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 iuch 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. I may applicatios, multicast data trasfer is more predomiat tha uicast data trasfer. I military etworks it is oftetated that multicast traffic domiates due to the ermissio 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 oervers 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. eed for group commuicatios. I the civilia cotext, a emergig applicatio that has already bee tested 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 obtaiuch 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. S S 3 3, 3 3,3 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. 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 S 3

2 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 maiources of iterferece that limit the multicast capacity of the etwork:. The chael iterferece costrait: While a large trasmissio radius may allow multiple receivers to receive packets from a multicast flow simultaeously, a larger radius meas a trasmittig ode iterferes with more odes, which limits the umber of simultaeous trasmissios. A good multicast structure should balace the tradeoff.. 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, of these odes are radomly chose as sources, ad each of these sources is associated with d radom destiatios, makig a total of d source-destiatio pairs i the system. We would like to uderstad the throughput scalig laws i the etwork as,, d. We assume that log d / 0 as, 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 beetudied 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, 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 works that deals with scalig laws i wireless etworks with multicast flows are [4, 8]. However, they use a Steier tree approach to costruct a multicast tree, ad they assume that every ode i the etwork is a multicast source. Our mai cotributio is to propose a simple architecture that achieves the same capacity i the order sese. Furthermore, our proof of the upperboud is differet from that i [4,8]. Last, we cosider the scalig law as a fuctio of the umber of multicast sources. Mai Results 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 {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 {c g( f( c g(} δ m(δ. We first derive a upper boud o the multicast capacity of a wireless ad hoc etwork. The mai techical challege is to take ito accout the broadcast ature of wireless media where multiple odes ca receive simultaeously. We show that the upper boud o the sum of the source rates that the etwork caupport is «O w.h.p, d log with a per flow throughput capacity of «O w.h.p. d log 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 d = Ω( log ; it is allowed to drop a arbitrarily small fractio of traffic; or 3 oe ca pose costraits o the locatios of the source ad destiatio odes. I compariso, whe there are d uicast source-destiatio pairs, the per flow throughput capacity for uicast is «Θ w.h.p. d log followig the Gupta-Kumar result. Therefore, by exploitig the properties of multicast, oe ca obtai multicast gai of for relatively large d. Θ( d Orgaizatio of the aper We begi i Sectio with a descriptio of the system model that we cosider. We derive the upper boud o multicast 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 choseuitably. However, sice our goal is to study the case of radomly choseources ad destiatios, i Sectio 5 we fid the lower boud o the achievable trasmissio rate. We coclude with possible extesios i Sectio 6.

3 . SYSTEM MODEL We cosider a square of uit area, i which wireless odes are located radomly. There are 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 d. 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 trasmissiouccessfully 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 []. A ode ca trasmit bit of data per uit of time if its trasmissio is successful. 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 = ad d = the model is the same as a uicast problem with Θ( sources. Whe = 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. I particular, we study the case where d = ǫ, ad 0 < ǫ <. I this paper, a multicast tree refers to a trasmissio structure that takes ito accout the broadcast ature of the wireless medium, which is differet from a miimum spaig tree. The differece is show i Figure. I the figure, A is the multicast source with B ad C as its receivers. The left figure shows a Steier tree i a wired etwork, where R is a relay ode. I this case, A trasmits oce to R, ad R trasmits to B ad C, respectively. The right figure shows a multicast tree i a wireless etwork where B ad C ca receive A s trasmissioimultaeously (i.e., oly oe trasmissio is eeded. The legth of the wireless multicast tree is larger tha or equal to that of the miimum Steier tree while the umber of trasmissios may be smaller. Therefore, takig ito accout such a effect is critical to derivig B R A a A miimum Steier tree C B R A C b A (wireless multicast tree Figure 3: A multicast tree i a wireless etwork. a upperboud ad to decidig the optimal trasmissio scheme for multicast i wireless etworks. I a multicast tree, all leaf odes are multicast receivers. 3. UER BOUND Our first objective is to derive a upper boud o the throughput capacity of multicast wireless etworks. We have the followig theorem. Theorem. The throughput of each multicast source i a radom wireless ad-hoc etwork is upper bouded by ««O mi, w.h.p. d log as, d = ǫ, 0 < ǫ <, ad is a fuctio of satisfyig where as. log d lim = 0, To proof this theorem, we still exploit the idea that commuicatios cosume space as i [, 4]. However, the focus is to take ito accout the properties of multicast. 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 trasmissios required to reach all destiatios? It has beehow i [] that the miimum trasmissio radius r mi has to be choseuch that r log + κ r mi = (where lim sup κ < + to guaratee coectivity of odes with high probability. Therefore, we assume that the trasmissio radius r r mi i the rest of the paper. A remaiig questio is whether or ot a larger commuicatio rage is beeficial for multicast flows to take advatage of the broadcastig ature of wireless media, which will be addressed i the proof of the theorem. Whe we have the aswer to the above two related questios, we ca derive the upperboud for capacity as follows. Let the trasmissio radius be r, sice our square is of size, the maximum umber of simultaeous trasmissios ca be bouded [] as S π ` r = 4 π r. ( Whe there are multicast flows ad each requires at least H i(r trasmissios, i, the capacity of a multicast

4 Circle II Circle I Note that there will be o ifiite loops because we reduce the umber of odes i the tree by at least i each step. To elaborate, before the process, A has at least 3 childre. We removed at least 3 childre odes ad added oly odes. A 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. V ar(l(m C log m (Lemma 4. from [6], flow satisfies Figure 4: New tree costructio. 4 λ π r ( i Hi(r. To prove the theorem, we first preset two lemmas. The first lemma is o the degree of multicast trees. The secod lemma is o the legth of multicast trees. Lemma. For a give multicast tree, oe ca costruct a correspodig multicast tree such that each ode has at most twelve childre odes that trasmit; ad the total umber of trasmissios is smaller tha or equal to that of the origial tree. roof. Cosider a multicast tree. First, we remove all leaf odes sice the two costraits do ot cocer leaf odes. I other words, i the rest of the proof, the degree of a ode does ot cout the leaf odes. If all remaiig odes i the origial tree have at most twelve childre odes, the the ew tree equals to origial tree. If ot, we eed to costruct a alterative tree that has a bouded degree of twelve ad a smaller umber of trasmissios compared to the origial tree T. The idea is show i Figure 4 ad discussed i the followig. Assume that the trasmit childre degree of ode A is greater or equal to 3 (i.e., ode A has at least 3 childre odes that eed to trasmit. The all receivers of its childre are withi Circle II (red. Assume we ca arbitrarily add relay odes. We add 6 square (blue oes ad 6 triagle (yellow oes. Replace Node A s trasmit childre odes by the ew odes. Node A plus these ew odes will cover all receivers i the origial tree that are covered by ode A ad its childre. The umber of total trasmissios is reduced to 3. To maitai the tree structure, A is the paret of all 6 square odes ad all leaf odes i Circle I i the origial tree. Oe square ode is the paret of oe triagle ode; ad all leaf odes betwee Circle II ad Circle I o the origial tree ca be coected to oe of the square odes. The ew tree has a bouded degree at ode A ad a smaller umber of trasmissios. Followig the same procedure to replace all odes with degree larger tha. The the fial tree has a maximum degree of. 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]. We ext preset a upper boud o the legth of multicast trees. Let M i( d be the legth of a multicast tree associated with source i. Lemma. The average legth of a multicast tree is Ω( d w.h.p.; i.e., ( X s M i( d > C d, (3 i= where C is a positive costat, whe log d lim = 0. roof. Let L i( d be the legth of the miimum spaig tree, ad S i( d be the legth of its Steier tree. We ca lower boud the total legth X i= X M i( d S i( d ( X 3/ L i( d, i= where ( follows from Result. Now, X L i( d E(L i( d ( δ i= ( V ar(li( d δ ( C log d δ 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 / 0 as. Thus, as, X L i( d ( i= ( s X i= E(L i( d < δ L i( d > E(L i( d δ i=.

5 Now, from Result, C, such that E(L i( d δ C d for large d. Hece, we have ( X s i= ( s X i= ( s X i= M i( d L i( d L i( d > C d > C d > E(L i( d δ Now we are ready to prove Theorem.. roof. Cosider (. We eed the trasmissio radius r ad the miimum umber of trasmissios i Hi(r. We cosider two situatios: Case : r = o(/ d. Give the trasmissio radius is r, we eed to obtai the miimum umber of total trasmissios eeded, i Hi(r, where Hi(r is the umber of trasmissios for multicast group i. Cosider a multicast tree for source i. Followig Lemma, we ca costruct a ew tree, T, that has bouded degree ad a smaller umber of trasmissios, deoted as H i(r. Let M i( d be the legth of T associated with source i. Followig Lemma, we have ( X s M i( d > C d. (5 i= I the tree T, there are at most d leaf odes i the tree. The legth to each leaf ode is at most r. Remove all leaf odes ad the legth of remaiig tree is Ω( d d r = Ω( d w.h.p. Each ode has at most K childre odes trasmittig, ad thus each trasmissio cotributes to at most Kr i terms of distace to the remaiig tree. Therefore, the miimum umber of trasmissios eeded satisfies: ( s X i= H i(r > C d, Kr where C is a costat ad C C. Because H i(r H i(r, usig the relatio (, we have λ (4 κ r d, w.h.p. (6 where κ is costat. I additio, by the capacity limit o each ode, we have λ. It is clear that the RHS of Eq. (6 is a decreasig fuctio of r. Therefore, upoubstitutig the miimum value of r to maitai coectivity, we have λ mi, d log «, w.h.p. (7 Case : r = Ω(/ d. I this case, similar to the proof i [8], we divide the area ito squares of size r. There are /r such squares. There exists a costat fractio of such squares that have at least oe multicast receiver. Oe trasmissio ca reach at most 9 eighborig squares. Therefore, we eed at least c /r trasmissios, where c is a costat. The umber of simultaeous trasmissios i the etwork is Θ(/r. Therefore, followig Eq. (, we have «κ λ mi,, where κ is a costat. I the case where d = ǫ, where 0 < ǫ <, we have d log. Combied with Eq. (7, the theorem follows. 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 ource destiatio pairs, we take d =. I particular, we take d = ǫ ad = 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 p w.h.p. ǫ log Notice that as ǫ 0, the multicast throughput capacity is essetially the same as the uicast capacity as it should be. Also otice that as ǫ, the beefit 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 choseources ad destiatios. 4. ORDER OTIMAL 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 while still keepig the average distace from a multicast source to its destiatios a costat. This locatio costrait is itroduced here to simplify the discussio ad highlight the comb structure. We will remove the assumptio i Sectio 5. As before, we study the system with multicast sources each with d destiatios. 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.

6 The schedulig algorithm will be choseuch that the squarelets that are Ks apart are scheduled simultaeously, where K is choseuch that the wireless iterferece costrait is satisfied. We costruct multicast comb structures usig the costructed squarelets. There is oe comb correspodig to each multicast flow. The combs are costructed so that at most flows pass through a squarelet, i order 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. 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 5, 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 6. 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. (8 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 Ks s S Figure 5: 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 choseo that all possible receiver odes i this squarelet are guarateed successful receptio. 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 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 throughput 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 lie o its correspodig comb. The average distace of a source ad its destiatios is still Θ(. For example, i Figure 6, 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. I other words, sice s = 3 log /, the proposed system ca accommodate a total of 3d 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 (9

7 d Figure 6: 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. /K. This results i a total source rate of K 3 d log, (0 which is of the same order of magitude as the upper boud calculated i the previous sectio. If > / 3 d log, the each comb is reused over multiple multicast sources. For example, if the umber of sources were set to be = ǫ ad d = ǫ, each comb would give each source a throughput of K p 3 ǫ log, 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 = ǫ 3 log. After this poit, the aggregated source rate remais costat as show i Figure 7, for a give value of d. 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 CAACITY 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 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 load of each squarelet does as well, leadig to reductio of throughput for the flows passig through the squarelet. I this sectio, we will study this effect. We will show that there exists two regimes. I the first regime, whe ad d are relatively large, all squarelets have the same amout of load asymptotically. I this regio, comb structure achieves the order optimal per-flow throughput. I the secod regime, whe ad d are relatively small, there are regios i which the traffic load is much larger tha that of the average. Therefore, to guaratee the same throughput for each flow, per flow throughput suffers. O the other had, we will thehow that the umber of such regios is a small percetage of the whole, ad we ca a arbitrarily small percetage of traffic to achieve order optimal throughput for the remaiig odes. I both regimes, the aggregated receive rate of the multicast flows is order optimal. d λ 3 log Figure 7: 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. Figure 8: 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 = / 3 d log with each source associated with d destiatios. Also recall that if the umber of sources is greater tha this value,

8 we have comb reuse with the total capacity remaiig costat. We idex the co-existig combs of Figure 6 from to s d = / 3 d log. We associate each source with oe of the combs. Whe there are more sources, the they evely share the combs. For a give multicast flow, both the source ad its destiatios choose the shortest path to reach the closest tooth o the chose comb. As show i Figure 8, 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 (8, we see that Euclidea legth of the tree is uaffected i the order sese by the brach paths. 5. Capacity-achievig Regio Theorem 3. Whe d = Ω( log, the comb scheme is order optimal for each flow. I other words, the comb scheme eables per flow throughput of «Θ w.h.p. d log for all flows. roof. Cosider Figure 9. 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. Let l be the umber of squarelets i a bi. We have l = /( d /s. We have idicated a few bis usig arrows i the Figure 9. The width of a bi is the width of a squarelet. d Figure 9: 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. Cosider a squarelet. 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 tooth. Let X k be the umber of braches passig through squarelet k. The value of X k decides the crowdedess. Let X ij be a radom variable, where X ij = 0 if squarelet j cotais o source or destiatio ode of multicast stream i ad X ij = otherwise. Let p = (X ij =, where p = ( s d+. I other words, X ij = j with probability p 0 with probability p, Let Br ij be the set of squarelets through which the brach for ode j to source i pass. X k = X X X ij i= k Br ij X s l m= X m, where X m has the same distributio as X ij ad is i.i.d. The iequality holds because a brach has at most l squarelets. Let ρ be the average load o a ode if source/destiatios are evely distributed. We have ρ = p l/ + s s d. The first term is the load o the squarelet from the radom braches ad the secod is the load due to evely distributed comb structure. The two terms are o the same order. Because the secod term is determiistic ad the same for all squarelet, we remove it from the followig discussios for simplicity. I particular, we focus o ρ = Θ(ρ, which deotes the average load due to radom braches. ρ = p l/. Giveome W R + ad usig the Cheroff boud, we have E e Xk (X k Wρ. e Wρ We calculate E `e X ext as follows:! E e Xk Xl k p k ( p sl k e k l Therefore, i=0 = ( + (e p sl (X W max(ρ, ( ( + (e psl e W max(ρ, esl(e p e W max(ρ, = e(e ρ e W max(ρ, e (e W max(ρ,. ( By the hypothesis d = Ω( log, we have d k log, where k is a costat, for large eough. Recall that p = ( s d+. Because d = ǫ, where ǫ <, we have p c d s for a costat c ad large eough. I this case, ρ c d ls k log. Let W = e + /k. We have (X k W max(ρ, = (X k Wρ. There are /3log quarelets. Use a uio boud, we have (max X k Wρ k 3 log (Xk Wρ.

9 Therefore, the load of each squarelet is at most a costat factor of that of the average w.h.p. The throughput of each flow is thus ««λ = Θ = Θ. ρ d log Compared with Theorem, the comb scheme is order optimal for each flow. 5. Whe ad d are small Theorem 3 shows that if ad d are relatively large, the comb scheme is order optimal for each flow. We ow cosider the case where ad d are relatively small. We have the followig theorem: Theorem 4. Whe d = o( log, the comb scheme is order optimal withi a factor of O(log for each flow. I other words, the comb scheme eables per flow throughput of Ω for all flows. log mi, d log ««w.h.p. roof. Cosider squarelet k. Let X k be the load of squarelet k. Followig Eq. ( i the proof of Theorem 3, we have (X k W max(ρ, e (e W max(ρ,. ( Let W = 3 log. For large eough, we have (max k (Xk W max(ρ,. where we use a uio boud. Therefore, the throughput of each flow is «λ = Ω W max(ρ, ««= Ω log mi, w.h.p. (3 d log Compared with Theorem, the comb scheme has a loss factor of at most log(. The proof idicates that there are certaiquarelets that have a large umber of iterferig flows compared to the average, 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. We show below that a ode ca achieve order optimal throughput with a probability of at least th, which is a costat, but ca be set arbitrarily close to. Theorem 5. Give ay threshold th > 0, with probability at least th, a ode ca receive the order optimal per flow throughput capacity usig the comb structure by droppig a small percetage of traffic. roof. The basic idea of the proof is as follows. We divide the etwork ito disjoit bis. Each bi has l squarelets. A brach may fall ito at most two disjoit bis. We say a bi is cogested if the most cogested squarelet i the bi has more tha W times of the average load, where W is a costat that is to be determied later. The traffic to/from a ode may be dropped if the ode s brach overlaps with a cogested bi. A ode ca receive the order optimal throughput if its brach does ot overlap with cogested bis ad either does its source ode. We will show that this probability is at least th. Next, we calculate the probability that a bi is cogested. Cosider a bi. Let Y be the umber of source ad destiatio odes whose traffic may use some squarelets of the bi. These source/destiatio odes could belog to differet multicast flows. Recall that X ij = is the evet that squarelet i cotais at least a ode associated with multicast flow j, where p = ( s d+. Let ρ be the average load of a squarelet as before. Note that odes i at most 3l squarelets ca cotribute to the traffic i a give bi of legth l. Therefore, we have Y 3l X s m= X m where X m has the same distributio as X ij ad is i.i.d. Note that Y upper bouds the load of the most cogested squarelet i the bi. Followig the proof of Theorem 3, we cahow that Let Let We have (Y W max(ρ, e (e W max(ρ,. (4 ( th 4 = th. W = e + l. th (Y W max(ρ, (e W max(ρ, e e (e W = th. I other words, with probability th, a bi is ot cogested. Recall that a ode may at most belog to two bis. Therefore, with probability at least ( th 4, the braches of a destiatio ode ad its source ode do ot to a cogested bi, ad thus ca receive order optimal throughput. We have thus show that by droppig a arbitrarily small fractio of the traffic, we ca achieve order optimal throughput for the rest of the odes. Note that as th approaches 0, the per-source throughput also approaches zero. However, the rate of decrease is slow: it is proportioal to the logarithm of / 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. We also ote that if a source ode passes a cogested area, it is ituitive to let the source ode trasmit first i the area. However, there exists a probability (take ito accout i th through the calculatio of p that multiple source odes cogest a bi. I this case, all receivers i

10 these multicast groups suffer. Noetheless, this probability is take ito accout i th. Thus, the statemet still holds. Next, we preset a geeral result o the aggregated multicast capacity for all ad d. Theorem 6. The comb scheme achieves order optimal aggregated multicast capacity. I other words, the comb scheme eables the total receivig rate of Θ mi d, ««d log w.h.p. roof. Followig Theorem 3, whe d = Ω( log, the comb scheme achieves the optimal throughput for each flow. Therefore, it also achieves the optimal aggregated throughput. O the other had, whe is small, followig Theorem 5, a ode achieves a order optimal throughput with probability th. So the aggregated receivig rate is at least that of th of the optimal throughput, ad thus order optimal. 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 beetudied 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 []. Gupta ad. R. Kumar, The capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. IT-46, o., pp , March 000. [] S. R. Kulkari ad. 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 roceedigs of IEEE INFOCOM 004, Hog Kog, March 004. [4] A. E. Gamal, J. Mamme, B. rabhakar, ad D. Shah, Throughput-Delay Trade-off i Wireless Networks, i roceedigs 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 erspective, i roceedigs 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 roceedigs of IEEE INFOCOM 000, Tel-Aviv, Israel, March 000. [0].Chaporkar ad S. Sarkar, Wireless Multicast: Theory ad Approaches, IEEE Trasactios o Iformatio Theory, vol. 5, o. 6, pp , Jue 005. [] E. M. Royer ad C. E. erkis, Multicast Operatio of the Ad Hoc O-Demad Distace Vector Routig rotocol, i roceedigs of the 5th aual ACM/IEEE Iteratioal Coferece o Mobile Computig ad Networkig (Mobicom, Seattle, WA, August 999. []. Siha, R. Sivakumar, ad V. Bharghava, MCEDAR: Multicast Core-Extractio Distributed Ad Hoc Routig, i roceedigs of the Wireless Commuicatios ad Networkig Coferece (WCNC, New Orleas, LA, September 999. [3] S.-J. Lee, M. Gerla, ad C.-C. Chiag, O-Demad Multicast Routig rotocol, i roceedigs of the Wireless Commuicatios ad Networkig Coferece (WCNC, New Orleas, LA, September 999. [4]. Jacquet ad G. Rodolakis, Multicast scalig properties i massively dese ad hoc etworks, i ICADS 05: roceedigs of the th Iteratioal Coferece o arallel ad Distributed Systems - Workshops (ICADS 05. Washigto, DC, USA: IEEE Computer Society, 005, pp [5] D. Z. Du ad F. K. Hwag, A proof of the Gilbert-ollak s cojucture o the Steier ratio, Algorithmica, o. 45, pp. 35, 99. [6] M. Steele, Growth rates of Euclidea miimal spaig trees with power weighted edges, The Aals of robability, vol., o. 6, pp , 988. [7] R. Motwai ad. Raghava, Radomized Algorithms. Cambridge Uiversity ress, Cambridge, UK, 995. [8] X. Li, S. Tag ad O. Frieder, Multicast Capacity of Large Scale Wireless Ad Hoc Networks i ACM MobiCom 007.