Strong Stochastic Stability for MANET Mobility Models

Transcription

1 trong tochatic tability for MAET Mobility Model R. Timo, K. Blacmore and L. Hanlen Department of Engineering, the Autralian ational Univerity, Canberra {roy.timo, Wirele ignal roceing, ICTA, Canberra Abtract At the core of any MAET imulation i a mobility model. To help enure reliable imulation reult, it i of interet to now if the mobility model i table: Will time-averaged meaurement of mobility model event converge? For example, doe the time-averaged ditance between a pair of node converge a imulation time increae? In thi paper, we tudy the tability of a cla of dicrete Random Waypoint Mobility Model RWMM. Thi cla include the claic Random Waypoint Mobility Model. We how that each mobility model in thi cla atifie a pointwie ergodic theorem a generalized trong law of large number; thu, all bounded time-averaged meaurement of mobility model event converge with probability one. A corollary of thi ergodic theorem how that each mobility model in thi cla alo poee a time-tationary regime. I. ITRODUCTIO rotocol for Mobile Ad-Hoc etwor MAET are contructed uing imulation. Mobility model [] are ued to generate node movement in imulation. It i well documented [2 5] that untable mobility model may lead to imulation reult that are unreliable. We define and tudy a cla of tochatically table mobility model, which include the claic Random Waypoint Mobility Model RWMM. Boudec and Vojnović [5] proved that a general verion of the claic RWMM the Random Trip Mobility Model i table. Thi tability property followed from the exitence and uniquene of a tationary ditribution for node location. We extend thi idea and ay that a mobility model i table if it atifie a pointwie ergodic theorem. If it i not table, then the relative frequency of event may not converge and imulation reult involving time average may change in time. A trong motivation for thi tudy i that a mobility model tability or intability may be paed up though the networ layer. For example, in [6] we how that the Dynamic ource Routing DR protocol preerve the tability of the mobility model; if the pointwie ergodic theorem hold for the mobility model, then it alo hold for the route election random proce. etwor imulation are baed on time average, o the ignificance of thi reult i immediate: If the mobility model i table and the routing protocol preerve thi tability, then the trong law of large number hold for networ layer imulation. A mobility model define a random proce. It i aid to be tationary if the probability law governing node movement doe not change in time. The trong law of large number hold for tationary mobility model, o many wor [4, 5, 7] have developed method which tranform non-tationary model into tationary model. tationary mobility model alo atify the tronger pointwie ergodic theorem. The following example illutrate how thi theorem may be ued. Example : uppoe a node i moved uing a tationary mobility model. Denote it location at time n with the random variable X n. In a imulation we record the node location over time with the equence x x 0, x, x 2,.... ince {X n } i tationary, Birhoff ergodic theorem [8, g. 30] aert that the limit f x lim f x n, x n+, x n+2,... exit almot everywhere a.e. for every bounded meaurement function f. For example, uppoe we are intereted in etimating the mean probability of ome mobility model event A. Chooe f to be the indicator function: fx n, x n+,... A x n, x n+,... {, if xn, x n+,... A The limit A x i the relative frequency of A in x. If we repeat thi imulation a large number of time and average over the reulting relative frequencie A x, the reulting average will converge to the mean probability of event A. Unfortunately, the probability law governing node movement for mot mobility model are not contant in time [4, 5, 7]. Model, uch a the claic RWMM, may be locally non tationary and diplay tarting tranient. The et of tationary mobility model i too mall for the tudy of MAET. In thi paper, we replace the tationary condition with a weaer aymptotically tationary condition. That i, only the long-run behavior of the mobility model need to be tationary. For example, we permit both tarting tranient and local non tationary propertie. A random proce which i aymptotically governed by a tationary proce i called Aymptotically Mean tationary AM. Gray and Kieffer [9, Thm. ] howed that a random proce atifie the pointwie ergodic theorem if and only if it i AM; hence, we hall ay that a mobility model i table if and only if it i AM.

2 In [0, Lem. ] we demontrated that a retricted verion of the claic RWMM i AM. In thi paper we extend AM theory to a general cla of RWMM, which include the claic RWMM a a pecial cae. In the claic RWMM [], each node elect a equence of waypoint w w 0, w,... uing an independent and identically ditributed i.i.d. random proce {W n }. For each pair w i, w i+, i 0,,..., the node elect a peed uniformly at random from the interval [min peed, max peed]. It then travel in a traight line between w i and w i+ at thi peed. The main reult of thi paper Theorem may be ummarized a follow: The claic RWMM i AM table and ergodic, and 2 A ufficient condition for table node movement i: a each node elect waypoint uing an AM random proce, b given a equence of waypoint, each node elect it peed in a tationary fahion. The paper outline i a follow. otation and terminology are introduced in ection II. ection III define table, tationary, ergodic, and AM mobility model. The general RWMM i developed in ection IV, and ection V preent our main reult in Theorem. The paper i concluded in ection VI, and Theorem i proved in the Appendix. II. OTATIO AD TERMIOLOGY Often we hall be intereted in random procee with the ame probability law, but with different notion of time. For clarity, we adopt the dynamical ytem model for a random proce [, 2] give excellent introduction. Let X {X n } be a dicrete time random proce with a dicrete finite alphabet X. We are intereted in the probabilitic behavior of mobility model, o we hall tudy the ditribution of {X n }. The th order joint ditribution of {X n } i the probability meaure on X defined by µ x 0 rob { } X 0 x 0,..., X x where X i0 X i, X i X. The et {µ : 0} i called the ditribution of {X n }. If ome conitency condition are atified 2, the Kolmogorov Repreentation Theorem [, Thm. I..2] aert that we may replace the ditribution with a probability meaure µ on the equence pace X i0 X i, X i X. An important ubet of X i the cylinder et: [x n m] { x X : x i x i, m i n} Let F X denote the σ-field generated [8, g. 2] by cylinder et. The action of time i captured with the hift tranform T n X x x n, x n+, x n+2,..., n Z The pair X, F X i a meaurable pace, the triple X, F X, µ i a probability pace, and the quadruple ode peed wa aumed to be contant. 2 The model tudied in thi paper atify thee condition. X, F X, µ, T X i a dynamical ytem. The original random proce {X n } i related to X, F X, µ, T X by {X n } {Π 0 TX nx} where Π 0x x 0. Why i thi model ueful? uppoe a node i moved according to ome mobility model, which may be decribed by the random proce {X n }. uppoe we record the node poition for the firt n time tep of a imulation: x n 0 x 0, x,..., x n. Thi imulation may be thought of a a probabilitic experiment where the node trajectory x n 0 i an event. The probability pace for thi experiment i X, F X, µ, and the trajectory x n 0 define the event [x n 0 ] F X. Thu, the probability that the trajectory i randomly choen to be x n 0 i µ[x n 0 ]. The ytem X, F X, µ, T X pecifie both the probability pace X, F X, µ and the hift T X. The hift pecifie what time cale i of interet. Thi model i ueful becaue one often need to conider different time cale for the ame mobility model. For example, uppoe we are given a mobility model to which we aociate the probability pace X, F X, µ. Let T X be given by, and let T X be a variable hift defined by T X x T fx X x x fx, x fx+,... where f : X Z. The propertie of the random proce {X n } aociated with T X are almot alway different to the random proce {Xn} aociated with T X. If one i intereted in the behavior of node with repect to T X, then one tudie X, F X, µ, T X. However, if one i intereted in the behavior of node with repect to T X for example, networ router reet their routing table at time at time defined by f then one tudie X, F X, µ, T X. III. TABLE MOBILITY MODEL uppoe the ytem X, F X, µ, T X decribe the mobility model and time cale of interet. A random proce i aid to be tationarity if it probability law doe not change in time. The following definition formalize thi tatement. Definition tationary: The ytem X, F X, µ, T X i aid to be tationary if, for all A F X, we have µa µ T X A, where T X A {x X : T X x A}. The pointwie ergodic theorem hold for tationary ytem: Lemma Birhoff ointwie Ergodic Theorem: [8, Thm. 24.] If X, F X, µ, T X i tationary and f i bounded and meaurable, then the limit f x lim f TX n x exit almot everywhere in µ a.e. [µ]. The value of f x in Lemma i, in general, random. A ytem where f x i contant a.e. [µ] i called ergodic. Definition 2 Ergodic: The ytem X, F X, µ, T X i aid to be ergodic if A T X A implie µa 0 or. Equivalently, a ytem X, F X, µ, T X i ergodic if and only if, for all bounded meaurable function f, the limit f f x lim f TX n x

3 i contant a.e. [µ]. Definition 3 table Mobility Model: A mobility model with dynamical ytem model X, F X, µ, T X i aid to be table if, for every bounded meaurable f, the limit f x lim f TX n x exit a.e. [µ]. From Lemma and Definition 2, it i clear that mobility model which are tationary and / or ergodic are table. However, mobility model do not need to be tationary or ergodic to be table; for example, the claic RWMM i nontationary but table [5]. A we hall ee, the following cla of random procee provide ueful tool for checing the tability of mobility model. Definition 4 Aymptotic Mean tationary AM: The ytem X, F X, µ, T X i aid to be aymptotic mean tationary if, for all A F X, the following limit exit µa lim µ T n X A 2 If X, F X, µ, T X i AM, then 2 define a probability meaure µ on X, F X. The probability meaure µ decribe the long-run average behavior of the ytem, and it i called the tationary mean of µ. Gray and Kieffer [9] howed that AM i both neceary and ufficient for the pointwie ergodic theorem. Lemma 2 AM ointwie Ergodic Theorem: [9, Thm. ] A ytem X, F X, µ, T X i AM if and only if, for every bounded meaurable f, the limit f x lim f TX n x exit a.e. [µ]. For u, Definition 4 and Lemma 2 provide an alternative decription for table mobility model. Lemma 3 table Mobility Model: uppoe we are given a mobility model to which we aociate the probability pace X, F X, µ. The model i table with repect to T X if and only if X, F X, µ, T X i AM. IV. A GEERAL RADOM WAYOIT MOBILITY MODEL In thi ection we preent a general dicrete RWMM. Conider a networ with V mobile node V {v, v 2,..., v V }, where each node i located in a dicrete finite pace. We ue four random procee to decribe the general RWMM. A. Waypoint Random roce er ode Each node v V elect a equence of waypoint w w 0, w,... in a random fahion from the pace. Let W {W i } i0 denote the random proce decribing how the waypoint are elected, and let W, F W, µ w, T W be the correponding dynamical ytem, where W. Example 2 Claic RWMM: In the claic RWMM, the waypoint are choen independently and identically ditributed p 2 5 p 3 6 p Fig.. Random peed election in dicrete pace and time i equivalent to random path election. The tate pace i {, 2,..., 3 }. uppoe the node mut travel from waypoint w to waypoint w 3. The path p, p 2 and p 3 where, for example, p 2, 7, 3, demontrate how the node location i ampled. i.i.d. from W. In thi cae, µ w i a product meaure, and T W i a Bernoulli hift [8, g. 3]. B. ath Random roce er ode Recall the claic RWMM. uppoe node v elect the equence of waypoint w. For each pair w i, w i+, i Z {0,,...}, the node elect a peed uniformly at random from [min peed, max peed]. It then travel in a traight line between w i and w i+ at thi peed. In dicrete pace and time, the node poition i ampled during it trip between w i and w i+ a lower peed will reult in more intermediate ample. The dicrete time equivalent to a random peed i a random path. At each time hift, the node move one tep along the path. Thi idea i depicted in Figure. For each pair w, w W label each path, which connect w and w, with a unique element from the finite et w,w {p, p 2,..., p w,w } Let denote the et of all path w,w W W w,w and i0 i, i, the collection of path equence. We hall now define the random proce { n }, which decribe the election of path. ince conditionally depend on W, the mot convenient way to define i via a channel. Given a equence of waypoint w, node v may randomly elect any equence of path p, which connect the element of w. Fix w W and let u define the collection of allowable path equence w by w {p : p i wi,w i+, i Z } Let F be the σ-field of ubet of generated by cylinder et. Let u define a family of probability meaure ν wp {ν w : w W } on, F uch that ν w w. The triple W, νwp, i called a channel.

4 We hall be intereted in a pecial type of channel: Definition 5 tationary Channel: The channel W, ν wp, i aid to be T W, T tationary if ν TW w A νw T A with w W and A F. Example 3 Claic RWMM: In the claic RWMM, when node v reache waypoint w i, it elect a peed uniformly at random from [min peed, max peed]. In dicrete time and pace, thi i equivalent to electing a path uniformly at random from the et wi,w i+. The dicrete RWMM define a channel W, ν wp,, where ν wp {ν w : w W, } and ν w [p n 0 ] n 2 i0 wi,w i+ if p i wi,w i+, i 0,..., n 2 ote: ν w w, for every w W. 2 ν ha finite input memory of order 2, ee [3, g ]. We now how that thi channel i tationary. Fix w W and conider any p n 0 n it i ufficient to conider cylinder et, then ν TW w [p n 0 ] n 2 i0 3 wi+,w i+2, if p i wi+,w i+2, i 0,..., n 2 4 Let p j+ p j for 0 j n, o that [ p n ] T [pn 0 ]. 0 ] ν w [ p n ] ν w T [pn n 2 i0 wi+,w i+2, if p i wi+,w i+2, i 0,..., n 2 Combine 4 and 5 to ee that the channel i tationary. We hall alo be intereted in the following type of channel. Definition 6 tationary and Ergodic Channel: A tationary channel W, ν wp, i aid to be ergodic if, for all A, B F W, we have lim ν wp T n W A B ν wp A νwp B Example 4 Claic RWMM: We now how that the channel 3 i tationary and ergodic. In thi cae, however, it i eaier to how that the channel atifie a tronger condition called output mixing. A channel i aid to be output mixing if, for all A, B F W and all w W, we have lim ν w T n A B ν w T n A ν w B 0 6 ince F i generated by cylinder et, it i ufficient to how 6 hold for any [p a 0 ], [p b 0 ] F. Let T n [pa 0 ] [ p n+a n ], where p i+n p i, 0 i a. For any n > b the cylinder et [ p n+a n 5 ] and [p b 0 ] depend on value of p i for indice i in dijoint et of integer. ee [, g. 8] for a imilar example. For any > b, we have 7, o 6 hold and the channel i mixing. Recall [4, Lem ] to ee that the channel i alo wea mixing and ergodic. The waypoint random proce W, F W, µ w, T W and the channel W, ν wp, induce a joint probability meaure µν wp on the input-output pace W, F W F µν wp A B µw ν w B w A where A F W and B F. The ditribution of the path random proce i given by µ p B µν wp W B µw ν w B w W where B F. Let, F, µ p, T be the correponding dynamical ytem for { n }. C. Location Random roce er ode uppoe node v elect the equence of path p to connect waypoint w. The total time n i required to travel from waypoint w 0 to waypoint w i i a function of the length of the firt i path p i 0 p 0, p,..., p i. Let lp i denote the length of the i th path p i, o that it tae lp i time tep to move from w i to w i+. We aume the length of each path i non zero and finite: 0 < lp <, p. Let { i n i j0 lp j, if i > 0 8 If we ue 8 to define time, the i th path p i connecting waypoint w i and w i+ tae the form ni, ni+,..., ni+, where j for n i j n i+. The equence of path p mut connect the connect the equence of waypoint w, o we mut have ni w i for all i 0,, 2,.... Given a equence of path p, the node actual movement i given by 0,,.... We let { n } denote the random proce decribing the movement of node v. Let, F, µ, T be the correponding dynamical ytem. D. Location Random roce All ode Let n,v denote the location of node v at time n. Let X n n,, n,2,..., n, V denote the location of every node in at time n. Thi random variable tae value from the dicrete finite pace X V V i i, i. The random proce {X n } repreent the location of all V node in over time n Z. Let X, F X, µ x, T X be the ytem for {X n }. V. MAI REULT Theorem : uppoe the node V {v, v 2,..., v V } are moved randomly uing the general dicrete RWMM defined in ection IV. Let W v and v denote the waypoint and path random procee for node v. Let W v, ν wp, v denote the channel connecting the path and waypoint procee, and let X denote the location proce for every node in the networ. a If, for all v V, W v i AM and W v, ν wp, v i tationary, then X i AM table. b If, for all v V, W v i ergodic, W v, ν wp, v i tationary and ergodic, then X i ergodic.

5 ν w T [pa +a 2 0 ] [p b 0 ] ν w b 2 j wj+,w j+2 [ p +a i0 ] [p b 0 ] wi+,w i+2, if p i wi+,w i+2, i 0,..., b 2, and p j wj+,w j+2, j,..., + a 2, ν w T [pa 0 ] ν µ [p b 0 ] 7 roof: [Theorem ] leae refer to the Appendix. Corollary.: The claic RWMM i AM table and ergodic. roof: [Corollary.] ee Example 2, 3 and 4. VI. COCLUIO In thi paper, a mobility model i aid to be table if it atifie a pointwie ergodic theorem. We how that a mobility model i table if and only if it i probabilitically decribed by an Aymptotically Mean tationary AM random proce. An AM random proce i a generalization of a tationary random proce, which permit tarting tranient and local non-tationary behavior. ince mot mobility model exhibit uch non-tationary propertie, the theory of AM random procee i ideal for the tudy of mobility model. We ue AM theory how that a general cla of dicrete Random Waypoint Mobility Model RWMM i table. Thi cla include, a a pecial cae, the claic RWMM which i ued throughout the MAET literature. The main reult of the paper how that any RWMM i table if node waypoint and peed are elected in a tationary fahion. Thi reult allow networ deigner to tailor table RWMM for individual networ requirement. AEDIX roof: [Theorem ] The proof ue three lemma: Lemma 4: If W v i AM and ergodic and W v, ν wp, v i tationary and ergodic, then v i AM and ergodic. Lemma 5: If v i AM, then v i AM. Lemma 6: If v i ergodic, then v i ergodic. The movement of each node i independent of all other node, o the theorem follow directly from Lemma 4, 5 and 6. We hall drop the ubcript v in the following three proof. roof: [Lemma 4] The channel W, ν wp, connect the waypoint random proce W, F W, µ w, T W to the path random proce, F, µ p, T. The Lemma i a direct conequence of [4, Lem and 9.3.3]. roof: [Lemma 5] Aume, F, µ p, T i AM. Each p i a label for ome path 0,,..., lp, where lp denote path length. Let f : C mar the connection between path label and path, and let L max{lp : p }. For each p, we have fp 0,,..., lp C, where C L i i. Let, F, µ, T be the ytem repreenting node movement. Thi ytem i formed by concatenating path: { n } f 0, f,... 0,..., l0, l0,..., l0+l,... Thi relationhip may be implified by defining a equenceto-equence coder: F :. et F p fp 0, fp, fp 2,.... For all A F, the meaure µ on, F i connected to the meaure µ p on, F by µ A µ p F A, where F A {p : F p A}. Let B denote the et of bounded meaurable function on, F. By the AM pointwie ergodic theorem [9, Thm. ], there exit a ubet of equence of full meaure µ p, uch that the limit g p lim g T n p 9 exit for every p and every g B. Let Ŝ denote the range of F when the domain of F i retricted to : Ŝ { : p, F p }. It Ŝ then follow, µ µ p F Ŝ µ p. Let {A } Ŝ be the partition of where A { p : F p }. ote: the et A are nonempty for every Ŝ. For each Ŝ, chooe a equence p A, o that we may define the peudo-invere F : Ŝ, where F p. For each p and n 0,, 2,..., let u define γ p n n i0 lp i o that we may define the variable-length hift T : : T n Γn T, where { γ Γ n F n, if Ŝ, if / Ŝ Let B denote the et of bounded meaurable function on, F. ic any h B and any Ŝ, then lim up h T n lim up lim up lim up h T Γn 0 h ΓnF p T F p h F Tp n

6 lim up g T n p g p 2 where follow by etting g h F, and 2 follow from 9 and g B. imilarly, we have lim inf h T n g p o that the limit lim h T n exit for every Ŝ, and every h B. ince µ Ŝ, by Lemma 2 the ytem, F, µ, T i AM. By Lemma 7 below we alo have that the ytem, F, µ, T i AM hence, i AM. Lemma 7: Let, F, µ, T be a dynamical ytem and conider the variable length hift T : T n γ T 3 where γ L. If, F, µ, T i AM with tationary mean µ, then, F, µ, T i AM. roof: [Lemma 7] how: if µ i T invariant, then µ i AM with repect to T. We follow [9, E.x. 4]. Define a new meaure µ A E µ [γ] L i0 µ T i A 4 where { : γ }. The et { }L i a partition of, o T A L T A. Alo, µ T A µ A L µ T A L µ A 5 ubtitute T A into 4 and ue 5 to how that µ i T invariant. µ T A L Eµ µ T i A [γ] i0 L µ A + E µ [γ] L µ A µ T A µ A 6 If µ A 0, then lim µ T n A 0. Here µ i aid to aymptotically dominate µ T under T, µ µ. The meaure µ and µ are defined on the ame meaure pace T, F. Ue µ µ with 6 and [9, Thm. 2] to ee that µ i AM with repect to T. 2 how: If µ i AM with repect to T, then µ i AM with repect to T. The ytem, F, µ, T i AM, and the ytem, F, µ, T i tationary. If µ A 0 and T A A, it follow that µ A 0. Furthermore, if A T A, then A T n A for any n, o A i alo T invariant: A T A. If µ A 0 and A T A, it follow that µ A 0. Hence, if µ A 0 and A T A, then µ A 0. Theorem [5, Thm. 2.2] complete the proof. roof: [Lemma 6] Aume, F, µ p, T i ergodic. uppoe A F i T invariant, A T A. By definition, we have that A T n A for all n. Fix A, then for every p where F p, we have F p A T n T n F p A. Thi implie F p A T lp0 F p A, and therefore F p A F T p A 7 ince 7 hold for each A and every p aociated with each, we have that p F A T p F A; hence, F A T F A. ince, F, µ p, T i ergodic, we have that µ p F A 0 or ; therefore, µ A 0 of. By definition,, F, µ, T i ergodic. REFERECE [] T. Camp, J. Boleng, and V. Davie, A urvey of Mobility Model for Ad Hoc etwor Reearch, IEEE Tran. Wirele Commun., vol. 2, no. 4, pp , eptember [2]. Kurowi, T. Camp, and M. Colagroo, MAET imulation tudie: The Incredible, ACM IGMOBILE Mobile Comp. and Commun. Review MC 2 R, vol. 9, no. 4, pp , October [3] T. Andel and A. Yainac, On the Credibility of MAET imulation, Computer, vol. 39, no. 7, pp , July [4] J. Yoon, M. Liu, and B. oble, ound Mobility Model, in roc. IEEE Intl. ymp. Mobile Ad Hoc et. and Comp., MobiHoc, eptember 2003, pp [5] J. Boudec and M. Vojnović, The Random Trip Model: tability, tationary Regime, and erfect imulation, IEEE/ACM Tran. etworing, vol. 4, no. 6, pp , December [6] R. Timo, K. Blacmore, and J. apandriopoulo, trong tochatic tability for Dynamic ource Routing, Tech. Rep. A006280, ICTA, Augut [7] J. Yoon, M. Liu, and B. oble, Random Waypoint Conidered Harmful, in roc. IEEE Annual Joint Conf. IEEE Comp. and Commun. ocietie, IFOCOM, March 2003, vol. 2, pp [8]. Billingley, robability and Meaure, Wiley erie in probability and mathematical tatitic. John Wiley & on, 3 rd edition, 995. [9] R. Gray and J. Kieffer, Aymptotically Mean tationary Meaure, J. Ann. rob., vol. 8, no. 5, pp , October 980. [0] R. Timo, L. Hanlen, and K. Blacmore, A Lower Bound on etwor Layer Control Information, in Joint ACoR/EWCOM Worhop, Vienna, Autria, eptember []. hield, The Ergodic Theory of Dicrete ample ath, vol. 3 of Graduate tudie in Mathematic, American Mathematical ociety, 996. [2] R. Gray, robability Random rocee, and Ergodic ropertie, pringer Verlag, 200 Reviion 987, gray/. [3] A. Feintein, On the Coding Theorem and It Convere for Finite Memory Channel, Inform. and Contr., vol. 2, no., pp , April 959. [4] R. Gray, Entropy and Information Theory, pringer Verlag, 2000 Reviion 990, gray/. [5] Y. Kaihara, Ergodicity and Extremality of AM ource and Channel, International Journal of Mathematic and Mathematical cience, vol. 2003, no. 28, pp , 2003.