MotionCast: On the Capacity and Delay Tradeoffs

Transcription

1 MotonCast: On the Capacty and Delay Tradeoffs Chenhu Hu, Xnbng Wang Dept. of Electronc Engneerng Shangha Jao Tong Unversty, Chna Feng Wu Mcrosoft Research Asa Bejng, Chna ABSTRACT In ths paper, we defne multcast for ad hoc networ through nodes moblty as MotonCast, and study the capacty and delay tradeoffs for t. Assumng nodes move accordng to an ndependently and dentcally dstrbuted (..d.) pattern and each desres to send pacets to dstnctve destnatons, we compare the capacty and delay n two transmsson protocols: one uses 2-hop relay algorthm wthout redundancy, the other adopts the scheme of redundant pacets transmssons to mprove delay whle at the expense of the capacty. In addton, we obtan the maxmum capacty and the mnmum delay under certan constrants. We fnd that the per-node capacty and delay for 2-hop algorthm wthout redundancy are Θ(/) and Θ(n log ), respectvely; and for 2-hop algorthm wth redundancy they are Ω(/( n log )) and Θ( n log ), respectvely. The capacty of the 2-hop relay algorthm wthout redundancy s better than the multcast capacty of statc networs developed n [3] as long as s strctly less than n n an order sense; whle when = Θ(n), moblty does not ncrease capacty anymore. The rato between delay and capacty satsfes delay/rate O(n log ) for these two protocols, whch s smaller than that of drectly extendng the fundamental tradeoff for uncast establshed n [] to multcast,.e., O(n 2 ). Categores and Subject Descrptors C.2. [Computer-Communcaton Networs]: Networ Archtecture and Desgn Wreless Communcatons General Terms Theory and Algorthms Keywords Multcast, Capacty, Delay, Scalng Law Permsson to mae dgtal or hard copes of all or part of ths wor for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. MobHoc 09, May 8 2, 2009, New Orleans, Lousana, USA. Copyrght 2009 ACM /09/05...$ INTRODUCTION Multcast n MANETs s predomnant n many practcal stuatons. For example, group communcatons n mltary networs and dsaster alarmng n sensor networs. Another example s the current moble multmeda servces. It s lely that there s a large number of moble users accessng the servces, whle they favor dfferent programmes provded by a varety of servce statons. These statons are then requred to multcast ther data stream to certan groups of users. Snce certan lns may be shared by several destnatons, one potental beneft of multcast s that t can reduce the total bandwdth requred to communcate wth all the destnatons. Thus, compared wth multple uncast capacty gan can be obtaned by usng multcast. L et al. n [3] study the capacty of a statc random wreless ad hoc networ for multcast where each node sends pacets to destnatons. Under protocol nterference model, they show that the per-node multcast capacty s Θ( n log n ) when = O( n ); the per-node multcast capacty s Θ( ) when log n n = Ω( n ). These results generalze the prevous capacty bounds on uncast by Gupta and Kumar [4] and broad- log n cast [5]. Other wors fallng nto ths class can be seen n [6] and [7]. Jacquet et al. [6] consder multcast capacty by accountng the rato of the total number of hops for multcast and the average number of hops for uncast. Shaotta et al. [7] propose a comb-based archtecture for multcast routng whch acheves the upper bound for capacty n an order sense. Whle the above studes are all based on statc networs, the effect of moblty on the capacty of wreless ad hoc networs has been frst explctly developed n [8], where Grossglauser and Tse demonstrate that per-node uncast capacty does not vansh as the sze of the networ grows by mplementng a 2-hop relay algorthm. Note that the prce of ths mprovng capacty s the ncreased delay. It has been shown n [] [9] that the 2-hop relay algorthm n [8] yelds a tremendous average delay of Ω(n). The relatonshps between capacty and delay are further nvestgated n the lterature of [] [2] [0] []. In the wor by Neely and Modano [], the authors present a strategy utlzng redundant pacets transmssons along multple paths n a cell parttoned MANET to reduce delay wth a sacrfce on the capacty. They establsh the followng necessary tradeoff: delay/capacty O(n), and develop schemes that can acheve Θ(), Θ(/ n) and Θ(/(n log n)) per-node capacty, when the delay constrant s on the order 289

2 of Θ(n), Θ( n) and Θ(log n), respectvely. In [], Toumps and Goldsmth construct a scheme that can acheve a pernode capacty of Θ(n (d )/2 / log 5/2 n) under fadng channels when the delay s bounded by O(n d ), whch turns out to be better than that n []. Afterwards, Ln and Shroff [2] search the optmal capacty-delay tradeoff and dentfy the lmtng factors of the exstng schedulng schemes n MANETs. Recently, Yng et al. [0] develop jont codng-schedulng algorthms to acheve the optmal delay-throughput tradeoffs. A ey feature of multcast n MANETs s that pacets can be delvered va nodes moblty, thus we refer t as Moton- Cast n ths paper. Intutvely, capacty and delay tradeoffs stll exst for MotonCast, but beng more complcated than the stuatons for an uncast scenaro. Pacets can also be delvered through the moblty of the relay nodes, thus a hgher per-node multcast capacty compared wth that of a statc wreless ad hoc networ s magnable. However, the schedulng desgn becomes harder because of the permanent change of the networ topology and now that more destnatons need obtan pacets from the source, t wll tae a longer tme to complete a multcast process, whch suggests a larger delay. Hence, some challengng questons rased naturally n ths context are as follows: What s the maxmum per-node MotonCast capacty? How long wll be the nduced delay to acheve ths capacty and what s the mnmum delay? How does the capacty and delay tradeoffs emerge for MotonCast? Answerng these questons s helpful for us to evaluate the performance and better understand the fundamental tradeoffs for multcast n large-scale ad hoc networs wth moblty. In our wor, we conduct a study on the scalng behavors n a cell parttoned MANET under multcast traffc pattern. To start wth, we propose a 2-hop relay algorthm wthout redundancy. Ths algorthm s a generalzed verson of the algorthm presented n [], whch corresponds to a decoupled queung model. The varaton s that more destnatons are assocated wth a source, and delay for a pacet wll be determned by the tme when t s delvered to all the destnatons. As for a specfc pacet, we clearly dvde nodes other than the source nto relays and destnatons (referred to as an non-cooperatve mode) frst. In ths case, the pacet may be carred to the destnatons ether through the relays or va the source, but wll not be passed from one destnaton to another. Once a pacet s sent to a relay, the relay wll be n charge of delverng t to all ts destnatons. Otherwse, f the source encounters a destnaton before a relay, t wll do the job tself. The expresson of MotonCast capacty and delay are calculated under ths model, and t turns out that capacty wll degenerate when s large. Then, we loose the constrants wthn our ntal model by permttng nformaton spread among destnatons (called a cooperatve mode). At ths moment, we do not dscrmnate destnatons and the remnant nodes except the source rgorously. We defne the frst node a source meets as the desgnated relay, whch n fact may possble be an ntended destnaton. Lewse, the desgnated relay should carry the pacet from the source untl t delvers ths pacet to all the destnatons that have not receved the message. Notce that only one relay relates to a specal pacet n the 2-hop relay algorthm, thus after a relay s desgnated current destnatons wll merely act as recevers for the pacet and do not help transmt the pacet to other destnatons. Next, we employ redundant pacets transmssons to reduce the delay. In a 2-hop relay strategy wth redundancy, a source sends a pacet to many dstnct relays before all the destnatons receve the pacet, whch ncreases the chance that a destnaton meets some of the relays at the expense of reduced capacty. If each tmeslot only one transmsson from a sender to a recever s permtted n a cell, we show that the expect delay n the networ s no less than Ω( n log ). Besdes, delay of O( n log ) s attanable n a proposed scheme wth per-node capacty of Ω(/( n log )). The man results of ths paper are summarzed as follows. For 2-hop relay algorthm wthout redundancy, the capacty for MotonCast s Θ(/) wth the average delay of Θ( n log ). Notce that the per-node capacty s better than the results of statc multcast scenaro n [3] as long as s strctly less than n n an order sense,.e., = O(n ɛ ) (0 ɛ < ). For 2-hop relay algorthm wth redundancy, the capacty s Ω(/( n log )) wth the delay scalng as Θ( n log ). Thus, capacty and delay tradeoffs emerge between these two algorthms,.e., we can utlze redundant pacets transmssons to reduce delay but the capacty wll also decrease. The tradeoff obtaned by us s better than that of drectly extendng the tradeoff for uncast to multcast. The rest of the paper s organzed as follows. In Secton II, we descrbe the networ model. In Secton III, we ntroduce the 2-hop relay algorthm wthout redundancy. In Secton IV, the 2-hop relay algorthm wth redundancy s presented. In Secton V, we dscuss the results. Fnally, we conclude n Secton VI. 2. NETWORK MODEL (a) Networ model. (b) Traffc pattern Fgure : A cell parttoned MANET model wth c cells and n moble nodes under multcast traffc pattern. Cell Parttoned Networ Model: The system model s based on the cell parttoned networ model exploted n [] and [3]. Suppose the networ s an unt square and there are n moble nodes n t. Then, we dvde t nto c nonoverlappng cells wth equal sze as depcted n Fgure. We assume nodes can communcate wth each other only when they are wthn a same cell (to locate the nodes, please refer to [2] and the references theren), and to avod nterference dfferent frequences are employed among the neghborng 290

3 cells. Addtonally, to bound the nterference nsde each cell, we assume that the number of the cells s on the same order as that of the nodes throughout ths paper. Thus, node per cell densty d = n/c scales as Θ(). Moblty Model: Dvdng tme nto constant duraton slots, we adopt the followng deal..d. moblty model. The ntal poston of each node s equally lely to be any of the c cells ndependent of others. And at the begnnng of each tme slot, nodes randomly choose and move to a new cell..d. over all cells n the networ. We do not account the tme a node moves from the exstng cell to a new one, hence ths model captures the characterstc of the nfnte moblty. Wth the help of moblty, pacets can be carred by the nodes untl they reach the destnatons. Traffc Pattern: We frst defne the source-destnaton relatonshps before the transmssons start. Numberng all the nodes from to n, we assume each node s a source node assocated wth = (n) randomly and ndependently chosen destnaton nodes D, D 2,..., D over all the other nodes n the networ. The relatonshps do not change as nodes move around. Then, the sources wll communcate data to ther destnatons respectvely through a common wreless channel. Defnton of Capacty: Frst, we defne stablty of the networ. Pacets are assumed to arrve at node wth probablty λ durng each slot,.e. as a Bernoull process of arrval rate λ pacets/slot. For the fxed λ rates, the networ s stable f there exsts a schedulng algorthm so that the queue n each node does not ncrease to nfnty as tme goes to nfnty. Thus, the per-node capacty of the networ s the maxmum rate λ that the networ can stably support. Note that sometmes the per-node capacty s called capacty for bref. Defnton of Delay: The delay for a pacet s defned as the tme t taes the pacet to reach all ts destnatons after t arrves at the source. The total networ delay s the expectaton of the average delay over all pacets and all random networ confguratons n the long term. Defnton of Redundancy: At each tmeslot, f more than one nodes are performng as relays for a pacet, we say there s redundancy n the networ. Furthermore, we say the correspondng schedulng scheme s wth redundancy or redundant. Otherwse, t s wthout redundancy. Defnton of Cooperatve: We adopt the term cooperatve here to refer a destnaton can relay a pacet from the source to other destnatons. Otherwse, the destnatons merely accept pacets destned for them, but do not forward to others, whch s called non-cooperatve mode. Notatons: In our wor, we adopt the followng wdely used order notatons n a sense of probablty. We say that an event occurs wth hgh probablty (w.h.p.), f ts probablty tends to as n goes to nfnty. Gven two functons f(n) and g(n), we say that f(n) = O(g(n)) w.h.p., f there exst a constant c such that lm P (f(n) cg(n)) =. () n We say that f(n) = Ω(g(n)) w.h.p., f g(n) = O(f(n)) w.h.p.. If both f(n) = Ω(g(n)) and f(n) = O(g(n)) w.h.p., then we say that f(n) = Θ(g(n)) w.h.p.. It s clear that only four frequences are enough for the whole networ. 3. CAPACITY AND DELAY IN THE 2-HOP RELAY ALGORITHM WITHOUT REDUN- DANCY In ths secton, we propose 2-hop relay algorthms wthout redundancy and compute the achevable capacty and delay both under non-cooperatve mode and cooperatve mode. Then, we explore the maxmum capacty and the mnmum delay n these stuatons. 3. Under non-cooperatve mode In ths subsecton, we descrbe a 2-hop relay algorthm wthout redundancy. Usually, a source sends a pacet to one of the relays, then the relay wll dstrbute the pacet to all ts destnatons. Whle as an ntal step, we consder the non-cooperatve mode, whch means a destnaton can not be a relay. Fgure 2: A decoupled queung model of the networ as seen by the pacets transmtted from a sngle source to multple destnatons. 2-hop Relay Algorthm Wthout Redundancy I: Durng a tmeslot, for a cell wth at least two nodes:. If there exsts a source-destnaton par wthn the cell, randomly select such a par unformly over all possble pars n the cell. If the source has a new pacet n the buffer ntended for the destnaton, transmt. If all ts destnatons have receved ths pacet 2, then t wll delete the pacet from the buffer. Otherwse, stay dle. 2. If there s no such par, randomly assgn a node as sender and ndependently choose another node n the cell as recever. Wth equal probablty, choose from the followng two optons: Source-to-Relay Transmsson: If the sender has a new pacet one that has never been transmtted before, send the pacet to the recever and delete t from the buffer. Otherwse, stay dle. Relay-to-Destnaton Transmsson: If the sender has a new pacet from other node destned for the recever, transmt. If all the destnatons who want to get ths pacet have receved t, t wll be dropped from the buffer n the sender. Otherwse, stay dle. 2 We assume that nodes can aware ths from the control nformaton passed over a reserved bandwdth channel. 29

4 The algorthm has an advanced decouplng feature between all n multcast sessons, as llustrated n Fgure 2, where nodes are dvded nto destnatons and relays for the pacets from a sngle source, and the pacets transmssons for other sources are modeled just as random ON/OFF servce opportuntes. Let p denote the probablty of fndng at least two nodes n a partcular cell, and q denote the probablty of fndng a source-destnaton par wthn a cell. From Appendx I, we obtan that p = c n n c c n q = + c (2) c + c +n + (3) When n tends to nfnty, t follows p (d + )e d and q e + d ( + ) + n. Thus, f = O(n ɛ ) (0 ɛ < ), c q 0; else f = Θ(n), q (d + )e d. Then, we have the followng theorem. Theorem. Consder a cell-parttoned networ (wth n nodes and c cells) under the 2-hop relay algorthm wthout redundancy I, and assume that nodes change cells..d. and unformly over each cell every tmeslot. If the exogenous nput stream to node whch maes the networ stable s a Bernoull stream of rate λ = O(/) and = O(n ɛ ) (0 ɛ < ), then the average delay W for the traffc of node satsfes where = p+q 2d. E{W } = O(n log ) (4) Proof. A decoupled vew of the networ as seen by a sngle source s shown n Fgure 2. In every tmeslot, a new pacet arrves wth probablty λ at source, and wth probablty the pacet s handed over a relay or transmtted to a destnaton. We frst show that the expresson = p+q stll holds. 2d Denote r for the rate at whch the source s scheduled to transmt drectly to one of the destnatons, and r 2 for the rate at whch t s scheduled to transmt to one of ts relays. Then, we have = r + r 2. Snce the relay algorthm schedules transmssons nto and out of the relay nodes wth equal probablty, hence r 2 s also equal to the rate at whch the relay nodes are scheduled to transmt to the destnatons. Every tmeslot, the total rate of transmsson opportuntes over the networ s thus n(r +2r 2). Meanwhle, a transmsson opportunty occurs n any gven cell wth probablty p, hence, cp = n(r + 2r 2) (5) Recall that q s the probablty that a gven cell contans a source-destnaton par. Snce the algorthm schedules the sngle-hop source-to-destnaton transmssons whenever possble, the rate r satsfes cq = nr (6) It follows from (5) and (6) that r = q p q, r2 =. The total d 2d rate of transmssons out of the source node s thus gven by = r + r 2 = p+q. 2d Next, we compute the average delay for the traffc of node. There are two possble routngs from a source to ts destnatons: one s the 2-hop path along source-relaydestnatons, the other s the sngle-hop path from source to destnatons drectly. As for the frst routng, pacet delay s composed of the watng tme at source and relay. In ths case, the source can be vewed as a Bernoull/Bernoull queue wth nput rate λ and servce rate, havng an expected number of occupancy pacets gven by L s = ρ( λ ) ρ, where ρ = λ. From Lttle s theorem, the average watng tme n the source s E{W s} = L s λ = λ λ. Besdes, ths queue s reversble, so the output process s also a Bernoull stream of rate λ. A gven pacet from ths output process s transmtted to r 2 (n ) the frst relay node wth probablty (because wth probablty r 2 the pacet s delvered to a relay, and each of the n relay nodes are equally lely). Hence, every tmeslot, ths relay ndependently receves a pacet wth λ probablty λ r = r 2. On the other hand, the relay (n ) node s scheduled for a potental pacet transmsson to a destnaton node wth probablty r = r 2 (because when n 2 t acts as a relay, t can transmt pacets to n 2 destnatons except the source of the gven pacet and tself wth equal probablty). Notce that pacet arrvals and transmsson opportuntes are mutually exclusve events n the relay node. However, dfferent from uncast, each relay node s n charge of sendng a same pacet to dstnct destnatons n the multcast scenaro abded by the algorthm. From a more delcate pont of vew, we model a relay node as n 2 parallel sub-queues (each of them buffers the pacets ntended for a certan destnaton), shown n Fgure 3. Then, when a new relay pacet arrves at the relay, t wll copy ths pacet nto vrtual-duplcates and add them nto respectve sub-queues assocated wth the destnatons. Hence for uncast, the ncomng rate of each sub-queue s λ r, whle for multcast t s tmes of that quantty. It follows that the dscrete tme Marov chan for queue occupancy n each sub-queue can be wrtten as a smple brth-death chan whch s dentcal to a contnuous tme M/M/ queue wth nput rate λ r and servce rate r. Each destnaton ( ) obtans the pacet from the relay though such a queue, thus the watng tme for t s an exponental dstrbuted varable wth expectaton of E{Wrd} = /( r λ r). The resultng watng tme W rd for multcast s determned by the maxmum value among all the watng tmes Wrd, Wrd, 2..., Wrd of the above queues. Observng that these watng tmes are..d exponental varables, by Lemma 2 (see the proof n Appendx II), we obtan that E{W rd } = log /( r λ r). Thus, f the pacet s delvered through the path source-relay-destnatons, the average delay s E{W s} + E{W rd }. Whle f the pacet s drectly sent to the destnatons by the source, t wll wat at the source for a tme W s frst, then the source dstrbutes ths pacet to the remnant destnatons. At ths tme, the source can be treated as a group of parallel M/M/ sub-queues correspondng to ts destnatons smlarly. The source wll copy ths pacet nto vrtual-duplcates and add them nto respectve sub-queues assocated wth the remnant destnatons. Snce the probablty that the source need send pacets drectly to destnatons s r, the ncomng data rate s thus ( )λ r for each sub-queue. Mean- 292

5 Fgure 3: A more delcate vew of a relay. Each of a relay node can be modeled as n 2 parallel subqueues bufferng pacets ntended for dfferent destnatons. In specal, sub-queues assocated wth destnatons of the current source are depcted n red n the fgure. whle, the servce rate at each sub-queue equals to the transmsson rate r between a source-destnaton par. Hence, the expectaton of the watng tme at each sub-queue s /( r ( )λ r ). And by Lemma 2, we have the expect watng tme for the pacet to reach all remnant destnatons s E{W sd } = log( )/( r ( )λ r ). Fnally, by weghtng the delay occurs n both two routngs, we acheve the total networ delay s E{W } = r E{Ws} + E{W sd} + rd} r2 E{Ws} + E{W = r2 λ log + λ r 2 λ r 2 n 2 (n ) + r λ log( ) + (7) λ r ( )λ r Loong upon the asymptotc behavors of the networ delay when, n, we have If = O(n ɛ ) (0 ɛ < ), then t follows r 0 and r 2, (d+)e d, whch means almost all the traffc 2d s carred along the path of source-relay-destnatons. To ensure the stablty of the networ, the ncomng rate should be less than the servce rate at any stage of the networ. Thus, λ > 0 r 2 n 2 λ r 2 (n ) > 0.e., λ < (n ) / (n ). Besdes, the (n 2) total networ delay s governed by the frst term of (7), whch s on the order of O(n log ) for a fxed traffc loadng value ρ r = λ (n 2) at each relay. (n ) If = Θ(n), then t follows r, (d+)e d and d r 2 0, whch means nearly all the pacets are delvered drectly from source to destnatons. Lewse, the networ capacty s lmted by r ( )λ r > 0,.e., λ <. Besdes, the total networ delay s ( ) governed by the second term of (7), whch scales as O( log ) = O(n log ) for a fxed traffc loadng value ρ s = ( )λ at the source. From the frst case of the above dscusson, we conclude the theorem. 3.2 Under cooperatve mode In the above subsecton, we propose a 2-hop relay algorthm wthout redundancy obtanng per-node capacty Ω(/) wth delay O(n log ), when = O(n ɛ )(0 ɛ < ). However, f = Θ(n) we fnd that wth the same amount of delay n an order sense, the per-node capacty decreases to Ω(/ 2 ). To avod ths degeneraton, n ths subsecton we brng forward a more general algorthm whch does not dscrmnate destnatons and the nodes other than the source,.e., under cooperatve mode. Ths algorthm acheves pernode capacty Ω(/) wth delay O(n log ) for any n, and t s descrbed as follows. 2-hop Relay Algorthm Wthout Redundancy II: For each cell wth at least two nodes n a tmeslot, a random sender and a random recever are pced wth unform probablty over all nodes n the cell. Wth equal probablty, the sender s scheduled to operate n the two optons below:. Source-to-Relay Transmsson: If the sender has a new pacet one that has never been transmtted before, send the pacet to the recever and delete t from the buffer. Otherwse, stay dle. 2. Relay-to-Destnaton Transmsson: If the sender has pacets receved from other nodes whch are destned for the recever and have not been transmtted to the recever yet, then choose the latest one, transmt. If all the destnatons who want to get ths pacet have receved t, t wll be dropped from the buffer n the sender. Otherwse, stay dle. The algorthm smply desgnates the frst node a source meets as the relay, no matter f t s a destnaton. Thus accordng to the schedulng scheme, all the pacets wll be delvered along a 2-hop path source-relay-destnatons. The dfference s that f the relay s a destnaton node, t need only relay the pacet to the rest destnatons; otherwse, t need relay the pacet to all destnatons. Snce we focus the performance n an order sense, we omt ths dfference between these two cases. Thus, followng the same analytcal steps as Theorem when s strctly less than n n an order sense, we summarze the next theorem. Theorem 2. Consder the same assumptons for the networ as Theorem, under the 2-hop relay algorthm wthout redundancy II. The resultng per-node capacty and the average delay are Ω(/) and O(n log ), respectvely, for all n. Snce the second algorthm s better than the frst one, we adopt ths algorthm and refer t as 2-hop relay algorthm wthout redundancy for bref n the rest of the paper. 3.3 Maxmum capacty and mnmum delay Although we have constructed the achevable capacty and delay f no redundancy s used, open questons stll leave 293

6 for the maxmum capacty and the mnmum delay of ths networ. We address these problems here by presentng the followng theorems. Theorem 3. The multcast capacty of a cell parttoned networ s O(/) f only a par of sender and recever s actve n each cell per tmeslot. Proof. We use hop argument to prove ths result. Consder the mnmum number of hops h mn that a source can send a pacet to all destnatons. Although nodes are moble, we can treat the process of the transmssons as a dynamc graph. Specfcally, we connect an edge along each two nodes f the pacet s transmtted between them. To form a connected graph among the source and ts destnatons, the mnmum number of edges s equal to ( + ) =, whch means that the graph s a tree. Thus, we get h mn =. Denote the nput rate at each source by λ, then the number of bts arrvng at these n nodes n an nterval [0, T ] s λt n. Thus, the total number of transmssons of all bts to ther destnatons s at least λt nh mn. On the other hand, the total number of possble transmssons at any tmeslot s upper bounded by that of the cells contanng at least two users, whch s no more than c. Hence, λt n T c (8).e., λ. Notce that d = Θ(), thus we have λ = d O(/). Theorem 4. Algorthm permttng at most one transmsson n a cell at each tmeslot, whch do not use redundancy cannot acheve an average delay of O(n log ). Proof. The mnmum delay of any pacet s calculated by consderng the stuaton where the networ s empty and node sends a sngle pacet to destnatons. Snce relayng the pacet can not help reduce delay, t can be treated as havng no relay at all. Denote p and W mn as the chance that node meets (.e., two nodes move nto a same cell) one of the destnatons n a tmeslot and the mnmum amount of tme t taes the source to meet all the destnatons, respectvely. We have that p = /c. Snce W mn = means that at the ( )th tmeslot the source has met destnatons and at the th tmeslot t meets the last one, thus the probablty W mn = can be wrtten as P {W mn = } = p ( p ) ( 2p ) (9) 2 + ( 3p ) 2 Theren the factor p denotes that the last destnaton D meets by the source can be any one of the destnatons. The frst term n the latter factor nfers that D has not been met n the former tmeslots. Because the frst term also ncludes the probablty that the source has not met D and any one of the other nodes from D to D, ths value should be subtracted from the frst term, so the second term attached and smlarly we have the followng E{W mn} + X terms. Hence, the expectaton of E{W mn} s = p ( p ) ( 2p ) = X ( 3p ) 2 = p ( p ) + 2 =+ X ( 2p ) = + X ( 3p ) = = p p 2 = p (2p ) 2 2 (3p ) = log p (0) + X wheren Lemma and the followng dentcal relaton for any x < are exploted = x = ( + X = x ) = ( x) 2. Fnally, notce that /p = Θ(n), we obtan that E{W mn} = Θ(n log ). Snce at any tmeslot, f there are more than one destnatons n a same cell as the source, only one destnaton could be selected as the recever, the actual delay E{W mn} for the pacet to be delvered to all the destnatons wll be larger or equal than E{W mn}, whch ponts out the theorem. Combnng these results wth the capacty and delay acheved by the 2-hop relay algorthm wthout redundancy, we fnd the exact order of the capacty and delay are Θ(/) and Θ(n log ), respectvely. 4. CAPACITY AND DELAY IN THE 2-HOP RELAY ALGORITHM WITH REDUNDANCY In ths secton, we adopt redundancy to mprove delay. The dea orgnates from a basc noton that f we send a partcular pacet to many nodes of the networ, the chances that some node holdng the pacet reaches a destnaton wll ncrease. Ths approach s also mplemented n [] and [4]. We frst consder the mnmum delay of 2-hop relay algorthms wth redundancy. Then, we desgn a protocol usng redundancy to acheve the mnmum delay. 4. Lower bound of delay In ths subsecton, we obtan lower bound of delay f only one transmsson from a sender to a recever s permtted n a cell n the below Theorem. Theorem 5. There s no 2-hop algorthm wth redundancy can provde an average delay lower than O( n log ), f only 294

7 one transmsson from a sender to a recever s permtted n a cell. Proof. To proof ths result, we consder an deal stuaton where the networ s empty and only node sends a sngle pacet to destnatons. Clearly the optmal scheme for the source s to send duplcate versons of the pacet to new relays whenever possble, and f there s a destnaton wthn the same cell as the source, t wll choose a destnaton as relay. And for a duplcate-carryng relay, t sends the pacet to be relayed to the destnatons as soon as t enters the same cell as a destnaton. Denote T N as the tme requred to reach the destnatons under ths optmal strategy for sendng a sngle pacet. In order to avod the nterdependency of the probablty that dfferent destnatons obtan a pacet from the source or the relay nodes, we addtonally assume that all the destnatons wthn a same cell as the source or a relay node can obtan the pacet durng the transmsson, whch s referred to as a mult-destnaton recepton style. Note that our assumpton dffers from the mult-user recepton ( []) n that usually each cell s permtted to have a sngle recepton except there are more than one ntended destnatons wthn a the cell, whle [] allows a transmtted pacet to be receved by all other users n the same cell as the transmtter. Denote T N as the tme to reach the destnatons when we add the mult-destnaton recepton assumpton. It s easy to see that E{T N } E{T N }. Then, let K t represent the total number of nodes that act as ntermeda relays (ncludng the source) at the begnnng of slot t. We have that every tmeslot the ncrease of the relays s merely due to the source sends the pacet to a new relay. Thus, we have for all t : K t t () Observe that durng slots {, 2,..., t} there are at most K t nodes holdng the pacet and wllng to help forward t to the destnatons. Hence, durng ths perod, the probablty that a destnaton meets at least a relay s at most ( c )tk t. We thus have P {T N > t} ( c )tk t ( d n )t2 = ( e d n t2 ) (n ) (2) Choosng t =pn log /d and lettng, t yelds that Thus: P {T N > t} ( e log ) = ( ) = e (3) E{T N } E{T N } E{T N T N > t}p {T N > t} ( e )Èn log /d (4) as, n. From (4), we prove the theorem. 4.2 Schedulng scheme In the above subsecton, we consder the mnmum delay of the networ f we mplement redundant pacets transmssons. In ths subsecton, for acqurng the upper bound of the delay, we propose a 2-hop relay algorthm wth redundancy to acheve the mnmum delay. Assume each pacet s labeled wth a Sender Number SN, and a request number RN s delvered by the destnaton to the transmtter just before transmsson. In the followng algorthm, we let each pacet be retransmtted n log tmes to dstnct relay nodes. 2-hop Relay Algorthm Wth Redundancy: In every cell wth at least two nodes, randomly select a sender and a recever wth unform probablty over all nodes n the cell. Wth equal probablty, the sender s scheduled to operated n ether source-to-relay transmsson, or relay-to-destnaton transmsson, as descrbed below:. Source-to-Relay Transmsson: The sender transmts pacet SN, and does so upon every transmsson opportunty untl n log duplcates have been delvered to dstnct relay nodes (possble be some of the destnatons), or untl the destnatons have entrely obtaned SN. After such a tme, the sender number s ncremented to SN +. If the sender does not have a new pacet to send, stay dle. 2. Relay-to-Destnaton Transmsson: When a node s scheduled to transmt a relay pacet to ts destnatons, the followng handshae s taen place: The recever delvers ts current RN number for the pacet t desres. The transmtter sends pacet RN to the recever. If the transmtter does not have the requested pacet RN, t stays dle for that slot. If all destnatons have already receved RN, the transmtter wll delete the pacet whch has SN number equal to RN n ts buffer. Next, we present the performance of ths algorthm. Theorem 6. The 2-hop relay algorthm wth redundancy acheves the O( n log ) delay bound, wth a per-node capacty of Ω(/( n log )). Proof. For the purpose of provng ths theorem, we consder an extreme case of the pacets transmssons. Note that when a new pacet arrves at the head of ts source queue, the tme requred for the pacet to reach ts destnatons s at most T N = T + T 2, where T represents the tme requred for the source to dstrbute n log duplcates of the pacet, and T 2 represents the tme requred to reach all the destnatons gven that n log relay nodes hold the pacet. The reason behnd ths clam s the submemoryless property of the random varable T N ([]), whch means the resdual tme of T N gven that a certan number of slots have already passed before t expres s stochastcally less than the orgnal tme T N. Now we bound the expectatons of T and T 2 by tang nto account the collsons among the multple sessons. The E{T } bound: For the duraton of T, there are at least n n log nodes who do not have the pacet, and hence every tmeslot the probablty that at least one of these nodes vsts the cell of the source s at least ( c )n n log. Gven ths event, the probablty that the 295

8 source s chosen by the 2-hop relay algorthm wth redundancy to transmt s expressed by the product α α 2, representng probabltes for the followng condtonally ndependent events: α s the probablty that the source s selected from all other nodes n the cell to be the transmtter, and α 2 represents the probablty that ths source s chosen to operate n source-to-relay transmsson. From Lemma 6 n [], we have α /(2 + d). The probablty α 2 that the source operates n source-torelay transmsson s /2. Thus, every tmeslot durng the nterval T, the source delvers a duplcate pacet to a new node wth probablty of at least φ, where φ ( ( n c )n log ) 2(2 + d) e d 4 + 2d The average tme untl a duplcate s transmtted to a new node s thus a geometrc varable wth mean less than or equal to /φ. It s possble that two or more duplcates are delvered n a sngle tmeslot, f we enable mult-user recepton. However, n the worst case, n log of these tmes are requred, so that the average tme E{T } s upper bounded by n log /φ. The E{T 2} bound: To prove the bound on E{T 2}, note that every tmeslot n whch there are at least n log nodes possess the duplcates of the pacet, the probablty that one of these nodes transmts the pacet to one of the destnatons s gven by the chan of probabltes θ 0θ θ 2θ 3. The θ values represent probabltes for the followng condtonally ndependent events: θ 0 represents the probablty that there s at least one other node n the same cell as the destnaton (θ 0 = ( c )n e d ), θ represents the probablty that the destnaton s selected as the recever (smlar to α, we have θ /(2 + d)), θ 2 represents the probablty that the sender s operates n relay-to-destnaton transmsson (θ 2 = /2), and θ 3 represents the probablty that the sender s one of the n log nodes who possess a duplcate of the pacet ntended for the destnaton (where θ 3 = n log /(n ) plog /n). Thus, every tmeslot, the probablty that each destnaton receves a desred pacet s at least e d 4+2dplog /n. Smlar to Theorem 4, snce T 2 completes when all destnatons receve the pacet, the value of E{T 2} s thus less than or equal to the log tmes of the nverse of that quantty. Hence, we have E{T 2} 4+2d e n log. d Fnally accordng to Lemma 2 n [], we bound the total networ delay E{W } = O( n log ), and obtan the achevable per-node capacty under ths algorthm s Ω(/( n log )) (Note that a new relay pacet arrvng at a relay wll occupy sub-queues n the model of Fg. 3 untl t reaches all destnatons, thus the capacty should be dvded by a factor n the expresson.). 5. DISCUSSION In Secton 3 and Secton 4, we present algorthms both wthout and wth redundancy to fulfll the tas of Moton- Cast. In ths secton, we draw a comparson of the capacty and delay wth the former results and dscuss the capacty and delay tradeoffs obtaned n ths paper. The capacty and delay tradeoffs between the 2-hop relay algorthm wthout and wth redundancy can be summarzed n the followng table. scheme capacty delay 2-hop relay w.o. redund Θ( ) Θ(n log ) 2-hop relay w. redund Ω( n ) Θ( n log ) log Compared wth the multcast capacty of statc networs developed n [3], we fnd that capacty of the 2-hop relay algorthm wthout redundancy s better when = O(n ɛ ) (0 ɛ < ); otherwse, capacty remans the same as that of statc networs,.e., moblty cannot ncrease capacty. However, capacty of the 2-hop relay algorthm wth redundancy s no better than that of statc networs f log = Ω(log n) due to the redundant pacets transmssons. Moreover, compared wth the results of uncast n [], we fnd that capacty dmnshes by a factor of / and /( log ) for the 2-hop relay algorthm wthout and wth redundancy, respectvely; delay ncreases by a factor of log and log for the 2-hop relay algorthm wthout and wth redundancy, respectvely. Ths s because we need dstrbute a pacet to destnatons durng MotonCast. Partcularly, f = Θ() we fnd the results of uncast s a specal case of our paper. Furthermore, we see that delay of the 2-hop algorthm wth redundancy s better than that of the 2-hop algorthm wthout redundancy, but ts capacty s also smaller than that of the no redundancy algorthm. Ths suggests that redundant pacets transmssons can reduce delay at an expense of the capacty. The rato between delay and capacty satsfes delay/rate O(n log ) for these two protocols. However, f we fulfll the job of MotonCast by multple uncast from the source to each of the destnatons, we fnd that capacty wll dmnsh by a factor of / and delay wll ncrease by a factor of for both algorthms wthout and wth redundancy, whch nfers the fundamental tradeoff for uncast establshed n [] becomes delay/rate O(n 2 ) n MotonCast. Thus, t turns out our tradeoff s better than that of drectly extendng the tradeoff for uncast to multcast. 6. CONCLUSION AND FUTURE WORK In ths paper, we study capacty and delay tradeoffs for MotonCast. We utlze redundant pacets transmssons to realze the tradeoff, and present the performance of the 2- hop relay algorthm wthout and wth redundancy respectvely. We fnd that the capacty of the 2-hop relay algorthm wthout redundancy s better than that of statc networs when = O(n ɛ ) (0 ɛ < ). And our tradeoff s better than that of drectly extendng the tradeoff for uncast to multcast. We have not taen nto account the mult-hop transmsson schemes and the effect of dfferent moblty patterns yet, whch could be a future wor. 7. ACKNOWLEDGMENT The authors wsh to than Janzhou Feng, Wentao Huang and other group members for the helpful dscussons. Ths wor s supported by NSF Chna (No , ); Chna Mnstry of Educaton (No ); Shangha Jaotong Unversty Young Faculty Fundng (No. 06ZBX800050); Qualcomm Research Grant; Chna Internatonal Scence and Technology Cooperaton Programm (No. 2008DFA630); PUJIANG Talents (08PJ4067); Shangha Innovaton Key Project ( ). 296

9 8. REFERENCES [] M. J. Neely, and E. Modano, Capacty and delay tradeoffs for ad hoc moble networs, IEEE Transactons on Informaton Theory, vol. 5, no. 6, pp , Jun [2] X. Ln and N. B. Shroff, The fundamental capacty-delay tradeoff n large moble wreless networs, Techncal Report, Avalable at lnx/papers.html [3] X. L, S. Tang and O. Freder, Multcast capacty for large scale wreless ad hoc networs, n Proceedngs of ACM MobCom, Sept [4] P. Gupta and P. R. Kumar, The capacty of wreless networs, IEEE Transactons on Informaton Theory, vol. 46, no. 2, pp , Mar [5] A. Keshavarz-Haddad, V. Rbero, and R. Red, Broadcast capacty n multhop wreless networs, n Proceedngs of ACM MobCom, Sept [6] P. Jacquet and G. Rodolas, Multcast scalng propertes n massvely dense ad hoc networs, n Proceedngs of Internatonal Conference on Parallel and Dstrbuted Systems, July [7] S. Shaotta, X. Lu and R. Srant, The multcast capacty of large multhop wreless networs, n Proceedngs of ACM MobHoc, Sept [8] M. Grossglauser and D. N. C. Tse, Moblty ncreases the capacty of ad hoc wreless networs, IEEE/ACM Transactons on Networng, vol. 0, no. 4, pp , Aug [9] A. E. Gamal, J. Mammen, B. Prabhaar, and D. Shah, Throughput-delay trade-off n wreless networs, n Proceedngs of IEEE INFOCOM, Mar [0] Le Yng, Schao Yang and R. Srant, Optmal delay-throughput trade-offs n moble ad hoc Networs, IEEE Transactons on Informaton Theory, vol. 54, no. 9, pp , Sept [] S. Toumps and A. J. Goldsmth, Large wreless networs under fadng, moblty, and delay constrants, n Proceedngs of IEEE INFOCOM, Mar [2] M. L and Y. Lu, Rendered path: range-free localzaton n ansotropc sensor networs wth holes, n Proceedngs of ACM MobCom, Sept [3] R. L. Cruz and A. V. Santhanam, Herarchcal ln schedulng and power control n multhop wreless networs, n Proceedngs of the Annual Allerton Conference on communcaton, Control and Computng, Oct [4] T. Spyropoulos, K. Psouns, and C. S. Raghavendra, Effcent routng n ntermttently connected moble networs: the mult-copy case, IEEE/ACM Transacton on Networng, vol. 6, no., pp , Feb [5] S. M. Ross, Stochastc processes. New Yor: John Wley & Sons, 996. Appendx I The dervaton of p and q Snce p represents the probablty of fndng at least two nodes n a partcular cell, the opposte event of t s there s no node (and ths happens wth a probablty of ( c )n ) or only one node n the cell (ths occurs wth a probablty of n ( c c )n, where n nfers that the node n the cell can be any one among all n nodes of the networ). Thus, we have the expresson of (2). As for q, t represents the probablty of fndng a sourcedestnaton par wthn a cell. Note that drectly calculatng the probablty can hardly obtan an ntegrate expresson, we temporarly adopt the followng assumptons here. Suppose the number of nodes n s dvsble by +. For smplcty, we unformly and randomly dvde the networ nto dfferent groups wth each of them havng + nodes. And assume pacets from each node n a specfc group must be delvered to all the other nodes wthn the group. Thus, any two nodes wthn a same group s a par of sourcedestnaton. The probablty that there s not any sourcedestnaton par belongng to any group wthn a partcular cell s + ( c c ) + ( c )+. Snce each group s ndependent wth others, the probablty that there s not any n + source-destnaton par n the cell s thus th power of the above quantty. Hence, the probablty of the nverse event q s gven by (3). Appendx II Useful lemmas Here we present useful lemmas n ths paper. Lemma. P= ( ) and r s Euler constant. =ln( + ) + r, where Proof. Denote the left-hand-sde of the equaton by A(), then we have A( ) = = +, t follows A() A( ) = Recall that ( ) = ( ) P= X X ( ) = ( ) = ( ) P=0 =. Notce that (5) =0, hence we obtan ( ) P= = ( ) P= X = P=0( ) =. Combnng wth (5), we get A() A( ) =, then A() = A() + A( ) =2A() = + X =2 = X = (6) Snce the rght-hand-sde of (6) s the harmonc seres, ths lemma holds. Lemma 2. Suppose X, X 2,..., X are contnuous..d exponental varables wth expectaton of /a, and denote X max = max{x, X 2,..., X }, then E{X max} = Θ(log /a) (for smplcty, we can treat E{X max} just as log /a), where. Proof. Consder the cdf of X max, F Xmax (t) = P {X max t} = ( e at ) (7) 297

10 Thus, the pdf of X max can be expressed as f Xmax (t) = dfxmax (t) dt Then, we obtan = ( e at ) ae at (8) Z E{X max} = ( e at ) ae at tdt = az = = = a = a 0 X X =0 0 X = X = X =0 a ( ) e a(+)t tdt ( ) a( + )2 a ( ) a 2 2 ( ) 2 = ( ) (9) Accordng to Lemma, we conclude ths lemma. 298