rate-distortion loer bounds on geographic routing overhead are derived. Compared to [], [] and other related orks, our main contributions may be summarized as follos: We present an information-theoretic formulation for evaluating routing overhead incurred by link state routing. As compared to our prior ork [], apart from the different netork structures studied flat and hierarchical, the routing protocol model used in [] specifies the exact mechanisms about hen and ho to update netork state information. Such specifications are not used in this ork and thus this formulation is more general. We evaluate loer bounds for the minimum rate at hich a node must receive link state information such that data Lt Y Z Z Z3 Y T T T3 T4 T5 T6 Fig.. An illustration of the relationship beteen random variables Lt, Y m and Z m associated ith node pair i, j. packets can be delivered ith guaranteed delivery ratio. 3 We characterize the effective transport capacity of a mobile ad hoc netork after taking into account the minimum routing overhead incurred by reliable link state routing. 4 We derive the threshold value η for the delivery error ratio, and point out that no link state routing protocols can achieve a delivery error ratio smaller than this threshold. The rest of this paper is organized as follos. The netork and routing protocol models are presented in Section II. The rate-distortion formulation and the evaluation of a loer bound on the minimum routing overhead are presented in Section III. With the results in Section III, the rate-distortion loer bounds on the overheads of link state routing protocols ith guaranteed delivery ratio are derived in Section IV-A, and a discussion of the capacity deficit caused by routing overheads is presented in Section IV-B. We present conclusions and future research directions in Section V. A. Netork Model II. MODEL The ad hoc netork consists of n mobile nodes all lying in a torus of unit area. The location of node i at time t is given by X i t, i =,...,n. The process X i } is stationary and ergodic ith stationary distribution uniform on the area of interest; moreover, the trajectories of different nodes are independent and identically distributed i.i.d.. Each node is equipped ith an omnidirectional antenna and the transmission radius is given by r. It is assumed that nπr >, hich is a necessary condition for netork connectivity. Let the random variable L ij t,} denote the link status beteen any to nodes i and j at time t, here represents the link state is up don. Formally,, if Xi t X L ij t = j t r,, otherise Note that L ij t is induced by X i t and X j t, thus it is also a stationary ergodic process. For the sake of notation simplicity, in the rest of this paper e ill use Lt instead of L ij t if there is no ambiguity. In order to facilitate the analysis, the folloing assumptions regarding L } are made: Assumption : All the processes L ij i j, i,j n} are assumed to be mutually independent. Although this assumption may not be true in general, theoretical analysis in [, Section III.C.4] and simulation studies presented in [6, Y3 t

3 Lt L ijt Encoder... Decoder ˆLk ij t t T T T3 T4 T5 T6 ˆLt Fig.. An illustration of encode-transmit-receive-and-decode procedure for node k to monitor the status of link i, j. ˆT ˆT ˆT3 = ˆT4 = T4 ˆT5 ˆT6 t Section 9] sho that the dependency beteen to neighboring links is eak and can be negligible. Assumption : The random process L is modeled as an alternating reneal process [9], hich is elaborated as follos. Let Y m } m= and Z m } m= denote the sequences of successive up and don time durations of Lt the relationship beteen Lt, Y m and Z m is briefly illustrated by Fig.. Both the sequence Y m } m= and the sequence Z m } m= are assumed to be independent and identically distributed i.i.d. ith probability density functions pdf f Y and f Z, respectively. Moreover, e assume that Y m } m= and Z m } m= are independent from each other. The expected values of Y m and Z m are denoted by µ Y and µ Z, respectively. For the convenience of further analysis, e define u as B. Link State Routing Protocol Model u = µ Y + µ Z A link state routing protocol requires each node to collect link state information from other nodes such that it can compute a path to any potential destination node in the netork. The process of collecting link state information produces a routing overhead. Specifically, the link state information of any given node pair i,j i j may be encoded and sent out as control packets by the routing protocol, and then may be received and decoded by some node k k =,...,n in order to allo node k to monitor the link status of i,j. The link state routing overhead associated ith node k is thus formed by bit streams of all such control packets received by node k, for any given k =,...,n. This encode-transmit-receive-and-decode procedure is illustrated by Fig.. The link status of i,j, hich is represented by Lt, is encoded by the link state routing protocol and then k sent to node k such that k is able to reproduce it as ˆL ij t for simplicity, e ill use ˆLt instead in the rest of this paper. Here e make to assumptions as follos: Assumption 3: It is assumed that the perceived link status ˆLt is updated and maintained based on the link state information collected by k from outside only. This means that node k does not make its on decisions to determine ˆLt at any given time t e.g., no prediction based schemes are used; link status is maintained and updated according to the received control packets only. Assumption 4: It is assumed that the positions of node i and j do not change significantly during the encode-transmitreceive-and-decode procedure described above. In other ords the time scale of forarding a packet is much smaller than that required for a significant change in position. Fig. 3. An example of i, j s link status Lt associated ith the sequence T m}, and the perceived link status ˆLt at node k associated ith the sequence ˆT m}. III. OVERHEAD ANALYSIS For any given nodes i, j and k, there are basically to types of errors associated ith i,j s link status Lt and the perceived link status ˆLt by k: type- error is the one that happens at time t if Lt = and ˆLt = ; type- error is the one that happens at time t if Lt = and ˆLt =. In this section e evaluate a loer bound on the minimum bit rate at hich a node must receive link state information, such that the time portions of these to types of errors are bounded from above by δ and δ, respectively. A. Notation and Rate-Distortion Formulation Definition : S is the set of time instants at hich type- error happens during the time interval [,], i.e., S = t t,lt =, ˆLt = }. Similarly, S is the set of time instants at hich type- error happens during the time interval [,], i.e., S = t t,lt =, ˆLt = }. Definition : D L, ˆL is the time portion of type- error in the link state information of node pair i,j at node k over a time period [,], i.e., D L, ˆL = Lt ˆLt dt 3 t S Similarly, D L, ˆL is the time portion of type- error in the link state information of node pair i,j at node k over a time period [, ], i.e., D L, ˆL = Lt ˆLt dt 4 t S Definition 3: T m } m= is the sequence of time instants at hich the link status of i,j changes i.e. the value of Lt changes. Based on the Assumption 3 and Assumption 4 made in Section II-B, it can be observed that every change in ˆLt follos after a change in Lt at some time instant T m and is folloed by a change in Lt at the time instant T m+. And there is at most one change in ˆLt beteen T m and T m+ for any given m. Let ˆT m denote such a change in ˆLt. Fig. 3 illustrates an example of Lt and ˆLt. It shos that ˆLt changes its value at ˆT and ˆT, and has T ˆT T and T ˆT T 3. It can be noted that ˆT m is not clearly defined for some values of m m = 3,4 for the example shon in

4 Fig. 3. To make the definition of ˆT m complete, e define ˆT 3 = ˆT 4 = T 4 in the example illustrated by Fig. 3. Formally, e have the folloing definition for ˆT m. Definition 4: Let ˆT m } m= denote a sequence of time instants hich is defined according to the folloing to cases: If ˆLt is not constant over T m,t m+, i.e., there exists c,} and τ T m,t m+ such that then let ˆLt = c, t Tm,τ; c, t τ,t m+, 5 ˆT m := τ; 6 If ˆLt is constant over T m,t m+, i.e., ˆLt = c,}, for any t T m,t m+, then let ˆT m := Tm, if Lt = c, t T m,t m+ ; T m+, if Lt c, t T m,t m+. Thus e have a sequence ˆT m : m =,,...}, ˆT =, here ˆT m denotes the time instant at hich ˆLt changes its value, and the folloing inequality 7 T m ˆT m T m+ 8 holds for m =,,,... Definition 5: Let N denote the number of changes in Lt during the time interval [, ]. Formally, Similarly, N is defined as N = supm T m } 9 N = supm ˆT m } For the sake of simplicity, e ill use N and N instead of N and N in the rest of this paper. By Definition 3-5, it is readily knon that N N N Then according to the netork model described in Section II- B, the average rate of i,j s link status changes is given by N lim = lim N = = µ Y + µ Z u Definition 6: T N and ˆT N are defined as random vectors such that T N = [T,T,...,T N ] and ˆT N = [ ˆT, ˆT,..., ˆT N ]. Define T := T = and ˆT := ˆT = for the trivial case N =. Definition 7: T N and ˆT N are defined as the sets of all possible vectors T N and ˆT N, respectively. Definition 8: P [t N ; ˆt N ] denotes the joint probability density function pdf of random vectors T N and ˆT N, ith t N T N and ˆt N ˆT N. Definition 9: D is the expected value of the time portion of type- error over the time period [,], i.e., [ ] [ = E = t N ; ˆt N ] D dt N dˆt N D D P t N T N,ˆt N ˆT N 3 is the expected value of the time portion of Similarly, D type- error over the time period [,], i.e., D [ = E D ] = P t N T N,ˆt N ˆT N [ t N ; ˆt N ] D dt N dˆt N 4 Definition : P δ,δ is defined as the family of pdfs P [t N ; ˆt N ] such that the folloing inequalities hold: T m ˆT m T m+ D δ D δ 5 No e ill present a rate-distortion based formulation to find the minimum bit rate required to represent the link state information such that the inequalities in 5 are satisfied. Definition : R δ,δ is defined as the rate-distortion function associated ith the time interval [,] the minimum bit rate at hich a node must receive link state information such that the inequalities in 5 are satisfied. According to [3], R δ,δ is given by R δ,δ = min P P δ,δ IT N ; ˆT N 6 here IT N ; ˆT N is the mutual information beteen T N and ˆT N. The minimum rate in bits/sec at hich a node must receive link state information of i,j such that a large fraction of packets are delivered, represented by Rδ,δ, is given by Rδ,δ = lim R δ,δ 7 In order to facilitate the analysis in the folloing section, e define another to rate-distortion functions as follos. Consider the i.i.d. sequence Y m } dran according to the pdf f Y y, y Y. Let Ŷm} denote the reconstruction of Y m }. We assume that for any given m, the folloing reconstruction constraint is satisfied: Ŷ m Y m 8 Let f Y, Ŷ y,ŷ be the joint pdf of Y and Ŷ, y,ŷ Y Ŷ. The distortion measure, d : Y Ŷ R+, is defined as dy,ŷ = ŷ y 9 Let F Y ǫ denote the family of the joint pdf f Y, Ŷ y,ŷ satisfying f Y, Ŷ y,ŷ =, for y > ŷ, and E[dy,ŷ] = y ŷ ŷ yf Y, Ŷ y,ŷdydŷ ǫ. Thus the rate-distortion function for Y is defined as follos. Definition : R Y ǫ is the rate-distortion function for source Y, ith reconstruction constraint 8 and distortion measure 9. According to [3], R Y ǫ is given by R Y ǫ = min IY ;Ŷ f Y, Ŷ y,ŷ FY ǫ

5 Similarly, e define R Z ǫ as the rate-distortion function for the i.i.d. sequence Z m } dran according to the pdf f Z z ith the same reconstruction constraint and distortion measure as in Definition. B. Link State Routing Overhead Lemma : The mutual information beteen T N and ˆT N satisfies the folloing relationship inf IT N ; ˆT N P P δ,δ N R Z δ N + R Y δ Proof: Note that = N N IT N ; ˆT N IT N ; ˆT N ITN ; ˆT N T N, if N = N + ;, otherise. 3 4 Then only need to sho that IT N ; ˆT N is no less than the right-hand side of equation 3. When N is odd, i.e., N = M + and M N, e have IT N ; ˆT N = IT M,T M+ ; ˆT M, ˆT M+ = IT M ; ˆT M, ˆT M+ + IT M+ ; ˆT M, ˆT M+ T M IT M ; ˆT M, ˆT M+ = IT M ; ˆT M + IT M ; ˆT M+ ˆT M IT M ; ˆT M 5 Thus only need to sho that 3 is true for N is even, i.e., N = M and M N. Using the standard definition of mutual information e get IT M ; ˆT M = ht M ht M ˆT M 6 here h denotes the differential entropy. Then e consider ht M ˆT M. Without loss of generality, e assume that L =, i.e. T = Y. Thus Then Y m = T m T m 7 Z m = T m T m 8 ht M ˆT M = M ht m T m, ˆT M 9 m= M m= ht m T m, ˆT m 3 = M ht m T m T m, ˆT m 3 m= = M hy m T m, ˆT m m= +hz m T m, ˆT m 3 Equation 3 follos from 9 since conditioning does not increase entropy, and 3 follos from 3 since ht m T m, ˆT m is already conditioned on T m, subtracting it from T m is similar to translating the random variable by a scalar. Equation 3 follos directly from 3, 7 and 8. Define a random vector U m such that U m = ˆT m T m 33 Since U m depends only on ˆT m and T m, e have hy m T m, ˆT m = hy m T m, ˆT m,u m hy m U m ; similarly e can get hz m T m, ˆT m hz m U m. Then e get the folloing upper bound for ht M ˆT M : ht M ˆT M M m= hy m U m + hz m U m 34 No consider ht M. Since Y,...,Y m,..., Z,...,Z m,... are independent, e have ht M = M m= By combining 6, 34 and 35, e get hy m + hz m 35 IT M ; ˆT M M IY m ;U m + IZ m ;U m 36 m= Define ρ Y m = E[U m Y m ] 37 ρ m Z = E[U m Z m ] 38 By equation 33 it is easy to verify that U m Y m and U m Z m. Thus by Definition e have IY m ;U m R Y ρ m Y 39 IZ m ;U m R Z ρ m Z 4 From 3, 8 and 3 e have [ ] D = E Lt ˆLt dt t S [ M ] E m Z m m=u = Similarly e can get D M ρ Z m= m 4 M = m= ρ Y m. By substituting 39 and 4 in 36 and noticing the fact that rate-distortion functions R Y and R Z are convex and non-increasing, e have IT M ; ˆT M M m= R Y ρ m Y + M R Z ρ Z MR Y M MR Y M m= ρ m Y m= M D MR Z δ M m M + MR Z M + MR Z + MR Y δ M M D ρ Z m m= 4

6 Hence e have proved this lemma. The folloing lemma provides a loer bound on the minimum rate in bits/sec at hich a node k must receive link state information about an arbitrary node pair i, j. Lemma : Under the distortion criterion parameterized by δ and δ, the loer bound on the minimum rate in bits/sec at hich a node k must receive link state information of an arbitrary node pair i,j is given by Rδ,δ hy + hz + log u e u 43 δ δ here u is defined in equation. Proof: From Equation, Lemma and the definitions of rate-distortion functions 6 and 7, it follos that Rδ,δ = lim R δ,δ N lim δ + RZ N N RY δ N = RZ uδ + R Y uδ 44 u According to Definition, the rate-distortion function R Y ǫ is given by R Y ǫ = min IY ;Ŷ 45 f Y, Ŷ y,ŷ FY ǫ Let Expoϕ denote the exponential distribution ith mean ϕ. Then IY ;Ŷ = hy hy Ŷ = hy hŷ Y Ŷ hy hŷ Y 46 hy h ExpoE[Ŷ Y ] 47 hy log eǫ 48 Here equation 47 follos from 46 since for a nonnegative random variable ith fixed mean, exponential distribution maximizes the differential entropy. 48 follos from 47 since f Y, Ŷ y,ŷ FY ǫ. Thus Similarly R Y ǫ hy log eǫ 49 R Z ǫ hz log eǫ 5 By substituting 49 and 5 in 44 e get 43 and hence have proved this lemma. Since there are totally nn node pairs and based on Assumption, e get the folloing theorem that provides a loer bound on the minimum rate at hich a node must receive link state information from the entire netork. Theorem : Under the distortion criterion parameterized by δ and δ, the loer bound on the minimum rate in bits/sec at hich a node must receive link state information from the entire netork, denoted by Φδ,δ, is given by nn Φδ,δ hy + hz + log u e u 5 δ δ Theorem implies that, apart from the distortion-related parameters δ and δ, mobility patterns impose a major impact on the minimum link state update rate, since it largely depends on hy, hz and u. C. Routing Overhead Under Markovian Mobility Pattern No e consider a special case for the mobility pattern, here e assume that both the up-time Z and don-time Y for each link follo exponential distributions. This assumption implies that the process Lt}, hich describes the link status changes, is a to-state Markov process. This allos us to derive closed-form results for the loer bound on Φ, and therefore e may gain some insights ithout involving ith complicated analysis. Since both Y and Z follo exponential distributions, then hy = log eµ Y 5 hz = log eµ Z 53 Since the process Lt is a stationary ergodic process see Section II-A, then it is readily knon that µ Z Pr[ i,j s link status is up ] = = πr 54 µ Y + µ Z By substituting, 5, 53 and 54 in 5, e get the folloing corollary. Corollary : When the netork is exhibiting Markovian link status changes, the loer bound on the minimum rate in bits/sec at hich a node must receive link state information from the entire netork under the distortion criterion parameterized by δ and δ, is given by Φ nn u log πr πr δ δ 55 IV. PRACTICAL IMPLICATIONS In the previous section, e evaluated the loer bound on the minimum rate of link state routing overhead such that the error in the link state information available at each node is bounded. In this section, e ill sho that this result can be applied to capture the interplay beteen control and data traffic, i.e., the tradeoff beteen the overhead incurred by control packets and the delivery ratio for data packets. Then e characterize the deficit caused by the routing overhead on the overall transport capacity of a mobile netork. Further e derive a threshold value for the delivery error ratio, and conclude that no link state routing protocols can achieve a delivery error ratio smaller than this threshold. A. The Tradeoff beteen Control and Data Traffic Consider the design of link state routing protocols so as to ensure that the delivery ratio for data packets is no less than η. In other ords, such protocols require each node to be able to compute a valid path for any possible destination ith probability no less than η. Let Ψ η denote the class of link state routing protocols capable of ensuring a delivery ratio of η. Then a loer

7 bound on the routing overhead of any protocol ψ Ψ η is given by the folloing theorem. Theorem : In order to ensure that the delivery ratio for data packets be no less than η, the loer bound on the minimum rate in bits/sec at hich a node must receive control packets, denoted by Φη, is given by Φη nn u hy + hz + log ln n πr lngn,r e πu r ln n πr η here gn,r is given by gn,r = nln 56 πr 57 Proof : Consider any protocol ψ Ψ η. For any given nodes k and l, let A t denote the event that k is able to compute a valid path to l at time t, and let σt denote the probability that A t happens, i.e., Pr[A t ] = σt 58 Since protocol ψ is able to route data packets ith guaranteed delivery ratio of η, then e have lim σtdt η 59 Let α t and α t denote the probabilities that type- and type- errors in i,j s link state information occur at node k at time t, respectively, here node k and node pair i,j are arbitrarily chosen. That is, α t = Pr[Lt =, ˆLt = ] α t = Pr[Lt =, ˆLt 6 = ] Then the posteriori probability that link i, j is don given that the perceived link status at node k is up, denoted by ζt, is given by ζt = Pr[Lt = ˆLt = ] = = Pr[Lt =, ˆLt = ] Pr[Lt =, ˆLt = ] + Pr[Lt =, ˆLt = ] α t α t + πr 6 α t Let α, α and ζ denote the time averages of α t, α t and ζt respectively, i.e., α = lim α tdt α = lim α tdt 6 ζ = lim ζtdt Note that ζt Pr[Lt =, ˆLt = ] = α t and α t Pr[Lt = ] = πr, then it follos from 6 that ζ α 63 α πr 64 Let St denote the hop distance of the actual shortest path from k to l given the precise link state information of the netork. Similarly let Ŝt denote the hop distance of the computed path from k to l Ŝt := if k is unable to compute a path to l. Let F S s := Pr[S > s] denote the complementary cumulative distribution function ccdf of St, and let FŜt s denote the ccdf of Ŝt. Then the probability mass function pmf for Ŝt, denoted by fŝt s}, is given by fŝt s = FŜt s FŜt s, S N. 65 According to Assumption, the probability that the computed path from k to l is a valid path given its hop distance Ŝt = s, denoted by p s t, is given by s p s t = Pr[Lt = ˆLt = ] = ζt s 66 Then the probability σt that k is able to compute a valid path to l at time t is given by σt = n fŝt sp s t s= = ζt s FŜt s FŜt s s= = ζt ζt s FŜt s 67 s= Note that if node k is able to compute a valid path from to l at time t i.e. the event A t happens, then obviously the hop distance of the computed path must be no less than the hop distance of the actual shortest path, i.e. Ŝt St. This fact implies that Pr[Ŝt > s A t] Pr[St > s A t ] 68 According to equation 58 and 68, e have FŜt s Pr[A t ]Pr[Ŝt > s A t] σtpr[st > s A t ] σt F S s Pr[A t ]Pr[St > s A t ] = σt F S s σtf St At s 69 here F St At s = Pr[St > s A t ] is the ccdf of St conditioned on the event A t. Note that S is the hop distance of the shortest path beteen to vertices in a random graph in hich there are n vertices and each of the nn possible edges occurs independently ith probability q := Pr[Lt = ] = πr > n. Then according to Lemma 3 in [, Appendix], the ccdf of S is loer bounded by F S s e gn,r s n πr 7 Notice that the ccdf of St i.e. F S s is time-invariant. This is implied by the fact that St is a stationary process. The stationarity of St can be deduced as follos. St is the hop distance of the shortest path from k to l at time t, hich is determined by the node positions X i t} n i=. Since X i t} are assumed to be stationary and i.i.d., then St must also be a stationary process.

8 It can be deduced from equation 67 and 69 that σt ζtσt ζt s F S s s= +σt σtζt ζt s F St At s 7 s= in hich ζt s F St At s ζt s = ζt ; 7 s= ζt s F H s s= n ln ζt ln gn,r lngn,r lngn,r Γ s= s n πr ds ζt s e gn,r ln ζt Γ, lngn,r ln ζt, lngn,r n πr n πr 73 here Γa,b = x a e x dx is the loer incomplete b Gamma function. It is easy to sho after some algebraic manipulations that ln ζt Γ, ln n πr 74 lngn,r n Combining 7, 7, 73 and 74 e get σt ln n πr ζtσt+σt σ t 75 lngn,r Rearranging 75: ζt lngn,r ln n πr σt σt 76 Since ζt,σt, then } lngn,r ζt min, ln n πr σt σt ln n πr ln n πr σt 77 lngn,r Taking the time average of ζt and then combining ith 59: ln n πr σt ζ lim ln n πr lngn,r dt ln n πr ln n πr lngn,r η 78 Substituting 63 and 64 in 78 e have α α ln n πr ln n πr lngn,r πr η 79 According to Theorem, the bit rate of the minimum routing overhead is no less than Φα,α, i.e., Φη Φα,α 8 Then the result follos directly after substituting 79 and 8 in 5. The next corollary follos directly from Theorem hen Markovian link status changes are assumed. Minimum Rate in Receiving Control Packetsbits/sec.5 x n= n=5 n=.... Delivery Error Ratio η Fig. 4. The minimum rate of receiving control packets vs. delivery error ratio η and the number of nodes in the netork n. Corollary : When the netork is exhibiting Markovian link status changes, the loer bound on the minimum rate in bits/sec at hich a node must receive control packets, hile ensuring that the delivery ratio for data packets be no less than η, is given by Φη nn u log πr ln n πr lngn,r ln n πr η 8 here gn,r is given by equation 57. Fig. 4 shos the plot of the loer bound on Φ against delivery error ratio η and number of nodes n according to 8. It is shon that for large n, the minimum rate at hich a node must receive control packets becomes very high. Moreover, it is observed that there is a linear relationship beteen the loer bound on Φ and the logarithm of η, and Φ can be arbitrarily large ith the decrease of η. This result characterizes the tradeoff beteen the routing overhead incurred by control traffic and the routing efficiency for data traffic. B. Capacity Deficit A ireless ad hoc netork is said to transport one bit-meter hen a bit is transmitted over a distance of one meter [4]. The transport capacity of a netork in bit-meters/sec is defined as the supremum over the set of feasible rate vectors of the distance eighted sum of rates []. The transport capacity is expressed as λnl, here λ is the average arrival rate in bits/sec at the nodes, n is the number of nodes and L is the average distance travelled by the bits. It is shon in [4] that the transport capacity of an arbitrary ireless netork is Θ W na, here n, W and A are the number of nodes deployed, the transmission rate of these nodes and the area over hich the netork is deployed respectively. It is shon in [4] that for a particular interference model knon as the Protocol Model, the upper bound on the transport capacity of an arbitrary ireless netork is given by λnl 8 π W na bit-meters/sec 8

9 here is some constant. Theorem shos that in order to ensure a delivery ratio as high as η, each node must receive control packets at the rate Φη in bits/sec. Since the a control packet received by a node travels at least one hop hich covers a circular area of radius r, the total routing overhead measured in bit-meters/sec is at least nr Φη. This leads to the folloing theorem. Theorem 3: For the Protocol Model, the upper bound on the residual transport capacity available to an arbitrary netork for transmitting data [ λnl ] is given by R [ 8 λnl ]R π W n nr Φη bit-meters/sec 83 Theorem 3 implies that it is impossible to make the delivery error ratio η arbitrarily small, since the transport capacity may be overhelmed by the overhead incurred by control packets. Combining Theorem and 3 lead to the folloing corollary. Corollary 3: For link state routing protocols, a delivery error ratio η is not achievable if it is smaller than some threshold value η. The loer bound on η is given by η lngn,r ln n πr 4 W u π n 3/ +hy +hz n r e πu r 84 The next corollary follos directly hen Markovian link status changes are assumed. Corollary 4: When the netork is exhibiting Markovian link status changes, for link state routing protocols, a delivery error ratio η is not achievable if it is smaller than some threshold value η. The loer bound on η is given by η lngn,r ln n πr πr 4 W u π n 3/ n r 85 Fig. 5 shos the plot of the upper bound on [ λnl ] R against delivery error ratio η according to 83. It is orth noting that any η, [ λnl ] R in the region above the curve cannot be achieved by any link state routing protocol. V. CONCLUSION AND FUTURE WORK In this paper e presented an information-theoretic frameork for analyzing the protocol overhead of link state routing in a mobile ad hoc netork. We formulated the minimum overhead problem as a rate-distortion problem, and then derived loer bounds on the minimum bit-rate at hich a node must receive link state information in order to route data packets ith a guaranteed delivery ratio. This result is then connected ith the results on the transport capacity of multi-hop ireless netorks in [4] in order to characterize the effective capacity left for users. We derive a threshold value for the delivery error ratio, and conclude that no link state routing protocols can guarantee a delivery error ratio smaller than this threshold. Directions of future research may include extending our proposed information-theoretic frameork to a ider range of routing protocols. Developing practical routing schemes that approach to the bounds indicated by the theoretical results could be another option. Upper Bound on Residual Transport Capacity bit meters/sec 5.5 x Unachievable Region for η, [ λnl ] R η.... Delivery Error Ratio η Fig. 5. The upper bound on the residual transport capacity available for transmitting data packets vs. delivery error ratio η. ACKNOWLEDGMENT This ork as funded in part by the National Science Foundation under grants CNS-3956 and CNS REFERENCES [] N. Zhou and A. Abouzeid, Routing in ad hoc netorks: A theoretical frameork ith practical implications, Proc. INFOCOM 5, Mar. 5. [] N. Bisnik and A. Abouzeid, Capacity deificit in mobile irless ad hoc netorks due to geographic routing overhead, Proc. INFOCOM 7, 7. [3] T. M. Cover and J. A. Thomas, Elements of Information Theory, rd ed. John Wiley & Sons, 5. [4] P. Gupta and P. R. Kumar, The capacity of ireless netorks, IEEE Trans. on Information Theory, pages , Mar.. [5] M. Grossglauser and D. N. C. Tse, Mobility increases the capacity of ad hoc ireless netorks, IEEE/ACM Trans. on Netorking, 4,. [6] G. Sharma, R. Mazumdar, and N. Shroff, Delay and capacity tradeoffs in mobile ad hoc netorks: A global perspective, Proc. INFOCOM 6, 6. [7] J. 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