Broadcast Throughput Capacity of Wireless Ad Hoc Networks with Multipacket Receptio Zheg Wag, Hamid R. Sadjadpour, J.J. Garcia-Lua-Aceves Departmet of Electrical Egieerig ad Computer Egieerig Uiversity of Califoria, Sata Cruz, 56 High Street, Sata Cruz, CA 95064, USA Palo Alto Research Ceter PARC), 3333 Coyote Hill Road, Palo Alto, CA 94304, USA Email:{wzgold, hamid, jj}@soe.ucsc.edu Abstract I this paper, we study the broadcast throughput capacity of radom wireless ad hoc etworks whe the odes are edowed with multipacket receptio MPR) capability. We show that with MPR, per ode throughput capacity of Θ R ) ) bits per secod ca be achieved as a tight boud i.e. upper ad lower bouds) for broadcast commuicatio, where R) is the receiver rage which depeds o the complexity of the odes. MPR achieves miimum gai of Θlog ) compared to poit-topoit routig whe the miimum value for R) is chose to guaratee coectivity i the etwork. I. INTRODUCTION Gupta ad Kumar i their semial work [] established that per ode throughput capacity of wireless ad hoc etwork with multi-pair uicast traffic scales as Θ / log ). Sice the, researchers have paid a lot of attetio to uderstad the throughput capacity of wireless ad hoc etworks for differet types of commuicatios such as multicast ad broadcast [3] [6]. Ufortuately, plai multihop routig i [] caot scale which leads to more ivestigatio i fidig ew ways for scalig wireless ad hoc etwork. Without exploitig ode mobility [7], i static etworks, the oly two possible approaches to icrease the order capacity of a ad hoc etwork cosist of ) icreasig the amout of iformatio i each relay by utilizig more badwidth, or ) avoidig iterferece by icreasig cooperatio amog odes. Work has bee carried out i both frots. If badwidth is allowed to icrease proportioally to the umber of odes i the etwork [8], higher trasport capacities ca be attaied for static wireless etworks. Ozgur et al. [9] proposed a hierarchical cooperatio techique based o virtual MIMO to achieve liear capacity scalig. Cooperatio ca also be exteded to the simultaeous trasmissio ad receptio by adjacet odes i the etwork, which ca result i sigificat improvemet i capacity []. MPR is aother cooperative techique that was first preseted by Ghez et al. [0], []. MPR allows multiple odes to trasmit their packets simultaeously to the same receiver ode ad the receiver decodes all these packets successfully. Ulike simple multi-hop routig [] that is based o avoidig R) is the receiver rage [] of MPR to distiguish from trasmissio rage r) i poit-to-poit commuicatios. The miimum value for receiver rage is R) = r) = Θ log /) that guaratees coectivity i the etwork []. iterferece, MPR techique embraces iterferece i wireless ad hoc etworks [], [3]. I this cotext, multiple odes cooperate to trasmit their packets simultaeously to a sigle ode usig multiuser detectio MUD), directioal ateas DA) [4], multiple iput multiple output MIMO) techiques, or physical or aalog etwork codig [3], [5]. Furthermore, Toumpis ad Goldsmith [6] have show that the capacity regios of ad hoc etworks are sigificatly icreased whe multiple access schemes are combied with spatial reuse i.e., multiple simultaeous trasmissios), multihop routig i.e., packet relayig), ad successive iterferece cacelatio SIC). I our previous work [], we have proved that the throughput capacity with MPR is Θ R)) i multipair uicast applicatios. Whe R) = Θ / to guaratee coectivity criterio, it achieves a gai of Θlog ) compared to multi-hop routig scheme of []. Broadcast capacity has also attracted the attetio of may researchers. Tavli [3] was first to show a boud of Θ/) for broadcast capacity of arbitrary etworks. Zheg [4] derived the broadcast capacity of power-costraied etworks together with aother metric called iformatio diffusio rate for sigle source broadcast. The work by Keshavarz et al. [5] is the most geeral case of computig broadcast capacity for ay umber of sources i the etwork. Our work was ispired by some of the cotributios i this paper. The mai cotributio of this paper is that with MPR, per source-destiatio broadcast capacity C B ) of wireless radom ad hoc etwork is tightly bouded by ΘR )) upper ad lower bouds) w.h.p., where R) is the receptio rage [], [] of a receiver. Furthermore, we ote that whe R) = r) Θ /, the broadcast capacity has the miimum gai of Θlog ) compared to the broadcast capacity of simple multi-hop routig [5]. This paper is orgaized as follows. Sectio II describes the etwork model ad defiitios. The upper ad lower bouds broadcast capacity of MPR will be preseted i Sectios III ad IV respectively. We coclude the paper i Sectio V. We say that a evet occurs with high probability w.h.p.) if its probability teds to oe as goes to ifiity. Θ, Ω ad O are the stadard order bouds.
II. NETWORK MODEL, DEFINITIONS, AND PRELIMINARIES We assume a radom wireless ad hoc etwork with odes distributed uiformly i the etwork area. Our aalysis is based o dese etworks, where the area of the etwork is a square of uit value. All odes use a commo trasmissio rage r) for all their commuicatios 3. Our capacity aalysis is based o extedig the protocol model for dese etworks itroduced i []. The Protocol Model i [] was defied as follows. Node X i ca successfully trasmit to ode X j if for ay cocurret trasmitter ode X k, k i, we have X i X j r) ad X k X j + )r). I wireless etworks with MPR capability, the protocol model assumptio allows multipacket receptio of odes as log as they are withi a radius of R) from the receiver ad all other trasmittig odes have a distace larger tha + )R). The differece is that we allow the receiver ode to receive multiple packets from differet odes withi its disk of radius R) simultaeously. Note that r) i poit-to-poit commuicatio is a radom variable while R) i MPR is a predefied value which depeds o the complexity of receivers. MPR model is equivalet of mayto-oe commuicatio. We assume that odes caot trasmit ad receive at the same time which is equivalet to half duplex commuicatios []. The data rate for each trasmitter-receiver pair is a costat value of W bits/secod ad does ot deped o. Give that W does ot chage the order capacity of the etwork, we ormalize its value to. The relatioship betwee receiver rage of MPR throughout i this paper ad trasmissio rage i [] is defied as ) log R) = r) Θ. ) Defiitio.: Throughput broadcast capacity: I a wireless etwork with odes i which each source ode trasmits its packets to all odes, a throughput of C B ) bits per secod for each ode is feasible if there is a spatial ad temporal scheme for schedulig trasmissios, such that by operatig the etwork i a multi-hop fashio ad bufferig at itermediate odes whe awaitig trasmissio, every ode ca sed C B ) bits per secod o average to all odes. That is, there is a T < such that i every time iterval [i )T, it ] every ode ca sed T C B ) bits to all odes. Defiitio.: Order of throughput capacity: C B ) is said to be of order Θf)) bits per secod if there exist determiistic positive costats c ad c c < c ) such that lim Prob C B) = cf) is feasible) = lim Prob C B) = c f) is feasible) <. Defiitio.3: Miimum Coected Domiatig Set MCDS R))): A domiatig set DS R))) of a graph G is defied as a set of odes such that every ode i the 3 I this paper, we assume receiver rage is equal to trasmissio rage ) graph G either belogs to this set or it is withi a rage R) of oe elemet of DSR)). A Coected Domiatig Set CDS R))) is a domiatig set such that the subgraph iduced by its odes is coected. A Miimum Coected Domiatig Set MCDS R))) is a CDSR)) of G with the miimum umber of odes. Defiitio.4: Maximum Idepedet Set MIS, r))): A Idepedet Set IS, r)) of a graph G is a set of vertices i G such that the distace betwee ay two elemets of this set is greater tha r). The MIS, r)) of G is a ISr)) such that, by addig ay vertex from G to this set, there is at least oe edge shorter tha or equal to r). We ote that MIS, r)) is a uique largest idepedet set for a give graph. Fidig such a set i a geeral graph G is called the MIS problem ad is a NP-hard problem. I [5], MIS, r)) is used to describe the maximum umber of simultaeous trasmitters i plai routig scheme. We defie a ew cocept, called Maximum MPR Idepedet Set MMIS), to describe the same cocept whe MPR scheme is used. Defiitio.5: Maximum MPR Idepedet Set MMIS, R))): A MPR set is a set of odes i G that cotais oe receiver ode ad all trasmittig) odes withi a distace of R) from this receiver ode. A Maximum MPR Idepedet Set MMIS, R))) cosists of the maximum umber of MPR sets that simultaeously trasmit their packets while MPR protocol model is satisfied for all these MPR sets. If we add ay trasmitter ode from G to MMIS, R)), there will be at least oe MPR set that violates the MPR protocol model. I this paper, T deotes the statistical average of T ad #T defies the total umber of vertices odes) i a tree T. The distributio of odes i radom etworks is uiform, so if there are odes i a uit square, the the desity of odes equals. Hece, if S deotes the area of regio S, the expected umber of the odes, EN S ), i this area is give by EN S ) = S. Let N j be a radom variable defiig the umber of odes i S j. The, for the family of variables N j, we have the followig stadard results kow as the Cheroff bouds [7]: Lemma.6: Cheroff boud For ay δ > 0, P [N j > + δ) S j ] < ) Sj e δ +δ) +δ For ay 0 < δ <, P [N j < δ) S j ] < e Sj δ Combiig these two iequalities we have, for ay 0 < δ < : P [ N j S j > δ S j ] < e θ Sj, 3) where θ = + δ) l + δ) δ i the case of the first boud, ad θ = δ i the case of the secod boud. Therefore, for ay θ > 0, there exist costats such that deviatios from the mea by more tha these costats occur with probability approachig zero as. It follows that we ca get a very sharp cocetratio o the umber of odes i a area. Thus, we ca fid the achievable lower boud w.h.p., provided that the upper boud mea) is give. I the
ext two sectios, we first derive the upper boud, ad the use the Cheroff boud to prove the achievable lower boud. III. UPPER BOUND BROADCAST CAPACITY WITH MPR I this sectio, we compute the upper boud of broadcast capacity for MPR. Note that #MCDSR)) equals the average miimum umber of retrasmissio required to broadcast a packet ad #MMIS, R)) is the average maximum umber of successful simultaeous trasmissios i the etwork. The followig Lemma computes a upper boud as the ratio of #MMIS, R)) to #MCDSR)). I [5], #MIS r)) is used to express the average maximum umber of simultaeous trasmissios without MPR istead of #MMIS, R)). Lemma 3.: Per ode broadcast capacity with MPR is upper bouded as O #MMIS,R)) #MCDSR)) Proof: We observe that #MCDSR)) represets the total average umber of chael usage required to broadcast iformatio from a source. By Defiitio., the total broadcast capacity i the etwork is equal to C B ) = i= λi ). Deote by N T the total umber of broadcasted bits i [0, T ], the C B ) = i= ). λ i N T ) = lim T T. 4) Also deote by N B b) the total umber of times ay bit b is trasmitted i order to broadcast to the etwork, the N B b) #MCDSR)). The total umber of retrasmissios for broadcast i [0, T ] is thus N T N B b). 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, #MMIS, R)) T + T max ) N T N B b) N T #MCDSR)).5) By combiig the two previous equatios, we have C B ) = lim N T T T = lim T N T T + T max #MMIS, R)) #MCDSR)), 6) which proves the lemma. We eed to compute the upper boud of #MMIS, R)) ad the lower boud of #MCDSR)) ad the combie them with Lemma 3. to compute the upper boud broadcast capacity for MPR. Lemma 3.: The average umber of odes i a broadcast tree with receiver rage R) has the followig lower boud: #MCDSR)) Θ R ) ) 7) Proof: We first divide the etwork area ito square cells. R) Each square cell has a area of which makes the diagoal legth of square equal to R) as show i Fig.. Uder this coditio, coectivity iside all cells is guarateed ad all odes iside a cell are withi receptio rage of each other. We build a cell graph over the cells that are occupied with at least oe vertex ode). Two cells are coected if there exist a pair of odes, oe i each cell, that are less tha or equal to R) distace apart. Because the whole etwork is coected whe R) = r) Θ /, it follows that the cell graph is coected [8], [9]. Fig.. 0 4 5 R ) R ) O R ) L R ) Receivers Receiver Circle Simultaeous Receive Cell 9 3 Cell costructio of wireless dese ad hoc etworks From the Defiitio.3 for MCDS, every ode has to be covered by MCDS. It has bee show [8] that if R) satisfies coectivity criterio, the each cell has at least oe ode w.h.p. which implies that all cells i the etwork are covered by MCDS. For ay receiver with R) as receiver rage, it ca cover at most i some literature they use 6) cells that is show i the upper right corer i Fig.. Therefore, to cover all R)/ ) cells i the etworks, the umber of odes i MCDS has to be at least R)/ ) /. Hece, the lower boud for MCDS is give by #MCDSR)) Θ R ) ), 8) which proves the Lemma. Lemma 3.3: The average umber of maximum MPR idepedet sets that trasmit simultaeously is upper bouded by #MMIS, R)) Θ). 9) Proof: We wat to fid out the maximum simultaeous MPR set trasmitters i this dese etwork. From the protocol model of MPR, the disk with radius R) cetered at ay receiver should be disjoit from the other disks cetered at the other receivers. We demostrate it by cotradictio. If the disks of differet receivers overlap, the there exists some trasmitters that are withi the receiver rage of two receiver odes. Based o the defiitio of MPR, these odes i the overlappig areas will sed two differet packets at ay time to their correspodig receivers, which is i cotradictio with the fact that each ode ca oly trasmit oe packet at a time. That meas the disk with radius R) cetered at ay receiver should be disjoit. 6 5 3 8 7 6 4
Thus, it is clear that i a dese etwork, there is πr ) trasmitters for each receiver ode with receiver rage of R). Sice MPR protocol model requires a miimum distace betwee receiver odes, it follows easily that each ). receiver ode requires a area of at least π R) + R) It is easy to show that there are a total of at most odes + ) i this etwork which provides the order upper boud of Θ) for #MMIS, R)). Combiig Lemmas 3., 3., ad 3.3, we state the upper boud of broadcast capacity with MPR i the followig theorem. Theorem 3.4: Per ode broadcast capacity for MPR is upper bouded as O R ) ). IV. LOWER BOUND BROADCAST CAPACITY WITH MPR I this sectio, we provide a achievable lower boud for broadcast capacity usig a TDMA scheme similar to the approaches preseted i [8], [9]. I order to satisfy the MPR protocol model, we should desig commuicatio for the etwork i groups of cells such that there is eough separatio for simultaeous trasmissio. Let L represets the miimum umber of cell separatios i each group of cells that allows simultaeous successful commuicatio as show for oe example of L = 4 i Fig.. Utilizig the MPR protocol model, L should satisfy R) + + )R) L = + R)/ = + + ). 0) Let s divide time ito L time slots ad assig each time slot to a sigle group of cells. If L is large eough, iterferece is avoided ad the MPR protocol model is satisfied. We kow from MPR protocol model that the miimum distace betwee two receiver ode should be + )R). By comparig this distace with L ) R) which is the distace betwee two receiver ode i our TDMA scheme ad usig Eq. 0, it is clear that the MPR protocol model is satisfied with this scheme. It ca be show i the upper middle two circles i Fig.. By usig this TDMA scheme, we ca derive a achievable broadcast capacity for MPR. The followig Lemma demostrates that this TDMA scheme with parameter L does ot chage the order of broadcast capacity. Lemma 4.: The capacity reductio caused by the proposed TDMA scheme is a costat factor ad does ot chage the order of broadcast capacity for the etwork. Proof: The TDMA scheme itroduced above requires cells to be divided ito L groups, such that oly odes i each group ca commuicate simultaeously. Eq. 0) demostrates that the upper boud of L is ot a fuctio of ad is oly a costat factor. Because the proposed TDMA scheme requires L chael uses, it follows that this TDMA scheme reduces the capacity by a costat factor. Next we will prove that whe odes are distributed uiformly over a uit square area, we have simultaeously at least LR)/ ) circular regios i Fig., each oe cotais ΘR )) odes w.h.p.. The objective is to fid the achievable lower boud usig Cheroff boud such that the distributio of the umber of odes i each receiver rage area is sharply cocetrated aroud its mea, ad hece i a radomly chose etwork, the actual umber of simultaeous trasmissio occurrig i the uit space is ideed Θ) w.h.p. similar to Lemma 3.3 for the upper boud aalysis. Lemma 4.: Each receiver i the cross sig i Fig. with circular shape of radius R) cotais ΘR )) odes w.h.p. ad uiformly for all values of j, j LR)/. ) This Lemma ca be expressed as lim P LR)/ ) j= N j EN j ) < δen j ) =, ) where N j is the umber of trasmitter odes i the receiver circle of radius R) cetered at the receiver j, E N j ) is the expected value of N j, ad δ is a positive small value arbitrarily close to zero. Proof: From Cheroff boud i 3), for ay give 0 < δ <, we ca fid θ > 0 such that P [ N j EN j ) > δen j )] < e θenj). ) Thus, we ca coclude that the probability that the value of the radom variable N j deviates by a arbitrarily small costat value from the mea teds to zero as. This is a key step i showig that whe all the evets LR)/ ) j= N j EN j ) < δen j ) occur simultaeously, the all N j s coverge uiformly to their expected values. Utilizig the uio boud, we arrive at LR)/ ) P N j EN j ) < δen j ) = P > j= LR)/ ) j= LR)/ ) j= N j EN j ) > δen j ) P [ N j EN j ) > δen j )] LR)/ ) e θenj). 3) Sice EN j ) = πr ), the we have LR)/ ) lim P N j EN j ) < δen j ) j= lim LR)/ ) e θπr ) 4) Whe R) satisfies coectivity criterio, the lim e θπr ) R ) 0, which cocludes the proof.
This lemma proved that w.h.p., there are ideed Θ) trasmitter odes iside LR)/ circles cetered aroud ) receiver odes with radius R). Combiig Lemmas 4. ad 4., we have the followig achievable lower boud. Theorem 4.3: The achievable lower boud for broadcast capacity with MPR is C B ) = Ω R ) ). 5) Proof: From the fact that our TDMA scheme does ot chage the order capacity Lemma 4.), we coclude that at ay give time there are at least ΩR )) simultaeous cells, each oe receives iformatio from ΩR )) trasmitters simultaeously. Hece, from the Lemma 4., the total umber of allowed simultaeous trasmissio is Ω). There are R)/ cells i the uit square etwork area. ) For broadcastig, every cell receives the broadcast packet from a eighbor cell ad relays it to the ext adjacet cell. The umber of relayig is at most Θ R ) ) i order to guaratee all the odes receive the packet from source. I other words, each broadcastig sessio requires Θ R ) ) relayig. Sice the etwork ca support Θ) simultaeous trasmissios, therefore, the total ) broadcast capacity for this etwork is give by Ω R ) Accordigly, per ode lower boud capacity is give by Ω R ) ) which proves the lemma. From Theorems 3.4 ad 4.3 ad coectivity criterio i Eq. ), the tight boud broadcast capacity of MPR ca be give i the followig theorem. Theorem 4.4: Per ode broadcast capacity of MPR is. C B ) = Θ R ) ), 6) where R) Θ /. This result implies that by icreasig the receiver rage, broadcast capacity for MPR system icreases. This is i sharp cotrast with simple routig techiques. The mai reaso for this differece is because whe odes are edowed with MPR capability, the strog iterferece from adjacet odes are embraced istead of combatig it. V. CONCLUSION I this paper, we show that with Multipacket Receptio MPR) techique, which is a cooperatio techiquemayto-oe), Θ R ) ) bits per secod broadcast capacity is achieved for both lower ad upper bouds. R) is the receiver rage which depeds o the complexity of the receiver ad coectivity coditio i the etwork. If the receiver rage has the miimum value to guaratee coectivity, MPR achieves Θlog ) gai compared with the plai routig [5]. By icreasig the receiver rage, the broadcast capacity with MPR icreases. This is a direct result of embracig the iterferece i wireless ad hoc etworks istead of competitio amog odes to access the chael. VI. ACKNOWLEDGMENTS This work was partially sposored by the U.S. Army Research Office ARO) uder grats W9NF-04--04 ad W9NF-05--046, by the Natioal Sciece Foudatio NSF) uder grat CCF-07930, by the Defese Advaced Research Projects Agecy DARPA) through Air Force Research Laboratory AFRL) Cotract FA8750-07-C-069, ad by the Baski Chair of Computer Egieerig. 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