Stochastic Modeling of Single-Hop Cluster Stability in Vehicular Ad Hoc Networks

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1 1 Stochastc Modelng of Sngle-Hop Cluster Stablty n Vehcular Ad Hoc Networks Khadge Abboud, Student Member, IEEE, and Wehua Zhuang, Fellow, IEEE Abstract Node clusterng s a potental approach to mprove the scalablty of networkng protocols n vehcular ad hoc networks (VANETs). Hgh relatve vehcle moblty and frequent network topology changes nflct new challenges on mantanng stable clusters. As a result, cluster stablty s a crucal measure of the effcency of clusterng algorthms for VANETs. Ths paper presents a stochastc analyss of the vehcle moblty mpact on sngle-hop cluster stablty. A stochastc moblty model s adopted to capture the tme varatons of ntervehcle dstances (dstance headways). Frstly, we propose a dscrete-tme lumped Markov chan to model the tme varatons of a system of dstance headways. Secondly, the frst passage tme analyss s used to derve probablty dstrbutons of the tme perods of nvarant clusteroverlap state and cluster-membershp as measures of cluster stablty. Thrdly, queueng theory s utlzed to model the lmtng behavors of the numbers of common and unclustered nodes between neghbourng clusters. Numercal results are presented to evaluate the proposed models, whch demonstrate a close agreement between analytcal and smulaton results. I. INTODUCTION A vehcular ad hoc network (VANET) s a promsng addton to our future ntellgent transportaton systems, whch s provsoned to support varous safety and nfotanment applcatons [2], [3]. Urban roads and hghways are hghly susceptble to a large number of vehcles and traffc ams. Therefore, networkng protocols for VANETs should be scalable to support such large scale networks. Node clusterng s a network management strategy n whch nearby nodes are grouped nto a set called cluster. In each cluster, a node, called cluster head (), s elected to manage the cluster. The remanng nodes are called cluster members (CMs), each belongng to one or multple clusters. Node clusterng, ust as n tradtonal ad hoc networks, s a potental approach to mprove the scalablty of networkng protocols such as for routng and medum access control n VANETs. For medum access control protocols, the can act as a central coordnator that manages the access of ts CMs to the wreless channel(s) [4]. For routng protocols, s can be made responsble for the dscovery and mantenance of routng paths, thus lmtng the control-message overhead n these processes [5]. Despte the potental benefts of node clusterng, formng and mantanng the clusters requre explct exchange of control messages. In VANETs, vehcles move wth hgh and varable speeds, causng frequent changes n the network topology, whch can sgnfcantly ncrease the cluster mantenance cost. Therefore, how to form stable clusters that last for a long tme s a maor ssue n node clusterng of VANETs. Copyrght c 2015 IEEE. Personal use of ths materal s permtted. However, permsson to use ths materal for any other purposes must be obtaned from the IEEE by sendng a request to pubs-permssons@eee.org. K. Abboud and W. Zhuang are wth the Center for Wreless Communcatons, Department of Electrcal and Computer Engneerng, Unversty of Waterloo, 200 Unversty Avenue West, Waterloo, Ontaro, Canada, N2L 3G1. E-mal: {khabboud, wzhuang}@uwaterloo.ca Ths work s presented n part n a paper at the 2014 ACM MSWM [1]. In a hghly dynamc VANET, vehcles on and leave clusters along ther travel route, resultng n changes n cluster structure. The temporal changes n cluster structure are ether nternal or external [6]. An nternal change n the cluster structure s concerned wth a change nsde the cluster such as when vehcles on or leave the cluster, resultng n a change n cluster-membershp. Frequent changes n the nternal cluster structure consume network rado resources and cause servce dsrupton for the cluster-based network protocols (e.g., n ntracluster resource allocaton, route dscovery, and message delvery). Therefore, analyzng the mpact of vehcle moblty on the rate at whch nodes enter and leave a cluster s an mportant measure of nternal cluster stablty. Ths metrc has been adopted by researchers to evaluate the performance of ther proposed clusterng algorthms through smulatons [7] [9]. A hgher rate of cluster-membershp changes ndcates a smaller tme perod of nvarant cluster-membershp and, therefore, lower nternal cluster stablty. On the other hand, an external change n the cluster structure s concerned wth the relatonshp of a cluster wth other clusters n a network. One metrc that evaluates the external relatonshp of a cluster s ts overlappng ranges wth neghborng clusters. The tme varatons of the dstance between two neghborng s, due to vehcle moblty, can cause the coverage ranges of the clusters to overlap. As the overlappng range between two clusters ncreases, the two clusters may merge nto a sngle cluster [4], [7], [10]. Frequent splttng and mergng of clusters ncrease the control overhead and dran the rado resources [9], [11], [12]. In general, a non-overlappng clustered structure produces a less number of clusters and lowers the desgn complexty of network protocols that run on the clusters. For example, two clusters may utlze the same rado resources at the same tme f they are non neghborng clusters [13] [14]. On the other hand, a hghly overlappng clustered structure may cause complexty n the channel assgnment, lead to broadcast storms, and form long herarchcal routes. Addtonal rado resources ought to be used to prevent ntercluster nterference due to overlappng, for example, assgnng dfferent tme frames for neghborng clusters [15] and assgnng dfferent transmsson codes to CMs located n a possbly overlappng regon [16]. Although researchers have favored formng non-overlappng (or reduced overlappng) clustered structure [12] [11] [10] [16], encounterng overlappng clusters durng the network runtme s nevtable, especally n a hghly moble network. Overlappng clusters have receved sgnfcant attenton snce the work by Palla et al. [17]. It s shown that real networks are better characterzed by well-defned statstcs of overlappng and nested clusters rather than dsont clusters. Addtonally, overlappng structure can provde a ground for cooperaton among the overlappng clusters. For example, n [18], overlappng s used for cooperatve nterference management for small cell networks. egardless of whether or not cluster overlappng s preferred, characterzng the overlappng state between neghborng clusters and ts changes over tme becomes crucal n the presence of node moblty. A hgher rate

2 2 of cluster-overlap state change ndcates a shorter tme perod of unchanged cluster-overlap state and, therefore, lower external cluster stablty. Despte the mportance of cluster stablty as a measure of clusterng algorthm effcency n VANETs, characterzng cluster stablty has taken the form of smulatons [7] [9] or case studes [19] n the lterature. In ths paper, we present a stochastc analyss of two cluster stablty metrcs: the change rate n the overlap state between neghborng clusters as a measure of external cluster stablty and the change rate n cluster-membershp as a measure of nternal cluster stablty. We adopt a stochastc vehcle moblty model that descrbes the tme varatons of ntervehcle dstances and accounts for the realstc dependency of these varatons at consecutve tme steps. Frstly, the dstance between two vehcles, separated by a number of vehcles on a hghway, s modeled as a dscrete-tme Markov chan wth a reduced dmensonalty. Usng the frst passage tme analyss, we derve the probablty dstrbutons of the tme before the frst change n the cluster-overlap state and the tme nterval between two successve changes n cluster-overlap state of two neghborng clusters. Secondly, the dstrbutons of the tme before the frst cluster-membershp change and the tme nterval between two successve cluster-membershp changes are derved. Thrdly, the overlappng regon between overlappng clusters and the unclustered regon between non-overlappng clusters are modeled as a storage buffer n a two-state random envronment. Usng G/G/1 queung theory, the steady-state dstrbutons of the numbers of common and unclustered nodes are approxmated. Fnally, we conduct MATLAB smulatons and demonstrate that the analytcal results of our model match well wth the smulaton results. II. SYSTEM MODEL Consder a connected VANET on a mult-lane hghway wth no on or off ramps. We focus on a sngle lane wth lane changes mplctly captured n the adopted moblty model. We choose a sngle lane from a mult-lane hghway nstead of a sngle-lane hghway, n order to be more realstc n a hghway scenaro. Assume that the hghway s n a steady traffc flow condton defned by a tme-nvarant ntermedate vehcle densty. All the vehcles have the same transmsson range, denoted by. Any two nodes at a dstance less than from each other are one hop neghbours. Tme s parttoned wth a constant step sze. Let X be the dstance headway between node and node + 1. The dstance headway s the dstance between two dentcal ponts on two consecutve vehcles on the same lane. Defne X = {X (m), m = 0, 1, 2... } to be a dscrete-tme stochastc process of the th dstance headway, where X (m) s a random varable representng the dstance headway of node at the m th tme step, = 0, 1, 2,..., m = 0, 1, 2,.... Furthermore, assume that X s are ndependent wth dentcal statstcal behavors for all 0. For notaton smplcty, we omt ndex when referrng to an arbtrary dstance headway. Throughout ths paper, F Y (y), P Y (y), and E[Y ] are used to denote the cumulatve dstrbuton functon (cdf), the probablty mass functon (pmf), and the expectaton of random varable Y, respectvely. A. Node Clusters We assume that s are selected accordng to some clusterng scheme, so that all the network nodes are grouped nto X 0 X 1 Overlappng range X 2 X c X 3 Fgure 1. Two neghborng s separated by N c = 4 nodes and X c = (X 0, X 1, X 2, X 3, X 4 ). possbly-overlappng, sngle-hop clusters (e.g., [10]). The range of each cluster extends one hop on both sdes of the. At the end of the cluster formaton, the vehcles are dstrbuted on the hghway accordng to a statonary probablty dstrbuton of the dstance headways. Let N CM be the number of CMs on one sde of a cluster and let N c be the number of nodes between two neghbourng s. The overlappng range between two neghbourng clusters s the common dstance covered by the transmsson range of both s. Defne the cluster-overlap state between two neghbourng clusters to be ) overlappng, when the dstance between the two s s less than 2; or ) nonoverlappng, otherwse. In our analyss, the 0 th tme step refers to the tme when the cluster formaton has ust completed. We assume that the clusters are ntally overlappng and the s reman the same over a tme nterval of nterest. B. Node Moblty The vehcles move accordng to the mcroscopc moblty model proposed n [20]. In ths model, a dstance headway, X, changes accordng to a dscrete-tme fnte-state Markov chan. The Markov chan has N max states correspondng to N max ranges of a dstance headway. Let X (m) s denote the event that the th dstance headway s n state s at the m th tme step 1, where s [0, N max 1] and, m 0. The dstance headway transts from one state to another accordng to a tr-dagonal state-dependent transton matrx, denoted by M. Wthn a tme step, a dstance headway n state can transt to the next state, the prevous state, or reman n the same state wth probabltes p, q, or r, 0 N max 1, respectvely, where q 0 = p Nmax 1 = 0 and r = 1 p q. III. EXTENAL CLUSTE STABILITY The cluster-overlap state s governed by the the dstance between two neghborng s. As ths dstance decreases, the s approach each other causng the two clusters to overlap. On the other hand, as the dstance between s ncreases, the s move apart from each other causng the two clusters to become dsont. The dstance between two neghborng s s equal to the sum of the dstance headways between the two nodes. Label the (N c + 2) nodes wth IDs 0, 1,..., N c + 1, where the followng has ID 0 and the leadng has ID (N c + 1). For notaton smplcty, 1 The length of the range covered by each state s a constant, denoted by L s n meters. The range s chosen such that L s τ v where v s the maxmum relatve speed between vehcles and τ s the constant tme step sze. The moblty model parameters are descrbed n detals n [20]. X 4 CM

3 3 Ω 6 Intertranston between Ω OV and Ω NOV Intratranston n Ω OV (Ω NOV ) Ω 5 Ω 2 Ω 4 Ω 8 Ω 10 Ω 1 Ω 3 Ω 7 Ω 9 OV 2 OV 1 NOV 1 NOV 2 OV NOV Fgure 2. An llustraton of a lumped markov chan for N = 2, N th = 4, N max = 3. A lne between two lumped states represents a non-zero two-way transton probablty n a sngle tme step between the lnked states. There exst non-zero transton probabltes between subsets of Ω OV 1 and Ω NOV 1. let X c = (X ) Nc be the sequence of dstance headways between the two s as llustrated n Fgure 1, where X c (m) = (X (m)) Nc, and {X c(m) (s 0, s 1,..., s Nc )} {X (m) s, [0, N c ]}. Consder ntally overlappng clusters,.e., N c X (0) < 2. Two neghborng s reman Nc overlappng untl X (m) 2 at some tme step m. The sequence of (N c + 1) ndependent and dentcally dstrbuted (..d.) dstance headways s an (N c +1)-dmensonal Markov chan, where each headway, X, s a brth and death Markov chan as descrbed n Subsecton II-B. For clarty, the term state refers to a state n the orgnal Markov chan, X, the term super state refers to a state n the (N c + 1)-dmensonal Markov chan, and the term lumped state refers to a set of super states (to be dscussed later n ths secton). Addtonally, parentheses ( ) are used for a sequence, whle curly brackets { } are used for a set. A super state n the (N c + 1)-dmensonal Markov chan s a sequence of sze N c + 1, n whch the th element represents the state (n the 1-dmensonal Markov chan) that the th dstance headway belongs to. That s, a super state, (s 0, s 1,..., s Nc ), means that dstance headway X s n state s [0, N max 1]. The sum of (N c + 1) dstance headways representng the dstance between the two s can be calculated from the (N c + 1)-dmensonal Markov chan. The state space sze of the (N c + 1)-dmensonal Markov chan s equal to N max (Nc+1), makng t subect to the state-space exploson problem when N c s large. However, snce we are nterested n the sum of the (N c + 1) dstance headways, the state space can be reduced accordng to the followng theorem. Theorem 1: Let X be a dscrete-tme, brth-death, rreducble Markov chan wth N max fnte states, and let set X = (X ) N 1 represent a system of N ndependent copes of chan X. The N-dmensonal Markov chan that represents the system, X, s lumpable wth respect to the state space partton Ω = {Ω 0, Ω 1..., Ω NL }, such that (s.t.) any two super states n subset Ω are permutatons of the same set of states [0, N L 1], where N L = (Nmax+N 1)! N!(N max 1)! s the state space sze of the lumped Markov chan. The proof of Theorem 1 and followng corollares are gven n the Appendx. Snce a lumped state, Ω = {(s 0, s 1,..., s N 1 )}, 0 N L 1, contans all super states that are permutatons of the same set of states, we can wrte the lumped state as a set of those states Ω = {s 0, s 1,..., s N 1 }. Snce the (N c + 1)-dmensonal Markov chan s rreducble, the lumped Markov chan s also rreducble [21]. The statonary dstrbuton of the lumped Markov chan can be derved from the statonary dstrbuton of the 1-dmensonal Markov chan accordng to the followng Corollary. Corollary 1: Consder a system of N ndependent copes of a fnte, dscrete-tme, brth-death, rreducble Markov chan, X, wth statonary dstrbuton (π ) Nmax 1. The statonary dstrbuton of the lumped Markov chan of Theorem 1, representng the system, X = (X ) N 1, follows a mult-nomal dstrbuton wth parameters (π ) Nmax 1. A. Tme to the frst change of cluster-overlap state Consder two overlappng clusters. At any tme nstant, the overlappng range between two neghbourng clusters s equal to 2 N c X (m), m 0. Therefore, accordng to Theorem 1, the tme varaton of the overlappng range between the two clusters can be descrbed by a lumped Markov chan wth lumped states Ω 0, Ω 1,..., Ω NL 1 whch represents the system, X c = (X ) Nc. Furthermore, dvde the lumped states nto two sets, Ω OV and Ω NOV. A lumped state Ω = {s 0, s 1,..., s Nc } belongs to Ω OV and to Ω NOV f Nc s < 2N and N c s 2N, respectvely, where N s the nteger number of the states that cover dstance headways wthn n the dstance headway s 1-dmensonal Markov chan. Let the system of the dstance headways between the two s be ntally n super state I c,.e., X c (0) I c, s.t. I c Ω k Ω OV, 0 k N L 1. Let the tme perod untl the clusters are no longer overlappng be T ov1 (Ω k ), gven that the dstance headways between them are ntally n states I c Ω k. Then, ths tme perod s equal to the frst passage tme for the system, X c, to transt from the lumped state Ω k to any lumped state Ω k, s.t. Ω k { Ω NOV. That s, T ov1 (Ω k ) = mn m > 0; X c (m) (k 0, k 1,..., k Nc ), N c k 2N X c (0) I c }. Let M Nc be the transton probablty matrx of the lumped Markov chan descrbng X c. One way to fnd the frst passage tme s to force the lumped states n Ω NOV to

4 4 become absorbng,.e., set the probablty of returnng to the same lump state, Ω, wthn one tme step to one Ω Ω NOV. Furthermore, let all the lumped states n Ω NOV be merged nto one sngle absorbng state and let t be the last (ÑL 1) th state, where ÑL s the number of states n the new absorbng lumped Markov chan. The transton probablty matrx of the new absorbng lumped Markov chan, MNc, s derved from M Nc as follows: MNc (Ω, Ω ) = M Nc (Ω, Ω ),, s.t. Ω, Ω Ω OV, MNc (Ω, Ω NL 1) = M N c (Ω, Ω ),, s.t. Ω Ω OV and Ω Ω NOV. Let T ov1 (Ω k ) denote the tme nterval from the nstant that the clusters are ntally formed tll the frst tme nstant that the cluster-overlap state changes, gven that the dstance headways are n super state I c Ω k. The cdf of T ov1 (Ω k ) s gven by F Tov1(Ω k )(m) = M Nc (Ω k, ΩÑL 1 ) + Ω Ω OV M Nc (Ω k,ω )F Tov1(Ω )(m 1), m 1 (1) where F Tov1(Ω k )(0) = 0. Equaton (1) calculates the cdf of T ov1 (Ω k ) recursvely. Snce F Tov1(Ω k )(m) = m n=1 P T ov1(ω k )(m), the frst term n (1) corresponds to the absorpton probablty wthn one tme step gven that the system s ntally n lumped state Ω k,.e., F Tov1(Ω k )(1) = M Nc (Ω k, ΩÑL 1 ). The second term n (1) corresponds to m n=2 P T ov1(ω k )(m) whch s the absorpton probablty wthn (m 1) tme steps gven that the system transted from Ω k to Ω Ω OV wthn one tme step. The sze of the state space of the lumped Markov chan can stll be large wth an ncreased number of nodes between (N max+n c)! (N c+1)!(n max 1)! the two s, snce N L =. However, the state space of the absorbng lumped Markov chan, needed to compute the tme perod untl the overlap state changes between the two neghborng s, s bounded accordng to the followng Corollary. Corollary 2: Consder a system of N ndependent copes of an rreducble Markov chan accordng n Theorem 1, and let the event of nterest be that the sum of the states of the N chans be larger than a determnstc threshold N th. The absorbng lumped Markov chan, requred to obtan the frst occurrence tme of the event of nterest, has a state space that s bounded by a determnstc functon of N th, when N > N th. Consder the scalablty of analyzng a system of N dstance headways, X N, to an ncreased number of dstance headways, N. Usng the lumped Markov chan, the scalablty of analyzng system X N s mproved for: ) the steady-state analyss - The problem of fndng the statonary dstrbuton of a system of dstance headway s of constant computatonal complexty wth respect to N (accordng to Corollary 1); and ) the transent analyss (.e, the frst passage tme analyss) - The computatonal complexty of the frst passage tme analyss s dependent on the state space sze of the consdered Markov chan. Accordng to Corollary 2, the state space sze of the absorbng lumped Markov chan s upper bounded by the total number of nteger parttons of all nteger that are less than N th as dscussed n Appendx A.3. Fgure 3 shows the state space reducton usng the proposed lumped Markov chan. In ths subsecton, we focus on the tme nterval from the nstant that two partally overlappng neghborng clusters are N Nmax ÑL NL 2 x x N Dmensonal Markov chan (a) Lumped Markov chan (b) Absorbng lumped Markov chan N (c) Fgure 3. The state space sze of a Markov chan representng a system of N Markov chans (dstance headways), X N, wth N max = 9 when the system X N s represented by (a) an N-dmensonal Markov chan, (b) a lumped Markov chan accordng to Theorem 1, and (c) an absorbng lumped Markov chan accordng to Corollary 2 wth N th = 8. formed tll the tme nstant that they no longer overlap. Gven an ntal super state of the two neghborng clusters at the end of the cluster formaton stage, consder the followng: ) a proactve re-clusterng procedure n whch re-clusterng s trggered after a fxed perod of tme, say t seconds from the cluster formaton; and ) a reactve re-clusterng procedure n whch re-clusterng s trggered when the cluster-overlap state changes. In ), the probablty that the overlap state changes between the two overlappng neghborng clusters before reclusterng s trggered s equal to F Tov1(Ω k )( t). In ), the reclusterng perod s equal to T ov1 (Ω k ) wth the cdf calculated by (1). Up untl now, we have consdered a par of neghborng clusters n a specfc super state when they are ntally formed. In realty, the ntal state of a par of neghborng clusters s a random varable. For a gven N c, snce the dstance headways are statonary when the clusters are formed, the probablty that two overlappng neghborng clusters are ntally n lumped / [ ] state Ω s gven by where s gven by,ω Ω OV (22) n Appendx A.2. Usng the law of total probablty, the cdf of the tme for the frst change n overlap state to occur between two ntally overlappng clusters s gven by F Tov1 (m) = F Tov1(Ω )(m) Ω Ω OV Ω Ω OV, m = 1, 2,.... (2)

5 5 B. Tme perod between successve changes of cluster-overlap state In the precedng subsecton, we have analysed the tme nterval durng whch two neghborng clusters reman overlappng snce the clusters are formed. Durng ths tme nterval, the cluster-overlap state remans unchanged. Suppose two neghborng clusters overlap n cluster formaton and the overlap state changes at tme T ov1 (< t) and becomes nonoverlappng. The cluster-overlap state may change agan before re-clusterng s trggered. As a result, the tme perod between two consecutve changes of cluster-overlap state equals ) the cluster-overlappng tme perod when the overlap state changes from overlappng to non-overlappng, plus ) the cluster-nonoverlappng tme perod when the overlap state changes from non-overlappng to overlappng. Durng a cluster-overlappng or cluster non-overlappng tme perods, the cluster-overlap state remans unchanged ndcatng how long the cluster remans externally stable. 1) Cluster-overlappng tme perod: The second cluster-overlappng tme perod may not be equal to T ov1, snce the ntal state may not be the same as that when the clusters are ntally formed. We refer to ths perod as cluster overlappng perod, denoted by T ov. To derve the dstrbuton of T ov, the same approach used to fnd the dstrbuton of T ov1 can be used. Notce that the absorbng lumped Markov chan s the same as that used to calculate the dstrbuton of T ov1. The only dfference s the dstrbuton of the ntal state, I c. One way to fnd the dstrbuton of I c at the tme when the second overlappng state occurs s as follows: Make the lumped states n set Ω NOV absorbng, wthout combnng them nto one absorbng state. The correspondng transton probablty matrx, M N c, s equal to M Nc wth M N c (Ω, Ω ) = 0 and M N c (Ω, Ω ) = 1,, s.t. Ω Ω NOV ; Calculate the absorbng probablty ψ for each absorbng lumped state Ω Ω NOV ψ = Ω Ω OV k k Ω k Ω OV by lm M (m) m N c (Ω, Ω ) (3) where M (m) N c (Ω, Ω ) denotes the (Ω, Ω ) th entry of the m th power of matrx M N c ; Form another absorbng Markov chan by makng the lumped states n set Ω OV absorbng, wthout combnng them nto one absorbng state. The correspondng transton probablty matrx, M N c, s equal to M Nc wth M N c (Ω, Ω ) = 0 and M N c (Ω, Ω ) = 1,, s.t. Ω Ω OV ; Calculate the absorbng probablty φ for each absorbng lumped state Ω Ω OV by φ = ψ lm M (m) m N c (Ω, Ω ). (4) Ω Ω NOV The probablty that the dstance headways between the two neghborng clusters are n state Ω Ω OV at the tme when the second overlappng state occurs s equal to φ. Therefore, the cdf of the cluster-overlappng perod s gven by F Tov (m) = φ F Tov1(Ω)(m), m = 1, 2,... (5) Ω Ω OV where F Tov1(Ω )(m) s gven by (1). However, usng ths approach, we lose the advantage of havng a sngle absorbng state and, therefore, a bounded state space (accordng to Corollary 2). We propose to approxmate the dstrbuton of the system ntal state at the tme when the second overlappng state occurs, φ, as follows MNc (Ω, ΩÑL 1 φ ) MNc (Ω, ΩÑL 1 ). (6) Ω Ω OV The approxmated φ for lumped state Ω ( Ω OV ) s equal to ts statonary probablty weghted wth the absorpton probablty wthn one tme step. Notce that ths weght elmnates all the lumped states Ω Ω OV that are not drectly accessble from states n Ω NOV. Fgure 2 llustrates an example for a lumped Markov chan, where the drectly accessble lumped states are those connected by sold lnes,.e. Ω OV 1 and Ω NOV 1. When the overlappng state of two neghborng clusters changes from non-overlappng to overlappng, the only possble states to be reached frst are those n Ω OV 1. 2) Cluster-non-overlappng tme perod: Consder two ntally overlappng clusters, the cluster state can change to become non-overlappng and agan to become overlappng. The tme perod between two consecutve changes of cluster-overlap state equals the cluster-non-overlappng tme perod when the state changes from non-overlappng to overlappng. Neghborng s may move apart from each other and the clusters become dsont. Ths may result n dsrupton to ntercluster and/or ntracluster communcatons and/or sezure of the cluster membershp status from edge CMs. Ths produces unclustered nodes that may create ther own cluster whch can trgger re-clusterng and ncrease the clusterng cost. Let T nov denote the cluster non-overlappng tme perod. The same procedure used to calculate the cdf of T ov can be used to derve the cdf of T nov, whch s gven by F Tnov (m) = ψ F Tnov1(Ω)(m), m = 1, 2,... (7) Ω Ω NOV where F Tnov1(Ω )(m) = M N c (Ω, ΩÑ L 1 ) + [ k Ω k Ω NOV M N c (Ω, Ω k )F Tnov1(Ω k )(m 1) ], m 1, ψ M Nc (Ω,ΩÑ ) L 1 Ω Ω NOV M Nc (Ω,ΩÑ L 1 ), and M Nc s the probablty transton matrx that corresponds to the lumped Markov chan wth all states n Ω OV combned nto one absorbng state. That s, M Nc s derved from M Nc as follows: M N c (Ω, Ω ) = M Nc (Ω, Ω ),, s.t. Ω, Ω Ω NOV, M N c (Ω, ΩÑ L 1 ) = M N c (Ω, Ω ),, s.t. Ω Ω NOV and Ω Ω OV. The average cluster-non-overlappng tme perod s gven by [22] ( E[T nov ] = Ψ I M ) 1M1 N c (8) where Ψ s a row vector of sze Ñ L n whch the th element equals ψ, I s the dentty matrx of sze equal to that of Ñ L, and M 1 s a column vector of ones wth sze Ñ L. The second moment of the cluster-non-overlappng tme perod s gven by 2 2 The frst and the second moments of the cluster-overlappng perod can be calculated smlarly by adustng (8)-(9) to correspond to the absorbng lumped Markov chan wth transton matrx M Nc.

6 6 CM e l e ol eo r e r CM Fgure 4. Illustraton of the events that cause changes n clustermembershp. X 0 X 1 X 3 [22] X CM X 2 E[T 2 nov] = 2Ψ M N c (I M N c ) 2M1 + E[T nov ]. (9) IV. INTENAL CLUSTE STABILITY Due to relatve vehcle moblty, two events result n changes to the cluster-membershp: ) a vehcle leavng the cluster, and ) a vehcle enterng the cluster. Let e or and e ol denote the events that a vehcle leaves the cluster from the rght sde and the left sde of the, respectvely. Let e r and e l denote the events that a vehcle enters the cluster from the rght sde and the left sde of the, respectvely. Fgure 4 llustrates these events. Consder the tme for the frst change n cluster-membershp to occur after cluster formaton, and denote ths tme by T CM1. Ths tme s equvalent to the frst occurrence tmes of one of the four events,.e., T CM1 = T (e or e r e ol e l ), where T (e) denotes the frst occurrence tme of event e. Furthermore, let T CM1r = T (e or e r ) and T CM1l = T (e ol e l ) be the frst occurrence tme of the frst change n cluster-membershp (after cluster formaton) due to a vehcle leavng and enterng the cluster from the rght and the left sde of the, respectvely. Therefore, T CM1 = mn {T CM1r, T CM1l }. Snce T CM1r and T CM1l are ndependent, the cdf of the tme for the frst change n cluster-membershp to occur after cluster formaton s gven by F TCM1 (m) = 1 (1 F TCM1r (m))(1 F TCM1l (m)). Notce that T CM1r and T CM1l are..d.. Therefore, we focus on calculatng only one of them, say T CM1r. A. Tme to the frst change of cluster-membershp Let N CM be the number of CMs on the rght sde of the, and assume that N CM > 0 3. Let X CM = {X } N CM be the set of dstance headways of the and the N CM nodes as llustrated n Fgure 5, where X CM (m) (s 0, s 1,..., s NCM ) [X (m) s, [0, N CM ]]. The system, X CM, can be represented by an (N CM + 1)- dmensonal Markov chan. Suppose that set X CM s n super state I CM = (k 0, k 1,..., k NCM ) when the NCM 1 clusters are ntally formed, s.t., k < N, NCM k N, and I CM Ω k. Let the tme perod untl a node enters/leaves the cluster from one sde be T CM1r(Ω k ), gven that X CM I CM Ω k. Then ths tme perod s equal to the frst passage tme for the system, X CM, to transt from super state I CM to a super state (k 0, k 1,..., k N CM ) such that N CM k < N 3 When N CM = 0, the problem reduces to a sngle dstance headway, wth only the event of a node enterng the cluster causng the cluster-membershp change. In ths case, the frst passage tme analyss for one dmensonal chan can be used. Fgure 5. A cluster wth N CM = 3 and X CM = {X 0, X 1, X 2, X 3 }. NCM 1 (.e., a node enters the cluster) or k N (.e.,{ a node leaves the cluster). That s, { T CM1r(Ω k ) = mn m > 0; X CM (m) (k 0, k 1,..., k N NCM CM ), k < N } } N CM 1 s N X CM I CM. Snce the change n cluster-membershp occurs at the edge of the cluster, the value of X NCM n the system, X CM, s crtcal to dentfy the change. Notce that, ntally, the dstance headway X NCM can only be n a state k NCM [N N CM 1 k, N max ]. Therefore, we propose to lump the (N CM + 1)-dmensonal Markov chan nto parttons (lumped states) Ω 0, Ω 2,... Ω N L 1, such that each lumped state Ω = {(s 0, s 1,..., s NCM )} contans all super states that have the frst N CM states,.e., (s 0, s 1,..., s NCM 1), as permutatons of each other 4. We refer to ths chan as edge lumped Markov chan. Furthermore, dvde the lumped states nto three sets, Ω I, Ω L and Ω E, such that a lumped state Ω = {(s 0, s 2,..., s NCM )} belongs to ) Ω I, f N CM 1 s < N, and N CM s N ; ) Ω L, f N CM 1 s N ; and ) Ω E, f N CM s < N. Let M NCM be the transton probablty matrx of the descrbed lumped markov chan. The tme for the frst cluster-membershp change to occur, T CM1r(Ω k ), s the frst passage tme for system X CM to transt from super state I CM Ω k Ω I to any state n Ω L (.e., when a node leaves the cluster) or Ω E (.e., when a node enters the cluster). To fnd the dstrbuton of T CM1r(Ω k ), we force the lumped states n Ω E and Ω L to become one absorbng state. Followng the same steps as n Secton III, the cdf of T CM1r(Ω k ) can be derved as F TCM1r (Ωk ) (m) = M NCM (Ω k, ΩÑL 1 ) + Ω Ω I M NCM (Ω k, Ω )F TCM1r (Ω (m 1), ) m 1 (10) where M NCM s the probablty transton matrx of the new absorbng lumped Markov chan wth ÑL states, such that the (ÑL 1) th state s the sngle absorbng state contanng all states n Ω E and Ω L. For a random ntal state of X CM, the probablty that X CM s ntally n lumped state Ω = {(s 0, s 2,..., s NCM )} s gven by Ω Ω πs N CM Nmax, K I k=k π = N N CM 1 k u=0 s u, 4 Snce the (N CM +1)-dmensonal Markov chan s lumpable nto parttons Ω 1, Ω 2,... Ω NL 1, Ω = {(s 0, s 1,..., s NCM )} contans all super states that are permutatons of each other accordng to Theorem 1. Then, t s lumpable nto parttons that are subsets of Ω 0, Ω 2,... Ω NL 1.

7 7 where s the statonary dstrbuton of lumped state Ω = {(s 0, s 2,..., s NCM 1)} of the N CM -dmensonal Markov chan lumped accordng to Theorem 1. Hence, the cdf of the tme nterval between the tme nstant when the cluster s ntally formed tll the frst cluster-membershp change s gven by F TCM1r (m) = 1 Ω Ω I Ω Ω I π (,sncm ) F TCM1r (Ω ) (m) Nmax k=k π k (11) where (, s NCM ) s the state ndex of the dstance headway of the NCM th CM n the th lumped state. B. Tme perod between successve changes of clustermembershp In the prevous subsecton, we have analysed the tme nterval from ntal cluster formaton to the frst cluster-membershp change. In order to have a better measure of nternal cluster stablty, we analyse the tme nterval between two successve cluster-membershp changes n ths subsecton. Let T CM denote the tme nterval between two consecutve membershp changes of a cluster. Notce that the cluster-membershp change rate,.e. the rate at whch nodes enter or leave the cluster, s the recprocal of T CM. We focus on one sde of the cluster n ths subsecton, snce a smlar dervaton for the other sde can be done. To derve the dstrbuton of T CM, the frst step s to fnd the dstrbuton of I CM at the tme when the frst clustermembershp change occurs. In order to do ths, frst we make the lumped states n sets Ω E and Ω L of the lumped Markov chan absorbng, wthout combnng them nto one state. The result s an absorbng markov chan and let M CM be ts probablty transton matrx. Then the probablty of absorpton n lumped state Ω e Ω E and the probablty of absorpton n lumped state Ω l Ω L are gven respectvely by and ψ Ee = ψ Ll = 1 Ω Ω I 1 Ω Ω I Ω Ω I Ω Ω I π (,sncm ) Nmax k=k π k π (,sncm ) Nmax k=k π k lm M (m) m N CM (Ω, Ω e ) lm M (m) m N CM (Ω, Ω l ) where M (m) N CM (Ω, Ω E ) denotes the (Ω, Ω E ) th m th power of matrx M N CM. Note that ψ Ee e Ω e Ω E entry of the and ψll l Ω l Ω L are the probabltes that the frst cluster-membershp change occurs due to a vehcle enterng the cluster and leavng the cluster, respectvely. When calculatng the tme nterval between successve cluster-membershp changes, the examned system changes. Let X CME and X CML be the systems of dstance headways of the and the nodes on one sde of the cluster when the frst cluster-membershp change occurs due to a node enterng the cluster and a node leavng the cluster, respectvely. For example, f system X CM s absorbed n lumped state Ω = {(s 0, s 1,... s NCM )}, then the ntal lumped state for system X CML s {(s 0, s 1,... s NCM 1)} f Ω Ω L and the ntal lumped state for system X CME s {(s 0, s 1,... s NCM, s NCM +1)} f Ω Ω E, where s NCM (seconds) Fgure 6. Illustraton of the alternatng renewal process between overlappng and non-overlappng tme perods. [N N CM s, N max ]. Let Ω e be the lumped state for system X CME correspondng to lumped state Ω e for X CM, and let Ω l be the lumped state for system X CML correspondng to lumped state Ω l for X CM. Addtonally, let ψ E e equal ψ Ee weghted by the statonary dstrbuton (21) to account for the added dstance headway n the system, X CME. The cdf of the tme nterval between two successve cluster-membershp changes s approxmated by F TCM (m) = ψ F E e T (m) CM1e(Ω e ) e Ω e Ω E + ψ F Ll TCM1l(Ω )(m). (12) l l Ω l Ω L V. NUMBES OF COMMON CMS AND UNCLUSTEED NODES BETWEEN CLUSTES In Secton III, the tme for the frst change n clusteroverlap state along wth the cluster-overlappng and clusternon-overlappng tme perods are studed. Despte the mportance of the change n overlap-state as a measure of external cluster stablty, t s a bnary metrc. A quanttatve metrc that descrbes n detal the level of external stablty s desred. One quanttatve measure s the number of nodes located between the clusters. That s, the number of nodes shared between overlappng clusters and the number of nodes left unclustered between dsont clusters. The number of common nodes between neghborng clusters s an ndcator of the level of ntercluster communcaton nterference that can occur durng the overlappng perod. On the other hand, durng the nonoverlappng perod, the number of unclustered nodes between dsont clusters s an ndcator of the porton of network nodes that are left unserved by the clustered structure. Gven ntally overlappng neghborng clusters, vehcles can enter and leave the overlappng/unclustered regon. Addtonally, the cluster-overlap state may change over tme. Therefore, n ths secton we nvestgate the system of two neghborng clusters n terms the change of the numbers of common CMs and unclustered nodes between the two clusters along wth the change n the cluster-overlap state. Snce the system of dstance headways between the neghborng clusters, X c, constructs a fnte rreducble lumped Markov chan, there exsts an nfnte sequence of cluster-overlappng and cluster-non-overlappng tme perods [22]. Therefore, the overlap state between clusters fluctuates between overlappng and non-overlappng scenaros. Let {η(m), m = 0, 1,... } be a stochastc process wth state space { 1, 1}. If N c X (m) < 2,.e., the clusters overlap, then η(m) = 1; otherwse, η(m) = 1. Denote by ζ 1, θ 1, ζ 2, θ 2,... the lengths of successve ntervals spent n states -1 and 1, respectvely, where ζ 1, ζ 2,... are..d. and θ 1, θ 2,... are..d.. The process {η(m)} alternates between states -1 and 1, as shown n Fgure 6, whch s referred to as alternatng renewal process [23]. Snce we assume that

8 8 Overlappng regon common CM (a) Unclustered regon CM NCM 1 s < N and N CM s N. Let {π E, } S E =1 be the statonary dstrbuton of the edge lumped Markov chan. Furthermore, dvde the lumped states of the fully lumped Markov chan representng system X CM nto two sets, Ω and Ω c. A lumped state Ω = {s 0, s 1,..., s NCM } belongs to Ω f N CM s < N and to Ω c otherwse. Let T (e r1, Ω k ) be the frst occurrence tme of event e r1 gven that system X CM s ntally n lumped state Ω k Ω. Usng the recursve formula (1), we have F T (er 1,Ω )(m) = M NCM (Ω, ΩÑL 1 ) + k MNCM (Ω, Ω k )F T (er 1,Ω k )(m 1). The cdf of Ω k Ω T (e r1) s approxmated by CM unclustered node (b) Fgure 7. Illustraton of the events that cause a vehcle to (a) enter the overlappng regon and (b) leave the unclustered regon between neghborng clusters. the clusters are ntally overlappng, then η(0) = 1 and ζ k = Tov, k and θ k = Tnov, k.e., k th cluster-overlappng perod and the k th cluster-non-overlappng perod, respectvely. We assume that the Tov s k are..d. wth cdf (5) and the Tnov s k tme perods are..d. wth cdf (7) and they are ndependent of one another 5. The k th cycle s composed of ζ k and θ k. A. Node nterarrval tme durng an overlappng/nonoverlappng perod Durng an overlappng/non-overlappng perod, vehcles enter and leave the overlappng/unclustered regon resultng n a change n the number of common/unclustered nodes between neghborng clusters. Consder two overlappng clusters. A vehcle can enter the overlappng regon from ether of the clusters. Let T and T I be the frst arrval tme and the nterarrval tme of nodes to the overlappng regon, respectvely. We are nterested n the arrval tmes that cause an ncrease n the number of common nodes n the two clusters. The tme for the frst node enterng the overlappng regon s T = mn(t (e r1), T (e l 2)), where e r1 s the event that a vehcle enters the followng cluster from the rght sde of ts, and e l 2 s the event that a vehcle enters the leadng cluster from the left sde of ts as llustrated n Fgure 7(a). Note that T (e r1) and T (e l 2) have the same probablstc behavors. We consder that T (e r1) and T (e l 2) are ndependent when the number of common nodes between clusters s a small fracton of the total number of nodes n the two clusters. The tmes, T (e r1) and T (e l 2), can be calculated ndependently by applyng the frst passage tme analyss on two edge lumped Markov chans, each dentfyng the hop edge node of ts correspondng cluster, as n Subsecton IV-A. However, we propose to approxmate the dstrbutons of T (e r1) and T (e l 2) by calculatng them from a fully lumped Markov chan wth the ntal dstrbuton calculated from the state space of the edge lumped Markov chan. Snce the dstrbutons of T (e r1) and T (e l 2) are the same, we wll focus on one of them only, say T (e r1). Let S E be a set of states of the edge lumped Markov chan for a cluster wth N CM nodes, such that a lumped state Ω = {s 0, s 1,..., s NCM } belongs to S E f 5 Index k s dropped from Tov k and T nov k to refer to an arbtrary overlappng and non-overlappng perod, respectvely. F T (er 1)(m) ω F T (er 1,Ω)(m), m 1 (13) Ω Ω where ω = π E, s the ntal probablty dstrbuton f O (Ω )=Ω of states Ω Ω and f O (Ω ) s a functon that maps a lumped state from edge lumped markov chan to the correspondng one n the fully markov chan, note that ω = 0 f Ω S E s.t.f O (Ω ) = Ω Ω S E and Ω Ω. In order to calculate the probablty dstrbuton of node nterarrval tme to the overlappng regon, the probablty dstrbuton of the state of the system when a node frst enters the cluster needs to be calculated. Consder a cluster wth N CM 1 nodes at tme zero. When a node enters the cluster, system X CM representng the N CM CMs can only be n an edge lumped state Ω = {s 0, s 1,..., s NCM } s.t. the frst N CM states construct a lumped state, Ω k = {s 0, s 1,..., s NCM 1}, n a fully lumped Markov chan for system X CM, that satsfes ) Ω k Ω and ) MNCM (Ω k, ΩÑL 1 ) > 0. That s, Ω k Ω s drectly accessble from a lumped state n Ω c. As a result, the node nterarrval tme to the overlappng regon from one cluster can be approxmated by F T (eir 1 )(m) where ω I = Ω Ω ω I F T (er 1,Ω)(m), m 1 (14) M NCM 1(Ω,ΩÑL 1 )π E, f O (Ω )=Ω M NCM 1(Ω,ΩÑL 1 )π E, Ω Ω f O (Ω )=Ω the probablty dstrbuton of the ntal state when a node ust entered the cluster, and MNCM 1 s the probablty transton matrx of the absorbng lumped Markov chan that represents system {X 0, X 1,..., X NCM 1}. The cdf of the node nterarrval tme to the overlappng regon s gven by F TI (m) = 1 (1 F T (eir 1 )(m)) 2. (15) When two clusters become dsont, vehcles enter and leave the unclustered regon. Let us consder the node nterdeparture tme from the unclustered regon that causes the number of unclustered nodes to decrease, denoted by T Io. Nodes can leave the unclustered regon and enter ether of the two clusters. It can be concluded that the tme for a node to leave the unclustered regon s equal to the mnmum of two tme ntervals T (e r1) and T (e l 2), as llustrated n Fgure 7(b). Notce that the events that cause the node departure from the unclustered regon durng a non-overlappng perod are the same as those causng the node arrval to the overlappng regon durng the overlappng perod. Therefore, the dstrbuton of T Io can be calculated accordngly. s

9 9 B. Steady-state dstrbutons of the numbers of common CMs and unclustered nodes In ths secton we nvestgate the lmtng behavor of the external cluster stablty. Consderng clusters ntally formed to be partally overlappng, we examne the external cluster stablty under the assumpton that cluster mantenance s not mplemented. That s, we want to answer two questons: After a long tme, what s the probablty that two neghborng clusters are overlappng (non-overlappng)? What s the probablty dstrbuton of the number of common CMs (unclustered nodes) n the overlappng (unclustered) regon? The frst queston can be answered usng the theory of alternatng renewal process. The lmtng overlappng and nonoverlappng probablty s gven by P ov = E[T nov] E[T ov]+e[t nov] E[T ov] E[T ov]+e[t nov] and P nov =, respectvely [23]. For the second queston, we propose to model the problem as a storage buffer wth a two-state random envronment [24]. The buffer content represents the number of nodes n the overlappng/unclustered regon between neghborng clusters. The two random states of the buffer are the overlappng and the non-overlappng states whch fluctuate accordng to the alternatng renewal process as descrbed earler. Let N (ζ k ) (N o (θ k )) be the numbers of nodes enterng (leavng) the buffer durng the k th overlappng perod (non-overlappng perod), respectvely. Let N ( t) (N o ( t)) be the numbers of nodes enterng (leavng) the overlappng (unclustered) regon durng an arbtrary tme perod, t, respectvely. The numbers N ( t) and N o ( t) are pont processes correspondng to the..d. nterrenewal perods T I and T Io, and representng the nput process (output process) of nodes to (from) the buffer, respectvely. The mean and the varance of the nput process durng an overlappng perod and the output process durng a non-overlappng perod are gven by [23] E[N (ζ k )] = E[T ov] E[T I ], V ar[n (ζ k )] = c2 T I E[T I ] E[T ov], (16) E[N o (θ k )] = E[T nov] E[T Io ], V ar[n o(θ k )] = c2 T Io E[T Io ] E[T nov], (17) respectvely, where c TI and c TIo are the coeffcents of varaton of T I and T Io, respectvely. Consder the k th cycle. The buffer content at the begnnng of the cycle s gven by 6 B k = [B k 1 + N (ζ k 1 ) N o (θ k 1 )] +. Assumng that the processes N (ζ k 1 ) and N o (θ k 1 ) are non-decreasng for all k, the buffer content model can be assocated wth a G/G/1 queue [24]. In the queueng model, the servce tme of customer k 1 s S k 1 = N (ζ k 1 ) and the nterarrval tme between customers k 1 and k s A k 1 = N o (θ k 1 ). Then the buffer content at the begnnng of the k th cycle s the watng tme of the k th customer. Therefore, the buffer content at an arbtrary tme step, m, s equal to the vrtual watng tme (or the workload) of ths G/G/1 queue [24] [25]. The vrtual watng tme depcts the remanng servce tme of all customers n the system at an arbtrary tme step. Let V (m) denote the vrtual watng tme (buffer content) at an arbtrary tme step m. The relaton between the vrtual watng tme at the m th tme step and the customer watng tme at the begnnng of a cycle s gven by [24] V (m) = B n(m) + S n(m) m + 6 y = [x] + s equvalent to y = max(0, x) n(m) 1 k=1 A k + (18) where n(m) = max{k 0 : k =1 A k m}, m 0. To fnd the lmtng probablty dstrbuton of the buffer content (.e., the number of common/unclustered nodes between two neghborng clusters) a dffuson approxmaton s used. The dffuson approxmaton s a second order-approxmaton that uses the frst two moments of the servce and nterarrval tmes of the G/G/1 queue [26]. Let ρ = E[S k ]/E[A k ] be the ntensty factor. A steady-state dstrbuton of the buffer content exsts f ρ < 1 and t s approxmated ( by) a geometrc dstrbuton wth parameter equal to 1 λ2 λ 2 2µ. The approxmated pmf s gven by [26] [27] ) ( ) P V (n) (1 λ2 λ 2 n λ 2 2µ λ 2, n 0 (19) 2µ where µ = ρ 1 and λ 2 = E[S2 k ] E[A k ] whch can be calculated from (16) and (17). The lmtng probablty dstrbuton of the numbers of common CMs and unclustered nodes between the two clusters can be descrbed by the pmf (19) wth probablty P ov and P nov, respectvely. Let P C0 and P U0 denote the lmtng probabltes that there are zero common CMs and zero unclustered nodes between neghborng clusters, respectvely. These probabltes are gven by P C0 = P ov P V (0) + P nov, and P U0 = P nov P V (0) + P ov. VI. NUMEICAL ESULTS AND DISCUSSION Ths secton presents numercal results for the analyss of the proposed external and nternal cluster stablty metrcs. The external cluster stablty metrcs are the tme to the frst change of cluster-overlap state, T ov1 and the tme nterval between successve changes of cluster-overlap state (clusteroverlappng perod, T ov and cluster-non-overlappng perod, T nov ). The nternal cluster stablty metrcs are the tme to the frst change of cluster-membershp, T CM1, and the tme between successve cluster-membershp changes, T CM. Addtonally, numercal results are presented for pmfs of the steadystate numbers of common CMs and unclustered nodes between two neghbourng clusters. We consder a connected VANET n three traffc flow condtons, uncongested, near-capacty, and congested, each correspondng to a set of parameters lsted n Table I. For values of N c and N CM at the 0 th tme step, we smulate a smple weghted clusterng algorthm, where s are chosen wth the mnmum average relatve speed to ts one-hop neghbors, such that each vehcle belongs to a cluster and no two s are one-hop neghbors (.e., smlar to the use of moblty nformaton for clusterng n [10], [11]). The dstance headways of vehcles on the hghway follow a truncated exponental, gamma, and Gaussan dstrbutons for the uncongested, near-capacty, and congested traffc flow condtons, respectvely. The vehcles speeds are..d. and are normally dstrbuted wth mean 100 klometer per hour and standard devaton of 10 klometer per hour [28]. Fgure 8 plots the probablty dstrbutons of N c and N CM for the resultng clusters from smulatng the clusterng algorthm. Intally, we Table I. System parameters n smulaton and analyss Traffc flow condton D(veh/km) E[N CM ] E[N c] Uncongested Near-capacty Congested (meter) N max X c(0) X CM (0) {0,1,1,1,1,2} {1,1,1,1,5}

10 10 Fgure 8. The pmfs of (a) the number of nodes between two neghborng s, N c and (b) the number of nodes n a cluster N CM, calculated from smulatng a smple weghted clusterng of vehcles Smulatons Theoretcal approxmaton Theoretcal exact φ Fgure 9. The pmf, φ = P (I c Ω ), of system X c beng n lumped state Ω Ω OV at the nstant when the second overlappng cluster state occurs. set N c to ts average value from the cluster formaton results. For D = 42 vehcles per klometer (veh/km), we set I c and I CM to the states wth hghest probablty of occurrence at the cluster formaton stage. The Markov-chan dstance headway model has the followng parameters: N max = 9, each state covers 20 meters range of dstance headways, the tme step s equal to 2 seconds, and the transton probabltes are tuned accordng to the results n [20]. Based on these parameters, we generate tme seres of dstance headway data accordng to the mcroscopc moblty model, usng MATLAB. Each smulaton conssts of 20,000 teratons. Fgure 9 compares the dstrbuton of the state of system X c, when the second overlappng state occurs, calculated usng the exact dervaton (4) and the proposed approxmaton (6). The values on the x-axs represent arbtrary IDs gven to the lumped states Ω Ω OV. The results from the proposed approxmaton shows close agreement wth the exact and the smulaton results. Fgure 10 plots the pmf of the tme nterval Fgure 10. The pmfs of (a) the tme to the frst change n cluster-overlap state, T ov1 (Ω k ), for I c = {0, 1, 1, 1, 1, 2} Ω k when the clusters are ntally formed; (b) the tme to the frst change n cluster-overlap state T ov1 ; and (c) the cluster-overlappng tme perod, T ov, when D = 26 veh/km. for the frst change n cluster overlappng state, for (a) a gven ntal state of X c and (b) when averagng over random ntal states, respectvely. The theoretcal results for the pmfs of the cluster-overlappng perod are calculated from the cdf n (5). The calculated pmf of T ov n Fgure 10(c) s based on the approxmaton gven n Fgure 9. The dstrbuton of T ov1 (Ω k ) changes wth I c belongng to dfferent lumped states Ω k. The dstrbuton of T ov1 descrbes the average tme before the frst cluster-overlap change for a randomly pcked cluster n the network. When clusters overlap, the cluster-overlappng perod s equal to the tme perod between two successve clusterovelap state changes (.e., the tme perod of nvarant clusteroverlap state). Note that the average tme for the frst change of cluster-overlap state s larger than the average tme perod between successve changes of cluster-overlap state. When the second overlappng state occurs between neghourng clusters, the clusters state s closer to non-overlappng than that when the clusters are ntally formed, on average. That s, the clusters state can only be n the accessble lumped states (Ω OV 1 n Fgure 2). Fgure 11 plots the pmf of the tme perod from the cluster formaton tll the tme step that a frst change n clustermembershp occurs for (a) a gven ntal state I CM Ω k

11 11 Fgure 11. The pmfs of (a) the tme to the frst change n clustermembershp, T CM1 (Ω k ), for I CM = {1, 1, 1, 1, 5} Ω k when the cluster s ntally formed; (b) the tme to the frst change n cluster-membershp, T CM1 ; and (c) the tme perod between two successve cluster-membershp changes, T CM, when D = 26 veh/km. and (b) a random ntal state, and (c) the pmf of the tme perod between successve cluster-membershp changes. The theoretcal results for the pmfs of T CM1 (Ω k ) and T CM1 are calculated from the cdfs n (10) and (11), respectvely. The pmf of the tme perod between successve cluster-membershp changes s calculated from the cdf n (12) and s plotted n Fgure 11. The smulaton results closely agree wth the theoretcal calculatons. It s observed that, when the frst change n cluster membershp occurs after the cluster formaton, the second change n cluster membershp has a hgher probablty of occurrng n a shorter tme perod. Ths reflects the effect of a wreless lnk between a CM and fluctuatng between connectng and dsconnectng states n a short perod of tme. The mpact of ths fluctuaton can lead to frequent re-clusterng that drans the precous VANET rado resources. Some clusterng algorthms for VANETs ams to localze the mpact of ths fluctuaton wthn the clusters [4], [7], [10] 7. Fgure 12 plots the pmf of the frst arrval tme of nodes nto the overlappng regon T er1, for a near-capacty traffc flow condton. The exact theoretcal value s calculated from the edge Markov chan as explaned n Appendx A.4, whereas the approxmated value s calculated from the fully lumped Markov chan usng (13). The results show that approxmatng the node-arrval tme to the overlappng/unclustered regon usng the fully lumped Markov chan s adequate. Fgure 13 plot the pmfs of the cluster-overlappng, T ov tme perod for dfferent vehcle denstes when N c s set to the average values n Table I 8. The tme nterval between successve changes of cluster-overlap state s equal to T ov (T nov ) when the two clusters are overlappng (dsont). Notce that the vehcle densty has lttle mpact on the dstrbuton of the overlappng/non-overlappng perods when N c s set to the average value. However, ths s not true for all N c. Fgure 14 plots the average cluster-overlappng and the average cluster-non-overlappng tme perods for dfferent numbers of nodes between neghbourng clusters, N c. The average values are calculated usng (8) and the values of N c are from the clusterng results n Fgure 8. For a fxed N c, the average cluster-overlappng perod s larger for a larger densty, whereas the average cluster-non-overlappng perod s smaller for a larger densty. The reason s that, n a congested traffc flow condtons, the dstance headways are small when compared to those n an uncongested traffc flow condton. Therefore, for the same N c, the cumulatve dstances are smaller for a hgh densty. It should be noted that the large values of average cluster-overlappng tme perods for N c = 1 are due to the connected network assumpton. Fgure 14 shows that, as N c ncreases, the average cluster-overlappng perod reduces and the average cluster-non-overlappng perod ncreases for the same traffc flow condton. To nvestgate the lmtng behavor of the number of vehcles n the overlappng/unclustered regon, we frst calculate the two parameters µ and λ 2 for the three vehcle denstes. Notce that the dstrbutons (5), (7), and (15) are all condtonal on the ntal cluster state n terms of N c and N CM. Therefore, n the calculaton of µ and λ 2, we use the law of total expectaton to calculate E[T ov ] = n P N c (n)e[t ov (n)] and E[TI 2 ] = n P N c (n)e[ti 2 (n)], where T ov(n) s the cluster-overlappng tme perod for two clusters separated by N c = n nodes and T I (n) s the node nterarrval tme for a cluster wth N CM = n nodes, respectvely. The calculatons are done for near-capacty and congested traffc flow condtons only. The reason s that the dffuson approxmaton assumes that the pont processes N (ζ k ) and N o (θ k ) are normally dstrbuted accordng to the central lmt theorem. Ths assumpton s not satsfed for an uncongested traffc flow, due to a relatvely small number of vehcles between two clusters as shown n Fgure 8. The ntensty factor s found to be ρ = and for D = 26 and 42 veh/km, respectvely. As a result, the steady-state dstrbuton does not exst. However, consder only N c E[N c ] for both cases, we fnd that ρ = 0.33, and 0.64 for D = 26, and 42 veh/km, respectvely. Fgure 15 plots the steady-state probablty dstrbutons for the non-zero number of vehcles n the overlappng/unclustered regon when N c E[N c ] for near-capacty and congested traffc flow condtons. The theoretcal results are normalzed to the value 1 P V (0), snce the probablty dstrbutons n Fgure 15 represent P TI (m) Smulatons Theoretcal exact Theoretcal approxmaton m (seconds) 7 Fgure 16, n the appendx, plots the pmfs of the tme nterval between two successve cluster-membershp changes for dfferent vehcle denstes. 8 The pmfs of the cluster-non-overlappng, T nov, follow the same trends. So, the plots are omtted due to space lmtatons. Fgure 12. The pmf of the nterarrval tme of nodes to the overlappng regon when N c = 5 and D = 26 veh/km.

12 12 Fgure 13. The pmf of cluster-overlappng tme perod wth vehcle densty of (a) D = 9, (b) D = 26, and (c) D = 42 veh/km. the non-zero number of common CMs wth probablty P ov and the non-zero number of unclustered nodes wth probablty P nov. The smulaton results closely agree wth the theoretcal calculatons. However, there exst slght dfferences between smulaton and theoretcal results especally at the values of n = 5 and n = 8, for D = 26 and 42 veh/km, respectvely. Ths s manly due to complete overlappng between neghborng clusters. When two clusters completely overlap,.e., become one hop neghbors, all the nodes between them become common nodes, however no addtonal nodes can enter the overlappng regon. Ths s not accounted for n our model. Accordng to many clusterng algorthms, when two s become one hop neghbors, they merge nto a sngle cluster [7], [8], [10]. Fgure 15 shows that the smulaton results excludng the complete cluster overlappng data are n closer agreement wth the theoretcal results n comparson wth smulaton results that nclude the complete cluster overlappng data. Addtonally, the numercal and smulaton E[T ov ],E[T nov ] (seconds) E[T ov ],D=9 E[T nov ],D=9 E[T ov ],D=26 E[T nov ],D=26 E[T ov ],D=42 E[T nov ],D=42 Fgure 15. The steady-state pmfs of buffer content,.e., the number of non-zero nodes n the overlappng/non-overlappng perod, for (a) D = 26 and (b) D = 42 veh/km. results for the lmtng probabltes of havng zero common CMs and zero unclustered nodes are gven n Table II. The probablty dstrbutons of T ov1, T ov, T nov, T CM, and V (m) derved n ths paper provde ndcators for the stablty of a cluster n terms of ts relaton wth ts CMs and ts relaton wth neghbourng clusters. Ths can be used to enhance network protocol desgn for VANETs. For example, the derved probablty dstrbutons of T ov and T nov can be used to update the transmsson codes assgned to dfferent clusters so that the hdden termnal problem caused by cluster overlappng s avoded wth a certan desred probablty threshold [16]. Addtonally, the dstrbuton of the clusteroverlappng perod can be utlzed to dynamcally choose the value of the tme threshold used to avod frequent mergng and splttng of neghbourng clusters n VANETs [4], [7], [10]. The tme perod between successve cluster-membershp changes, T CM, provdes a lower bound on the clustermembershp duraton. Ths s extensvely used n the lterature for performance evaluaton of clusterng algorthms [7] [9]. The probablty dstrbuton of T CM can be used to choose the tme threshold value that determnes when an unclustered node can create ts own cluster after t has dsconnected from ts, thus mnmzng re-clusterng frequency [4], [7], [10]. The lmtng probablty dstrbuton of the number of common CMs between neghbourng clusters can help determne the amount of addtonal rado resources (e.g., tme slots) that should be allocated to a cluster n order to avod ntercluster nterference N c Fgure 14. Average cluster-overlappng and cluster-non-overlappng tme perods for dfferent N c values wth vehcle densty D = 9, 26, and 42 veh/km. The values of N c are those n Fgure 8. Table II. Lmtng probabltes of zero common CMs/unclustered nodes D(veh/km) Smulaton Theoretcal P C P C P U P U

13 13 VII. CONCLUSION Ths paper presents a stochastc analyss of sngle-hop cluster stablty n a hghway VANET wth focus on a sngle lane. The tme perods of nvarant cluster-overlap state and clustermembershp are proposed as measures of external and nternal cluster stablty, respectvely. A stochastc moblty model that descrbes the tme varatons of ndvdual dstance headways s adopted n the analyss. The system of dstance headways that govern the changes n the overlap state and the cluster membershp s modeled by a dscrete-tme lumped Markov chan. The frst passage tme analyss s employed to derve the dstrbutons of the proposed cluster stablty metrcs. The analyss provdes nsghts about the tme perods durng whch a cluster s lkely to reman unchanged n terms of ts clustermembershp and ts overlap state wth neghborng clusters. Addtonally, the lmtng probablty dstrbutons of the numbers of common and unclustered nodes between neghborng clusters are approxmated usng queung theory and dffuson approxmaton. The probablty dstrbutons derved for the proposed cluster stablty metrcs can be utlzed n the development of effcent clusterng algorthms for VANETs. APPENDIX A.1 Proof of Theorem 1 Let M N = {M N (S, S )}, 0 S, S Nmax N 1, be the transton matrx of the N-dmensonal Markov chan that represents the system of N ndependent copes of the 1- dmensonal Markov chan, X, wth transton matrx M = {M(u, u )}, 0 u, u N max 1. A dscrete-tme Markov chan wth stochastc transton matrx M N s lumpable wth respect to the partton Ω f and only f, for any subsets Ω and Ω n the partton, and for any super states S 1 and S 2 n subset Ω [21], S Ω M N (S 1, S) = S Ω M N (S 2, S). (20) Consder the left hand sde (LHS) of (20). Snce X s a brth-death process, the super state S 1 = (u 0, u 1,..., u N 1 ), 0 u N max 1, can transt to any super state n set A = {(u 0, u 1,..., u N 1 )}, where state u {u 1, u, u + 1},.e., A 3 Nmax. Let subsets A = A Ω and A = A Ω. Snce M N (S 1, S) = 0 S A, the LHS of (20) reduces to S A M N (S 1, S). Smlarly, for the rght hand sde (HS) of (20), the super state S 2 = (v 0, v 1,..., v N 1 ), 0 v N max 1, can transt to any super state n set B = {(v 0, v 1,..., v N 1 )}, where state v {v 1, v, v + 1},.e., B 3 Nmax. Let subsets B = B Ω and B = B Ω. Snce M N (S 2, S) = 0 S B, the HS of (20) reduces to S B M N (S 2, S). Consder two sequences, S and S, that are permutatons of each other, and defne ϱ(s, O ) = S to be the permutaton operator on sequence S under ndex order O that gves S,.e., S = (S (O (k))) S k=1. For example, f S = (1, 0, 2) and S = (0, 2, 1), then O = (2, 3, 1). Let S 1 = (u 0, u 1,..., u N 1 ) be a super state n subset A. Therefore, M N (S 1, S 1) = N 1 n=0 M(u, u ). Snce S 1, S 2 Ω, there exsts an ndex order O 12, s.t. ϱ(s 1, O 12 ) = S 2. Addtonally, S 2 = (v 0, v 1,..., v N 1 ) s.t. S 2 = ϱ(s 1, O 12 ). Note that S 2 B 2. As a result, M N (S 2, S 2) = N 1 n=0 M(v, v ) = N 1 n=0 M(u O 12(n), u O 12(n) ). Snce the product operaton s commutatve, we have M N (S 2, S 2) = M N (S 1, S 1). In general, S 1, S 2 Ω s.t.ϱ(s 1, O 12 ) = S 2 and S 1 A, S 2 B s.t.s 2 = ϱ(s 1, O 12 ) and M N (S 2, S 2) = M N (S 1, S 1). Hence, S A M N (S 1, S) = S B M N (S 2, S), whch ends the proof. A.2 Proof of Corollary 1 Consder the tr-dagonal probablty transton matrx of the Markov chan, X, as descrbed n Subsecton II-B. The statonary dstrbuton of the chan, X, s gven by π = [ where π 0 = 1 k=0 ( pk q k N max 1 =1 ) π 0, 1 N max 1 (21) ( )] 1 pk 1. k=0 q k+1 Consder the th lumped state Ω = {s 0, s 1,..., s N }. Let N D be the number of dstnct states n {s 0, s 1,..., s N 1 } n whch (u 1, u 2,... u ND ) and (n u1, n u2,..., n ND ) are the sequences of dstnct states and ther correspondng frequences, respectvely, where 0 u N max 1 and N D =1 n u = N. Note that the sze of the lumped state s equal to the number of super states that are permutatons of each other,.e., 1 Ω N!, 0 N L 1. Therefore, the lumped states result from all possble outcomes of choosng N states from N max dfferent states ndependently, where choosng state s has the probablty π, 0 s N max 1. Ths s a generalzaton of the Bernoull tral problem. Hence, the statonary dstrbuton for the lumped state Ω s gven by = N! N D ND k=1 n π nu k u k. (22) u k! k=1 That s, the statonary dstrbuton of the lumped Markov chan s mult-nomal, whch ends the proof. A.3 Proof of Corollary 2 Let the lumped state Ω = {s 0, s 1,..., s N 1 } be a lumped state such that, f the system enters ths state, the event of nterest occurs. Then, {s 0, s 1,..., s N 1 } s an N-restrcted nteger partton of an nteger that s greater than or equal to N th. In combnatorcs, an nteger partton of a postve nteger n s a set of postve ntegers whose sum equals n. Each member of the set s called a part. An N-restrcted nteger partton of an nteger n s an nteger partton of n nto exactly N parts. Therefore, Ω = {s 0, s 1,..., s N 1 } Ω OV, {s 0, s 1,..., s N 1 } s an nteger partton of an nteger that s less than N th. Snce, an nteger N th can be parttoned nto at most N th parts (.e. when all the parts equal to one) and the order of the N states n the lumped state s not mportant, the number of lumped states Ω OV when N > N th s equal to that when N = N th. Notce that Corollary.2 apples only on the lumped Markov chan and not the orgnal N-dmensonal one. Ths ends the proof. ACKNOWLEDGMENT Ths work was supported by a research grant from the Natural Scences and Engneerng esearch Councl (NSEC) of Canada and BlackBerry Lmted.

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Wehua Zhuang (M 93-SM 01-F 08) has been wth the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Canada, snce 1993, where she s a Professor and a Ter I Canada esearch Char n Wreless Communcaton Networks. Her current research focuses on resource allocaton and QoS provsonng n wreless networks, and on smart grd. She s a co-recpent of several best paper awards from IEEE conferences. Dr. Zhuang was the Edtor-n-Chef of IEEE Transactons on Vehcular Technology ( ), and the Techncal Program Symposa Char of the IEEE Globecom She s a Fellow of the IEEE, a Fellow of the Canadan Academy of Engneerng, a Fellow of the Engneerng Insttute of Canada, and an elected member n the Board of Governors and VP Moble ado of the IEEE Vehcular Technology Socety. She was an IEEE Communcatons Socety Dstngushed Lecturer ( ).