Throughput-Delay Trade-off for Hierarchical Cooperation in Ad Hoc Wireless Networks

1 Throughp ut-Delay T rade-of f for Hierarc hical Cooperation in Ad Hoc W ir eless Networks A yfer ¨ Ozg ¨ ur , Oli vier L ´ ev ˆ eque, Member , IEEE Abstract — Hierarchical cooperation has r ecently been shown to achiev e better thro ughput scaling than classica l multihop schemes under certain assumpti ons on the channel model in static wireless networks. Howe ver , the end-to-end delay of th is scheme turns out to b e significantly larger than those of multihop schemes. A modification of the scheme is proposed h ere that achieves a throughput-delay trade-off D ( n ) = (log n ) 2 T ( n ) for T ( n ) between Θ( √ n/ log n ) and Θ( n/ log n ) , wh ere D ( n ) and T ( n ) are respectively t he a verag e delay per bit and the aggr egate throughput in a network of n nodes. This trade-off complements the previous results of El Gamal et al., which show that the throughput-delay trade-off for multihop schemes is given by D ( n ) = T ( n ) where T ( n ) lies between Θ(1) and Θ( √ n ) . Meanwhile, the present p aper consid ers th e n etwork multip le- access problem, wh ich may be of interes t in its own right. Index T erms — Ad hoc Wireless Netwo rks, Hiera rchical Coop- eration, Scaling Laws, Thro ughput-Delay T rade-off. I . I N T R O DU C T I O N Scaling laws offer a way of stu dying f undam ental trade- offs in wireless networks as well as o f highlighting the qu alitativ e and arch itectural prop erties of specific d esigns. Such stu dy has been initiated by th e work [1] of Gu pta and Kumar in 2000. Their by now familiar model con siders n nodes random ly distributed on a unit area, each of which w ants to communicate to a random destination a t a c ommon rate R ( n ) . They ask what is the m aximally ach iev able scaling of the agg regate throug hput T ( n ) = nR ( n ) and show that co operation be- tween nodes can dram atically improve p erform ance. Instead of usin g the simple scheme o f time-shar ing between direct transmissions fr om source nod es to de stinations, which o nly achieves aggr egate th rough put Θ (1) , the nodes can cooper ate and re lay the packets b y multih opping from one nod e to th e next, in wh ich c ase an aggregate throug hput scaling of Θ( √ n ) is achieved. T he price to pay , howe ver , is in term s of delay . In the multi-hop s cheme, the packets ne ed to be r etransmitted many times before th ey reach their ac tual destinations, which results in larger end -to-end delay . Mor e precisely , a s sho wn later in [ 2], [3], in a multi-hop sch eme, b its are delivered to their destinatio ns in Θ ( √ n ) time- slots on average a fter they leav e their sou rce nodes, while the a verage d elay for th e simple TDMA sch eme remains only Θ(1) . Note tha t this accou nts only for on-the-fligh t d elay; the queuing delay at the source node is n ot considered . Manuscript recei ve d ??; re vised ??, and ??. The work of A yfer ¨ Ozg ¨ ur was supported by Swiss NSF grant Nr 200020-11 8076. The material in this paper was presented in part at the IEEE Internat ional Confere nce on T eleco mmunicatio ns, Saint -Petersb urg, June 2008. The a uthors are wit h the Ecol e Polyt echniqu e F ´ ed ´ erale de Lausan ne, Facul t ´ e Informatiqu e et Comm unicat ions, Buildi ng INR, Station 14, CH - 1015 Lausanne, Switzerland (e-mails: { ayfer .ozgur , oli vier .leveque } @epfl.ch). Recently , i t has been shown in [4] th at under certain assump- tions on the channel m odel, a much better th rough put scaling is achievable in wireless networks th an the one achieved by classical multi-hop schemes. T he authors exhibit a hierar chical cooper ation scheme that uses d istributed MIMO co mmun ica- tion to achieve agg regate thro ughp ut scaling arbitrar ily c lose to linear, i.e. T h ( n ) = Θ( n h h +1 ) for any integer h > 0 . The parameter h corr esponds to the n umber of hiera rchical lev els used in the scheme an d by increa sing h , one can get arbitrarily close to linear scaling. A natural question is whether there is a price to pay for this superior scaling of the throug hput. In p articular, what is the thro ughpu t-delay trade- off for h ierarchica l cooperatio n? In this paper, we analyz e the delay perfor mance of the hie rarchical cooperation scheme and show that the structure suggested in [4] is very suboptima l from the d elay po int of view . W e propose a mo dification of the scheme in this paper, that achieves much better delay perfor mance for the same throug hput. More precisely , we show that one impo rtant drawback o f the scheme in [4], that is not immediately clear from the presentation in there, is that it uses an extremely large bulk-size, where the bulk-size o f a scheme refer s to the minimu m number of bits th at shou ld be communicated between each source-de stination p air under this sch eme. T he bulk-size used b y th e sch eme in [4] scales as B h ( n ) = Θ( n h 2 ) ; in o ther w o rds, it grows arb itrarily fas t as the throug hput appro aches linear scaling. Note that the bulk-size immediately imposes a lower bou nd on the end-to -end dela y of each com municatio n; even if there is no transmission delay from the source n ode to the d estination node, receiving a bulk of B ( n ) bits will take a t least Θ( B ( n ) / log n ) channel uses for a destination node, sin ce a simple application of the c ut-set bound upper bounds the rate of r eception by (or transmission from) a no de with log n bits per ch annel use. The ba sic idea behin d the hierarch ical cooper ation scheme in [4] is to first d istribute the bits of a source nod e among its neighbo ring n odes, so that these bits can then b e simul- taneously transmitted to a gr oup o f nodes in the vicinity of the destination node. By collecting the obser vations of the receiving nodes to t he actual destinatio n no de, th e destination node is ab le to recover the bits intended for itself. This kind of tra nsmission is often referred to as distributed MIMO since it resembles the multiple -input-m ultiple-ou tput transmis- sions between a tr ansmitter and receiver pair with multip le transmit and receive an tennas re spectiv ely . The ef ficiency o f the distributed MIMO transmission increases with the size of (nu mber of nodes con tained in) th e transmit and r eceiv e clusters, f ormed around the source n ode and the destination node respectively . However , if the size of the transmit cluster is large, the b ulk of data to be commu nicated by each source node has to be large e nough to be chopped off an d distributed among the many nod es in the cluster . Hence, the size of th e transmit cluster imp oses a lower bo und on the bulk size that needs to b e commun icated betwee n ea ch source-d estination pair . Moreover , distributing the bits o f the source node b efore the MIMO transmission and co llecting the o bservations to the destina tion nod e following the MIMO transmission bring s another traffic requirement. It has been shown in [4] that this cooper ation traffic can be hand led efficiently if de composed into multiple prob lems of the or iginal kin d, i.e., of com- municating between n source-d estination pairs in a network of n n odes an d r eusing the ide a o f d istributed MIMO. This recursion builds a h ierarchical a rchitecture that is shown to be efficient from thro ughp ut poin t of view . Howe ver, since distributed MIMO based com municatio n imp oses a lower bound on the bulk-size, repeatin g the idea recu rsiv ely yields a sche me with even larger bulk-size. This is th e r eason why the bulk size of th e hierar chical coop eration scheme increases as Θ( n h 2 ) with h hierarch ical le vels. In this paper, we suggest a mod ification of the hier archi- cal cooper ation scheme in [4] th at handles the p roblem of cooper ation more efficiently . In order to do this, we study the problem of cooperatio n more carefully b y posing it as a network multiple a ccess problem, instead o f separating it into multiple unicast prob lems as was originally don e in [4]. I n the network mu ltiple access prob lem, each of the n nodes in the ne twork is interested in conv eying indepen dent informa tion, say L bits, to each of th e other node s in th e network. W e pr opose a two-phase hierar chical sche me that solves the mu ltiple access p roblem in Θ( n h +1 h ) time-slots for any h > 0 . Using this scheme for coo peration , the m odified hierarchica l cooperatio n scheme achieves the sam e ag gregate throug hput T h ( n ) = Θ( n h h +1 ) by using a mu ch smaller bulk-size B h ( n ) = T h ( n ) . W e show that reduc ed bulk size consequen tly reduces the delay and the modified hie rarchical cooper ation scheme achiev es D h ( n ) = Θ( n ) . W e pro ceed by optim izing scheduling in this scheme to further reduce the en d-to-en d delay . T o do this, we need to consider a genera lized version o f the multiple access problem where each node in the network is interested in co n veying indepen dent in formatio n to each of the no des in a subset of A ( n ) nodes, wh ere the A ( n ) < n nodes a re chosen unifor mly at rand om amon g the n nodes in the network. W e show t hat this task ca n be acco mplished in Θ( A ( n ) n n h h +1 log n ) channel uses f or any h > 0 if A ( n ) ≥ n h h +1 . This allows us to achieve a throu ghpu t delay trade-o ff of ( T ( n ) , D ( n )) = ( n b / lo g n, n b log n ) for any 0 ≤ b < 1 . T his n ew re sult is depicted in Figure 1, together with previous results from th e literature. A related line of resear ch (see e.g. [2], [5], [6], [7]) is the characterizatio n o f the through put-de lay tr ade-off f or mobile n etworks, wh ere no des m ove over the duration of commun ication according to a certain m obility p attern. I n general, mob ility sch emes achieve an agg regate thro ughpu t scaling comparable to that of hierar chical co operatio n (i.e. up Mobility Sche mes 1 1 Multi-hop √ n T(n) D(n) n Cooperation Hierarchical √ n n log n √ n log n √ n log n n log n Fig. 1. Throu ghput-del ay performance achie ved by hierarchic al cooperati on togethe r with known results from the lit erature . to linear in n ), but the delay scaling perfo rmance of such schemes may vary sign ificantly , depen ding on the chosen mobility model. For instance, under the c lassical random walk mobility mo del con sidered in [2], the p erform ance is quite poor, as illustrated in Figur e 1. But from the delay p oint of view , a more prominent disadvantage which is common to all mobility models and which d oes n ot appear on th e g raph in Figure 1, is the con stant th at prece des th e delay scaling law . Roughly spea king, this p re-con stant relates to the speed of nodes in the case of mobility schemes, wh ereas it relates to the speed of light in th e case of hier archical coopera tion. I I . S E T T I N G A N D M A IN R E S U LT S There are n nodes uniformly and indep endently d istributed in a square of unit ar ea. E very n ode is both a source a nd a destination. The sources and destinations are paired up one- to-one in a random fashion withou t any consideration on respective locations. E ach source has the same traffi c rate R ( n ) to s end to it s destination n ode. (In the following text, we will sometimes refer to this tra ffic pattern as the unicast pro blem in order to disting uish it fr om the multicast problem s that will b e discussed in Sections IV and V -B.) The aggregate thr ough put of the system is T ( n ) = nR ( n ) . The co mplex b aseband- equiv alent chan nel gain between node i and node k at time m is given b y: H ik [ m ] = r − α/ 2 ik exp( j θ ik [ m ]) (1) where r ik is the distance between the no des, θ ik [ m ] is the random pha se at time m , un iformly distributed in [0 , 2 π ] and { θ ik [ m ] , 1 ≤ i ≤ n, 1 ≤ k ≤ n } is a collection of i.i.d . random processes. Th e θ ik [ m ] ’ s and the r ik ’ s are also assumed to b e indepen dent. Th e constant α ≥ 2 is c alled the power path loss exponent of the environment. Note that the ch annel is random, dep ending o n the location of the users and the ph ases. The loca tions are assumed to be fixed over th e duration of the communication . The phases are assumed to vary in a stationary e rgodic manner (fast fading). W e assume that th e channel g ains are known at all th e n odes. 2 The signal r eceived by n ode i at time m is giv en by Y i [ m ] = n X k =1 H ik [ m ] X k [ m ] + Z i [ m ] where X k [ m ] is the sign al sent by no de k at time m and Z i [ m ] is white circularly sym metric Gau ssian noise o f variance N 0 per symbo l. Every nod e is subject to a tr ansmit power constraint that we denote by P . 1 The delay D ( n ) of a communicatio n scheme fo r this net- work is define d as the average time it takes fo r a b it, o r p acket of constant size, to reac h its destination n ode af ter it leaves its sourc e node, where the average is taken over all bits or packets tra veling in the network. So d efined, the delay of a scheme quantifies the average time spent by the bits tra veling inside the ne twork while operated un der this scheme. This definitio n o f delay is consistent with [2] , [3] and therefor e the comparison in Figure 1 of the mu ltihop scheme and hier archical coop eration is fair . Howe ver, note that this definition does not include the queuing d elay at the source node, as the clo ck starts when a packet leaves its sour ce node. Th e delay at th e source node can be accou nted f or b y assuming a particu lar packet arrival process and stud ying the overall delay of a packet fro m its arriv al at the sour ce queu e to the decoding at the destination node. Th e transmission delay s giv en in Figure 1 can be regard ed as lower bo unds to this overall delay . C onsider for examp le th e simple T DMA sch eme, one at a time transm ission b etween the source- destination pairs, that cor respond s to the o rigin in Figure 1. Assum e indepen dent Poisson packet a rriv al at each so urce nod e of approp riate rate. If we assume round -robin fashion, backlog unaware scheduling b etween the transmissions, the overall delay o f the TDMA sche me will be Θ( n ) much larger th an the Θ(1) delay predic ted by Figu re 1. Howev er , it is known that this delay can b e reduced to O (log n ) with backlog aw are scheduling [8]. In gene ral, how larger is the overall delay f rom the transmission delay g i ven in Figur e 1 depe nds o n how well we can match the packet ar riv al process with backlog aware s chedulin g schemes. In this paper, o ur aim is to quantif y the transmission d elay of the discu ssed schem es; the secon d question regardin g the queuing delay at the source is left ope n. The following theorem is the ma in result of th is paper . Theor em 2.1: Using h ierarchical coo peration , the following points are achievable o n the thr ough put-delay s caling curve, ( T ( n ) , D ( n )) = Θ  n b / lo g n, n b log n  where 0 ≤ b < 1 (see Figur e 1). I I I . O V E RV I E W O F T H E H I E R A R C H I C A L C O O P E R AT I O N S C H E M E In this section, we give a brief overview of the hierar- chical coo peration scheme as p resented in [ 4] and establish 1 W e present the low-l e vel assumptions on the chan nel and network model in this section for the sake of complete ness. Ho we ver , most of the discussions in the follo wing sectio ns will re ly on intermediate results established i n [4], hence the dependenc e of the re sults on the low le vel assumptions m ight not be al ways clear . the throug hput-d elay trade-o ff fo r th is scheme. Some of the discussions presented here directly build on results already established in [4]. The hierarchical cooperation schem e is based on clustering the nod es in the network and performin g long-r ange MI MO transmissions between the clusters. The long -rang e MIMO transmissions shou ld be pro ceeded and fo llowed by coop- eration p hases establishing transmit and receive co operation respectively , w hich y ields three su ccessi ve phases in the oper- ation of the network. I f simple TDMA is used fo r establishing cooper ation in ph ase 1 an d 3, the overall schem e achieves a √ n -scaling of the aggregate th rough put. This is th e three phase scheme discussed in Sectio n III-A. Higher th roug hputs can be achieved b y setting the coo peration pro blem as multiple com- munication pro blems and u sing the three phase sch eme as a solution to each of those communication problems. This y ields the idea of recursion and results in a hierarch ical a rchitecture, where increasing the numb er of l ev els in the hierarchy yields an aggregate thro ughpu t scaling arbitrarily close to linear . T he hierarchica l co operatio n sch eme is d iscussed in mo re detail in Section III-B. A. The Thr ee Pha se Scheme The network is di vided into clusters of M 1 nodes and a particular source nod e s sends M 1 bits to its destination n ode d in thr ee steps: (1) Node s first distrib utes its M 1 bits among the M 1 nodes in its cluster, one bit for each no de; (2) These n odes tog ether can th en f orm a distributed transmit antenna array , sending the M 1 bits simultaneously to the destination cluster where d lies; (3) Each n ode in the destina tion cluster gets one o bservation from the MI MO t ransmission, and it qua ntizes and ships the observation to d , wh ich can then do joint MIMO processing o f a ll the o bservations and d ecode th e M 1 transmitted bits. From th e network po int of view , all source-destina tion pairs have to eventually accomplish these three steps. Step 2 is long- range comm unication and only one sou rce-destinatio n pair can operate a t a time. Steps 1 and 3 inv o lve local communica- tion and can be parallelized ac ross sou rce-destinatio n pairs. Combining a ll this leads to th e following three phases in the operation of the n etwork: Phase 1: Setting Up T r ansmit Cooperatio n Clusters work in para llel. W ithin a cluster , each source n ode distributes M 1 bits to the oth er no des, 1 bit f or each n ode, such that at the end of the ph ase, each n ode has 1 bit f rom each of the other nodes in its cluster . (Recall our assumption that ea ch n ode is a source for some commu nication request and a destination for another .) Thus, since there are M 1 source n odes in each cluster , this g iv es a traf fic demand of exchanging M 1 ( M 1 − 1) ∼ M 2 1 bits. Usin g TDMA, on e-at-a-time tr ansmission b etween pairs 3 of nodes, th ese M 2 1 bits can be exchange d in M 2 1 time slots. 2 Phase 2: MIMO T ransmissions Successiv e long- distance MIMO transmissions ar e performed b etween source- destination p airs, o ne at a time. I n e ach one of the MIMO transmissions, say the one between s and d , the M 1 bits of s are simultaneou sly transmitted by th e M 1 nodes in its cluster to the M 1 nodes i n the cluster of d . Each of the lon g-distance MIMO transmissions are rep eated for ea ch source-destination pair in the network, hence we need n time-slots to complete the ph ase. Phase 3: Cooperate to De code Clusters work in par allel. Since there are M 1 destination nodes inside the clusters, each cluster received M 1 MIMO transmissions in ph ase 2 , one intended for e ach of the destination n odes in the c luster . Thus, each node in th e cluster has M 1 received ob servations, one from each o f the MIMO tran smissions, and each ob servation is to be conv eyed to a d ifferent node in its cluster . No des quantize each o bservation in to fixed Q b its, so there are a total of QM 2 1 bits to be exchanged in side each cluster . Using TDMA as in Phase 1 , the p hase can be completed in QM 2 1 time slots. 3 In [4], it is shown that each destination n ode is ab le to decode the transmitted bits from its source no de from the M 1 quantized sign als it gathers by the end o f Phase 3 . The throu ghput achiev ed by the scheme can be calcula ted as follows: each sou rce node is able to transm it M 1 bits to its destination n ode, h ence n M 1 bits in total are delivered to th eir destinations in M 2 1 + n + Q M 2 1 time slots, yield ing an agg regate throug hput of nM 1 M 2 1 + n + QM 2 1 bits per time-slot. Choosing M 1 = √ n to maximize this expression yields an agg regate throughput T ( n ) = 1 2+ Q √ n . Note that as o pposed to multiho p, this three phase scheme allows only b ulk tra nsmission between any sou rce-destinatio n 2 Note that although because of the broadcasting nature of T DMA, e very bit of a source node can be con veyed to all other nodes in the cluster for free, thi s is not wha t we require here. In the MIMO transmissio ns that are follo wing in the next phase, e very node is independentl y encoding its data and it does not need to kno w the bits transmitted by the other nodes. A standard referenc e on the capa city of MIMO channels is [10]. The deri vat ions f or the current case can be found in [4]. 3 In order to be able to con vey the salient feat ures of the hierarchic al coopera tion scheme to the reade r in the simplest way , a rather informal approac h is tak en in this sect ion and some t echnic al details are omitte d. A rigorous description of the scheme can be found in [4]. For example, it is not necessary to ha ve e xactly M 1 nodes in each cluste r but it suf fices to have Θ( M 1 ) nodes for a scaling law anal ysis. It is shown in [4] that by di viding the ne twork into cells of certa in area, we can en sure ha ving Θ( M 1 ) no des in all cell s with high probabilit y . Moreov er, the case when a source node and its destina tion lie i n the same cluste r should be treated separately . Similarly , assuming that each source node is sending e xactly 1 bit to each of the other nodes in its cluster in pha se 1 is a s implifica tion. A rigorous a rgument will assume t hat each sourc e node is sending L bits to eac h of the other nodes where L is a large enough constant independen t of M 1 and n . The rates of the TDMA transmissions in phase 1 and phase 3 and the per node rate for the MIMO transmission in phase 2 are assumed to be 1 for simplicit y , so that 1 bit is transmitte d in 1 time s lot. The actual rates of these transmissions can be sho wn to be constant s depending on the system parameter s a nd independ ent of M 1 and n . Also, in phase 1 and phase 3 not all clusters should be allo wed to operate simul taneousl y but a TDMA scheme be tween the clu sters sho uld be employed so that the resultant inter-cl uster interference is bounded and each clust er becomes acti ve a constan t fraction of time. pair in the ne twork; i.e . one can n ot arbitrarily co mmunica te one bit (or L bits with L co nstant) using the three-ph ase scheme, but M 1 = √ n bits should be tran sferred between ev ery source-destina tion pair with each use of the scheme. The end -to-end delay of this scheme is simply the to tal time for the th ree p hases, since the bits are leaving their so urce nodes at the beginn ing o f the first ph ase and are only decod ed by their respectiv e d estination nodes at the end of the third phase. W ith the ch oice M 1 = √ n , we see that th e delay of the three phase scheme is D ( n ) = (2 + Q ) n . Note that th is delay scaling is muc h worse when c ompared to the d elay o f the multi-ho p scheme achieving same aggregate th roug hput. B. The Hierar chical Coop eration S cheme Higher aggregate thr oughp ut scaling can be ach iev ed by using better n etwork co mmunicatio n schemes th an TDMA to establish the tr ansmit and receive co operatio ns in th e first and third phases of the three p hase scheme d escribed in the previous section. Recall that there ar e M 2 1 and QM 2 1 bits to be exchanged in side each cluster in phases 1 and 3, respecti vely . This traffic demand o f exchan ging M 2 1 bits (or QM 2 1 bits) can be han dled by setting u p M 1 sub-pha ses, and assign ing M 1 pairs in each sub-phase to communicate their 1 b it (or Q bits). The traffic to be hand led at eac h su b-phase now lo oks similar to the origin al network co mmun ication p roblem (the unicast n etwork p roblem defined in Section II), with M 1 users instead of n . Any schem e suggesting a go od solu tion for the original problem can now be u sed inside the sub-phases as an alternative to TDMA; for examp le, the multi-ho p scheme and the three-phase scheme itself would b e two alternativ es both a chieving an agg regate through put scaling Θ( √ M 1 ) (in a network of size M 1 ) as opp osed to the Θ(1) scaling ach iev ed by TDMA. Consider using the three p hase sch eme fo r c ooperatio n as suggested in [4]. More p recisely , we want to handle the traffic of commun icating 1 bit (or Q bits) between the M 1 pairs assigned in each sub -phase of ph ase 1 (or pha se 3 ), by fu rther dividing the c lusters into smaller clu sters of size M 2 and reusing the three phase scheme (TDMA-MI MO-TDMA). Note that this will create a hierarchical structure with two levels . See Figure 2. Note howe ver that the three phase scheme in Section III-A, allows o nly bulk tran smissions between source- destination pairs. In th is particu lar case, one will have to commun icate M 2 bits b etween the so urce-de stination pairs assigned a t eac h sub -phase, as oppo sed to the original req uire- ment o f communicating only 1 bit ( or Q bits). F o r the overall scheme, this in turn inc reases the bulk si ze to b e co mmunic ated between every source- destination pair in the network f rom M 1 bits to M 1 × M 2 bits. So fo r th e 2-le vel hiera rchical sche me, we have to start by assum ing that eac h sou rce node in the network has M 1 × M 2 bits to co mmunica te to its destination node. It can be seen that th ese M 1 × M 2 bits pe r source destination pair , or a total n × M 1 × M 2 bits in the network, can be commun icated in M 1 ( M 2 2 + M 1 + QM 2 2 ) + M 2 n + M 1 Q ( M 2 2 + M 1 + QM 2 2 ) (2) time slots u sing th e 2-lev el hier archical schem e. T he first term M 1 ( M 2 2 + M 1 + QM 2 2 ) is th e completion time of phase-1 4 of the three phase sche me. It is divided into M 1 sessions; in ea ch session, M 1 source-d estination p airs are assigned to commun icate their M 2 bits u sing a thr ee p hase schem e o f TDMA-MIMO- TDMA. Recall fr om the computation s o f the three p hase scheme in Sectio n II that this takes M 2 2 + M 1 + QM 2 2 time slots ( M 1 and M 2 here corresp ond to the n and M 1 , respectively , of the previous section). A similar argu ment holds for th e third term M 1 Q ( M 2 2 + M 1 + QM 2 2 ) in (2) wh ich is the completio n time for phase-3 with the extra Q factor . Note that at th e end of the first phase, each sour ce n ode has distributed its M 1 × M 2 bits among the M 1 nodes in its cluster, hence M 2 bits fo r each node. These bits can be r elayed to the destination cluster in M 2 successiv e MIMO transmissions. Since th e MIM O transmissions have to b e rep eated fo r each o f the n sourc e-destination pairs in the network, the completion time of the seco nd phase is M 2 n in ( 2). Therefo re, the ag gregate throug hput of the 2- lev el scheme is given by the expression M 2 M 1 n M 1 ( M 2 2 + M 1 + Q M 2 2 ) + M 2 n + M 1 Q ( M 2 2 + M 1 + Q M 2 2 ) (3) and the optimal choices of M 1 = n 2 / 3 and M 2 = n 1 / 3 maximize the ag gregate throughput scaling to T 2 ( n ) = M 1 = n 2 / 3 , while the de nominato r dictating the d elay of the scheme is o f order D 2 ( n ) = M 2 × n = n 4 / 3 . Note th at with the 2 -lev el hierarchical schem e, we improve the aggregate th rough put scalin g fr om √ n f or th e th ree-ph ase scheme in the previous section to n 2 / 3 , at the cost of in creasing the bulksize from √ n to n , which , in turn, increa ses the delay from n to n 4 / 3 . The argumen t can be applied recu rsiv e ly to build an h - lev el h ierarchical scheme. The optimal cluster size at the k ’th lev el of an h -lev el hier archical scheme can be compu ted as M k = n h +1 − k h +1 . Th e aggregate th rough put a chieved by an h - lev el hierarchical coope ration scheme is given by T h ( n ) = M 1 = n h h +1 . The bulk-size is B h ( n ) = M h × . . . × M 1 = n h 2 and the end -to-end delay is D h ( n ) = M h × M h − 1 × · · · × M 2 × n = n h 2 + h +2 2( h +1) where w e o bserve that for large h , the delay expo nent is linear in h . The re sults o btained in this section estab lish th e po or delay perfor mance of hiera rchical co operatio n. Note that the delay is mostly due to the large bulk-size used by the sch eme. This is d ifferent from multi-hop schem es since their bulk-size is constant ( Θ(1 ) ) indep endent of the throug hput achiev ed. The delay in multihop is rath er due to the time spent in relayin g the messages inside the network . In the next section , we suggest a mod ification o f the scheme so that it achieves the sam e throug hput using much smaller bulk-size. I V . H I E R A R C H I C A L C O O P E R AT I O N W I T H S M A L L E R B U L K - S I Z E In this sectio n, we treat the pro blem o f coo peration in the three p hase schem e with more care. W e start by defining th e network multiple access pro blem to be the fo llowing. Definition 4.1 (Th e Network Multiple Acce ss Pr oblem) : Consider the assumptio ns on the network and chan nel model giv en in Section II. Let e ach node in th e n etwork be interested in commun icating indep endent info rmation to each of the other n odes in the network . In particu lar , let us assum e that each node has an indep enden t 1 bit message (or L indepen dent bits, with L co nstant) to send to each of the other nod es in th e network an d the quan tity of interest is the smallest time F ( n ) required to accomp lish this task. This problem we refer to b e the network multip le access problem. The fo llowing th eorem provid es an achiev able solution to this prob lem. Theor em 4.1: For any integer h > 0 , the network MAC problem can be solved in F ( n ) ≤ K n h +1 h time-slots w .h.p. 4 , for some con stant K > 0 indepen dent o f n . Pr oof of Theor e m 4.1: Let us start by assuming that th ere exists a schem e that solves the multiple access problem in F ( n ) = n b time-slots with b > 1 . Note that o ne such scheme is simple TDMA that yield s b = 2 . Using this existing scheme, we will con struct a new scheme that yields sm aller F ( n ) . As before, let us start b y di vid ing th e network into clusters of M nodes. Let us first f ocus on o ne specific cluster S an d one node d located outside of this cluster . In particu lar , all nodes in S have 1 bit to send to d . These bits can be com municated to d in two step s: (1) The nod es in S simultaneou sly transmit th eir 1 bit messages destine d to d for ming a distributed transm it antenna array fo r MIMO tr ansmission. The no des in the destination cluster where d lies, for m a distributed receiv e antenna array for this MIMO tran smission. (2) Each n ode in the d estination cluster obtains one observa- tion from the MIMO transmission in the previous phase; it quantizes and ships this observation to d , which can do joint MIMO proce ssing o f all the observations and decode the M transmitted b its from the nodes in S . As a first step tow ard s handling the whole n etwork problem, note th at these two steps should be accomp lished between S and all other nodes in the netw ork. This can again be done in two steps: Phase 1: MIMO transmissions W e perf orm successive long-d istance MIMO transm issions between S an d all other nodes in th e ne twork. In each of the MIM O tran smissions, say b etween S and d , the M n odes in S are simultaneou sly transmitting the 1 bit messages they would like to com mu- nicate to d and th e M nodes in the cluster where d lies are observing the MIMO transmission. The MIMO tran smissions 4 with high probability 5 PHASE 1 PHASE 2 PHASE 3 PHASE 1 PHASE 2 PHASE 3 PHASE 1 PHASE 2 PHASE 3 PHASE 1 PHASE 2 PHASE 3 PHASE 1 PHASE 3 PHASE 2 PHASE 3 PHASE 1 PHASE 2 Fig. 2. The salie nt fe atures of the three phases and the time di vision in a hierarchic al sche me are illu strated. Figure tak en from [4]. should be r epeated for each n ode in the ne twork, hen ce we need n time -slots to comp lete the phase. Phase 2: Cooperate to decode Clusters work in parallel. Since there are M n odes inside each cluster , each clu ster received M MIMO transmissions from S in the p revious phase, o ne inten ded f or each nod e in the cluster . Thus, each node in the cluster has M observations, on e from each of the MIMO tran smissions, a nd each o bservation is intended for a different n ode in the cluster . Each of these o bservations can be qu antized into Q bits, with a fix e d Q , wh ich yie lds exactly the original network multip le acc ess pro blem, with M nodes instead of n . Using the scheme we assumed to exist in the beginning of the pr oof, this task can be co mpleted in QM b time slots. The total time we have spe nt d uring the two phases f or handling the traffic orig inated f rom cluster S is given b y n + QM b . Fro m the network p oint of view , the above two step s should be comp leted fo r all n/ M clusters in the network. T hus, the multicasting task can be completed in n M ( n + QM b ) time slots in total. Choosin g M = n 1 b in order to minimize this quantity yields F ( n ) = (1 + Q ) n 2 − 1 b . Note th at 2 − 1 b < b for b > 1 . In other words, we have established a solution for the multiple access prob lem that is better than the one we started with . Ind eed, the two phase scheme described above can b e used recursively yielding a better schem e at each step of the recur sion. In par ticular, starting with TDMA achieving b = 2 and app lying the idea recursively h times, one g ets a scheme that solves the multiple access prob lem in Θ( n h +1 h ) time slots. Th e operation of this scheme is illustrated in Figure 3.  The interest in the multiple access problem ar ises from the fact that it exactly models the required tr affic for cooperation i n the three ph ase sch eme. R ecall the co mmunic ation requir ement inside th e clusters in Phase 1 an d 3 descr ibed in Section III-A. This communication req uirement, eq uiv alen t to a n etwork multiple access prob lem, is h andled using TDMA in the PHASE 1 PHASE 2 PHASE 1 PHASE 2 Fig. 3. The figure illustra tes the time-di vision in the hierarchica l s cheme that solv es the netwo rk multiple access proble m. three p hase scheme wh ich ha s been seen to be sub optimal from throug hput point of view in th e Section II I-A. In the hierarchica l cooperatio n sch eme de scribed in Section I II-B, this multiple access problem is handled by decomp osing it into a num ber of unicast network prob lems. The resultant scheme is optim al in terms of thro ughp ut, but not very satisfying in terms of bulk-size. By using the so lution to the mu ltiple access prob lem suggested in this section, one can mo dify the hierarchica l coope ration sche me, so as to achieve the same throug hput with smaller bulk-size and c onsequen tly smaller delay . The r esultant m odified h ierarchical scheme is illustrated in Figure 4. Note that the gain is comin g from treating the cooper ation pr oblem a s it is and n ot necessarily as multiple unicast prob lems as was previously done in Section III-B. Cor ollary 4 .1: A m odified hierarchical cooperation scheme can achieve an agg regate throug hput T h ( n ) ≥ K 1 n h h +1 with bulk-size B h ( n ) = K 2 n h h +1 and d elay D h ( n ) ≤ K 3 n w .h.p. , for a ny integer h ≥ 0 and so me positi ve con stants K 1 , K 2 , K 3 indepen dent of n . Pr oof of Cor olla ry 4 .1: Consider the th ree ph ase hierar chical scheme described in Section III- A. By Theore m 4.1, the required tr affic fo r tr ansmit and receive cooper ation in ph ase 1 an d phase 3 can be handled in K M h +1 h and K QM h +1 h time slots r espectively . The exp ression for the ag gregate throug hput then becom es M n K M h +1 h + n + K Q M h +1 h 6 PHASE 2 PHASE 1 PHASE 3 PHASE 2 PHASE 2 PHASE 2 PHASE 2 PHASE 3 PHASE 3 PHASE 3 PHASE 3 Fig. 4. T he figure illustrate s the time-di vision in the modified hier archic al sc heme that uses the scheme in Figure 3 for cooper ation. Note the differen ce in operati on of the phases between the modified hie rarchic al coop eration scheme and the origina l hierarchic al coop eration scheme of [4] in Figure 2. which is maximized by the ch oice M = n h h +1 , yielding aggre- gate throug hput T h ( n ) = 1 1+ K + K Q n h h +1 , bulk-size B h ( n ) = n h h +1 and delay D h ( n ) = (1 + K + K Q ) n .  V . H I E R A R C H I C A L C O O P E R AT I O N W I T H B E T T E R S C H E D U L I N G In the pre vious sectio n, we presented a mo dified hierarchical scheme th at achieves throu ghput T h ( n ) = Θ( n h h +1 ) u sing bulk-size B h ( n ) = Θ ( n h h +1 ) . Howe ver, the de lay of this scheme is still D h ( n ) = Θ( n ) . I n this section, we o ptimize the schedulin g in the scheme to furthe r impr ove the delay perfor mance to D h ( n ) = Θ( n h h +1 log n ) . W e fir st start by improving the s chedulin g in the three phase scheme with h = 1 discussed in Sec tion III-A. W e then con sider the modified hierarchica l scheme with h ≥ 2 d iscussed in Section I V . Before starting, we state the following binn ing lemma, similar in spirit to Lemm a 4.1 and L emma 5.1 in [4] and can be proven using similar techn iques. Th e lemma will be used repeated ly througho ut the rest of the pap er . Lemma 5.1: Let us assume that f ( n ) balls are thrown in to n bins, in depend ently and unifo rmly at rand om. The following proper ties are satisfied with failure pro bability expon entially small in n . (a) If lim n →∞ f ( n ) n log n = ∞ , then there are Θ( f ( n ) n ) nodes in each bin. (b) If lim n →∞ f ( n ) n = c with c ≥ 0 a constant inde penden t of n , then ther e are at most O (lo g n ) nodes in each bin. A. Better Schedu ling for the Thr ee Phase Scheme Recall the operation of the three p hase scheme from the point of v iew of a single so urce-de stination pair s - d as described in Section II I-A: a step (1) where s distributes its M bits among the M nodes in its cluster , followed by a step (2 ) w here these M b its are simultaneou sly transmitted to the destination cluster via MI MO transmission , and a step (3) wh ere the quantized MIMO observations are collec ted at the d estination node d . These th ree steps need to be e ventually accomplished for each source- destination p air in the network. In this section , we improve the sched uling in accom plishing this task: we organize M su ccessi ve sessions an d allow only n/ M source-d estination pairs to co mplete the thr ee steps in each session. In the beginn ing of each session we random ly choose on e source node from each cluster, thus n/ M sou rce nod es in total. In gen eral, the n/ M destination nodes cor respond ing to these randomly ch osen sou rce n odes can b e located anywh ere. Howe ver, f rom L emma 5.1, we kno w that no m ore th an log n of these destination no des are located in th e same cluster with high probability . W e proceed by acco mplishing th e three steps for these chosen source-d estination pairs: Phase 1: Setting Up T r ansmit Cooperatio n Clusters work in parallel. The ch osen so urce node in each cluster distributes its M bits to the othe r nod es by using TDMA, which ta kes M time-slots in total. Note that as op posed to the scheme described in Sec tion III -A, there is on ly one sou rce nod e operating in each clu ster . Phase 2 : MIMO T ransmissions Successive MIMO trans- missions originated from each c luster are per formed , transmit- ting the bits of th e acti ve source node in each cluster to its respective d estination cluster . Note that in the current case, there is only one MIMO transmission or iginated fro m each cluster , so there are only n/ M MIMO transmissions that need to be per formed in total. Th is will requ ire total time n/ M . Phase 3: Cooperate to Decode Clusters work in pa rallel. Each cluster receiv ed at most log n MI MO transmissions in phase 2 by Lemma 5.1-b , each MI MO transmission intend ed for a different destination node in the cluster . The r eceived observations at each node are q uantized into Q bits and are to be conveyed to the actual destination nod es. The tra ffic inside each cluster is at most o f exchan ging QM log n bits a nd can be completed using TDMA in a t most QM log n time slots. (See Figure 5.) The opera tion continues with the next session b y choosing a new set o f n/ M source no des rando mly among th e n odes that have not yet ac complished th e above steps. Note that all source-d estination p airs will acc omplish th e three steps in a total of M sessions. W ith this rath er smoo ther op eration on the network le vel, we accomplish to serve n/ M sour ce-destination pairs in each session, that is transfer M × n M bits in total to their destinations in M + n M + QM log n time slots yielding agg regate thro ughp ut M × n M M + n M + Q M lo g n (4) which is max imized by the choice M = √ n y ielding ag- gregate throu ghput T ( n ) = 1 2+ Q √ n log n . T he delay experien ced by each bit is now much less co mpared to the three phase scheme in Section III-A, since it is again dictated by th e to tal time spent in the three phases (d enomin ator o f (4)), which is now less than D ( n ) = (2 + Q ) √ n log n . Note that instead of choosing M = √ n , which is the optimal choice to max imize th e through put ach iev ed by the scheme, o ne can choo se M = n b with 0 ≤ b ≤ 1 / 2 . In th is case, we also restrict the n umber of source-destin ation pair s to be served in each sess ion to M , which can b e less than the 7 1 s s 2 d 2 2 d d 1 d 1 Fig. 5. The three phase scheme with better scheduli ng. T he figure illustrate s the operation in one session. total number of clu sters n/ M . Indeed, we o perate one sou rce node in each of the M ( ≤ n/ M ) clusters and simply keep the remaining clusters in activ e. The expression f or the ag gregate throug hput becomes M × M M + M + QM lo g n which implies that the scheme achieves ag gregate throughpu t T ( n ) = n b / lo g n and delay D ( n ) = n b log n fo r any 0 ≤ b ≤ 1 / 2 . Note that th is throug hput- delay trade- off dif fer s only by log n from the trade- off achieved by multi-hop schemes. B. Better Scheduling for the Hierarc h ical Co operation S cheme In th is section, we ado pt the scheduling ide a of Section V -A to the m odified hierarch ical scheme pr esented in Sectio n IV. Howe ver, th is mo dification is not trivial and req uires us to consider a generalized version of the network multiple access problem . Definition 5.1 (Th e Generalized Network MAC Pr ob lem): Consider th e assumptio ns on the n etwork and c hannel mo del giv en in Section II. Let each of the n nod es in the ne twork be interested in con veying indep endent informa tion to a su bset A ( n ) o f the n odes ( A ( n ) ≤ n ), wher e the A ( n ) nod es are chosen random ly amo ng th e n nodes in the network. In particular, let u s assume that each node in the network has an indepen dent 1 b it message (or L independ ent bits, with L co nstant) to send to ea ch of these A ( n ) nodes and the quantity of interest is the minima l time G ( n ) require d to accomplish this task. W e define this to be the g eneralized network multiple access pro blem. The following theor em provides an achiev ab le solution to this prob lem. W e skip the p roof of the theorem since it is similar in spirit to the pr oof of Theor em 4.1. Theor em 5.1: For any integer h > 0 , if A ( n ) ≥ n h h +1 , then the network multiple acc ess proble m can be solved in G ( n ) ≤ K A ( n ) n n h +1 h log( n ) time-slots w .h.p ., fo r some constant K > 0 independent of n . Note that the g eneralized network multiple access prob lem contains the network multiple access problem discussed earlier as a specia l case with A ( n ) = n . Plu gging A ( n ) = n in Theorem 5 .1, we recover the re sult of Theorem 4.1 with an extra log n factor . Indeed, wh en th e condition A ( n ) ≥ n h h +1 is satisfied with strict ineq uality in o rder, t he extra log n factor in Theorem 5.1 is not needed . Ho wev er , it is n eeded to account for the case A ( n ) = n h h +1 , in which case it arises due to p art-b of Lemma 5. 1. W e are now read y to ap ply the schedu ling idea in Sec- tion V -A to the hierarchic al coopera tion scheme. Consider dividing the network into clusters of M 1 nodes and then further d ivide these clu sters in to smaller clusters of size M 2 . Follo wing th e sch eduling id ea in Section V -A, let us organize M 1 / M 2 sessions and for each session rando mly ch oose one small cluster inside every large cluster . On ly the source nodes located in the chosen small clusters an d their correspo nding destination nodes will be served at each s ession. As usual, we are operatin g in three successiv e phases in each session: Phase 1: Setting U p T ransmit Coo peration T he a ctiv e small clusters operate in par allel. Note th at there is a sing le activ e cluster of size M 2 inside e very large c luster of size M 1 . Let S be the chosen small cluster inside the larger cluster L that will operate in th e current session . In this p hase, each of the M 2 source nodes in S ne ed to distribute their M 1 bits among the M 1 nodes in the larger cluster L , ea ch of the M 1 bits goes to a different nod e. This can be accomplished in two sub-pha ses: • Sub-Phase 1: MIMO t ransmissions Successiv e MIMO transmissions are performed between nodes in S and each node in L . In e ach of these MIMO transmissions, say the one be tween S and a n ode d in L (located ou tside o f S ), the M 2 nodes in S are simultan eously tra nsmitting the 1 bit messages th ey would like to communicate to d . The M 2 nodes located in the same small cluster with d are acting as a d istributed receive antenna array fo r this MIMO transmission. Since these MIMO tra nsmissions should b e r epeated for every nod e in L , th is su b-phase takes a total of M 1 time-slots. See Figur e 6. • Sub-Phase 2: Coo perate to Decode All small clusters in the network work in parallel. In particular, each small cluster in L has received M 2 MIMO transmissions fro m S in the previous phase, one MIMO transmission for each node in this small cluster . Thu s, ea ch no de in the small cluster has M 2 observations, one fro m each of th e MIM O transmissions and each o bservation is to be co n veyed to a different no de in th e cluster . Quantizing each obser vation into Q bits, we get the network mu ltiple access problem defined in Section IV in a ne twork o f size M 2 , an d by Theorem 4.1 this prob lem can be handled in QM h 1 +1 h 1 2 time-slots for any integer h 1 > 0 . 8 Fig. 6. The figure illustrates sub-phase 1 of phase 1 of the modified hierarc hical scheme with bett er scheduling. Note that there is only one small cluster distributi ng bi ts inside eve ry large cluster . Phase 2: MIMO T ransmissions A t the en d of the first phase, all source no des in the active small clusters hav e distributed their M 1 bits am ong th e nodes in the larger cluster . Now , suc cessi ve lo ng-distan ce M 1 × M 1 MIMO transmissions between large clu sters ar e perfo rmed. Durin g each MI MO transmission, th e M 1 bits of a particular sou rce no de in the activ e small c luster a re transferred to the destination cluster where its d estination nod e is located. The nu mber of MI MO transmissions to be perfo rmed in this p hase is equ al to the total nu mber of source nodes a ctiv e in this session. Hence the total pha se can be completed in n M 1 × M 2 time-slots. Phase 3 : C ooperate to Decode By pa rt-a of Lemma 5.1, there are or der M 2 destination n odes loc ated in each of the large clusters. Thus, each large cluster has r eceiv ed M 2 MIMO transmissions in th e previous phase, and the quantized MIMO observations spread over the M 1 nodes of the large cluster should be collected at the corre sponding M 2 destination nodes. This is the generalize d network multiple acc ess p roblem of size M 1 with A ( M 1 ) = M 2 . By Theor em 5.1, it can be solved in Q M 2 M 1 × M h 2 +1 h 2 1 log M 1 time-slots f or any in teger h 2 > 0 provided that A ( M 1 ) ≥ M h 2 h 2 +1 1 . Gathering ev erything together, at ev e ry session of this modified hierarch ical coop eration sch eme, we deliver M 1 × M 2 × n M 1 bits to th eir destinations in a total of  M 1 + Q M h 1 +1 h 1 2  + n M 1 × M 2 + Q M 2 M 1 × M h 2 +1 h 2 1 log M 1 time-slots. The agg regate throughpu t is given by n M 1 × M 2 × M 1 M 1 + Q M h 1 +1 h 1 2 + n M 1 × M 2 + Q M 2 M 1 × M h 2 +1 h 2 1 log M 1 which is maxim ized by the ch oice h = h 2 = h 1 + 1 , M 1 = n h h +1 and M 2 = M h − 1 h 1 , yielding ag gregate thr ough put T ( n ) = n h h +1 log n and d elay D ( n ) = n h h +1 log n . Note that these choices f or M 1 and M 2 satisfy the constrain t A ( M 1 ) = M 2 ≥ M h 2 h 2 +1 1 . Note that at this point, we have proven that all p oints on the th roug hput-d elay scaling curve ( T ( n ) , D ( n )) = ( n h h +1 / lo g n, n h h +1 log n ) with h b eing a p ositiv e integer are a chiev ab le. In order to show that a ll points o n the line ( T ( n ) , D ( n )) = ( n b / lo g n, n b log n ) with 0 ≤ b < 1 are achiev ab le, we can choose M 1 = n b with 0 ≤ b ≤ h h +1 in the ab ove d iscussion, while main taining the re lationships M 2 = M h − 1 h 1 and h = h 2 = h 1 + 1 . Extendin g the argu ment at th e end o f Section V -A, we also restrict the number of small cluster s to be served in each session to M 1 /h 1 which can now be less than the total number of large clusters n/ M 1 ( ≥ M 1 /h 1 ) . Indeed, we operate on e small cluster in e ach of the M 1 /h 1 large clu sters and simply keep the remaining large clusters inacti ve. The expression f or the a ggregate th rough put becomes M 1 h 1 × M 2 × M 1 M 1 + Q M h 1 +1 h 1 2 + M 1 h 1 × M 2 + Q M 2 M 1 × M h 2 +1 h 2 1 log M 1 which shows that we can achie ve aggregate through put T ( n ) = M 1 / lo g M 1 and d elay D ( n ) = M 1 log M 1 . Recalling that M 1 = n b , we get the po ints o n the throug hput-d elay scaling curve ( T ( n ) , D ( n )) = ( n b / lo g n, n b log n ) f or any 0 ≤ b ≤ h h +1 and h > 0 . This conclud es the proo f of the main result of this pape r .  V I . C O N C L U S I O N The present work shows that hierarch ical co operatio n not only can lead to high er th rough put in ad ho c n etworks, but also to reasonable end- to-end delay , giv en that some extra care is taken in setting up coop eration at the lower levels and scheduling comm unication s. Mean while, we have discu ssed the n etwork multip le-access problem in the presen t paper, which is of inte rest in its own right. R E F E R E N C E S [1] P . Gupta and P . R. Kumar , The Capacity of W irele ss Networks , IEEE Tra ns. on Information Theory 42 (2), pp. 388–404 , March 2000. [2] A. El Gamal, J. Mammen, B. Prabhakar , D. Shah, Optimal Throughpu t- Delay Scaling in W ireless Networks-P art I: The F luid Model , IEEE Trans. on Information Theory 52(6), pp. 2568-2592, 2006. [3] A. El Gamal, J. Mammen, B. Prabhakar , D. Shah, Optimal Throughpu t- Delay Scaling in W irele ss Networks-P art II: Constant-Si ze P acke ts , IEEE Tra ns. on Information Theory 52(11), pp. 5111-5116, 2006. [4] A. ¨ Ozg ¨ ur , O. L ´ ev ˆ eque, D. T se, Hiera rc hical cooperati on achie ves optimal capacit y scaling in ad hoc ne tworks , IEEE Transact ions on In formation Theory , V ol. 53 (10), pp. 3549-3572, October 2007. [5] M. Grossglauser and D. Tse, Mobil ity Increases the Capacity of Ad- hoc W ir eless Netwo rks , IEEE/A CM T ransacti ons on Networki ng 10(4), pp. 477-486, 2002. [6] M. J. Neely and E. Modiano, Capacity and Delay T radeof fs for Ad-Hoc Mobile Networks , IEEE Transact ions on Information Theory , V ol. 51 (6), pp. 1917-1937, June 2005. [7] G. Sharma, R. R. Mazumdar and N. B. Shroff, Delay and Capacity T rade-of fs in Mobile Ad Hoc Networks: A Global P erspective , 2006 IEEE INFOCOM Conference, Barcelona , Spain, April 2006. [8] M. Neely , E. Modiano and Y . S. Cheng, Logarit hmic Delay for N × N P ack et Switc hes Under the Cr ossbar Constraint , IE EE/A CM Trans. on Networ king, V ol. 15 (3), June 2007. [9] Gallager R. G. , Information Theory and Reliable Communication , Wil ey & Sons Inc., 1968. [10] E. T elatar , Capacity of Multi-Ant enna Gaussian Channels” E uropean Tra ns. on T elecommunic ations, ET T , vol.10 (6), pp. 585-596, Nov . 1999. 9 A yfer ¨ Ozg ¨ ur recei ved B. Sc. degree s in electri cal enginee ring and physics from Middle E ast T echnical Univ ersity , Turk ey , in 2001 and M.Sc. degree in elect rical engineering from the same uni versit y in 2004. She is now a Ph.D. student at the Laboratory of In formation Theory , Swiss Federal Instit ute of T echnolo gy-Lausanne. Her researc h interests incl ude wireless communic ations and information theory . O l i vier L ´ ev ˆ eque was born in Switzerland i n 1971. H e recei ved the physic s diploma from E PFL in 1995 and complete d his Ph.D. in mathemati cs at EPFL in 2001. Si nce then, he has been with the Laboratory of Information Theory at E PFL. He spent the a cademic al year 2005-2006 at the Electrical Engineeri ng Department of Stanford U ni versity , where he was appo inted as lectu rer . His rese arch inte rests incl ude stochastic analy sis, random m atrice s, wireless communic ations and informa tion theory . 10