The Multicast Capacity of Large Multihop Wireless Networks

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1 The Multicast Capacity of Large Multihop Wireless Networks Sriivas Shakkottai Dept. of ECE, ad Coordiated Sciece Laboratory Uiversity of Illiois at Urbaa-Champaig Xi Liu Dept. of Computer Sciece Uiversity of Califoria at Davis Davis, CA R. Srikat Dept. of ECE, ad Coordiated Sciece Laboratory Uiversity of Illiois at Urbaa-Champaig ABSTRACT We cosider wireless ad hoc etworks with a large umber of users. Subsets of users might be iterested i idetical iformatio, ad so we have a regime i which several multicast sessios may coexist. We first calculate a upper-boud o the achievable trasmissio rate per multicast flow as a fuctio of the umber of multicast sources i such a etwork. We the propose a simple comb-based architecture for multicast routig which achieves the upper boud i a order sese uder certai costraits. Compared to the approach of costructig a Steier tree to decide multicast paths, our costructio achieves the same order-optimal results while requirig little locatio iformatio ad o computatioal overhead. Categories ad Subject Descriptors C.. [Computer-Commuicatio Networks]: Network Architecture ad Desig wireless commuicatio Geeral Terms Theory ad Algorithms Keywords Ad hoc etworks, multicast flows, Steier tree, asymptotic capacity. INTRODUCTION Wireless ad hoc etworks are of use whe there is a lack of fixed commuicatio ifrastructure. Such situatios might arise i calamity eviromets, sesor etwork applicatios, ad a variety of other civilia ad military cotexts. Most of Research supported i part by DARPA CBMANET grat, NSF grat CNS-05969, NSF CAREER Award CNS , ad NSF grat CNS Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. MobiHoc 07, September 9 4, 007, Motréal, Québec, Caada. Copyright 007 ACM /07/ $5.00. the research i large ad hoc etworks has focused o uicast data trasfers, either betwee users, or betwee a user ad fixed ifrastructure. Oe area that has received little attetio is the use of such ad hoc etworks for multicast data trasmissio. I may applicatios, multicast data trasfer is more predomiat tha uicast data trasfer. I military etworks it is ofte stated that multicast traffic domiates due to the eed for group commuicatios. I the civilia cotext, a emergig applicatio that has already bee tested i may uiversities is the use of wireless ad hoc etworks to broadcast replays durig football games. A situatio like a football game would have a large umber of spectators, each havig a mobile device ad a desire for a replay of a importat momet i the game. There is almost o ifrastructure available from which they could obtai such data, ad there is a strog icetive to form a ad hoc etwork for this purpose. Some of the users might be close to data sources (perhaps if they were close to a Iteret access poit), ad they would act as sources for the multicast traffic. Other odes would act as relays ad siks for the data. The questios arise as to how may multicast sessios ca be supported by such a etwork, what the total capacity of the etwork would be, ad how to achieve the capacity i a simple ad practical maer. Cosider Figure. There is a fiite area with a umber of wireless odes. There are 3 multicast flows i progress, with odes receivig each multicast flow labeled, ad 3, respectively. The sources of these flows are labeled as S, S, ad S 3, respectively. The hops are labeled with the sessios that they carry. Some odes may be either sources or destiatios ad merely act as relays (ulabeled odes). Suppose each hop, if scheduled, could carry oe bit per time slot. The first costrait o the system capacity is that we may ot be able to schedule all the hops simultaeously due to iterferece. Eve if all hops could be scheduled simultaeously, we see that the throughput of each source is at most 0.5 bits per time slot i this example, sice all the multicast flows cotai at least oe ode that is shared with oe other multicast flow. The total throughput of the system would the be.5 bits per secod. Thus, we see that there are two mai sources of iterferece that limit the multicast capacity of the etwork:. The chael iterferece costrait: while we must have a sufficietly large trasmissio radius for the etwork to be coected, a larger radius meas a trasmittig ode iterferes with more odes, which lim-

2 S S 3 3 3, 3 S,3 3 Figure : Example of multicast flows i a wireless ad hoc etwork. Receivers are labeled with the flows that they are iterested i. Some odes act as pure relays, i which case they are ot labeled. The system throughput is determied by the umber of flows that pass through each ode. its the umber of simultaeous trasmissios. A good multicast structure should therefore limit the total umber of hops for each multicast stream.. The flow iterferece costrait: if multiple flows pass through the same ode, the rate that each flow obtais is a fractio of the ode s trasmissio capacity. I other words, system capacity is divided amog the flows. Therefore, we cosider the multicast capacity as a fuctio of the umber of multicast sources. I this paper we will study how the throughput of multicast wireless ad hoc etworks scales with the umber of sources, the umber of destiatios per source, ad the total umber of odes preset i the system. We will cocetrate o the case whe the umber of odes i the system is large. We cosider a uit square i which odes are dropped at radom, s of these odes are chose as sources, ad each of these sources is associated with d destiatios, makig a total of s d source-destiatio pairs i the system. We would like to uderstad the throughput scalig laws i the etwork as, s, d. We assume that log d / s 0 as s, d. There are a umber of studies that cosider such questios with regard to uicast flows, startig with the semial paper by Gupta ad Kumar [], where achievable upper bouds are derived for wireless ad hoc etworks i a fiite regio with a large umber of odes. The Gupta-Kumar result was re-derived i a much simpler maer i []. Related ideas ca also be foud i [3,4]. A umber of papers have also looked at delay-throughput tradeoffs i mobile models for such etworks [4 8], although we do ot study this issue i our paper. Multicast i wireless etworks has bee studied i [9,0], but these papers do ot deal with scalig laws. Some examples of multicast protocols are preseted i [ 3]. I compariso, our focus i this paper is ot to develop a protocol, 3 but to show that a simple routig structure ca be capacity achievig i a order-optimal sese. To the best of our kowledge, the oly prior work that deals with scalig laws i wireless etworks with multicast flows is [4]. However, they use a Steier tree approach to costruct a multicast tree, ad their results oly apply to a sigle multicast flow i a large etwork. Our mai cotributio is to propose a simple architecture that achieves the same capacity i the order sese. I additio, we also cosider the scalig law as a fuctio of source odes. Mai Results We first derive a upper boud o the multicast capacity of a wireless ad hoc etwork alog the lies of [4]. I the Gupta-Kumar model, the umber of source-destiatio pairs is. To compare ad cotrast our result with theirs, we too assume that the umber of S-D pairs i our etwork with multicast flows is. I particular, we assume that the umber of multicast sources is ǫ for some ǫ > 0, ad the umber of receivers per multicast flow is ǫ. We express our results i the order sese. We say f() O(g()) with high probability (w.h.p) if give δ > 0, c, ad m(δ) such that P{f() c g()} δ m(δ). Similarly, we say f() Θ(g()) with high probability (w.h.p) if give δ > 0, c, c, ad m(δ) such that P{c g() f() c g()} δ m(δ). Uder the above model we show that the upper boud o the sum of the source rates that the etwork ca support is «ǫ O w.h.p, log with a per flow throughput capacity of «O w.h.p. ǫ log Note that this is a direct geeralizatio of the Gupta-Kumar upper boud for the per flow throughput capacity for uicast, which is «O w.h.p. log Our upper boud is a simple extesio of the argumets i [,4]. The achievability of the upper boud is the mai cotributio of this paper. To achieve the upper boud, we propose a simple routig architecture to trasfer multicast data i the etwork. The architecture cosists of a tree called the multicast comb, which is costructed idepedet of the receiver locatios. The receivers the complete the multicast tree by attachig themselves to the comb usig shortest path routig i a small viciity. Usig this simple architecture, we show that the achievable throughput matches the upper boud i a order sese if each flow is allowed to drop a arbitrarily small fractio of its receivers or we ca pose costraits o the locatios of the source ad destiatio odes. Orgaizatio of the Paper We begi i Sectio with a descriptio of the system model that we cosider. We derive the upper boud o multicast

3 capacity ad compare it to the uicast upper boud i Sectio 3. We the proceed i Sectio 4 to desig a simple architecture that we call the multicast comb structure that achieves the upper boud i the order sese if the sources ad destiatios ca be chose suitably. However, sice our goal is to study the case of radomly chose sources ad destiatios, i Sectio 5 we fid the lower boud o the achievable trasmissio rate. We coclude with possible extesios i Sectio 6.. SYSTEM MODEL We cosider a square of uit area, i which wireless odes are located radomly. There are s multicast sources chose radomly from the odes, ad each of these sources is associated with d destiatio odes also chose radomly. Thus, the total umber source-destiatio pairs is s d. We assume that time is slotted, ad the capacity of the wireless chael is such that a hop, if scheduled, ca successfully trasmit bit per slot. We use the protocol model to model iterferece betwee trasmissios, as proposed i []. The model is illustrated i Figure. We assume that all odes choose idetical trasmit radii r, which is large eough for the etwork to be coected. Suppose that ode i trasmits to ode j. Node j receives the trasmissio successfully if every other ode that trasmits simultaeously is at a distace of at least (+ )r from j. This implies that circles of radius r aroud each receiver must be disjoit []. T (+ ) r R r Figure : Illustratio of the protocol model for wireless trasmissio. The assumptio is that a trasmissio would be successful as log as there is o other trasmissio i a circle of radius (+ )r from the iteded receiver. Whe s = ad d = the model is the same as a uicast problem with Θ() sources. Whe s = ad d = the source ca broadcast its data i oe hop to all the destiatios with a trasmissio radius of. We would like to uderstad the capacity scalig law ad its practical achievability betwee these two extremes. 3. UPPER BOUND ON MULTICAST CAPAC- ITY IN RANDOM NETWORKS Our first objective is to derive a upper boud o the throughput capacity of multicast wireless etworks. The derivatio is similar to that i [,4], with a small modificatio to accout for the fact that we have multiple multicast flows. Cosider a bit b origiatig at source i. Let the umber of hops required by b to reach all its fial destiatios be deoted H i(b). The the total umber of hops used by all bits to reach their respective destiatios is P i,b Hi(b). Note that this could take several time slots. We would like to kow if these bits ca be trasmitted i some time iterval T. Now, sice our assumptio is that each trasmissio ca support the trasfer of oe bit, the umber of trasmissios required for this to happe is the same as the umber of hops required. Let the total umber of simultaeous trasmissios possible i the system be S. Note that S is idepedet of time as we assume that system does ot chage with time. From the above argumet, we require that for the bits to reach their destiatios i a iterval T, X H i(b) TS. () i,b Thus, if the rate at which sources ca geerate bits is λ (commo to all sources), the umber of bits geerated by these sources i time iterval T is simply λt s. If we ca show that the total umber of hops required to support these bits i time iterval T is at least λt sh(b) with high probability for some H(b), the our ecessary coditio is ow λt sh(b) TS. () Now, if we take the radius of trasmissio to be r, sice our square is of size, the maximum umber of simultaeous trasmissios ca be bouded as S 4 π r. (3) Hece, from the above ad () the source rate i of a radom ad hoc etwork i both uicast ad multicast cases is upper bouded as λ 4 π r sh(b). (4) The questios that eed to be aswered to obtai a useful boud o the throughput capacity are: (i) what should the radius r be?, ad (ii) what is the umber of hops required to reach all destiatios? It has bee show i [] that the radius r has to be chose such that r log + κ r = (where lim sup κ < + ) to guaratee coectivity of odes with high probability. We ow eed the followig results from [5, 6] for the legth of the multicast tree i radom etworks. Result Suppose we drop m odes i a uit square, where the positio of each ode is chose uiformly at radom i the square. The as m the legth of the miimum spaig tree L(m) satisfies. E(L(m)) C m (Lemma 3.3 from [6]), ad

4 . V ar(l(m)) C log m (Lemma 4. from [6]), where C ad C are positive costats, ad the otatio is used to deote asymptotic equality. Also, the legth of the optimal Euclidea Steier tree coectig all the m odes S(m) 3 L(m) [5]. Theorem. The throughput of each multicast source i a radom wireless ad hoc etwork is upper bouded by ««O mi, w.h.p. s d log as, where s ad d are iteger fuctios of satisfyig s d, ad s, d as. log d lim = 0, s Proof. Cosider (4). We eed the radius of trasmissio r ad the total q umber of hops sh(b). Now, as i [], we log choose r > so as to maitai coectivity w.h.p. We use Result to lower boud the total legth of the hops required by bits from all the sources to reach their destiatios ( sh(b)r). Let M i( d ) be the legth of the multicast tree associated with source i, L i( d ) be the legth of the miimum spaig tree, ad S i( d ) be the legth of its Steier tree. We would like to lower boud the total legth X s i= X s M i( d ) S i( d ) () X s 3/ L i( d ), i= where () follows from Result. Now, ) X s L i( d ) E(L i( d )) P ( s δ i= () V ar(li( d)) sδ () C log d sδ 0 as, where () follows from Chebyshev s iequality ad the i.i.d ature of the multicast trees, ad () follows from Result. Also we have used the hypothesis that log d s 0 as. Thus, as, ) X s L i( d ) P ( P i= ( s X i= E(L i( d )) s < δ L i( d ) s > E(L i( d )) δ ) i=. Now, from Result, C, such that E(L i( d )) δ C d for large d. Hece, we have ( X s L i( d ) P > C ) d i= P s ( s X i= L i( d ) s > E(L i( d )) δ ). w.h.p. The by use of the rela- Thus, sh(b)r C s d tio (4), we have λ κ r s d, where κ is costat. I additio, by the capacity limit o each ode, we have λ. Upo substitutig the value of r, the above yields the desired result. I order to cotrast the multicast case with the uicast case, we take the umber of source destiatio pairs to be the same i both cases. This would eable us to characterize the gai that could potetially be achieved usig multicast. Sice the uicast regime cosists of source destiatio pairs, we take s d =. I particular, we take d = ǫ ad s = ǫ, where 0 ǫ. We have the followig corollary: Corollary. The throughput of each multicast source i a radom wireless ad hoc etwork is upper bouded by «Θ w.h.p. ǫ log Notice that as ǫ, the multicast throughput capacity is essetially the same as the uicast capacity as it should be. Also otice that as ǫ 0, the poit of usig multicast is lost sice all odes could be reached by oe broadcast hop. As a example, cosider the case where = 0, 000 ad ǫ = / (i.e., there are 00 sources, each with 00 destiatios). The per-source throughput that ca achieved by usig multicast is 0 times that of uicast. We ow move to the problem of desigig a simple schedulig ad routig algorithm that could be used to implemet the multicast idea. We first show that i a etwork where we are allowed to choose the sources ad destiatio, the multicast capacity ca easily be attaied. We the cosider the case of radomly chose sources ad destiatios. 4. ORDER OPTIMAL COMB STRUCTURE We develop a simple architecture, whereby we may achieve the upper boud foud above (i the order sese), whe we are allowed to select the source ad destiatio odes appropriately. As before, we study the system with s multicast sources each with d destiatios, ad the cosider the case whe s = ǫ ad d = ǫ. The mai features of our costructio are as follows. We first divide the regio ito squarelets i the maer of []. The squarelet size is large eough so that there is at least oe ode i each squarelet with high probability, ad odes i adjacet squarelets are capable of commuicatig with each other. Let the legth of a squarelet be s. The schedulig algorithm will be chose such that the squarelets that are Ks apart are scheduled simultaeously, where K is chose such that the wireless iterferece costrait is satisfied. We costruct multicast comb structures usig the costructed squarelets. There is oe comb correspodig

5 to each multicast flow. The combs are costructed so that at most flows pass through a squarelet, so as to keep iterferece betwee flows miimal. I the comb structure, the width betwee two cosecutive comb teeth is determied by d, the umber of destiatios per multicast source. Each multicast destiatio ode ca reach ay oe tooth o the comb to receive multicast data. This ca be doe i the viciity of the destiatio ode usig ay routig algorithms, such as shortest path routig. To costruct the comb, the locatio of the destiatio odes is ot required ad o cetral cotrol is eeded. I additio, the cost of the comb structure is the same as a optimal Steier tree i a order sese. Ks s S We ow preset the details of the architecture ad derive its throughput capacity. Recall that the legth of a squarelet is s. We have the followig useful result from []: Result For a squarelet size 3 log s =, o squarelet is empty with probability at least. We eed to esure that a ode i oe corer of a squarelet ca trasmit to a ode i the opposite corer of a adjacet squarelet, i.e., the trasmissio radius is chose as r = 5s. We ca guaratee successful receptio at a receivig ode, if o other trasmissio takes place withi a distace of 5s ( + ). Recall that Ks is the distace such that squarelets that are this distace apart ca be scheduled simultaeously. The, as illustrated i Figure 3, we have (K )s ( + ) 5s K = + ( + ) 5. We call the subset of squarelets a distace of K squarelets from each other, capable of simultaeous trasmissio as a equivalece class. So the umber of such equivalece classes is K. We the have a system i which the periodicity with which ay squarelet is scheduled is K time slots. Usig the above idea, we costruct the followig comb structure for multicast traffic. Suppose that we are able to select the source ad receiver odes of the multicast. We first costruct multiple combs, oe for each multicast flow as show i Figure 4. I the figure we have illustrated two multicast combs (oe lightly shaded (cya), ad the other dark (mageta)) correspodig to two multicast flows. The distace betwee the teeth of each comb is chose to be d. The Euclidea legth of comb routig for a particular multicast comb is d + = Θ( d ). (5) From Result, we have that the Euclidea legth of the optimal Steier tree is Θ( d ), which is the same as our comb structure i a order sese. The costructio of the Steier tree requires global locatio iformatio ad a cetralized cotroller. I compariso, the costructio of the comb structure oly requires iformatio o the umber of destiatio odes ad does ot require a cetral cotroller to compute the tree. Each comb is positioed oe squarelet farther to the right ad below to the previous oe. Note that by usig this structure, we have esured that the maximum umber of Figure 3: Illustratio of schedulig costraits with = 0.5. The source is i the squarelet labeled S, ad the squarelet with the X is where the receiver is located. K is chose so that all possible receiver odes i this squarelet are guarateed successful receptio. multicast flows that use ay particular squarelet is. Sice each squarelet is scheduled with a periodicity of K slots ad the chael has a capacity of bit per time slot (if scheduled), if oly multicast flows share a squarelet, their idividual throughputs would be /K bits per time slot. Sice we ca choose the locatios of the source ad destiatio odes as desired, we let the source ad destiatios of a particular multicast flow to lie o its correspodig comb. For example, i Figure 4, the sources ad destiatios associated with the lightly shaded (cya) comb would lie somewhere o the comb. As the distace betwee the teeth of a comb is d, the umber of such combs that ca be costructed is s d. Choosig d = ǫ ad s = 3 log /, the proposed system ca accommodate a total of ǫ (6) 3 log sources without usig oe comb for multiple flows. As explaied above, sice there are a maximum of flows usig each squarelet, the sources ca each trasmit at a rate of /K. This results i a total source rate of ǫ K 3 log, (7) which is of the same order of magitude as the upper boud calculated i the previous sectio. If s > ǫ / 3 log, the each comb is reused over multiple multicast sources. For example, if the umber of sources were set to be s = ǫ as i the previous sectio, each comb would have to carry the traffic of 3 ǫ log sources, givig each source a throughput of K 3 ǫ log,

6 d case the both sources ad destiatios must reach the comb i a multi-hop maer. Oce they reach the comb, they would have access to the multicast traffic o that comb. The limitig factor is that as the umber of such access paths icrease, the itersectios betwee them does as well, leadig to reductio of throughput for the itersectig flows. I this sectio, we will study this effect. We will first show that there are squarelets i which the umber of such itersectios is large. We will the show that the umber of such squarelets is a small percetage of the whole, ad we ca drop odes i these squarelets to achieve order optimal throughput for the remaiig odes. d Figure 4: The comb idea for multicast data trasfer. Each comb carries a data from a differet source. The size of the squarelet places a fudametal limit o the umber of possible coexistet combs. which is of the same order as the per flow throughput of the previous sectio (See Corollary ). Notice that i this scheme, the throughput of the system icreases with the umber of multicast sources at a costat rate util s = ǫ 3 log. After this poit, the aggregated source rate remais costat as show i Figure 5, for a give value of d. λ s 3 log s Figure 5: The upper boud for multicast usig the comb method. We see that the boud icreases liearly with the umber of multicast sources as the comb gets filled. The throughput capacity is costat afterward. We have proposed a simple multicast architecture, ad show that its throughput i ideal circumstaces is idetical (i the order sese) to the upper boud. However, we have yet to study its performace whe we are ot at liberty to place the sources ad destiatios. We proceed to aswer this questio i the followig sectio. 5. ACHIEVABLE MULTICAST CAPACITY IN RANDOM NETWORKS We have just see how the comb architecture is capacityachievig i the case where source-destiatio placemets ca be made as desired. We will ow study the case where the sources ad destiatios are radomly chose. I this Figure 6: Usig the comb method for radomly placed sources ad destiatios. A source would be assiged a comb ad would coect to the closest possible tooth. Destiatios would do likewise. Recall that i the previous sectio we showed that the capacity achievig umber of sources was s = ǫ / 3 log with each source associated with d = ǫ destiatios. Also recall that if the umber of sources is greater tha this value, we have comb reuse with the total capacity remaiig costat. We will study the system with the capacity achievig parameters above, although our derivatios are valid for a geeral s ad d. We idex the co-existig s d combs of Figure 4 from to = ǫ / 3 log. We associate each source with oe of the combs, ad both the source ad its destiatios choose the shortest path to reach oe of the teeth of the chose comb. As show i Figure 6, these paths are simple to costruct (they are either above or below the teeth). Note that the maximum legth of the path is d, which meas that the icrease i legth of the multicast tree due to these brach paths would be just d d = Θ( d ). Thus, comparig with (5), we see that Euclidea legth of the tree is uaffected i the order sese by the brach paths. As explaied i the example i the itroductio, the trasmissio rate of the sources (such that all their destiatios ca receive all the bits that they trasmit) depeds o the umber of multicast flows that share the same squarelets. To fid the achievable regio, we eed to kow the maximum umber of differet multicast flows that share the same squarelet. We ca the divide the throughput capacity by

7 this umber to fid the achievable throughput. We first eed a stadard result o the so-called occupacy problem (See [7] for example): Result 3 Suppose that we have m balls, ad we drop them uiformly at radom ito m bis. The the maximum umber of balls i ay bi is upper bouded as O log m log log m «, with probability /m as m. We are ow ready to fid the maximum umber of multicast flows sharig each squarelet. We have the followig theorem: Theorem 3. Give that the etwork is coected, the maximum umber of other iterferig multicast flows for ay particular multicast flow is O log log log 3 ǫ log 3 ǫ log with probability 3 ǫ log as. Proof. We already kow that the comb structure itself requires that multicast flows share some squarelets. Give a particular multicast flow, we would like to kow how may others share squarelets with it. Cosider Figure 7. Sice both sources ad destiatios of a particular multicast flow would use the shortest vertical path to coect to the relevat comb, we ca divide the area ito bis associated with a particular flow. The size of the bi is half the distace betwee the teeth of the comb, ad is hece /( d ). We have idicated a few of the bis usig arrows i the Figure 7. The umber of iterferig multicast flows is the umber of odes belogig to differet multicast flows that fall ito each bi.!, d Figure 7: The umber of available bis for droppig odes whe cosiderig a particular source. The arrows show the extet of a few bis associated with the comb. The umber of odes associated with differet multicast flows i each bi should ot be too large if the throughput capacity is to be ear optimal. We first fid the umber of bis preset. Clearly, the umber of strips (areas betwee the dark horizotal lies ad the earest tooth of the comb) i Figure 7 is d. The the umber of bis associated with each multicast flow is just d /s, which simplifies to 3ǫ log. (8) We ow drop odes ito these bis at radom. Although we really wat to kow the umber of types of odes i each bi (each type correspodig to a differet multicast flow), we ca fid a upper boud by assumig that each ode belogs to a differet multicast flow. Thus, the umber of ode types is upper bouded by ( s )( + d ) s d, sice d. Whe s is take to be capacity achievig value (see (6)) of ǫ 3 log, ad d take as ǫ, the umber of ode types that are dropped is upper bouded by 3ǫ log, (9) which is idetical to the umber of bis give i (8). Thus, we ca look at the system as a occupacy problem with equal umbers of balls ad bis. The result follows from Result 3. We have just see that multicast flows might have a large umber of other iterferig flows, causig a loss of throughput. But how may squarelets would actually have such overcrowdig? If the fractio of overcrowded squarelets were small, the we could simply drop the destiatios belogig to these squarelets. Thus, we propose to use a majority rule, which ca provide a high multicast rate (o the same order of the capacity upper boud) to a majority of users if we are allowed to sacrifice a arbitrarily small percetage of users i overcrowded areas. Let the probability that odes i ay particular squarelet do ot receive the order optimal throughput be P th. We show below that we ca achieve the goal of order optimal throughput for the remaiig odes for ay P th > 0. Theorem 4. Give ay threshold P th > 0, the probability that odes i a squarelet receive the order optimal per flow throughput capacity of «O ǫ log is at least P th. Proof. Cosider Figure 7 agai. Cosider ay squarelet that is o a multicast tree. There would be a umber of braches passig through this squarelet, where each brach is a path take by some ode to reach its desired comb. The legth of ay brach is at most / d, which is the same as ǫ / 3log squarelets. Call this brach legth i squarelets as l. Let X be the umber of radom braches passig through the squarelet. The value of X decides the crowdedess. Now, the probability that ay particular ode

8 lads i ay particular squarelet is the umber of odes divided by the umber of squarelets, ad is give by p = (3 d log )/. Let X ij be a radom variable, where j with probability p X ij = 0 with probability p, which is the probability that squarelet j is o multicast stream i. We the have X s lx X = + i= X s l + X, i= j= X ij X ij where we have used the fact that the legth of a brach is at most l squarelets. Let ρ be the average load o a ode if destiatios are evely distributed. We have ρ = s l p. Substitutig s = ǫ, d = ǫ, ad the values of l ad p as calculated above, we get ρ = 3 ǫ log /. Give some W R + ad usig the Cheroff boud, we have E `e X P(X Wρ) e. Wρ We calculate E `e X ext as follows: E e X E e X Therefore, = X sl i=0! k p k ( p) sl k e k sl = ( + (e )p) sl P(X W max(ρ, )) () ( + (e )p)sl e W max(ρ,) esl(e )p e W max(ρ,) = e(e )ρ e W max(ρ,) e () P th, (e W) max(ρ,) where () holds for large eough s. Therefore, for a give P th, we ca fid a fiite W large eough so that () holds. I other words, with probability P th, we ca guaratee that a ode ca achieve a multicast rate of /(W max(ρ, )), where W e lp th max(ρ, ). Sice ρ = 3 ǫ log /, this value is just W 3 ǫ log, Thus, the achievable rate is the same as the upper boud withi a costat factor /W = Θ((l /P th ) ). We have thus show that by droppig a arbitrarily small fractio of the odes, we ca achieve order optimal throughput for the rest of the odes. Note that as P th approaches 0, the per-source throughput also approaches zero. However, the rate of decrease is slow: it is proportioal to the logarithm of /P th. This kid of policy would be acceptable i real situatios where it is importat that most of the users obtai high rate trasmissio, rather tha havig to cut dow the rate for all users so as to satisfy a small percetage of overcrowded areas. 6. CONCLUSIONS I this paper we have developed a aalytical framework for studyig the multicast capacity of wireless ad hoc etworks. We started with a compariso of the uicast case that has bee studied i detail earlier, ad showed how the multicast capacity is a fuctio of the umber of multicast sources ad destiatios. We developed a ew ad simple scheme that we called the comb architecture that would achieve this upper boud if we were at liberty to place the sources ad destiatios. We also studied the radom etwork case ad showed that the price paid i terms of throughput capacity for the simple ad robust architecture is ot high i the order sese. I the future we would like to study experimetal wireless multicast etworks. 7. REFERENCES [] P. Gupta ad P. R. Kumar, The capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. IT-46, o., pp , March 000. [] S. R. Kulkari ad P. Viswaath, A determiistic approach to throughput scalig i wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 50, o. 6, pp , 004. [3] S. Toumpis ad A. J. Goldsmith, Large Wireless Networks uder Fadig, Mobility, ad Delay Costraits, i Proceedigs of IEEE INFOCOM 004, Hog Kog, March 004. [4] A. E. Gamal, J. Mamme, B. Prabhakar, ad D. Shah, Throughput-Delay Trade-off i Wireless Networks, i Proceedigs of IEEE INFOCOM 004, Hog Kog, March 004. [5] M. Grossglauser ad D. Tse, Mobility Icreases the Capacity of Adhoc Wireless Networks, IEEE/ACM Trasactios o Networkig, vol. 0, o. 4, pp , August 00. [6] X. Li ad N. B. Shroff, Towards Achievig the Maximum Capacity i Large Mobile Wireless Networks, Joural of Commuicatios ad Networks, Special Issue o Mobile Ad Hoc Wireless Networks, vol. 6, o. 4, December 004. [7] M. J. Neely ad E. Modiao, Capacity ad Delay Tradeoffs for Ad-Hoc Mobile Networks, IEEE Trasactios o Iformatio Theory, vol. 5, o. 6, pp , Jue 005. [8] G. Sharma, R. R. Mazumdar, ad N. B. Shroff, Delay ad Capacity Trade-offs i Mobile Ad Hoc Networks: A Global Perspective, i Proceedigs of IEEE INFOCOM 006, Barceloa, Spai, April 006. [9] J. E. Wieselthier, G. D. Nguye, ad A. Ephremides, O the Costructio of Eergy-Efficiet Broadcast ad Multicast Trees i Wireless Networks, i Proceedigs of IEEE INFOCOM 000, Tel-Aviv, Israel, March 000.

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