Broadcast Capacity in Multihop Wireless Networks

Transcription

1 Broadcast Capacity i Multihop ireless Networks Alireza Keshavarz- Haddad alireza@rice.edu Viay Ribeiro viay@rice.edu Rudolf Riedi riedi@rice.edu Departmet of Electrical ad Computer Egieerig ad Departmet of Statistics Rice Uiversity, 6100 Mai Street, Housto, TX 77005, USA ABSTRACT I this paper we study the broadcast capacity of multihop wireless etworks which we defie as the maximum rate at which broadcast packets ca be geerated i the etwork such that all odes receive the packets successfully i a limited time. e employ the Protocol Model for successful packet receptio usually adopted i etwork capacity studies ad provide ovel upper ad lower bouds for the broadcast capacity for arbitrary coected etworks. I a homogeeous dese etwork these bouds simplify to Θ(/ max(1, d )) where is the wireless chael capacity, the iterferece parameter, ad d the umber of dimesios of space i which the etwork lies. Iterestigly, we show that the broadcast capacity does ot chage by more tha a costat factor whe we vary the umber of odes, the radio rage, the area of the etwork, ad eve the ode mobility. To address the achievability of capacity, we demostrate that ay broadcast scheme based o a backboe of size proportioal to the Miimum Coected Domiatig Set guaratees a throughput withi a costat factor of the broadcast capacity. Fially, we demostrate that broadcast capacity, i stark cotrast to uicast capacity, does ot deped o the choice of source odes or the dimesio of the etwork. Categories ad Subject Descriptors Computer Systems Orgaizatio [Computer- Commuicatio Network]: Network Architecture ad Desig ireless commuicatio; Data [Codig ad Iformatio Theory]: Formal models of commuicatio Geeral Terms Performace, Theory Keywords Broadcast Capacity, Broadcast Scheme, Ad Hoc Networks, Multihop ireless Networks, Uicast Capacity 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. MobiCom 06, September 3 6, 006, Los Ageles, Califoria, USA. Copyright 006 ACM /06/ $ INTRODUCTION I wireless etworks, broadcast plays a particularly importat role, relayig a message geerated by oe ode to all other odes. Broadcast is a itegral part of a variety of protocols that provide basic fuctioality ad efficiecy to higher-layer services. Coordiated ad distributed computig, a prime task i sesor etworks, provides but oe example. Also, multicast protocols ad a host of uicast ad hoc routig protocols rely o broadcast, such as Dyamic Source Routig (DSR), Ad Hoc O Demad Distace Vector (AODV), Zoe Routig Protocol (ZRP), ad Locatio Aided Routig (LAR) [1]. There has bee a growig iterest to uderstad the fudametal capacity limits of wireless etworks [ 9]. Results o etwork capacity are ot oly importat from a theoretical poit of view but also provide guidelies for protocol desig i wireless etworks. Hitherto, most research o etwork capacity has focused o the capacity of uicast coectios betwee radom source ad destiatio odes. I this paper we study the capacity of wireless etworks for broadcastig. e defie the broadcast capacity λ B of a multihop wireless etwork as the maximum rate of geeratio of broadcast packets by a set of odes B i the etwork such that all odes receive the packets successfully. To the best of our kowledge, so far oly oe paper studies the capacity of wireless etworks for broadcastig [10]. That paper models the locatios of the odes by a Poisso poit process ad the wireless chael as Gaussia ad cosiders λ B whe B cosists of a sigle geeratig ode. It proves that the broadcast capacity is a costat factor of a computed upper boud whe the Poisso itesity of the odes is fixed ad the umber of odes goes to ifiity. The paper does ot derive bouds for the broadcast capacity whe the umber of odes is fiite or whe the odes have a differet distributio i the plae tha Poisso. As the first cotributio of this paper, we develop ovel bouds of the broadcast capacity λ B for a geeral wireless etwork with a fiite umber of odes, a arbitrary topology, ad for a arbitrary set of geeratig odes B. e assume that the wireless chael capacity has a fixed rate (bits per secod) ad model successful packet receptios with the Protocol Model, a popular model for wireless chaels i the literature o etwork capacity [, 11 13]. Surprisigly, we fid that the broadcast capacity does ot chage by more tha a costat factor whe we vary the umber of odes, the radio rage ad the area of the etwork. For the special case of large homogeeous etworks we fid that the broadcast capacity is Θ( ) where max(1, d ) 39

2 is the iterferece parameter of the wireless chael ad d is the umber of dimesios of the space i which the etwork lies. e adopt stadard otatio from complexity theory; O(.), Ω(.), ad Θ(.) describe asymptotic upper, lower, ad tight bouds respectively. Iterestigly, we prove that mobility caot sigificatly icrease the broadcast capacity of wireless etworks. I cotrast to uicast capacity which ca icrease by as much as Ω( ) ( is the umber of etwork odes i a fixed area) with the help of mobility [5], the broadcast capacity chages at most by a factor of O(max(1, d )) with mobility. As our secod cotributio, we study the throughput of broadcast schemes of wireless multi-hop etworks. The simplest approach to broadcastig is blid floodig, i which every ode rebroadcasts the packet. Blid floodig, however, produces redudat broadcasts ad wastes precious badwidth ad power [14]. May broadcast schemes have bee proposed for ad hoc etworks that are far more efficiet tha floodig (see [15 17] ad refereces therei). So far, o theoretical aalysis of the achievable throughput of these schemes has bee performed. e defie the maximum throughput λ S,B of a give broadcast scheme, S, as the maximum aggregate rate of geeratio of broadcast packets by a set of odes B i the etwork such that the scheme ca dissemiate the packets to all odes successfully. e establish a close tie betwee λ S,B ad the backboe size of a broadcast scheme S. Ideed, for λ S,B to be withi a costat factor of the broadcast capacity it is ecessary ad sufficiet that its backboe uses a bouded umber of odes per radio rage area. Such backboes always have size withi a costat factor of the size of the Miimum Coected Domiatig Set (MCDS). As our third cotributio, we highlight the fudametal differeces betwee uicast ad broadcast capacity, i particular we compare ad cotrast how they are iflueced by chages i the radio rage (R) ad the iterferece parameter ( ). As metioed earlier, for ay etwork topology the broadcast capacity does ot chage more tha by a costat whe the radio rage varies. However, varyig the radio rage strogly iflueces the uicast capacity for etworks i two or three dimesioal space. Curiously i oe dimesioal space, because uicast ad broadcast are very similar, the radio rage does ot chage the uicast capacity more tha by a costat just like for the broadcast capacity. e fid that whe the iterferece parameter ( ) is large it has the same effect o both uicast ad broadcast capacity; both vary accordig to Θ( d ). However, if approaches zero the i some etworks the uicast capacity ca become as large as Θ(), ulike the broadcast capacity which will vary at most by a costat factor. The paper is orgaized as follows. I Sectio we summarize existig work o the etwork capacity. e itroduce a wireless etwork model ad defie relevat terms i Sectio 3. I Sectio 4 we compute upper ad lower bouds for broadcast capacity for geeral wireless etworks. e also derive specific results for the broadcast capacity for homogeeous dese etworks. Sectio 5 studies the maximum throughput of broadcast schemes ad compares them to the broadcast capacity. I Sectio 6 we compare the variatio of broadcast ad uicast capacities with differet etwork parameters. Fially, we coclude the paper i Sectio 7. All proofs are placed i the Appedix.. RELATED ORK Gupta ad Kumar [] study the etwork capacity for uicast coectios betwee radom sources ad destiatios i static wireless etworks cosistig of odes distributed i a circle of area A with wireless chael capacity. They defie the trasport capacity of a wireless etwork with uits of bit-meters per secod as the maximum rate of the packets times the distace they travel betwee the source ad the destiatio. Their mai result is that the aggregate trasport capacity of uicast coectios is Θ( A) i a arbitrary etwork with optimally placed odes ad Θ( A/log()) i a radom etwork where the odes are placed uiformly. As a result, the capacity of the etwork per ode is Θ( A/) (i radom etworks Θ( A/ log())) whe grows i a fixed area. I order to achieve a throughput withi a costat factor of the capacity, the radio rages of the odes must be set equal to Θ( A/) (i radom etworks Θ( A log()/)). The same authors also aalyze three dimesioal etworks [11]. They prove that if the odes are distributed i a sphere with volume V the the aggregate trasport capacity is Θ( 3 V ). Several other papers have ehaced the theory of etwork capacity. The results of Gupta ad Kumar were geeralized for a more accurate chael model [3]. Usig percolatio theory techiques [18] it was proved that Θ( A/) is achievable i radom etworks with high probability [4]. For wireless mobile etworks, Grossglauser ad Tse [5] show that per ode capacity ca be icreased to Θ(1) if packet delay is left ubouded. They propose a mobilitybased routig method i which the umber of retrasmissios of the uicast packets betwee source ad destiatio is reduced to. May other efforts demostrate that there is a trade-off betwee the capacity ad the delay i wireless mobile etworks, for differet mobility patters ad costraits o delay [19 5]. Itroducig a ew directio i etwork capacity research, Zheg studies the broadcast capacity of static wireless etworks [10]. The paper models the locatios of the odes through a Poisso poit process ad the chael as a Gaussia wireless chael whose capacity is give by the Shao Sigal to Iterferece plus Noise Ratio (SINR). It the computes the asymptotic bottleeck of the etwork by usig the distace properties of a Poisso poits process ad the SINR of the Gaussia chael. The idea is as follows. There exists a ode with high probability that is at a large distace from all other odes. Sice the capacity of the chael betwee ay pair of odes decreases with the icreasig distace betwee them, the maximum receivig rate is low for that ode. This receivig rate provides a upper boud for broadcast capacity. Zheg proves that the broadcast capacity is a costat factor of the computed upper boud whe the Poisso itesity of the odes is fixed ad umber of odes goes to ifiity. However, the paper does ot address the issue whether the upper boud O(log ) is achievable or ot, i the case whe the itesity teds to ifiity as the umber of odes grows, ad leaves the problem for the future study. This work does ot derive bouds for the broadcast capacity whe the umber of odes is fiite or whe the odes have a differet distributio tha Poisso i the plae. Our work differs from Zheg s work i several ways. First, 40

3 we compute bouds of the broadcast capacity for geeral wireless etworks with a fiite umber of odes ad a arbitrary topologies. Secod, we also take ito accout the iterferece faced by packet trasmissios of oe ode from trasmissios of its eighbors which was eglected by Zheg i her derivatio of a upper boud for capacity [10]. Our results show that this factor has a strog effect o broadcast capacity. Note that all the above metioed papers as well as this paper assume oly poit-to-poit codig at the receivers. If the odes are allowed to cooperate ad use sophisticated multi-user codig the a per-ode capacity of a higher order tha that described above ca be achieved [6 8]. A full discussio of these results is beyod the scope of this paper 3. IRELESS CHANNEL MODEL AND BASIC NOTIONS I this sectio we describe the wireless etwork model ad defie several terms relevat to our aalysis of broadcast capacity. 3.1 Network ad Coectivity e cosider a wireless etwork cosistig of wireless odes. Let X i for i = 1,,..., deote the locatio of the differet odes. For simplicity we also use X i to refer to the i th ode itself. All odes use the same bit-rate to trasmit data. e deote by G(R) the geometric graph formed by the odes whe each ode has trasmissio radius R. The vertices of G(R) are the odes of the etwork. Two odes X i ad X j are adjacet, that is joied by a edge, i G(R) if ad oly if X i X j R. Note that icreasig R ca oly icrease the umber of edges i G(R). A domiatig set of a etwork graph is defied as the set of odes such that every ode i the etwork is either i the set or has a adjacet ode which is i the set. I other words, a domiatig set is the set of odes i a wireless etwork which cover all odes. A Coected Domiatig Set (CDS) is a domiatig set such that the subgraph iduced by its odes is coected. A Miimum Coected Domiatig Set (MCDS) is a CDS of the graph with the miimum umber of odes. If a broadcast packet is received by all odes i the etwork the the set of odes which trasmit the packet build a CDS. Clearly, a MCDS uses the miimum umber of trasmissios to dissemiate the broadcast packet to all odes. A Idepedet Set is defied as a set of odes such that o two of them are adjacet i the graph. Nodes of a idepedet set are spaced far apart from each other, that is they are sparse i the etwork. A Maximum Idepedet Set (MIS) is a idepedet set of the graph with the maximum size. Note that the differet sets we have defied above will chage with the radio rage. I this paper, we thus iclude the radio rage R i the otatio for the defied sets. For example MCDS(R 1) is a MCDS of G(R 1) ad MIS(R ) is a MIS of G(R ). e use the symbol # for represetig the size of a set. I this paper we assume that for the radio rage R the etwork is coected. he we vary the radio rage of the odes we assume that the rage does ot get so small that the etwork becomes discoected. 3. Chael Model e employ the Protocol Model for modelig successful trasmissios [, 11 13]. The rules for successful receptio of a packet are as follows. Assume that ode X i trasmits a packet to ode X j. The the trasmissio is successfully received by X j if ad oly if 1. the Euclidea distace betwee X i ad X j is less tha R X i X j R, (1). ad for every ode X k that trasmits durig trasmissio of X i to X j X k X j (1 + )R. () e refer to as the iterferece parameter. The circular area with radius R ad ceter of X i is called trasmissio area of X i. Oly odes located withi this area ca receive packets from X i successfully. The larger circular area with radius (1 + )R ad cetered at X i is called the iterferece area of X i. Durig a trasmissio from X i, ay ode X j withi this area is blocked from receivig a packet from ay ode other tha X i. The aular part of the iterferece area that lies outside of trasmissio area is called the shadow area. It follows that the odes i the shadow area of X i do ot receive ay packet successfully while X i is trasmittig. The Protocol Model allows us to aalyze the broadcast capacity for geeral etwork topologies ad apply graph theoretic methods for the aalysis. Furthermore, this model is easier to uderstad tha Physical Models such as the oes of [, 3] ad it allows to study the effects of iterferece i terms of the simple parameter o the etwork capacity. 3.3 Broadcast Capacity ad Maximum Throughput e defie broadcast capacity for a subset B := {B 1, B,...} of odes that geerate broadcast packets. The reaso we do so is that i some etworks oly a few odes may be required to broadcast packets. I such cases we are iterested to kow the maximum rates at which this particular subset of odes ca successfully broadcast packets rather tha the maximum broadcast rates whe all odes (or oly a sigle ode) geerate broadcast packets. Assume that B i geerates packets at rate λ Bi 0. e say that the rate vector [λ Bi ] i is achievable if all odes of the etwork receive all geerated broadcast packets successfully withi some give time T max <. I this paper we study the maximum achievable broadcast rates for origiatig odes B whe the fractio of the aggregate rate that each ode uses is prespecified. That is, give a vector of weights g = [g i] #B i=1, gi > 0 such that i gi = 1 we study the broadcast capacity λ B(g) := sup{a : λ Bi = g ia, [λ Bi ] i is achievable} (3) ad the maximum throughput of scheme S λ S,B(g) := sup{a : λ Bi = g ia, [λ Bi ] i is achievable for S}. (4) The broadcast capacity, λ B(g), ad maximum throughput of scheme S, λ S,B(g), are related: λ S,B(g) sup λ S,B(g) = λ B(g). (5) S 41

4 e later derive bouds for broadcast capacity λ B(g), ad maximum throughput of scheme S, λ S,B(g). e fid that the correspodig lower ad upper bouds are always withi a costat factor of each other ad idepedet of g. e thus subsequetly drop the argumet g ad simply refer to broadcast capacity as λ B ad maximum throughput of S as λ S,B. 4. BROADCAST CAPACITY OF IRELESS NETORKS I this sectio we compute bouds for the broadcast capacity of wireless etworks which determie the capacity up to a small costat factor. e first prove bouds that apply to arbitrary coected etworks. e the improve these bouds for homogeeous dese etworks ad fially discuss whether mobility ca improve the broadcast capacity. e assume throughout that the trasmissio radio rage R of odes is large eough to esure that static etworks are coected ad that mobile etworks stay coected with probability oe as the odes move. 4.1 Broadcast Capacity for Geeral ireless Networks e ow determie differet upper ad lower bouds for the broadcast capacity that apply to ay arbitrary coected wireless etwork. The accuracy of ay oe of these bouds varies with the etwork sceario. hile oe boud may be more accurate tha aother boud i oe particular etwork, i a differet etwork the opposite may be true. Ideally we would like to compute the broadcast capacity up to a costat factor. Oe way to do this is to determie upper ad lower bouds that are tight, that is if they differ by at most a costat factor. Our first bouds are summarized i Theorem 1. Sice i a successful broadcast every ode must receive the data, the broadcast capacity caot be higher tha the maximum data rate at which a ode ca receive data. e thus obtai the upper boud,, which is a hard upper boud for ay etwork (small, large, static or mobile). e obtai the lower boud with the help of the MCDS of a etwork. Note that give a MCDS, we ca broadcast a message throughout the etwork by makig each MCDS ode retrasmit it oce. As we demostrate i the proof, this ca always be performed with less tha #MCDS(R) +1 trasmissios which gives the lower boud. Theorem 1. For a arbitrary coected wireless etwork λb. (6) #MCDS(R) + 1 The lower boud i Theorem 1 ca be difficult to evaluate i practice for ay arbitrary etwork. e preset bouds that are easier to evaluate i the ext theorem. Theorem shows that if the odes ca be covered by a domiatig set of size M the the broadcast capacity is larger tha times a factor that depeds oly o M. To use this lower boud all we have to do is fid a domiatig set ad the compute its size M. Clearly the boud approaches whe M becomes smaller. Oe practical way to reduce M is to icrease the radio rage of odes, R. Theorem. If i a coected wireless etwork all odes are covered by M trasmissios, the λb. (7) 3M 1 Followig a differet lie of reasoig, Theorem 3 gives aother lower boud for the broadcast capacity which oly depeds o the iterferece parameter. This boud outperforms the lower boud of Theorem 1 for very large etworks. e derive the result usig a TDMA schedulig method that reduces the iterferece betwee simultaeous trasmissios. As a cosequece this boud improves o the oe i Theorem 1 which was derived by allowig oly oe trasmissio at ay give time. Theorem 3. For a arbitrary coected wireless etwork i the plae λb. (8) ( + ) Note that the iterferece parameter is usually a small umber i wireless ad hoc etworks. Thus the deomiator of the lower boud i Theorem 3 will i practice ot get very large. The bouds we fid for the etwork capacity i Theorems 1 ad 3 are eough to determie the etwork capacity up to a costat. e ext go further ad study more carefully the effect of iterferece ad topology of the etwork o the broadcast capacity. e assume that the etwork is large eough to cotai at least two odes which are outside each other s iterferece area. Theorem 4 computes a upper boud i terms of the ratio of the size of MCDS(R) to the size of MIS( R). Essetially, #MCDS(R) equals the miimum umber of retrasmissio required to broadcast a packet ad #MIS( R) the maximum possible umber of successful simultaeous trasmissios i the etwork possible. e call a idividual trasmissio of the broadcast packet successful if at least oe ode receives the trasmitted packet successfully. Ituitively, the ratio #MCDS(R)/#MIS( R) thus approximately represets the umber of packet trasmissio time uits required to broadcast a message. Cosequetly, the broadcast capacity is iversely proportioal to this quatity. Theorem 4. For a arbitrary coected wireless etwork λ B #MIS( R) #MCDS(R). (9) Corollary 5 shows that if > ad the etwork is large eough that there are at least two odes which do ot iterfere with each other the the broadcast capacity has a smaller upper boud tha that i Theorem 1. I additio, if the etwork size is so small as compared to the iterferece rage that oly oe successful trasmissio per time ca occur, the the broadcast capacity upper boud becomes /#MCDS(R). Corollary 5. I a coected wireless etwork, if #MIS( R) > 1 ad > the λ B 4

5 ad if #MIS( R) = 1 the. λ B #MCDS(R) The results of the Theorems 1, 3 ad 4 ca be summarized as follows: Case (i): If #MIS( R) = #MCDS(R) λb #MCDS(R). (10) Case (ii): If #MIS( R) > 1 ad max( 1 + #MCDS(R), ) λb. ( + ) (11) Case (iii): If #MIS( R) > 1 ad > max( 1 + #MCDS(R), ( + ) ) λb. (1) e have already provided some ituitio for cases (i) ad (ii) listed above. e ow discuss the third case which makes more explicit the effect of the iterferece parameter o the broadcast capacity. Case (iii) cosiders a etwork that is large compared to the iterferece area of a sigle ode. I additio it assumes that the iterferece parameter is very large. e observe from (1) that the upper boud decreases at a markedly slower rate with tha the lower boud. hile the upper boud has order Θ( ), the lower boud has order Θ( ). Iterestigly, there exist two very differet classes of etwork topologies such that for oe the broadcast capacity tracks the upper boud of (1) ad for the other the broadcast capacity tracks the lower boud of (1). Cosider the topology i which the odes are distributed o a lie. For this topology the capacity is a costat factor of the upper boud. Now cosider the topology i which the odes are homogeeously distributed ad dese i a plae. Here the capacity is a costat factor of the lower boud. The broadcast capacity for these two cases will be studied i greater detail i Theorem 8. This observatio motivates us to seek a more accurate relatioship betwee topology ad broadcast capacity. Let us suppose that the topology is such that durig a broadcast we are forced to retrasmit the same packet K i times withi the iterferece rage of a particular ode X i. The X i ca receive broadcast data at a maximum rate of /K i. e coclude that / max i(k i) is a upper boud of broadcast capacity, which is caused by iterferece. e ow approximate the quatity max i(k i) usig a MIS(R). Because a MIS(R) is a idepedet domiatig set, the umber of its elemets that lie i the iterferece rage of X i is approximately equal to the umber of times we retrasmit the same packet withi its iterferece rage, K i. Cosequetly, we approximate max i(k i) by the bottleeck factor, K(R), which we defie for a particular MIS(R) as the maximum umber of its elemets that lie i the iterferece rage of ay sigle ode. Note that K(R) is a fuctio of radio rage R. Theorem 6 provides tight bouds o the capacity i terms of the bottleeck factor. Theorem 6. Assume that i a coected wireless etwork #MIS( R) > 1. The, there are positive costat umbers c 1 ad c idepedet of the etwork parameters such that c 1 λb c K(R) K(R). (13) Moreover, we ca show that +1 3 K(R) 4( +1.5). 4. Broadcast Capacity for Homogeeous ireless Networks e ow study the broadcast capacity for a wireless etwork model which is widely employed i studyig uicast etwork capacity. I this model, the odes are distributed uiformly i a circle or square of area A, the umber of odes grows to ifiity. The broadcast capacity bouds have a simple form i homogeeous dese etworks because the maximum umber of simultaeous trasmissios ad the MCDS size ca be estimated i terms of area ad radio rage for these etworks. Moreover, the bouds we compute here are closer tha the bouds which we computed before for arbitrary wireless etworks i Sectio 4.1. Theorem 7 gives tight bouds for the broadcast capacity i a homogeeous dese etwork where the odes are distributed withi a square area. Theorem 7. If the odes are uiformly distributed i a square with area A 1 4 R 15A log(), ad R the, + (1 + ) 5 λb mi(1, 96 π ) (14) almost surely for large. Theorem 8 geeralizes this result for the homogeeous etworks where the odes are distributed i a d-dimesioal cube. Theorem 8. If the odes are uiformly distributed i a d-dimesioal cube with volume V c d R d (c > 0 a costat umber) ad R d + 3 d 3V log(), the there exist positive umbers c 1 ad c idepedet of etwork parameters, such that c 1 almost surely for large. max(1, d λb c ) max(1, d ) (15) It is well kow that the radio rage R must be larger tha R c := d V log() i order to have coectivity i the etwork with high probability [9]. e see that R d + 3 d 3R c satisfies the coditio of Theorem 8. Oe might woder if (15) holds whe R c < R d + 3 d 3R c. Ideed, it ca be show that this is the case. The proof of this result is however quite ivolved ad is hece ot icluded i this paper. 43

6 4.3 Broadcast Capacity for Mobile Networks Previous work by Grossglauser ad Tse [5] showed that mobility ca cosiderably icrease the aggregate uicast capacity of homogeeous dese etworks to Θ(). The questio is: Ca mobility similarly icrease the broadcast capacity of homogeeous dese etworks? Our results i previous sectios reveal that mobility ca icrease capacity by at most a factor that depeds o. From Theorem 1 we see that is a hard upper boud for the broadcast capacity of ay wireless etwork icludig mobile oes. Usig this result ad the lower boud of Theorem 8 we ca boud the maximum icrease i broadcast capacity that mobility ca deliver. he < 1, mobility ca icrease the broadcast capacity by at most a costat factor of 1/c 1 ad whe 1 by at at most a factor d /c 1. Ituitively, a mobile etwork ca have a higher capacity tha a static etwork if mobility moves odes away from each other s shadow areas. I mobile etworks, if we ca egieer (cotrol) the mobility of the odes the the broadcast capacity ca always be icreased to. For example, oe way of doig this is to move all odes i the etwork ito the trasmissio area of oe particular ode of the etwork. The this ode ca broadcast at rate successfully. However, agai the mobility does ot icrease the broadcast capacity more tha a factor which oly depeds o ad d, i cotrast to uicast capacity which ca be icreased by factor of Θ( 1 d ) usig the mobility. 5. MAXIMUM THROUGHPUT OF BROADCAST SCHEMES I this sectio we study the maximum throughput of differet broadcast schemes. Several schemes have bee proposed i the literature that efficietly broadcast a packet (see [15 17] ad refereces therei). Although the efficiecy of these schemes has bee aalyzed extesively, o work has bee performed to aalyze their maximum throughput ad compare it with the broadcast capacity. e show that the maximum throughput of a broadcast scheme depeds o properties such as the size ad topology of the broadcast backboes it uses. The size of the backboe has a strog ifluece o the throughput because it represets the umber of retrasmissios for every broadcast packet. Ituitively, the broadcast schemes with smaller backboe sizes are more efficiet i utilizig the capacity of the etwork ad so ca have higher throughput. Theorem 9 provides a lower boud i terms of backboe size for the maximum throughput of the broadcast scheme. Theorem 9. Let Backboe(R) be the broadcast backboe of a give broadcast scheme with the radio rage R i a coected wireless etwork. The, λs,b λb. #Backboe(R) + 1 Theorem 10 computes a upper boud for the maximum throughput of a give scheme i terms of the backboe size. Theorem 10. Let Backboe(R) be the broadcast backboe of a coected wireless etwork. The, λ S,B #MIS( R) #Backboe(R). (16) From Theorem 10 we see that if #MIS( R) is bouded above by a costat umber as the umber of odes grows the the upper boud of capacity i (16) ad the lower of Theorem 9 are tight. I practice this coditio will hold if the area of the etwork remais costat as the umber of odes icreases. For the rest of the discussio we require otios of efficiet ad iefficiet broadcast schemes. A efficiet broadcast scheme is a scheme that bouds the umber of broadcast odes per radio rage. As a result it always builds a backboe of size withi a costat factor of the size of the MCDS. Some schemes which build broadcast backboes usig a MIS have bee proved to be efficiet [30 37]. e refer to all other broadcast schemes as iefficiet broadcast schemes. e prove i Theorem 11 that for efficiet broadcast schemes the maximum throughput is always withi a costat factor of the broadcast capacity. Note that i our aalysis we oly take ito accout the effect of backboe size o capacity ad ot other factors such as the schedulig of the packet trasmissios from differet odes which is a MAC layer operatio. Theorem 11. For a efficiet broadcast scheme there is a positive umber c which depeds oly o the scheme (ad ot the etwork parameters) such that for a arbitrary coected wireless etwork cλ B λ S,B λ B. e show i Theorem 1 that for iefficiet broadcast schemes, there exists a sequece of etworks of icreasig size for which the maximum broadcast throughput teds to zero. A simple example of this situatio is whe umber of odes i a bouded area goes to ifiity ad we use blid floodig for broadcast. However, recall from Theorem 3 that the broadcast capacity λ B has a positive lower boud. e coclude that λ S,B/λ B caot be lower bouded by a positive costat for iefficiet broadcast schemes. Thus iefficiet schemes must ot be employed for broadcast i dese etworks. Theorem 1. If i a coected wireless etwork broadcast scheme > 0 ad 0 as umber of #MCDS(R) #Backboe(R) odes the λ S,B 0. Theorem 1 ot oly emphasizes the eed for efficiet broadcast algorithms i dese etworks, it also clearly idicates the dager ad potetial pitfalls for eglectig iterferece whe aalyzig of broadcast protocols. Ideed, e see that if > 0 the efficiecy is a ecessary ad sufficiet coditio for a scheme to have maximum throughput withi a costat factor of the broadcast capacity. However, ote that if = 0 the efficiecy is ot a ecessary coditio ay more. Figure 1 shows a etwork with iterferece parameter = 0. e see that the backboe size is 1 ad all backboe odes o the small circle ca sed simultaeously to the odes o the large circle. Hece whe the ode i the ceter is the origiatig ode of broadcast the throughput equals ; half the time the ceter ode trasmits ad durig the other half all odes o the small circle trasmit 44

7 I order to better uderstad this similarity betwee broadcast ad uicast capacity i oe dimesioal etworks, observe that whe the maximum umber of simultaeous trasmissios is proportioal to the average umber of uicast (resp. broadcast) retrasmissios the the uicast (broadcast) capacity becomes proportioal to. Sice for a broadcast packet the average umber of retrasmissios is proportioal to the maximum umber of simultaeous trasmissios, the capacity of the etwork for broadcastig packets will remai costat ad ot deped o the radio rage. The same thig happes to uicast packets whe the odes are distributed o a lie. Figure 1: The radii of the small ad large circles are ar ad (1 + a)r respectively, where 0 < a < 1. Sice the iterferece parameter is zero, if all odes o the small circle trasmit simultaeously the the odes o the large circle will receive successfully. simultaeously. However, the MCDS size for this etwork is less tha 13 irrespective of. Thus we see that although 0, the throughput is positive. #MCDS(R) #Backboe(R) 6. COMPARISON OF THE BROADCAST AND UNICAST CAPACITIES I this sectio we highlight some fudametal differeces betwee broadcast capacity ad uicast capacity. e do so by comparig how the broadcast ad uicast capacity of wireless etworks vary with radio rage R ad iterferece parameter. Fially, we study the capacity per ode for both uicast ad broadcast. All the aalysis i this sectio pertais to homogeeous dese etworks. 6.1 Effect of Radio Rage As discussed i Sectio 4, chagig the radio rage does ot chage the broadcast capacity by more tha a costat factor. Uicast capacity, i stark cotrast, has bee show to deped strogly o R [,11]. It has bee show that for odes distributed uiformly i a circle with area A, the maximum A umber of simultaeous trasmissios is Θ( ) ad uicast packets eed to be retrasmitted o average Θ( A ) R R times. Therefore the aggregate capacity of the etwork is A limited to Θ( / A ) = Θ( A ). The same techique ca be applied whe the odes are distributed i a R R R sphere with volume V to prove that etwork capacity has order Θ( V )/ 3 V ) = Θ( 3 V ). Both cases show that 3 R 3 R 3 R the radio rage must be miimized i order to maximize throughput. e ow cosider a oe dimesioal space ad assume that the odes are distributed uiformly o a lie segmet of legth D. The maximum umber of simultaeous trasmissios is Θ( D ) ad the average umber of retrasmissios R for uicast packets is Θ( D ). Cosequetly, the etwork capacity becomes Θ( D / D ) = Θ( ). Just like for broad- R R R cast capacity we see here that uicast capacity becomes idepedet of the radio rage ad the size of the area. 6. Effect of Iterferece Parameter The broadcast capacity of large homogeeous etworks is Θ( ) which does ot deped o whe < 1. max(1, d ) he > 1 the broadcast capacity decreases by the factor d. Earlier studies of capacity set the iferece parameter ( ) to a costat value [,11]. The impact of this parameter o capacity, i particular if it is very small or very large, has ot bee studied. Addressig this issue is importat because appears i uicast capacity bouds. I fact, a very small or large value of affects the uicast capacity sigificatly. he the iferece parameter is large ( >> 1), the uicast capacity will decrease by a factor of Θ( 1 ). I some cases, whe the iterferece parameter d is small ( << 1) the uicast capacity icreases largely. Earlier work presets the upper boud 8A ad 3 6V 3 π π for the aggregate uicast capacity i ad 3 dimesioal space [,11]. Figure shows a wireless etwork i the ad 3 dimesioal space ad a particular traffic patter. If the horizotal ad vertical spacig betwee odes i Fig. are related through ε 3, we ca easily show that the aggregate trasport capacities are ad 3 ε(1+ ) 4ε (1+ ) for sufficietly large. If we set ε = 3 the we see that the aggregate uicast capacities of these etworks are proportioal to 4 1 ad 3 1. I the special case of 0, we ca choose ε = O(1/) 3 ad the aggregate uicast capacity becomes Θ(). 6.3 Capacity per ode e here cosider the broadcast capacity per ode, that is λ B divided by the umber of odes. If B cosists of all odes i the etwork ad all odes get a equal share of the broadcast capacity, the the broadcast capacity per ode is Θ( ) which is less tha the uicast capacity per ode by a factor of Θ( ) i static etworks ad by a factor of Θ() i mobile etworks. However, B ad g (see Sectio 3) ca be chose arbitrarily, that is oly a few odes ca geerate broadcast packets ad these ca also share the badwidth uequally betwee them. For arbitrary B ad g we proved that we ca broadcast at aggregate rate Θ( ), that is withi a costat factor of sup B λ B. That meas the capacity for broadcastig is flexible for ay choice of B ad g i the etwork. I the case of uicast, however, we do ot ejoy such flexibility. I order to achieve the uicast capacity, the source ad destiatio odes must have special locatios i the etwork ad sed data at appropriate rates. 45

8 (a) (b) Figure : ireless etworks i ad 3 dimesioal space. The flows do ot iterfere with each other if ε 3. The aggregate trasport capacities of these etworks are ad 3 for large. ε(1+ ) 4ε (1+ ) 7. CONCLUSION AND FUTURE ORK e have proved usig the Protocol model that the broadcast capacity of wireless etworks is proportioal to the wireless chael capacity ( ). Furthermore, it does ot chage by more tha a costat factor whe the radio rage, the area of the etwork ad the umber of odes i the etwork vary. However, we explicitly computed the effect of the iterferece parameter o the capacity which depeds o the topology of geeral etworks. I the particular case of homogeeous dese etworks the broadcast capacity decreases 1 by a factor of O( ). Iterestigly, we showed that i max(1, d ) cotrast to the impact of mobility o uicast capacity, mobility caot chage the broadcast capacity by more tha a factor which oly depeds o the iterferece parameter. I additio, we studied the maximum throughput of broadcast schemes for wireless multihop etworks. e proved that a ecessary ad sufficiet coditio for a broadcast scheme to achieve a maximum throughput withi a costat factor of the broadcast capacity is that it relay o a backboe which has a bouded umber of odes per radio rage area. Fially, we studied the fudametal differeces betwee uicast ad broadcast capacity ad the effects of etwork parameters o them both. e foud that broadcast capacity does ot deped o the choice of source odes or the dimesio of the etwork ulike uicast capacity which does. For future work, we will study the broadcast capacity i mobile wireless etworks uder differet mobility models, for example radom uiform mobility ad egieered (cotrolled) models. I additio, we will ivestigate how differet chael models like the physical model [] ad Shao chael model [3, 10] ifluece the broadcast capacity. Ackowledgmet Fiacial support comes i part from NSF, grat umber ANI , ad from Texas ATP, project umber REFERENCES [1] X. Hog, K. Xu, ad M. Gerla, Scalable routig protocols for mobile ad hoc etworks, IEEE Network, vol. 16, pp. 11 1, 00. [] P. Gupta ad P. R. Kumar, The capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 46, o., pp , 000. [3] A. Agarwal ad P. R. Kumar, Capacity bouds for ad hoc ad hybrid wireless etworks, Computer Commuicatio Review, vol. 34, o. 3, pp , 004. [4] M. Fraceschetti, O. Dousse, D. Tse, ad P. Thira, O the throughput capacity of radom wireless etworks, IEEE Trasactios o Iformatio Theory, 004. [5] M. Grossglauser ad D. N. C. Tse, Mobility icreases the capacity of ad-hoc wireless etworks, i INFOCOM, 001, pp [6] R. Negi ad A. Rajeswara, Capacity of power costraied ad-hoc etworks, i INFOCOM, 004. [7] S. Toumpis ad A. J. Goldsmith, Large wireless etworks uder fadig, mobility, ad delay costraits, i INFOCOM, 004. [8] B. Liu, Z. Liu, ad D. F. Towsley, O the capacity of hybrid wireless etworks, i INFOCOM, 003. [9] S. Toumpis, Capacity bouds for three classes of wireless etworks: asymmetric, cluster, ad hybrid, i MobiHoc. ACM Press, 004, pp [10] R. Zheg, Iformatio dissemiatio i power-costraied wireless etwork, i INFOCOM, 006. [11] P. Gupta ad P. Kumar, Iterets i the sky: The capacity of three dimesioal wireless etworks, Commuicatios i Iformatio ad Systems, vol. 1, o. 1, pp , 001. [1] 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. [13] P. Kyasaur ad N. H. Vaidya, Capacity of multi-chael wireless etworks: impact of umber of chaels ad iterfaces, i MobiCom, 005, pp [14] S.-Y. Ni, Y.-C. Tseg, Y.-S. Che, ad J.-P. Sheu, The broadcast storm problem i a mobile ad hoc etwork, i MobiCom. ACM Press, 1999, pp [15] B. illiams ad T. Camp, Compariso of broadcastig techiques for mobile ad hoc etworks, i MobiHoc. ACM Press, 00, pp [16]. Lou ad J. u, Localized Broadcastig i Mobile 46

9 Ad Hoc Networks Usig Neighbor Desigatio. i Mobile Computig Hadbook, CRC Press, 005, ch. 8. [17] I. Stojmeovic ad J. u, Mobile Ad Hoc Networkig. IEEE Press, 004, ch. 7, pp [18] G. Grimmett, Percolatio. Secod editio, Spriger Verlag, [19] G. Sharma, R. R. Mazumdar, ad N. B. Shroff, Delay ad capacity trade-offs i mobile ad hoc etworks: A global perspective, i INFOCOM, 006. [0] M. J. Neely ad E. Modiao, Capacity ad delay tradeoffs for ad hoc mobile etworks, IEEE Trasactios o Iformatio Theory, vol. 51, o. 6, pp , 005. [1] S. N. Diggavi, M. Grossglauser, ad D. N. C. Tse, Eve oe-dimesioal mobility icreases the capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 51, o. 11, pp , 005. [] X. Li ad N. Shroff, Towards achievig the maximum capacity i large mobile wireless etworks. uder delay costraits, Joural of Commuicatios ad Networks, vol. 6, o. 4, p , 004. [3] G. Sharma ad R. Mazumdar, O achievable delay/capacity trade-offs i mobile ad hoc etworks, i iopt, 004. [4] A. E. Gamal, J. P. Mamme, B. Prabhakar, ad D. Shah, Throughput-delay trade-off i wireless etworks, i INFOCOM, 004. [5] N. Basal ad Z. Liu, Capacity, delay ad mobility i wireless ad-hoc etworks, i INFOCOM, 003. [6] P. Gupta ad P. R. Kumar, Towards a iformatio theory of large etworks: a achievable rate regio, IEEE Trasactios o Iformatio Theory, vol. 49, o. 8, pp , 003. [7] M. Gastpar ad M. Vetterli, O the capacity of wireless etworks: The relay case, i INFOCOM, 00. [8] G. A. Gupta, S. Toumpis, J. Sayir, ad R. R. Müller, O the trasport capacity of gaussia multiple access ad broadcast chaels, i iopt, 005, pp [9] R. B. Ellis, J. L. Marti, ad C. Ya, Radom geometric graph diameter i the uit ball, to appear Algorithmica, 005. [30] C. R. Li ad M. Gerla, Adaptive clusterig for mobile wireless etworks, IEEE Joural o Selected Areas i Commuicatios, vol. 15, o. 7, pp , [31] S. Basagi, Distributed clusterig for ad hoc etworks, i ISPAN. IEEE Computer Society, 1999, p [3] K. Alzoubi, P.-J. a, ad O. Frieder, Message-optimal coected domiatig sets i mobile ad hoc etworks, i MobiHoc. ACM Press, 00, pp [33]. Lou ad J. u, A cluster-based backboe ifrastructure for broadcastig i maets, i IPDPS. IEEE Computer Society, 003, p [34] P.-J. a, K. Alzoubi, ad O. Frieder, Distributed costructio of coected domiatig set i wireless ad hoc etworks, Mob. Netw. Appl., vol. 9, o., pp , 004. [35] J. u ad F. Dai, A distributed formatio of a virtual backboe i maets usig adjustable trasmissio rages, i ICDCS. IEEE Computer Society, 004, pp [36] G. Caliescu, I. I. Madoiu, P.-J. a, ad A. Z. Zelikovsky, Selectig forwardig eighbors i wireless ad hoc etworks, Mob. Netw. Appl., vol. 9, o., pp , 004. [37] A. Keshavarz-Haddad, V. Ribeiro, ad R. Riedi, Color-based broadcastig for ad hoc etworks, i iopt, 006. Appedix Proof of Theorem 1: Cosider a arbitrary ode X i i the etwork. The maximum rate of trasmissio or receptio of data by X i is. Sice X i must either receive or geerate all broadcast packets, the broadcast capacity has a hard upper boud of, irrespective of the mobility of the odes. To prove the lower boud, we desig a TDMA scheme that provides a broadcast rate equal to. Call #MCDS(R)+1 the differet odes of B, B i (i = 1,,..., #B). Set m = #MCDS(R). First B i (i = 1) trasmits w i bits at rate to its MCDS eighbor with the lowest idex. The i each of the ext m time slots of legth w i/ the MCDS ode with the lowest idex that has ot yet trasmitted the received w i bits rebroadcasts them to all odes i its radio rage. e thus broadcast w i bits to the etire etwork i time (m+1)w i/, that is at rate /(m+1). e repeat the same procedure for i =, 3,..., #B, 1,... of set B. Node B i thus geerates broadcast data at rate λ Bi = m+1 w i i wi = m + 1 w i (17) i wi ad λ B = /(m + 1). From (17), we see that by choosig the w i s appropriately we ca support ay rates λ Bi that sum to /(m + 1). Proof of Theorem : Cosider M odes which cover all odes of the etwork. These odes build a domiatig set. e ca build a CDS by coectig -hop ad 3-hop away odes of the domiatig set o a spaig tree [3]. Therefore at most (M 1) additioal odes are eeded i order to build a CDS. Thus #MCDS(R) 3M. Theorem 1 the gives the result. Proof of Theorem 3: e desig a TDMA scheme for ay arbitrary wireless etwork which has a broadcast throughput equal to the lower boud. e do so i three steps. Step 1: Divide the etwork ito cells ad build a cell graph by coectig occupied cells. Divide the plae ito square cells with diagoal R such that the coordiates of their ceters are (i R, j R ) for i, j Z (see Fig. 3). Note that by desig every pair of odes i the same cell are withi each other s radio rage. Next, we build a cell graph over the occupied cells (the cells which cotai at least oe ode); these cells are colored grey i Fig. 3. The vertices of the cell graph are the occupied cells ad two cells are coected (adjacet) if there exists a pair of odes, oe i each cell, that are less tha R distace apart. Because the etwork is coected it follows that the 47

10 Figure 3: TDMA scheme collisio free for broadcastig. It uses k colors to schedule the cells trasmissios cell graph is coected. e the build a spaig tree over the cell graph which we use to route broadcast packets. Step : Color the cells with k colors L(0, 0), L(0, 1),..., L(k 1, k 1) such that cells with the same color are far apart. e assig color L(r i, r j) to the cell with ceter (i R, j R ), where r i = i(mod)k. The value of k is chose large eough such that whe two odes i differet cells with the same color trasmit simultaeously, all of the odes i their trasmissio rage ca receive successfully. It is easy to show by geometry that k = 1+ R+(1+ )R R/ = 1+ (+ ) is large eough to have this property. e the divide time ito k time slots which correspod to k differet colors. I each time slot oly odes i the cells with the correspodig color are allowed to trasmit. Step 3: Schedule packet trasmissios amog the odes. For every pair of adjacet cells o the spaig tree of the cell graph we choose two odes which coect the cells to be relays. he a packet eeds to be forwarded from cell S1 to adjacet cell S, the relay i cell S1 forwards the packet to the relay i cell S. The relay i S rebroadcast the packet to all odes i S. If S is ot a leaf vertex o the spaig tree of the cell graph the its relays which coect it to other cells will forward the packet ad the process cotiues this way till the broadcast packet dissemiates to all odes. By geometry, it is easy to show that every cell has at most 0 adjacet cells. There are thus at most 0 relays i each cell. e divide the time slot correspodig to each color ito 1 equal time slots of legth T. I the first time slot correspodig to every color, each ode B i B geerates T λ Bi /(1k λ B) bits for broadcast, if more tha oe B i s are located i the same they broadcast the packets i some order after each other. I the remaiig 0 time slots of ay particular color, the relays of cells with that color trasmit (to other odes the cell or to the correspodig relay of adjacet cell) ay broadcast data that they have received but ot yet forwarded. 1k Note that every relay ca forward packets at rate with this setup. Thus whe λ B <, the broadcast backlog at every relay will always be less tha T bits. 1k Proof of Theorem 4: First, we prove that the umber of simultaeous successful trasmissios i the etwork is bouded by #MIS( R). e call a broadcast trasmissio successful if at least oe ode receives the trasmitted packet successfully. Cosider two simultaeous trasmitters X i, X j ad assume that X i is trasmittig a bit b to a ode X r. For a successful trasmissio, it is ecessary that X i X r R ad X j X r (1 + )R. The X j X i X j X r X r X i R. This meas that simultaeous trasmitters eed to be at least distace R apart from each other. Sice MIS( R) is the largest set of odes with the property that every pair of its elemets are at least R apart from each other, #MIS( R) is a upper boud o the umber of simultaeous trasmissios. If the umber of simultaeous trasmissio becomes larger tha this, the some of trasmissios become redudat because o ode receives their packets successfully. Assume that the broadcast packets are geerated with the rate of λ B bits per secod. Deote by N T the umber of geerated broadcast bits i [0, T ]. By defiitio λ B = N lim T T T. Also, deote by N B(b) the umber of times ay bit b is trasmitted i order to be received by all odes. The total umber of bits that must be trasmitted for the broadcast bits geerated i [0, T ] is thus N T NB (b) b=1 i=1 1. Sice all broadcast packets are received i a limited time (T max), at time T + T max all trasmissios of N T bits are fiished. Therefore, N T b=1 N B (b) i=1 Sice N B(b) #MCDS(R) 1 #MIS( R)(T + T max) (18) N T N T #MCDS(R) b=1 N B (b) i=1 1 (19) By combiig the two pervious iequalities we have N T λ B = lim T T #MIS( R) #MCDS(R) lim T T + T max. (0) T Proof of Corollary 5: If #MIS( R) > 1 cosider the circles with radius R cetered at odes of a fixed MIS( R). Sice every two odes of MIS( R) are at distace larger tha R from each other, the circles are disjoit. Now cosider a fixed MCDS(R). Its odes build some paths betwee the odes at the ceters of the circles. Sice the MCDS odes coect the ceter to some odes outside the circle, we ca show by geometry that the circle cotais at least 1 MCDS odes whe >. Thus #MIS( R) By combig (1) with Theorem 4, we have λ B #MIS( R) #MCDS(R) 1 #MCDS(R). (1). () 1 The case #MIS( R) = 1 is straightforward. Proof of Theorem 6: First, we compute the upper ad lower bouds for K(R). Cosider a ode X u MIS( R). The circle C(X u, R) (X u is the ceter ad R is the radius) does ot cotai all odes i the etwork because #MIS( R) > 1. Now cosider the CDS built by coectig -hop ad 3-hop away odes i MIS(R) [31, 3]. e ca show by geometry that the CDS which coects X u to the odes outside of the circle cotais at least