Scheduling Algorithms and Bounds for Rateless Data Dissemination in Dense Wireless Networks

Transcription

1 Scheduling Algoithms and Bounds fo Rateless Data Dissemination in Dense Wieless Netwoks Kan Lin, David Staobinski, Ai Tachtenbeg Depatment of Electical and Compute Engineeing Boston Univesity Boston, Massachusetts, USA Sachin Agawal NEC Euope Heidelbeg, Gemany Abstact Many applications in wieless cellula netwoks ely on the ability of the netwok to eliably and efficiently disseminate data to a lage client audience. The stochastic natue of packet loss acoss eceives and channel intefeence constaints between cells complicate this task, howeve. In this pape, we analyze the poblem of minimizing the delay of data dissemination in dense multi-channel wieless cellula netwoks, using ateless coding tansmission. We begin with an exteme value analysis of the delay in a single cell setting, and show that the gowth ate of this andom vaiable becomes deteministic as the client audience scales up. Next, we extend the analysis to multi-cell, multi-channel settings and deive tight pefomance bounds on the delay. Ou analysis eveals that the availability of moe channels does not always educe delay popotionally. This sub-linea gain effect is guaanteed to occu if the diffeence between the chomatic numbe and the factional chomatic numbe of the gaph is geate than one. I. INTRODUCTION The polifeation of affodable mobile devices has significantly boosted deployments of wieless cellula netwoks (e.g., Wi-Fi, 3G, LTE, etc.). Such wieless cellula netwoks typically consist of two components: base stations, which povide backbone communication, and wieless clients; these ae illusted in Fig. 1 with a five-cell wieless netwok, whee each base station (hencefoth clustehead) is suounded by clients within its adio ange (collectively called a cluste). The client population within one cluste can be quite high fo moden wieless netwoks; fo example, the statistics bueau of Japan epots a population density of 5751 pesons pe sq km in Tokyo [1], many of whom cay smatphones. Indeed, gowing density in wieless netwoks is an inevitable tend, which in tun, aises challenges fo some data tansmission applications. This pape focuses on the fundamental poblem of designing efficient data dissemination stategies in such dense wieless netwoks, whee communication occus though a boadcast channel, as may be witnessed, fo example, when smatphones eceive updates of events happening aound thei location o when wieless sensos schedule peiodic fimwae updates. The aim of data dissemination is to boadcast a set of identical data packets fom a media souce, to all tageted wieless clients in the netwok though the clusteheads in an efficient manne (i.e., shot time duation). Seveal intinsic factos of wieless netwoks make this objective paticulaly challenging. Fig. 1. Illustation of a wieless cellula netwok. Wieless clients ae connected to base stations (clusteheads), which ae inteconnected via eliable wieline/wieless links themselves. The dotted iegula shapes contou adio signal ange of each clustehead. It is assumed that one o moe of the clustehead is connected to geate netwoks (not shown in the figue). The fist difficulty ises fom high ates of packet losses typically expeienced by wieless clients. Given a big client audience, base stations can be ovewhelmed by the numbe of etansmission equests, a phenomenon called boadcast stom [2], making taditional (N)ACK-based etansmission mechanism infeasible. Instead, a ateless code, which is a fom of packet-wise fowad eo coection (FE scheme, allow each wieless client to decode packets fom any sufficiently lage subset of encoded packets. Yet, thee still emains the question of how many boadcasting packets ae needed fo the data dissemination to complete, and especially how this numbe is affected by the stochastic envionment of lossy wieless channels. The second difficulty esults fom the divesity of wieless netwok topologies. This difficulty emeges mainly because of the adio channel conflicts among clustes: simultaneous opeation on the same fequency by neighboing clusteheads will cause intefeence and can affect data packet eception. As

2 such, boadcasting must be caefully scheduled to avoid this type of intefeence poblem, potentially educing dissemination efficiency. The thid difficulty aises when one wishes to exploit channel divesity; fo example, in the U.S., IEEE wieless nodes can tansmit ove 11 diffeent channels in the 2.4 GHz band, thee of which ae non-ovelapping. In an effot to maximize dissemination efficiency while avoiding intefeence, one needs to solve the allocation poblem of these channel esouces ove both space and time. Ou goal is to detemine optimal stategies fo minimizing the time equied fo all the wieless clients to eceive a disseminated file (i.e. maximizing dissemination efficiency) fo geneal multi-cell topologies and multiple channels, given all the factos descibed above, using ateless coding tansmission. Ou contibutions ae as follows. Fist, we conduct a study of the fluctuation of the completion time in a single cluste. Specifically we pefom an exteme value analysis of the completion time, asymptotic in inceasing client audience N, and find that the completion time scales popotionally to log(n) + o(log(n)) in pobability. This esult implies that, despite the stochastic natue of packet loss, the poblem lends itself to a deteministic optimization poblem. We futhe ague that this deteministic phenomenon also speads out to multiple clustes, which substantiates the feasibility of employing a deteministic scheduling policy. Thus, we popose asymptotically optimal tansmission policies fo geneal multicluste, multi-channel topologies and povide tight lowe and uppe bounds on thei completion time. As pat of ou analysis, we find that dissemination efficiency is subject to a sub-linea gain with espect to gowing channel availability, unlike the case fo cluste chains [3] whee linea gain can be achieved. This sublinea cuve stands as the absolute bound that any asymptotical scheduling policy can achieve, a fact that is valuable fo those who design and opeate cellula netwoks. The est of the pape is oganized as follows: Section II discusses elated wok and intoduces gaph coloing backgound that is necessay to ou late analysis. In Section III we descibe ou models of the poblem and its fomulation. Section IV deals with single cluste ateless coding tansmission, whee the fluctuation of dissemination delay is studied and its deteministic tend is evealed. We expand ou analysis to geneal multiple cluste topologies in Section V with details of ou optimal policy and pefomance bounds. Finally, we conclude this pape in Section VI. A. Related Wok II. PRELIMINARIES Consideable effot has been invested in the liteatue in seach of solutions to some of the issues descibed above. In [4] the authos conduct a quantitative analysis on the pefomance of FEC codes, such as ateless codes, in the pesence of a lage client population. An asymptotic cumulative distibution function (CDF) of the completion time is deived. Howeve, this wok only consides the analysis of a single cluste, wheeas ou wok conside the fluctuation of the completion time itself fo its asymptotic behavio, and take multiple clustes oganized along an abitay topology into consideation as well. The wok in [3] analyzes plaintext data dissemination ove simple cluste topologies, namely, linea chains of clustes. The limitation of this wok is that it does not conside the scenaio of ateless coding, no geneal multi-cluste topologies. Moeove, unlike ou wok, the esults of [3] apply to the expected completion time, while ou wok conside the behavio of the completion time andom vaiable itself. The wok in [5] analyzes the gain of netwok coding (anothe FEC coding scheme) in a layeed, multi-tansmitte multi-eceive systems, whee packets ae passed ove laye by laye using a andom tansmission scheme. Howeve, this model is inconsistent with ous, whee packets ae consideed to be eady in all base stations at fist and spead out simultaneously instead of being passed down laye by laye. Thee has been substantial wok on esouce allocation fo maximizing the thoughput of multi-hop wieless netwoks (see [6] fo a suvey). In [13] the authos show that maximizing thoughput fo a single channel ove one-hop is equivalent to the classical Maximum Weighted Matching poblem, solvable in polynomial time. The poblem of obtaining an efficient intefeence-awae link scheduling scheme fo a wieless netwok is also consideed in [12]. The authos in [14] pesent low-complexity (although not optimal) algoithms fo distibuted scheduling of wieless nodes in a netwok. In [7] the authos study the multi-flow poblem and showed the NPhadness of finding Maximum Independent Set (MIS). Ou pape is distinct fom this pevious body wok in seveal aspects. Fist, we conside delay as the pimay optimization metic athe than thoughput. Since netwok taffic is typically busty, the delay of a bust of data (geneically efeed to as a file in this pape) is often the pimay metic of inteest. Second, we explicitly model packet loss athe than assuming that communication is loss fee even if cetain intefeence constaints ae satisfied. This leads to a moe ealistic model since wieless links ae notoiously pone to packet losses due to fading, etc. Finally, ou analysis focuses on the case of densely populated netwoks, the key fact that significantly simplifies the design of scheduling algoithms fo lossy wieless netwoks, as detailed in the sequel. B. Gaph Coloing Backgound Ou analysis in this pape is elated to some fundamental gaph coloing poblems, the most significant of which we list heein fo sake of completeness. Definition 1: An independent set of a gaph G is a set of vetices, such that no two of them ae adjacent. A maximal independent set (MIS) of G is an independent set that is not a subset of any othe independent set of G. We use a matix I to epesent a collection of independent sets (o maximal independent sets) of a gaph G(V, E). Each ow index coesponds to one vetex and each column epesents a distinct independent set. Evey element in this matix takes on a value of eithe 0 o 1, with 1 indicating

3 the existence of the coesponding vetex in the associated independent set. Definition 2: A vetex coloing of a gaph G is a labeling of the gaph s vetices with colos such that no adjacent vetices have the same colo. The chomatic numbe χ(g) of G is the smallest numbe of colos among all vetex coloing scheme of the gaph. Definition 3: A b-fold coloing of a gaph G is an assignment of colo sets of size b to vetices, such that each vetex is assigned one set and adjacent vetices eceive disjoint sets. An a : b-coloing is a b-fold coloing, whose sets ae subsets of a univese of a available colos. The b-fold chomatic numbe χ b (G) of G is the smallest a fo which an a : b-coloing exists. Definition 4: The factional chomatic numbe χ f (G) is defined to be χ b (G) χ f (G) = lim = inf b b b χ b (G). b The diffeence between χ f (G) and χ(g) is unbounded in geneal. Moe details about b-fold and factional coloing poblems can be found in [9]. III. MODELS AND PROBLEM FORMULATION A. Netwok Topology We assign the scatteed wieless clients (also simply called nodes) to a set of clustes based on physical poximity to base stations (also called clusteheads), as shown in Fig. 1. Specifically, all the nodes associated with a given base station ae consideed as belonging to the this cluste. A node is associated with only one base station, as is typically the case in pactice, even if it is unde the ange of multiple base stations. Base stations tansmit data packets via boadcast, that is, each packet is simultaneously tansmitted to all the nodes within adio ange. Physical intefeence constaints induce a logical gaph G(V, E), whee the vetex set V epesent base stations, and edge set E epesent intefeence constaints i.e., an edge between vetex i and j means that the the coveage aeas (clustes) of base stations i and j ovelap. Fo example, the cluste gaph coesponding to Fig. 1 is a pentagon, one cluste pe node. We assume that the cluste topology is abitay but fixed, i.e., it does not change duing data dissemination. We also assume that cluste i contains N i = α i N nodes, whee α i is a constant coefficient, with V i=1 α i = 1, and N efes to the scaling vaiable. In this pape we focus on the asymptotic egime of a lage client population. Theefoe, ou analysis applies to the case whee N. B. Tansmission Model We conside the poblem of disseminating a file consisting of M packets to all the wieless clients via boadcasts fom base stations. We assume that links between base stations ae fast and eliable, meaning that the entie file is available at evey base station befoe data dissemination poceeds. The time axis is slotted and each boadcast packet tansmission takes one time slot. Base stations ae equipped with a single adio, as is typically the case, meaning that a node can eithe tansmit o eceive (but not both) on one channel at a point of time. Futhemoe, we conside the use of ateless coding fo tansmitting the file. Instead of tansmitting the oiginal M packets, the base station tansmits a long sequence of encoded packets (e.g., using andom linea codes) of these M packets. In each slot, the base station geneates and tansmits a new packet. The geneating algoithm is devised in such a way that as soon as a node eceives M diffeent coect packets 1, it can estoe the oiginal file. C. Channel Intefeence Model We assume that thee ae C non-ovelapping channels available in total that can be opeated on by base stations, all of them having identical, but independent statistical chaacteistics. The packet loss pobability on each channel is denoted p. Thus, the pobability that a node within a cluste does not coectly eceive a packet boadcast by the base station is p, independently of all othe events (e.g., coect eception o not of that packet by othe nodes in the cluste). In geneal, the pesence of intefeence o cosstalk between neighboing clustes esults in jamming. We assume that a packet tansmission in a cluste on a specific channel will bing intefeence to its neighboing clustes if they communicate on the same channel, i.e., all cluste nodes will be unable to coectly eceive packets. As a esult, tansmission on a channel in a cluste bas all its neighboing clustes fom opeating on the same channel duing the same time slot. D. Poblem Fomulation Ou objective hencefoth is to detemine a contol policy that minimizes the completion time (i.e., the total numbe of tansmission time slots needed) until all the N nodes eceive the entie file (M distinct decoded packets) in a netwok with C channels and intefeence constaints modeled by a gaph G. We denote TN C (G) the minimum completion time obtained using the optimal contol policy. IV. SINGLE CLUSTER TOPOLOGY In this section, we investigate the pefomance of data dissemination in single cluste topology as the fist step to develop analysis fo geneal multi-cluste topologies. Specifically, we conside a single cluste containing N nodes. Since thee is no intefeence constaints fom neighboing clusteing in this case, we do not expect benefit of using multiple channels. Hence, we simplify the notation of completion time in this section to T N. Because of the fact that ateless coding is designed to handle multiple clients, the optimal contol policy of this case is nothing but letting the clustehead keep tansmitting new ateless encoded packets until completion. In the est of this section, we analyze the completion time of data dissemination T N unde this policy and show that its gowth becomes 1 This is tue fo an ideal ateless code. In pactical scenaios, a eceive may need slightly moe packets fo decoding.

4 deteministic fo lage N. Ou analysis use the standad asymptotic O( ) and o( ) notations [11]. Conside fist the completion time fo a specific node n (1 n N), which we denote T (n). The pobability that the completion time fo this node takes exactly M + k slots is given by a negative binomial distibution. The CDF of this pobability distibution can be expessed as P{T (n) M + k} = 1 I p (k + 1, M), whee I p ( ) coesponds to the so-called egulaized incomplete beta function [10]: I p (k + 1, M) = k+m j=k+1 (k + M)! j!(k + M j)! pj (1 p) k+m j. (1) Given N nodes, the completion time of the entie cluste is the maximum value of the completion time of all its nodes, namely, T N = max{t (1), T (2),, T (N)}. Next, we examine how T N scales as N gows. The elationship between them is descibed with the help of a function k(n). When consideing a specific function k(n), the following lemma, whose poof is omitted, povides us with a sufficient condition on the convegence of pobabilities. Lemma 1: If lim N N I p (k(n) + 1, M) = 0, then lim N P{T N M + k(n)} = 1. On the othe hand, if lim N N I p (k(n) + 1, M) =, then lim N P{T N M + k(n)} = 0. As a esult, the following theoem pedicts the completion time of data dissemination. Theoem 1: Fo fixed M and p, the completion time of data dissemination using ateless coding in a single cluste is, with pobability one, T N = log λ N + o(log λ N), whee λ = 1/p. In othe wods, log λ N 1 in pobability. Poof: The poof follows fom bounding the expession N I p (k(n) + 1, M) followed by showing that the choices of k(n) = log λ N + M log λ log λ N and k(n) = log λ N log λ log λ N espectively make N I p (k + 1, M) go to 0 (in the plus sign case) and 1 (in the minus sign case). The poof is fulfilled by applying this esult to Lemma 1. The key insight offeed by this theoem is that, despite the stochastic natue of the envionment, the ode of gowth of the completion time becomes deteministic, when N gets lage. We conducted simulation of the delay fo paametes M = 10, p = 0.2 ove 4500 iteations and found that 35% of instances occus at the mode of 21 packets when N = 10 3, while 43% of instances occus at the mode of 29 packets when N = Futhe, in both cases, ove 90% of the samples have a completion time that diffe at most by two packets fom the mode. This popety geatly simplifies the design of efficient scheduling policies fo multiple clustes, as explained in the next section. T N V. MULTIPLE CLUSTER TOPOLOGIES In this section, we extend ou analysis to the geneal case of multiple clustes with multiple channels. As descibed in Section III, we use the notation of N i = α i N whee V i α i = 1 to epesent the numbe of nodes in the i-th cluste. As a esult, when consideing the tansmission demand fo an individual cluste, the esults of Section IV emains applicable by eplacing N with α i N. We fist show that asymptotically the same time is needed fo each cluste to complete dissemination to all nodes. Ou poblem is then educed to a scheduling poblem among the clustes. Ou pimay goal is to detemine the optimal contol policy and deive its pefomance, and seconday goal is to assess the impact of inceasing the numbe of channels available fo data dissemination on the completion time pefomance. All the asymptotic esults of this section hold with pobability one as N. A. Stuctue of the Optimal Tansmission Policy Denote T Ni to be the total numbe of tansmissions equied fo cluste i. Though these numbes diffe amongst diffeent clustes fo finite N, they scale identically as N gets lage, as the following lemma (whose poof is also omitted) demonstates, based on Theoem 1. Lemma 2: Fo a fixed set of coefficients α i, i = 1, 2,, V, the total numbe of tansmission slots equied by diffeent clustes scale identically as N, namely T Ni = log λ N + o(log N), 1 i V. This lemma shows that as N the tansmission demand of diffeent clustes becomes identical to T N, the numbe given by Theoem 1. Accoding to ou channel intefeence model, we futhe have the following lemma (poof is omitted). Lemma 3: Fo any tansmission policy that induces intefeence, thee exists anothe policy that does not induce intefeence and causes no geate tansmission delay. In summay, to solve the poblem of finding an asymptotically optimal tansmission policy, it is sufficient to find fo each time slot an appopiate maximal independent set (MIS) fo each channel. Note that the selected MIS may change fom one time slot to the next. B. Analysis In this section, we deive the optimal policy fo geneal multi-cluste topologies with multiple channels. Then, we povide bounds on the optimal completion time that eveal an inteesting elationship with both χ(g) and χ f (G). 1) Optimal Tansmission Policy: The availability of multiple channel esouce allows us to select C MISs fo each time slot (a diffeent channel is assigned to each MIS). As a esult, some clustes may belong to multiple MISs and have multiple channels available fo tansmission, wheeas only one of them can be used. We ecall that I coesponds to be the matix of all maximal independent sets of G. We also intoduce a matix P in which each ow index i epesents a diffeent MIS (coesponding to a diffeent column of I) and each column j epesents a diffeent tansmission schedule coesponding to

5 a diffeent combination of C MISs chosen out of available. Theefoe the dimensions of this matix ae (. Note that the sum of the elements in each column of P is equal to C. An enty P ij is set to 1 if MIS i belongs to tansmission schedule j. Note that we assume > C, since the case of C is tivial: each MIS can be assigned a channel so the completion time is the same as fo a single cluste. The poduct I P yields a V ( matix. The value of the (i, j) enty of this matix coesponds to the numbe of channels allocated to cluste i using tansmission schedule j. Since each cluste can only use one channel in a given time slot, we intoduce a cutoff function f( ), which takes an intege matix as its agument and etuns a binay matix of the same dimension. f(a) ij = { 1 if Aij 1 0 if A ij = 0. In this way, f(i P) gives the numbe of effective channels usable in each cluste (i.e., 0 o 1). Ou optimization poblem is thus: Minimize Subject to ( x i (2) i=1 f( I P x 1,, x ( 0. ) x 1. x ( 1. 1 whee x i (i = 1,, x ( ) stands fo the potion of time that the netwok should tansmit using a specific tansmission schedule, i.e., a specific combination of C MISs out of available as indicated by the i-th column of matix P. Fo convenience, we denote x and 1 the column vecto [x 1,, x ( ]T and [1,, 1] T espectively. We denote θ(g, the optimal objective value etuned by (2), i.e., θ(g, = min( ( i=1 x i). This numbe efes to the amount of time nomalized by T N. Clealy, θ(g, is no less than 1. The minimal dissemination delay achievable is TN C(G) = θ(g, T N by applying the following policy: Solve the pogamming poblem (2). Tansmit in MISs indicated by the j-th column P fo x j / ( i=1 x i potion of time, until evey cluste finishes. The ode does not matte. 2) Pefomance bound: We now bound θ(g,. The pupose of these bounds is to assess the gain achieved by the availability of additional channels and establish a elationship between θ(g, and the chomatic and factional chomatic numbes of the gaph. Computing the optimal policy involves solving pogamming poblem (2). To compute a lowe bound, we fist eplace f(i P) by I P in the constaint, namely, Minimize ( x i (3) i=1 Subject to I P x 1 x 0. Because f(i P) I P by definition of f( ), the solution of (2) also satisfies all constaints of (3) but does not necessaily achieve the minimum of (3). Denoting θ (G, the solution of (3), we have θ (G, θ(g,. Using this elaxation, the following lemma shows the completion time fo C channels is at most C times smalle than that fo a single channel if C χ f (G). The poof of this lemma is omitted fo space consideation. Lemma 4: Fo C χ f (G) θ(g, χ f (G) C. Note that this bound is tight when C = 1. The following lemma shows that the (nomalized) completion time fo C channels is smalle o equal to χ(g)/c if C χ(g). Moeove, by the definition of the chomatic numbe, this bound is tight when C = χ(g). The poof of this lemma is omitted fo space consideation. Lemma 5: Fo C χ(g), θ(g, χ(g) C. Combining Lemmas 4 and 5, we obtain the following bounds on the pefomance of the optimal policy in geneal topologies with multiple channels: Theoem 2: max( χ f (G), 1) θ(g, max(χ(g) C C, 1) o equivalently, max( χ f (G) C T N, T N ) TN C (G) max( χ(g) C T N, T N ). 3) Sub-linea pefomance gain: Theoem 2 shows that the completion time deceases with the numbe of channels. With one channel the lowe bound in Theoem 2 can always be achieved. Howeve, with inceasing channels this lowe bound is no longe guaanteed. Geneally, fo topologies whose factional coloing numbe χ f (G) diffes fom the vetex coloing numbe χ(g), pefomance will divege fom the lowe bound χ f (G)/C as C inceases. We efe to this phenomenon as to the sub-linea gain effect. Theoem 2 implies a sufficient condition on C fo this type of sub-linea gain effect to occu, namely χ f (G) < C < χ(g). This esult holds because we know that if χ(g)/c > 1 then θ(g, 1 (since θ(g, = 1 only when C χ(g)). Note that the condition χ f (G) < C < χ(g) is sufficient but not necessay fo the sub-linea effect to occu as shown by the next example. To illustate, we conside a specific andom gaph G of 12 clustes (not shown fo space consideation), with factional

6 channel multi-cluste topology using the optimal policy is about χ f lage than the completion time in a single cluste, whee χ f is the factional chomatic numbe of the gaph epesenting the inte-cluste topology. As the numbe of channels made available fo data dissemination is inceased, the completion time is educed until the numbe of channels is equal to χ, the chomatic numbe of the gaph. Since the factional chomatic numbe is always smalle o equal to the chomatic numbe, the gain in completion time gows less than linealy in the numbe of channels. Moe pecisely, we show that if thee exists a value C such that χ f < C < χ, then the sub-linea effect is guaanteed to occu. Thus, this condition always holds if χ χ f > 1. These esults could be useful fo netwok povides who ae gauging the usefulness of acquiing additional spectum fo thei netwok. Fig. 2. The pefomance gain atio θ(g, 1)/θ(G, of a andom gaph of 12 clustes. The hoizontal line indicates the highest (linea) gain achievable. The case C = 4 (see cicled egion) is whee the sub-linea gain effect occus. chomatic numbe χ f (G) = and chomatic numbe χ(g) = 5. By solving (2), we obtain θ(g, fo diffeent values of C. As it tuns out, the optimal pefomance fo C = 1 is θ(g, 1) = χ f (G) = Fo C = 2 and C = 3, we espectively have θ(g, 2) = χ f (G)/2 = and θ(g, 3) = χ f (G)/3 = Howeve, fo C = 4, we have θ(g, 4) = > χ f (G)/4 = Ultimately, fo C = 5, θ(g, 5) = χ(g)/5 = 1. Fig. 2 depicts the atio of θ(g, 1) to θ(g, and illustates the divegence of the pefomance gain fom the linea line when C = 4. The sub-linea effect esults fom the natue of ateless coding, i.e., having multiple channels available in a given cluste does not help in educing the completion time pefomance of its membe nodes. Thus, the cases of linea pefomance gain fo small C ae achieved by going to geat lengths to seach fo altenative MISs, in an effot to balance channel coveage. The sub-linea effect occus when such othe MIS options fail. The occuence ate of the sublinea gain effect geatly depends on the netwok size (in tems of clustes) and intefeence constaints. In fact, we conducted a simulation using 1000 Edős-Rényi andom gaphs (pobability of connection between any two vetices is set 0.3). When the netwok size is V = 10 clustes, less than 2% of all gaphs showed sublinea gain effect, but when V inceases to 16 clustes, this pecentage inceased to close to 15%. VI. CONCLUSION We conside the poblem of achieving minimum delay ateless data dissemination in lossy, multi-channel wieless cellula netwoks with abitay topologies. We show that the tansmission demand (i.e., the numbe of tansmissions equied in each cluste) gows almost deteministically when the numbe of clients gows to infinity. This enables the use of linea pogamming to povide asymptotically optimal tansmission policies. Ou analysis shows that the completion time in a single ACKNOWLEDGMENT This wok was suppoted in pat by the National Science Foundation unde gant CCF and a gant fom Deutsche Telekom Laboatoies. This wok was completed when Sachin Agawal was employed by Deutsche Telekom Laboatoies. REFERENCES [1] Statistical Handbook of Japan 2010, The Statistics Bueau and the Diecto-Geneal fo Policy Planning of Japan, [2] Y-C Tseng et al., The boadcast stom poblem in a mobile ad hoc netwok, Wieless Netwoks, vol. 8, no. 2/3, 2002 [3] D.Staobinski and W.Xiao, Asymptotically Optimal Data Dissemination in Multichannel Wieless Senso Netwoks: Single Radios Suffice IEEE/ACM Tansactions on Netwoking, vol. 18, no.3, pp , Jun [4] W. Xiao and D. Staobinski, Exteme Value FEC fo Reliable Boadcasting in Wieless Netwoks, IEEE JSAC (Special Issue on Simple Wieless Senso Netwoking Solutions), Vol. 28, No. 7, pp , Sept [5] A. Eyilmaz, A. Ozdagla, M. Medad, E. 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