Distribution of Path Durations in Mobile Ad-Hoc Networks and Path Selection

Transcription

1 Distribution of ath Durations in Mobie Ad-Hoc Networks and ath Seection Richard J. La and Yijie Han Abstract We investigate the issue of path seection in mutihop wireess networks with the goa of identifying a scheme that can seect a path with the argest epected duration. To this end we first study the distribution of path duration. We show that, under a set of mid conditions, when the hop count aong a path is arge, the distribution of path duration can be we approimated by an eponentia distribution even when the distributions of ink durations are dependent and heterogeneous. Secondy, we investigate the statistica reation between a path duration and the durations of the inks aong the path. We prove that the parameter of the eponentia distribution, which determines the epected duration of the path, is reated to the ink durations ony through their means and is given by the sum of the inverses of the epected ink durations. Based on our anaytica resuts we propose a scheme that can be impemented with eisting routing protocos and seect the paths with the argest epected durations. We evauate the performance of the proposed scheme using ns- simuation. I. INTRODUCTION Muti-hop wireess ad-hoc networks have been the focus of active research in recent years. Unike a wireine network with a fied infrastructure, a wireess ad-hoc network can be depoyed with no infrastructure and mobie nodes can estabish and maintain a network in an autonomous manner. Due to nodes mobiity, inks are epected to be set up and torn down much more frequenty than in a wireine network. As a resut, a network topoogy varies with time as the connectivity between nodes changes dynamicay. Frequent ink faiures and network topoogy changes in mobie ad-hoc networks (MANETs) render the routing protocos designed for wireine networks (e.g., the Internet) rather inefficient. A suite of new routing agorithms have been proposed for MANETs to dea with frequent network topoogy changes 8,, 4, 5. A detaied discussion of avaiabe routing protocos is provided in the monographs 3. Due to nodes mobiity, inks aong a provided path may become unavaiabe in an unpredictabe manner. When one or more inks aong a path in use become unavaiabe (which we ca a path faiure), the path is no onger vaid and a path recovery procedure is triggered to find an aternate path. Detecting and recovering from a path faiure can take a nonnegigibe amount of time (from appications viewpoint), during which service to on-going traffic wi be disrupted. Such a disruption in service can degrade the performance of time-critica appications. Furthermore, an initiation of path recovery incurs additiona overhead. Therefore, from the perspective of providing reiabe network service and Authors are with the Department of Eectrica & Computer Engineering, University of Maryand, Coege ark. E-mai: {hyonga,hyijie}@isr.umd.edu. minimizing contro overhead, a good routing agorithm shoud take into consideration the epected duration as we as other requirements when seecting a path. The duration of a path refers to the amount of time for which the path remains avaiabe after its set-up unti one of the inks aong the path fais for the first time. Intuitivey the duration of a path shoud depend on the durations of the inks aong the path and their dependence structure. Therefore, there is much interest in better understanding the statistica properties of ink and path durations and their reation. Better understanding of their statistica properties wi aow us to approimate the frequency of disruption in service and resuting overhead. Hence, it wi hep us evauate the performance of on-demand routing protocos and the adverse effects of potentiay frequent disruptions in service on the performance of upper ayers (e.g., Transmission Contro rotoco) without having to run time-consuming detaied simuations. A numerica eampe using the Random Waypoint (RW) mobiity mode is given in 5, Section 8. To the best of authors knowedge there is very itte known about the distribution of path durations and its reation with those of the inks that provide them. Consequenty, most of eisting routing protocos seect a path based on some heuristic argument; the Dynamic Source Routing (DSR) protoco seects the minimum hop path, whereas the Ad-hoc On-demand Distance Vector (AODV) routing protoco seects the first discovered path. Associativity Based Routing (ABR) protoco seects the path with maimum average age of the inks. However, it is not cear how the hop count or the average age of the inks aong a path is reated to its (epected) duration. Aong this ine Sadagopan et a. 7 presented a simuation study of the distribution of muti-hop path durations under various mobiity modes. Their study shows that the distribution of path duration can be accuratey approimated by an eponentia distribution when the number of hops is arger than 3 or 4 for a mobiity modes considered. However, no cear epanation was offered for the emergence of an eponentia distribution. In order to correct the current state of affairs Han et a. 5 deveoped an approimate framework for studying the distributions of path and ink durations. They showed that, under certain conditions, the distribution of path duration (under appropriate scaing) converges to an eponentia distribution when the number of hops becomes arge. This resut is in ine with the simuation resuts provided in 7, and is obtained as a simpe appication of am s Theorem 7, Thm. 5-4, p. 57. In addition, they epored the connection between the epected duration of a path and the epected durations of the

2 inks aong the path. To be more precise, they showed that when the number of hops is arge, the inverse of the epected duration of a path is approimatey given by the sum of the inverses of epected durations of the inks aong the path. The resuts reported in 5 provide the first evidence that when the hop count is arge, the distribution of path duration can indeed be approimated by an eponentia distribution. The appication of am s theorem in 5, however, requires that the ink ecess ives be mutuay independent, which is in genera not true. The ecess ife of a ink in a path refers to the amount of time the ink remains avaiabe unti it is torn down for the first time after the path set-up. Two neighboring inks aong a path, for eampe, share a common node. Ceary, the ecess ives of these two neighboring inks depend on the mobiity of the shared node, introducing some eve of dependence in them. Moreover, such oca dependence in ink ecess ives may be more evident under group mobiity modes where the mobiity of a set of nodes may be correated. Therefore, it is of much interest to see if the same distributiona convergence to an eponentia distribution hods without the independence assumption and how the parameter of the imit distribution (which decides the epected duration) is affected by the dependence. In this paper we etend the resuts in 5 by reaing the independence assumption on the ink ecess ives imposed in 5; we ony require that the dependence of ink ecess ives go away asymptoticay with increasing hop distance between the inks. This assumption can be stated using what is known as a miing condition (Section V-A). It aows the possibiity of reativey strong oca dependence in ink ecess ives which can be ehibited, for eampe, under group mobiity modes. We demonstrate that, under some mid conditions (to be stated precisey), the same distributiona convergence to an eponentia distribution reported in 5 hods under this much weaker condition (Section V-B). We point out that reaing the independence assumption demands a new technique for proving the distributiona convergence and compicates the proof consideraby; a suitabe etension of am s theorem that deas with dependent processes is not avaiabe. We aso show that the parameter of the emerging eponentia distribution is the same whether the ink ecess ives are mutuay independent or not. In other words, the parameter of the eponentia distribution is given by the sum of the inverses of the epected ink durations. This suggests that for a path with a sufficienty arge hop count the dependence of ink ecess ives does not significanty affect the path duration distribution. Based on this observation, we outine a scheme that can be impemented in eisting routing protocos to seect the path with the argest epected duration with minima communication overhead (Section VI). We impement this scheme in AODV and evauate the performance gain (Section VII). The paper is organized as foows. A basic framework for modeing path durations is given in Section II. Section III introduces the set-up under which the asymptotic distribution of a path duration with increasing hop count is studied. In Section IV we study a simper case in which ink durations have the same distribution and their dependence is imited to a finite neighborhood. This is foowed by a study of more genera cases in which the ink duration distributions may be heterogeneous and the dependence is not imited to a finite neighborhood in Section V. We outine how our resuts can be used to impement a scheme for seecting a path with the argest epected duration during a path discovery phase in Section VI. Simuation resuts are presented in Section VII to demonstrate the performance gain from our proposed scheme. A word on the notation and convention used throughout: We define a the random variabes (rvs) of interest on some common probabiity space (Ω, F, ). Two IR vaued rvs X and Y are said to be equa in aw if they have the same distribution, a fact we denote by X = st Y. The independence between two rvs X and Y is denoted by X Y. If G is a probabiity distribution on IR +, et m(g) denote its first moment which is aways assumed to be finite. Convergence in distribution (with n going to infinity) is denoted by = n. For any in IR, with components (, ), set = +. II. A BASIC FRAMEWORK This section describes the same basic framework that we borrow from 5 for our anaysis: Consider a MANET where a set of nodes creates and maintains network connectivity. We assume that an on-demand agorithm is used and a path between a source node and a destination node is set up ony when a request is made. Let V = {,..., N} denote the set of N mobie communicating nodes. Each node moves across a domain D of R or R 3 according to some mobiity mode. Due to nodes mobiity, inks between nodes are set up and torn down dynamicay. We assume that a ink between two nodes is either up or down. Two nodes without a ink between them estabish such a ink as soon as they become reachabe, e.g., when they come within a transmission range of each other or when the signa to interference and noise ratio (SINR) at the receiver eceeds certain threshod, and packets from each other can be successfuy decoded. The atter case captures the characteristics of the physica ayer (e.g., path oss and channe fading) more accuratey. Athough this is not needed for the anaysis, communication inks are assumed bidirectiona since such bidirectiona communication is typicay required between two nodes for reiabe forwarding of packets, for instance, by means of acknowedgments for each transmission. Estabishing a path from a source node to a destination node requires simutaneous avaiabiity of a number of communication inks that are up at the time of path request and coectivey provide the desired connectivity between the source and the destination. The duration of a path provided by the underying routing protoco is then defined as the amount of time that eapses unti one of the inks aong the path goes down for the first time after the path set-up. A ink may go down (which we ca a ink faiure) due to either mobiity or degradation in channe condition. For simpicity of anaysis, path set-up deays are assumed negigibe. A. Reachabiity processes We mode the situation outined above as foows: For a pair of distinct nodes i and j in V, we introduce a {0, }-vaued

3 3 reachabiity process {ξ ij (t), t 0} with the interpretation that ξ ij (t) = (resp. ξ ij (t) = 0) if the unidirectiona ink from node i to node j, denoted by ink (i, j), is up (resp. down) at time t 0. Since the communication inks are assumed bidirectiona, we must have ξ ij (t) = ξ ji (t). The process {ξ ij (t), t 0} is simpy an aternating on-off process, with successive up and down time durations given by the rvs {U ij (k), k =,,...} and {D ij (k), k =,,...}, respectivey. The reachabiity processes can be defined in a number of ways. For eampe, for each i in V, et {X i (t), t 0} describe the trajectory of node i, i.e., X i (t) denotes the position of node i at time t 0. If we do not epicity mode channe fading between nodes, it is reasonabe to assume that two nodes can communicate with each other reiaby if the distance between them is smaer than some fied transmission range r min > 0. Hence, a ink between two distinct nodes i and j in V eists at time t 0 if and ony if their distance is smaer than r min, eading to the definition ξ ij (t) := X i (t) X j (t) r min, t 0. () Aternative modes can take into account the physica ayer characteristics of the channe. For instance, two nodes i and j in V can maintain a ink between them at time t 0 if and ony if ( j F ji (t) min, ) i F ij (t) > Γ () Ψ i (t) Ψ j (t) for some threshod Γ > 0, where i is the maimum transmission power of node i, and F (t) = (F ij (t)) denotes the channe gain matri (incuding fading) at time t with F ji (t) 0 and F ii (t) = 0, i, j =,... N. Different choices of Ψ i (t) in () ead to different physica ayer modes. In the simpest form, one can assume that a node i can decode the packets from node j if and ony if the received signa power eceeds some threshod Γ > 0, 6. In this case the reachabiity process between nodes i and j is given by () with Ψ i (t) = as the numerators give the argest achievabe received signa power at the nodes. Simiary, if one assumes that packets can be successfuy decoded if and ony if the achieved SINR eceeds the threshod Γ 3, 4, then the reachabiity process between nodes i and j is again given by () with Ψ i (t) = W i + k (t) F ki (t), (3) k T X(t)\{j} where W i is the noise variance at node i, T X(t) is the set of transmitters at time t and k (t) denotes the transmission power of node k. The right hand side of (3) represents the sum of noise power and interference at node i at time t. This impies that nodes i and j have connectivity if and ony if the achieved SINR vaue using the maimum transmission power eceeds Γ in both directions. B. ath duration Net we endow V with a time-varying graph structure by introducing a time-varying set E(t) of directed edges through the reation E(t) := {(i, j) V V : ξ ij (t) = }, t 0 (4) where by convention we set ξ ii (t) = 0 for each i in V and a t 0. Thus, a path can be estabished (in principe) between nodes s and d at time t 0, if node d is reachabe from node s by a path in the undirected graph derived from the directed graph (V, E(t)). Let sd (t) E(t) denote the set of paths from node s to node d providing this reachabiity. This set of paths is empty when the nodes s and d are not reachabe from each other at time t. When non-empty, this set sd (t) may contain more than one path since mutipe paths may eist between nodes s and d. In such a case, the routing protoco in use seects one of the paths in sd (t) and et L sd (t) denote the set of inks in the seected path. For each ink in L sd (t), et T (t) denote the time-to-ive or ecess ife after time t, i.e., T (t) is the amount of the time that eapses from time t onward unti ink is down. The time-to-ive or duration Z sd (t) of the estabished path from node s to node d using the inks in L sd (t) is defined as the amount of time that eapses from time t unti one of the inks in L sd (t) goes down, at which point a path recovery procedure is initiated. This quantity is simpy given by Z sd (t) := min (T (t) : L sd (t)), t 0. (5) III. THE SET-U AND MODELING ASSUMTIONS In this paper we are interested in studying the distribution of path duration as the number of hops becomes arge. In the foowing subsection we first describe the set-up used to mode this scenario. Then, we state the modeing assumptions under which the distributiona convergence of path duration is estabished with increasing hop count. A. The set-up In order to study the distribution of path duration with a arge hop count, we investigate the asymptotic distribution of path duration (under appropriate scaing of ink ecess ives) as the number of hop count increases. This is done by introducing a parametric scenario with a sequence of networks in which both the number of communicating nodes and the domain across which they trave increase: For each n =,,..., et V (n) = {,..., N (n) } and D (n) denote the set of mobie nodes and the domain across which the nodes move, respectivey. For each node i in V (n), the (t), t 0} denotes the trajectory of node i in D (n). The stochastic process that governs the arriva of path requests is assumed to be independent of these trajectory processes. D (n) -vaued process {X (n) i. Scaing We are interested in the situation where N (n) nn () and Area(D (n) ) n Area(D () ) (6) as n goes to infinity; it is customary to reparameterize so that N (n) = n. When in force, the scaing (6) guarantees N (n) Area(D (n) ) N () Area(D () ), From now on we omit this quaifier in a asymptotic equivaences.

4 4 so that the density of nodes, i.e., the number of nodes per unit area, is asymptoticay constant.. Stationarity As the system is epected to run for a ong time, we can assume that steady state has been reached. This possibiity is captured by taking the N (n) trajectory processes to be jointy stationary. Joint stationarity of the trajectory processes aso impies that the N (n) (N (n) ) reachabiity processes are jointy stationary. For distinct i < j in V (n), et the rvs {(U (n) ij (k), D (n) ij (k)), k =,,...} denote the sequence of up and down times for the reachabiity process {ξ (n) ij (t), t 0}. Writing ( (k) = (U (n) ij (k), D (n) ij (k)), i < j, i, j V (n)), k =,,..., we require that the sequence of rvs { (k), k =, 3,...} be stricty stationary. In particuar, for distinct i < j in V (n), the sequence {(U (n) ij (k), D (n) ij (k)), k =, 3,...} constitutes a stationary sequence with generic marginas (U (n) ij, D (n) ij ). We denote by G(n) ij the cumuative distribution function (CDF) of U (n) ij. This mode is genera enough that ink dynamics due to both mobiity and channe fading can be captured by a suitabe choice of the CDFs for U (n) ij. We-known resuts for renewa processes and independent on-off processes in equiibrium 7, Sections 5-6 can be generaized as foows: With = (i, j), in the notation introduced in Section II, we have T (n) (0) ξ (n) ij (0) = = F (n) (), R (7) where the conditiona probabiity F (n) () is given by { ( ) F (n) G (n) = m(g (n) ) 0 (y) dy if > 0 0 if 0 (8) for some ink duration CDF G (n) with support in IR +. In other words, F (n) is simpy the distribution of the forward recurrence time associated with U (n). From (8) it is easy to see that the duration of a one-hop path has a non-increasing probabiity density function (DF). If X (n) denotes any R + - vaued rv distributed according to F (n), then the reation (7) simpy states, with a itte abuse of notation, that T (n) (0) ξ (n) ij (0) = = st X (n). The rv (5) can now be viewed as the rv Z (n) defined by ( Z (n) := min X (n) : =,..., H (n)) (9) where H (n) = L (n) sd (0). Due to the underying stationarity assumptions, it ceary suffices to consider ony the case t = 0 as we do from now on. B. Modeing assumptions There are a few sources of difficuty in modeing and studying the distribution of path durations: First, the set L sd (0) of inks in the seected path is a random subset of E(0), which depends on the reachabiity processes at t = 0. Second, the reachabiity processes are usuay not mutuay independent. This is cear from either formuation () or (). In this subsection we epain how we hande these issues.. Asymptotics of the random set L (n) sd (0) With increasing network size under scaing (6) the average number of hops in a path between two randomy seected nodes is epected to increase with n. For eampe, consider the mode with a fied domain, in which the connectivity between two nodes is determined by (). We first seect the ocations of a source and a destination according to some stationary spatia distribution of the nodes. Then, for each n = 3, 4,..., pace the remaining n other nodes on the domain according to the same stationary distribution whie decreasing the transmission range of the nodes as / n. If minimum hop routing is empoyed, the number of hops aong the shortest path wi increase approimatey as n. Thus, we assume that a pair of nodes s and d in V (n) can be seected such that im L (n) sd (0) =, where for convenience the sequence { L (n) sd (0), n =,,...} is assumed to be deterministic.. Dependence of the reachabiity processes and ink ecess ives As mentioned earier, the ink ecess ives {X (n), =,..., H (n) } in (9) are not mutuay independent in genera. The authors of 5 skirted this difficuty by assuming that the reachabiity processes {ξ ij (t), t 0} are mutuay independent so that the rvs {X (n), =,..., H (n) } are mutuay independent. They provided a simuation study (Section 9 in 5) using the RW mobiity mode without pause to justify this assumption; it shows that the correation coefficient of ink ecess ives in (9) decays rapidy with increasing hop distance between the inks. More specificay, it indicates that the correation coefficient of ink ecess ives between two neighboring inks is sma and that of two inks separated by intermediate ink(s) is amost negigibe. This observation provides some evidence that the dependence of ink ecess ives may indeed decrease quicky with hop distance in some cases. However, the observed fast decrease of correation in hop distance may be a consequence of the fact that the mobiity of a node in the RW mode is independent of other nodes, and if the mobiity of a set of nodes is strongy correated (e.g., sodiers in a patoon partaking in a mission), this may no onger be true. In the foowing sections we rea the independence assumption of the reachabiity processes in 5 and repace it with what are commony known as miing conditions. These conditions impose a form of asymptotic independence as the hop distance between inks increases, whie aowing dependence in an (unbounded) neighborhood. IV. FINITE DEENDENCE WITH HOMOGENEOUS LINK DURATION DISTRIBUTION In this section we consider a simper case where ink durations have the same CDF G with support in IR + and the dependence in ink ecess ives is imited to a finite oca Decreasing the transmission range whie keeping the domain fied has the same effect as increasing the domain size whie keeping the transmission range fied.

5 5 neighborhood. First, in order to mode the ink ecess ives, we introduce a stationary sequence of rvs {X i, i =,,...} whose CDF is given by F () = { m(g) 0 ( G(y)) dy, if > 0 0, if 0. (0) We et X (n) = X for a n + := {,,...} and a, i.e., rv X is used to mode the ecess ife of the - th ink in an -hop path. The path duration of an -hop path is modeed by rv Z (n) := min(x (n) : =,..., ). The rvs X i, i =,,..., are identicay distributed from the stationarity assumption, but may not be mutuay independent. The aforementioned assumption of finite dependence of ink ecess ives is given by the foowing: Assumption : (m-dependence 8) The rvs X i, i =,,..., satisfy X X if > m, where m is a finite positive integer. This assumption is consistent with the findings in 5, Fig. 9, where the dependence in ink ecess ives under the RW mobiity mode appears to be imited to a very sma neighborhood. Assumption : For every 0 and any given ɛ > 0, there eists an integer n = n (; ɛ) such that ( G ɛ, n = n n), n +,... Assumption is equivaent to saying that a ink duration is stricty positive with probabiity one, i.e., im G(/n) = G(0) = 0. It is obvious that this assumption hods triviay if the CDF G is continuous (i.e., ink durations can be modeed as continuous rvs). Therefore, it is a reasonabe assumption. Theorem : Suppose that Assumptions and hod for the stationary sequence {X i, i =,,...} and the CDF G. If the condition im ma c 0 X i < c X i < c, X j < c i j m = im ma X j < c X i < c () c 0 i j m = 0 hods, then im Z (n) = { e λ, if > 0 0, if 0 () where λ = (m(g)). roof: A proof is provided in Appendi III of the suppementa document due to a space constraint. Theorem tes us that as the number of hops aong a path increases the distribution of path duration can be we approimated by an eponentia distribution with parameter λ for a sufficienty arge. Note that rv Z (n) = min(x (n) : =,..., ) tends to decrease with increasing. This is aso obvious from the fact that λ as n. Thus, in order to keep Z (n) from converging to a constant rv with vaue zero as increases, rv Z (n) is scaed by the hop count in (). It is interesting to note that the parameter of the emerging eponentia distribution is given by the same λ = /m(g) whether the rvs {X i, i =,,...} are assumed to be ocay dependent as here or mutuay independent as assumed in 5. The condition in () impies that as c 0, the rare events {X j < c} do not occur in custers in a oca neighborhood of node i. One interpretation of this condition is as foows: Assume a very sma c. Rare events of ink ecess ives being smaer than c are primariy caused by nodes being cose to the edge of their transmission ranges and about to move out of the transmission ranges at the time of path set-up (rather than one or both of the nodes moving at an etremey high speed). Condition () impies that one pair of neighboring nodes being about to eave the transmission range of each other at the time of path set-up, does not mean the same is true for other pairs of neighboring nodes aong a path, which is reasonabe. This condition is vaidated in the case of the RW mobiity mode in Appendi V of the suppementa document. V. GENERAL DEENDENCE WITH HETEROGENEOUS LINK DURATION DISTRIBUTIONS In the previous section we considered the simper case where the dependence in ink ecess ives is imited to a finite neighborhood. As mentioned earier, this may be reasonabe in some cases. However, we show that it can be reaed consideraby. To be precise, the same distributiona convergence can be obtained even when the dependence of ink ecess ives is not bounded to any finite neighborhood and the ink duration distributions are heterogeneous. In this section we first define the miing conditions that describe the manner in which the dependence of ink ecess ives decays with the hop distance between the inks. Then, we estabish the distributiona convergence of path duration in more genera cases under the miing conditions. A. Miing conditions Suppose that W := { i, n =,,... ; i =,..., h(n)} is an array of IR-vaued rvs, where {h(n), n } is a sequence of positive integers with im h(n) =. Denote the joint CDF of rvs { i, i,..., i n } by J (n) i i n. For notationa simpicity we write J (n) i i n (u) for J (n) i i n (u,..., u). Let {u n, n } be a sequence of rea numbers (which typicay increases with n) and A := {α n,m, n =,,... ; m =,..., h(n)} be an array of non-negative rea numbers such that, for any integers < i < < i p < j < < j q h(n) where j i p > m, we have J (n) i...i pj...j q (u n ) J (n) i...i p (u n )J (n) j...j q (u n ) α n,m. (3) Definition : (D(u n ) condition 0, ) Suppose that we can find a sequence {m(n), n =,,...} of non-negative integers and an array A of rea numbers satisfying the condition in (3) such that (i) im m(n) =, (ii) m(n) = o(h(n)), m(n) i.e., im h(n) = 0, and (iii) im α n,m(n) = 0. Then, we say that the array W satisfies the condition D(u n ).

6 6 The condition D(u n ) imposes a form of dependence decay : As n increases, the dependence of two events { i u n,..., i p u n } and { j u n,..., j q u n } decreases as the distance j i p between the two sets of rvs increases. However, since m(n), it aows dependence in an unbounded neighborhood. One can easiy verify that a sequence that satisfies the m-dependence condition in Assumption satisfies the condition D(u n ) with any sequence {u n, n }. In order to state the definition of the second miing condition, we first need to introduce some notation. Let k be a fied positive integer. We divide the interva {,,..., h(n)} into k + disjoint subintervas 3 : The first k subintervas have a ength n := h(n)/k, where denotes the integer part of, and the ast interva has a ength smaer than k. For j =,,..., k, define and I (n) = {(j ) n +,..., j n }, I (n) k,k+ = {k n +,..., h(n)}. Note that I (n) = n for j =,..., k, and 0 I (n) k,k+ < k, where I denotes the cardinaity of I. Definition : The array W is said to satisfy the condition D (u n ) if, for a j =,,..., ( ) ( ) im, i > u n = o k i,i I (n) :i<i for a k > j. (4) A sufficient condition for the condition D (u n ) to hod is ( im h(n) k ( ) = o k ) sup, i > u n i,i I (n) :i<i for a k > j. (5) The interpretation of the condition D (u n ) in the contet of our probem wi be provided shorty. B. Distributiona convergence Define = (X (n) ), =,...,. Let W := {, n =,,... ; =,..., }. We denote the CDF of rv by J (n). We first make the foowing two assumptions. They are the same assumptions imposed in 5, Assumptions and for independent ink ecess ives cases. Assumption 3: For every 0, 4 ( ( )) im ma G (n) =,...,H (n) = 0. 3 We ca a finite set of consecutive integers {i,..., i } an interva with ength i i +. 4 In 5 the rvs X (n) are impicity scaed by, whie in this paper this scaing is carried out epicity. A more concrete way to epress Assumption 3 is as foows: For every 0 and any given ɛ > 0, there eists an integer n = n (; ɛ) such that ma G (n) =,...,H (n) ( ) ɛ, n = n, n +,... It is cear that the interpretation of this assumption is the same as that of Assumption and states that a ink duration is stricty positive with probabiity one. (. Assumption 4: (scaing) Let λ (n) = m(g (n) )) There eists some positive constant λ such that im = λ (n) = λ. (6) Assumption 4 simpy means that the ink ecess ives are scaed (by the average of the inverses of epected ink durations divided by λ) so that we can define the parameter of the imit distribution. Under Assumption 3, one can show that Assumption 4 is equivaent to the foowing assumption. Assumption 4A: There eists some positive constant λ such that, for any fied (0, ), we have = > λ as n. Lemma : Suppose that Assumption 3 hods. Let be some positive rea number and u n = /, n =,,.... Then, for any sequence {m(n), n =,,...} that satisfies conditions (i) - (ii) of Definition, we can find an array A of rea numbers which satisfies (3) and condition (iii) in Definition. roof: A proof is provided in Appendi IV of the suppementa document due to ack of space. Lemma impies that, under Assumption 3, the array W = {, n =,,... ; =,..., } aways satisfies the condition D(u n = /) for any in (0, ). In fact, in the proof we prove a sighty stronger resut: For any integers < i < < i p < j < < j q, the events { i,..., i p (n) } and {W j,..., j p } become asymptoticay independent as n increases. In other words, J (n) i...i pj...j q (u n ) J (n) i...i p (u n )J (n) j...j q (u n ) 0. For the cases with dependent ink ecess ives, we introduce two additiona assumptions. Assumption 5: For any sequence {Î(n), n =,,...} of sets of consecutive positive integers, where Î (n) {,..., }, λ (n) Î(n) = O ( Î(n) A sufficient condition for Assumption 5 to hod is that there eists some arbitrariy sma positive constant ε such that the epected ink durations satisfy m(g (n) ) ε for a n =,,... and =,...,. The interpretation of this assumption is that the epected ink durations do not ).

7 7 decrease to zero with increasing network size. Since the ink durations are ikey to depend on the nodes mobiity and their transmission ranges but not directy on the network size, this is a reasonabe assumption. Assumption 6: The array W = {, n =,,... ; =,..., } satisfies the condition D (u n ) with u n = for any (0, ). ( ) The condition D u n = impies that the rare events { } X (n) j in a neighborhood are not strongy correated as n (hence 0). The roe and interpretation of this condition are simiar to those of condition () in the m- dependence case (stated at the end of Section IV). Theorem : Suppose that Assumptions 3-6 hod. Then, we have { im Z (n) e = λ, if > 0 0, if 0 (7) roof: The proof is given in Appendi I. Theorem states that the distribution of an h-hop path can be we approimated by an eponentia distribution for a sufficienty arge h. As a byproduct it aso tes us that if the ink duration distributions are given by G, =,..., h, the epected duration of the path can be approimated by /( =,...,h (m(g )) ). Somewhat surprisingy, the parameter of the emerging eponentia distribution in (7) is the same as that of the eponentia distribution with independent ink ecess ives 5, Theorem. This hods even with somewhat strong oca dependence that may eist, and is consistent with the simiar observation made in Section IV. This again suggests that the distribution of path duration is not significanty affected by the dependence of the reachabiity processes and ink ecess ives when the hop count is sufficienty arge. VI. AN OUTLINE OF A ROOSED SCHEME Detecting a ink faiure and finding an aternative path can take a non-negigibe amount of time in practice. This is because ink faiures are often detected through a faiure to receive/echange a contro message over a pre-determined period. When oca recovery is unsuccessfu after a ink faiure, packets queued at the originator of the faied ink and additiona packets on the way to the node which were to be routed using the ink, wi eventuay be dropped by the node and must be retransmitted by their senders. These dropped packets ead to a waste of wireess resources. Moreover, osses of consecutive packets cause the transport ayer protoco to back off, reducing its transmission rate. This may cause senders to rey on timeout to detect the packet osses, which can take more than a few seconds. Hence, frequent ink faiures aong the paths in use wi resut in disruptions in service and degrade the performance of appications, especiay that of time critica appications. For these reasons a routing agorithm shoud consider its epected duration in addition to other quaities (e.g., estimated avaiabe bandwidth or congestion eve) when choosing a path. In a arge scae MANET the hop distance between a source and a destination is ikey to be arge 4. Our resuts in the previous sections te us that when hop counts are arge, (i) the distribution of path duration can be we approimated by an eponentia distribution and (ii) the inverse of the epected duration of a path is approimatey given by the sum of the inverses of the epected durations of the inks aong the path. Thus, in order to approimate the epected duration of a path, a source needs to know ony the sum of the inverses of the epected ink durations. Unfortunatey, accurate estimation of the epected ink durations aong a path is difficut in practice. Instead, we approimate them using average ink durations eperienced by the nodes: Under our scheme each node maintains the average duration of the inks that it estabishes with other nodes. These average ink durations are used as estimates to the epected ink durations aong a path and are provided to the source during a path discovery phase. Suppose that a node has routing information for a requested destination. Then, it generates a repy message and specifies the inverse of its estimate of the epected duration of the path to the destination in a fied Inverse ath Duration (ID) in the repy. A node that receives a repy message, first adds to the ID vaue the inverse of its average of ink durations, and then forwards it to the net upstream node. Finay, when the source receives the repy message, it adds the inverse of its average ink duration to the ID vaue. Then, the source chooses a path with the smaest ID vaue, i.e., the argest estimated epected duration. Fig.. λ (n) + λ (n) + λ (S) ath Request S n n Repy Repy λ (n) + λ (n) ath Request λ (n) An eampe of an estimation of epected path duration. Let us epain this procedure using the eampe shown in Fig.. The source node S wants to find a path to destination node D and broadcasts a path request to its neighbors. Assume that node n does not have routing information for D and forwards the request to its neighbor, node n. Assume that node n has routing information for D. It generates a repy with the initia ID vaue of λ (n), which is the inverse of its average ink duration. Here node n s average ink duration is used as an estimate of the epected duration of its ink with D. Then, it forwards the repy to node n. Upon receiving the repy, node n adds the inverse λ (n) of its average ink duration to the ID vaue and forwards the repy to source node S. Again, node n s average ink duration is used in pace of the epected duration of its ink with n. When S receives the repy, it first adds λ (S) to the ID vaue of λ (n) + λ (n) in the repy. Then, it uses the inverse of the fina ID vaue as an estimate of the epected duration of the discovered path {(S, n), (n, n), (n, D)}. As ony the sum of the inverses of average ink durations is coected, this proposed modification can be easiy impemented in avaiabe on-demand routing agorithms with minima overhead. It is aso possibe with our scheme that a node cassifies its neighbors and maintains a separate average ink duration for D

8 8 each type of neighbors. The reason for maintaining separate averages is as foows. A arge scae MANET is ikey to comprise many different types of nodes. For eampe, a Future Combat System (FCS) wi incude different types of vehices (e.g., jeeps, tanks, etc.), sodiers, and possiby aeria vehices. Ceary, the duration of a ink between two nodes wi depend on their mobiity and communication capabiities. Thus, the durations of inks a node estabishes with its neighbors over time wi be dependent on their mobiity and communication capabiities, as we as its own. By maintaining separate averages, we can improve the accuracy of the estimates of epected ink durations. This is demonstrated using ns- simuation resuts in Section VII. A. Impementation in AODV We impemented our proposed scheme in the AODV routing protoco to evauate the potentia gain in path durations. Most of our changes are imited to the path request and repy messages and their handing. The rest of the AODV scheme is eft intact. Since our primary interest is to study the potentia benefits of the path seection scheme (rather than proposing a compete routing protoco), we did not attempt to minimize the overhead it generates. Such overhead is ikey to depend on the routing protoco in which the proposed scheme is impemented, and the overa overhead of the integrated routing protoco shoud be minimized. Each node maintains two separate counters - a sequence number and a broadcast ID. The sequence number is incremented when (i) the node generates a new path request message or (ii) it is the requested destination in a path request message and generates a path repy message. This sequence number is used to indicate the freshness of the path request or repy message. Every node aso keeps an average of the ink durations it eperienced in the past. As mentioned earier, a node can maintain a separate average for each type of neighbors by cassifying them. Route entry A node creates and maintains a route entry for each known destination node. 5 However, unike in AODV, the node can keep up to k p paths (instead of a singe path), where k p is a design parameter. In our simuation the vaue of k p is set to three. The information of each path is recorded in a subentry with four fieds: (i) destination sequence number, (ii) net hop to the destination, (iii) hop count, and (iv) Inverse ath Duration (ID). The ID fied contains the the sum of the inverses of epected ink durations approimated using average ink durations and reported in a path repy message during path discovery. In order to avoid keeping redundant path information we require that the net hop to the destination of these paths be different. If there is more than one path avaiabe, one of them is seected as the primary path as foows. The others are used as backup paths in case the primary path fais. Ranking route subentries The paths in a route entry are ranked first based on the destination sequence numbers, and then based on the ID vaues. Ties are broken using the 5 The routing protoco running at a node may not be aware of node s neighbors at the beginning when the route entries are empty. hop counts. To be more precise, when more than one path to a destination are discovered, the paths are first ranked by decreasing destination sequence number. If two or more paths have the same sequence number, then the one with a smaer ID vaue takes a higher preference as it has a arger estimated epected duration than the others. Finay, if the first two are the same, the path with a smaer hop count is preferred. This is shown in Fig.. The path with the highest preference is used as the primary path for routing packets when needed. Fig.. Dest Seq. # ID vaue hop count Dest Seq. # ID vaue hop count Before ranking Ranking of discovered paths After ranking ath or route request When a source does not have routing information for a desired destination, it initiates a path discovery process; it first increments its sequence number and broadcasts a path request message to its neighbors. A path request message contains (i) source ID and its sequence number, (ii) source broadcast ID, (iii) destination ID and its sequence number, and (iv) hop count to the source. The hop count is initiay set to one. Each path request message is distinguished by its source ID and broadcast ID. When a node receives a path request message, it carries out the foowing: ) It checks if there is a subentry for the source in which the net hop is the neighbor that forwarded the request message. If there is no such subentry, it creates a subentry using the information contained in the request message. The ID vaue is set to infinity, indicating the information is not avaiabe, because the request message does not contain an ID vaue. If there is such a subentry and the sequence number in the request message is arger than the one in the subentry, it updates the source sequence number and the hop count. After the update, it re-ranks the paths to the source based on the new sequence number and the hop count. ) If the node had received another copy of the same request message, it terminates the message. Otherwise, if it is not the requested destination node in the message and has no routing information to the destination, it increases the hop count in the message and rebroadcasts the message to its neighbors. ath or route repy If an intermediate node has a route entry for the destination with the destination sequence number no smaer than the destination sequence number in the request message, it generates a path repy message. The repy message contains (i) destination ID and its sequence number, (ii) ID vaue, and (iii) hop count. For the destination sequence number, ID vaue, and hop count, the information in the subentry for the primary path to the destination node is used. The repy message is then sent to the neighbor node that forwarded the request message, and the request message is terminated. If a copy of the path request message reaches the destination node, the destination generates a repy message. It first increases its sequence number to the maimum of its current 3 4

9 9 sequence number and the destination sequence number in the request message. Then, it generates a repy message with the same three fieds in the repy message generated by an intermediate node. The initia vaue of the ID and the hop count are set to zero. If the destination node receives mutipe copies of the same path request message from different neighbor nodes, it can generate more than one repy message. When an intermediate node receives a repy message, it first adds the inverse of its average ink duration to the ID vaue in the repy message. Reca that the node may cassify the neighbor that forwarded the repy message and add the average ink duration corresponding to the neighbor s type when a separate average is kept for each type of neighbors. After incrementing the ID vaue, the intermediate node updates its routing information to the destination: It creates a temporary subentry for the newy discovered path to the destination with the information provided in the repy message. Then, it checks if the new path is better than the k p -th ranked path in the entry for the destination. If it is, then it re-ranks the paths, incuding the newy discovered path, and inserts the new subentry in the route entry for the destination. Otherwise, it discards the temporary subentry. If the repy message is the first repy the node receives for the corresponding request, then it forwards the repy to its upstream neighbor(s). Here we aow the node to forward the repy to up to k r ( k p ) neighbors that reayed a copy of the request message to the node. This is done to increase the number of discovered paths (than in AODV) so that the source can choose the best path among them. These k r upstream nodes are seected based on their ranking. The parameter k r may be adapted based on the average degree of the nodes. 6 In our simuation k r is set to three. If the repy message is not the first repy, but the new path has a smaer ID vaue than the previous primary path to the destination, then the node can sti forward the repy message. This is to advertise the discovery of a more reiabe path, i.e., a path with a arger epected duration. However, we did not use this feature in the simuation, and ony the first repy is forwarded. Finay, when the source receives a repy message, it first updates the hop count and the ID vaue using its average ink duration and then its route entry for the destination. Data transmission can begin after the first path to the destination is discovered. However, after processing a newy arrived repy message, the source may switch its primary path to the destination if the newy discovered path has a smaer ID vaue than the previous primary path. When a ink faiure occurs, the node that detects the faiure first attempts a oca recovery if there is a cached backup path in the route entry for the destination. The detais of our oca recovery scheme are described in 6 and are omitted in this paper. If the oca recovery fais, it broadcasts a route error message. The error message contains a ist of destination nodes affected by the ink faiure. Each node that receives an error message checks if any of the paths in its route entries traverses the faied ink. If there is any such path, the corresponding 6 When the average degree is arge, even a sma vaue of k r wi aow the source to discover mutipe avaiabe paths to choose from. If the network is sparsey connected, then a arger vaue of k r may be preferred. subentry is removed. If a primary path is affected, it initiates a path recovery procedure when necessary. VII. SIMULATION RESULTS In this section we evauate the performance gain (in path duration) from our scheme outined in the previous section. The simuation is run with 00 nodes moving in a km km rectanguar region. The RW mobiity mode is empoyed. The transmission range of the nodes is set to 50 m. There are two casses of nodes; the speed of a cass and cass node is uniformy distributed in, 5 m/s and 0, 30 m/s, respectivey. The reason for having two casses of nodes with different speed ranges is to create a scenario with heterogeneous nodes that eperience diverse ink durations in the fied (e.g., sodiers vs. tanks or jeeps). Sower moving cass nodes in genera eperience onger ink durations than faster moving cass nodes in our simuation due to the same fied transmission range. Speeds of the nodes are chosen independenty of the seected waypoints and previous speeds. Each run of simuation is for,00 seconds, and a tota of 6 runs are carried out with different random seeds. 7 However, in order to reduce the effects of transient period data are coected ony in the ast 800 seconds of each run. A tota of approimatey 5,000 connections are set up between randomy seected source and destination nodes. The interarriva times of connection requests are given by independent and identicay distributed rvs, each of which is a sum of 5 seconds and an eponentia rv with a mean 5 seconds. Each connection request generates a path request message, triggering a path discovery phase. Therefore, in the ast 800 seconds of each run we generate on the average 40 path request messages. We simuate three different scenarios by varying the number of cass nodes (hence cass nodes as we) to iustrate the benefits of our scheme and a trend that emerges. We first begin with 40 cass nodes and increase it to 60 and then to 80. Our scheme is run under two different modes: In the first mode, each node maintains a singe average ink duration for a neighbor nodes. In the second mode each node cassifies the neighbors and maintains two separate averages - one for cass neighbors and the other for cass neighbors. The CDFs of ink duration under our scheme for the 60 vs. 40 scenario are shown in Fig. 3. In the first mode, we pot the distributions seen by cass nodes and cass nodes. In the second mode, we pot the distributions of the inks between (i) two cass nodes, (ii) a cass node and a cass node, and (iii) two cass nodes. As epected, in the first mode ink durations seen by cass nodes are much arger than those seen by cass nodes in the usua stochastic order. However, note that the discrepancy in the ink duration distribution (a) between two cass nodes and (b) between a cass node and a cass node is not very arge. The CDFs of the path duration under AODV and our scheme (both with and without separate averages) are potted in Fig. 4. The median vaues of path durations are given in Tabe I. In the first two scenarios there is a 60 percent increase in the median vaues over AODV when nodes maintain separate 7 This is done to reduce the size of the mobiity fie needed in ns- simuation.