Converge-Cast with MIMO

Transcription

1 Coverge-Cast with MIMO Luoyi Fu, Yi Qi 2, Xibig Wag 3, Xue Liu 4,2,3 Depart. of Electroic Egieerig, Shaghai Jiaotog Uiversity, Chia 4 Depart. of Computer Sciece & Egieerig, Uiversity of Nebraska Licol, USA ,2,3 {yiluofu, qiyi 33xwag8}@sjtu.edu.c, 4 xueliu@cse.ul.edu November 2, 200 Abstract This paper ivestigates throughput ad delay based o a ewly predomiat traffic patter, called coverge-cast, where each of the odes i the etwork act as a destiatio with k radomly chose sources correspodig to it. Adoptig Multiple-Iput-Multiple-Output (MIMO) techology, we devise two may-to-oe cooperative schemes uder coverge-cast for both static ad mobile ad hoc etworks (MANETs), respectively. I a static etwork, our scheme highly utilizes hierarchical cooperatio MIMO trasmissio. This feature overcomes the bottleeck which hiders coverge-cast traffic from yieldig ideal performace i traditioal ad hoc etwork, by turig the origially iterferig sigals ito iterferece-resistat oes. It helps to achieve a aggregate throughput up to Ω( ɛ ) for ay ɛ > 0. I the mobile ad hoc case, our scheme characterizes o joit trasmissio from multiple odes to multiple receivers. With optimal etwork divisio where the umber of odes per cell is costat bouded, the achievable per-ode throughput ca reach Θ() with the correspodig delay reduced to Θ(k). The gai comes from the strog ad itelliget cooperatio betwee odes i our scheme, alog with the maximum umber of cocurret active cells ad the shortest waitig time before trasmissio for each ode withi a cell. This, to a great extet, icreases the chaces for each destiatio to receive the data it eeds with miimum overhead o extra trasmissio. Moreover, our coverge-based aalysis well uifies ad geeralizes previous work sice the results derived from coverge-cast i our schemes ca also cover other traffic patters. Last but ot least, our cooperative schemes are of iterest ot oly from a theoretical perspective but also shed light o future desig of MIMO schemes i wireless etworks. Itroductio Fueled by the semial work of Kumar [] et al., who showed that the optimal static uicast capacity is Θ( ) ad Θ( ) for radom etwork, capacity aalysis of ad hoc etworks have triggered log We use the followig otatio throughout our paper: f() f() = o(g()) lim f() = ω(g()) lim f() = O(g()) lim sup g() = 0, g() f() = 0, f() g() <,

2 great iterest. Later o, Grossglauser ad Tse [2] demostrated that Θ() capacity per sourcedestiatio (S-D) pair is achievable if takig mobility of the etwork ito accout, but packets have to edure a larger delay. Due to the pheomeo that larger capacity is at the cost of a larger delay, some aalysis o capacity-delay tradeoffs arises. Oe iterestig work is from Neely ad Modiao [3] who itroduced redudat packets trasmissio through multiple opportuistic paths to reduce delay while a decrease o capacity is also icurred. Uder i.i.d. mobility, the per-ode capacity is show to be T () = Θ() ad delay D() yielded to scale as Θ( T ()) [3]. Later work also studied the tradeoff betwee capacity ad delay, where odes either perform traditioal operatios such as storage, replicatio ad forwardig ( [4]- [6]) or trasmit through codig or ifrastructure support ( [7]- [9]). However, all the results above strogly rely o the assumptio that all the cocurret trasmissios are always iterferig with others. This becomes a limitatio which largely costrais the capacity. I cotrast, MIMO eables odes to perform cooperative commuicatio by turig mutually iterferig sigals ito useful oes, where the gai of capacity ca the be obtaied. The gai is well demostrated by Aero et al. [0] who preseted a MIMO collaborative strategy which achieves a per-ode capacity of Θ( /3 ). Followig that, Özgür et al. [6] costructed a hierarchical cooperative scheme relyig o distributed MIMO commuicatios to achieve a liear capacity scalig. It turs out that early all the iterfereces ca be caceled through hierarchical cooperatio. Thereo, multicast scalig is take ito accout i [5] uder hierarchical cooperatio which achieves a aggregate capacity of Ω kš ɛfor ay ɛ > 0. This also achieves a gai o capacity compared with previous works o multicast such as []- [4]. While the tradeoff for uicast ad multicast traffic patter have bee extesively studied i previous work, coverge-cast is still a relatively ew cocept ad uder active research. Covergecast refers to a commuicatio patter i which the flow of data from a set of odes trasmit to a sigle ode, either directly or over multi-hop routes. Recetly, there appeared may ew applicatios such as real-time multimedia, battlefield commuicatios ad rescue operatios that impose striget capacity-delay requiremets o coverge-cast. I this paper, we joitly cosider the effect of coverge-cast ad cooperative strategies o asymptotic performace of etworks. The motivatios come from the followig reasos:. Although there have bee some researches o coverge-cast (such as [7], [8], [9], [20]), their major cocer is limited to the extreme case where all odes flow data to a sigle sik i the etwork. However, a wide rage of applicatios such as machie failure diagosis, pollutat detectio ad supply chai maagemet may require multiple such coverge-cast groups existig i parallel i the etwork rather tha a sigle oe. 2. Ulike multicast where the trasmissio process becomes more ad more diverse, vast space of further improvemet o its performace ca be discovered i covergecast, due to its coverget process. 3. Sice distictive sources may trasmit differet data to their commo destiatio, such traffic patter ca be treated as a geeralized reversed multicast. This esures a coverage o other kids of traffic modes such as uicast, multicast ad broadcast, sice f() = Ω(g()) lim if f() g() <, f() = Θ(g()) f() = O(g()) ad g() = O(f()) f() =eθ( ): The correspodig order Θ( ) which cotais a logarithmic order. 2

3 all of them ca be regarded as special cases of coverge-cast. To our best kowledge, there are o previous study o the etwork performace uder coverge-cast with MIMO. Cocetratig o throughput ad delay performace i this paper, we propose a ew type of may-to-oe cooperative schemes with MIMO i both static ad mobile etworks, from the perspective of coverge-cast. First, we desig a may-to-oe cooperative scheme i a static etwork, where the whole etwork is divided ito clusters with equal umber of odes i each of them. Commuicatios betwee clusters are coducted through distributed MIMO trasmissios combied with multi-hop strategy while withi a cluster it is operated through joit trasmissio of multiple odes to others oce a time. Each cluster ca be treated as a subetwork ad further divided ito smaller clusters. This process is carried out through hierarchical operatio. The multiple-trasmissiomultiple-receptio feature of MIMO suits the may-to-oe characteristic of coverge-cast well. I a traditioal ad hoc etwork, oly oe trasmissio ca be active at a time while all the adjacet trasmissios are treated as iterferece. This imposes a sigificat bottleeck o coverge-cast ad makes it impossible to achieve ideal performace. However, this bottleeck ca be removed with the adoptio of MIMO. The gai comes from the smart trasformatio from iterferig sigals ito useful oes to the receivers through hierarchical cooperative trasmissios i the scheme. Uder MANETs where hierarchical cooperatio caot be established due to the mobility of odes, we devise aother may-to-oe cooperative scheme where the etwork is still divided ito equal cells. I each time slot, multiple odes that possess iformatio for the same destiatio are allowed for joit trasmissio to other odes withi the cell. Other odes will receive a combiatio of the iformatio from these trasmitters due to the effect of MIMO through fadig chaels. This procedure cotiues, with the umber of odes that hold such mixed iformatio icreases, utill all the destiatios receive sufficiet mixed iformatio that ca be decoded with high probability. Our mai cotributios ca be summarized as follows: Our may-to-oe cooperative scheme i a static etwork breaks the bottleeck hiderig coverge-cast from achievig ideal performace i a traditioal etwork by covertig adjacet iterferig sigals ito useful oes. The achievable aggregate throughput ca be up to Ω( ɛ ) for ay ɛ > 0, which early approaches the upper-boud. For our may-to-oe cooperative scheme uder MANETs, the optimal choice for etwork divisio is costat-bouded umber of odes per cell. Whe combied with MIMO, this allows the maximum umber of cocurret trasmittig cells as well as the shortest waitig time before a trasmissio for each ode withi a cell. This leads to a per-ode throughput of Θ() with the correspodig delay reduced to Θ(k). Our results well uify ad geeralize the previous work (such as [7] i MANET ad [6], [7]- [20] i static etwork) sice all of them ca be easily applied to other traffic modes. Furthermore, our ovel may-to-oe cooperative schemes provide useful guidelies for future desig of MIMO schemes i wireless etworks. Especially, our scheme i MANETs breaks the vacacy of such MIMO scheme desig remaiig i mobile etworks before. The rest of the paper is orgaized as follows. I Sectio 2, we preset the models ad defiitios. I Sectio 3 ad Sectio 4, we describe our cooperative schemes uder static ad mobile ad hoc 3

4 etworks, respectively. The correspodig throughput ad delay achieved based o the two schemes are also preseted i detail i these two sectios. All the results are further discussed i Sectio 5. Fially, we preset cocludig remarks ad outlie the directios for future work i Sectio 7. Some proofs are provided i-lie ad others are i Appedix. 2 Models ad Defiitios 2. Network Model I this paper, we cosider a ad hoc etwork where odes are radomly positioed i a uit square. Traffic Patter: I coverge-cast sceario, we assume odes located i the etwork with each oe servig as a destiatio. For each destiatio ode, there are k radomly ad idepedetly chose sources. These k odes the sed packets to their commo destiatio. I multicast, all the packets set out from a source ode are the same while i covergecast, the packets from those k sources may be totally differet ad all of them are idispesable to form the complete iformatio. Moreover, the data rates of each edge of the spaig tree i multicast are all same while they are differet i each edge i covergecast. Physical Layer model: We assume that commuicatio takes place over a chael with limited badwidth W. Each ode has a power budget P. The chael gai betwee two odes v i ad v j at time t is give by: h ij [t] = Gd α/2 ij e jθ ij[t], () where d ij is the distace betwee the odes, θ ij [t] is the radom phase at time t, uiformly distributed i [0, 2π). {θ ik [t]} are i.i.d. radom processes across all i ad k, idepedet of each other. G ad the path loss propagatio α 2 Xare assumed to be costats. The, the sigal received by ode i at time t ca be expressed as Y i [t] = h ij [t]x j [t] + Z i [t] + I i [t], (2) j T[t] where Y i [t] is the sigal received by ode v i at time t, T[t] represets the set of active seders trasmittig sigals to v i, which ca be added costructively, Z i [t] is the additive white Gaussia oise at v i with variace N 0 per symbol ad I i [t] is the iterferece from the odes. Moreover, we assume each ode is equipped with oe atea. We do ot cosider the case where each ode has multiple ateas for the followig two reasos:. If each ode is assumed to have costat bouded umber of ateas, say, c ateas, the the throughput is c times that achieved i sigle-atea case, which does ot chage the throughput order; 2. If each ode has r ateas where r scales with, the the throughput achieved i order sese is r times that of sigle-atea case. This is trivial ad assumig r ateas o oe ode is ot realistic. 2.2 Defiitios Coverge-cast Sessio: a coverge-cast sessio is defied as the set composed of oe destiatio ad its correspodig k sources. 4

5 Delay: Delay is defied as the time a destiatio takes to receive all the packets from its correspodig k sources. The averagig is over all bits (or packets) trasmitted i the etwork. Throughput: A throughput λ > 0 is said to be feasible if each sources i a coverge-cast sessio ca sed at a rate of at least λ bits per secod to their commo destiatio. Deotig m(t) as the umber of packets from sources that a destiatio receives i t time slots. The, the log term per-ode throughput is defied as Ad the aggregate throughput is Λ = λ. 2.3 Notatios λ = lim if t m(t). t I table, we list all the parameters that will be used i later aalysis, proofs ad discussios. Notatio Defiitio Table : Notatios The total umber of odes i the etwork. k The umber of sources for each destiatio i the etwork. h The umber of layers a etwork is divided ito. i The ith layer of the etwork, where i h. i k i ci k ci t i T i D(i, k) B i M The umber of odes i the ith layer. The umber of sources for each destiatio ode i the ith layer. The umber of clusters i the ith layer. The umber of source clusters i the ith layer. The umber of coverge-cast sessios i the ith layer. The aggregate throughput at layer i i static etwork. The average delay to complete a coverge-cast sessio for a destiatio at layer i. The miimum amout of data a ode eeds to sed at layer i. We ca also call it bulk size. The average umber of odes i each cell. 3 May-to-oe Scheme uder Static Networks I this sectio, we will desig a cooperative scheme with MIMO uder static etworks. The we will aalyze the throughput ad delay achieved uder the scheme. 3. May-to-oe Cooperative Scheme uder Static Networks As is show i [6], hierarchical cooperatio ca achieve better throughput scalig tha classical multihop schemes uder certai assumptios o the chael model i static wireless etwork. This 5

6 motivates us to desig a hierarchical scheme which ca be applied to coverge-cast. Layer i step step 2 step 3 Layer i- step --> step 2 --> step 3 Figure : A global view of our hierarchical may-to-oe cooperative scheme. The algorithm starts from the bottom layer ad keeps executig util it reaches layer h. 3.. Schedulig Algorithm Uder hierarchical schemes, a etwork is divided ito clusters with equal umber of odes i each oe. Each cluster is the treated as a subetwork ad we ca further divide the subetwork ito smaller clusters. With recursio operatio, the procedure goes o util the etwork is divided ito h layers with the origial etwork at the hth layer ad the st layer at the bottom oe. A schedulig algorithm ca be desiged o each subetwork at each layer. The algorithm keeps executig from layer to layer, the process of which is similar per layer per cluster but with a larger scale as the umber of layer i icrease from to h. The procedure cotiues util all the layers have fiished the algorithm. Sice the algorithm is similar at each layer but with differet scale, we will preset our recursive cooperative scheme at a particular layer i. At each layer, the scheme is divided ito three steps, which are described as follows. Ad Figure shows the hierarchy ad recursio betwee layers i our algorithm. Step. Preparig for Cooperatio with Recursio: Sice there are k i source odes belogig to oe sessio at layer i, uder coverge-cast, they must distribute their packets to other odes i the same cluster. For each ode i the cluster, the k i sources joitly trasmit their packets to other odes i the cluster, which receives a liear combiatio of that bit mixed with chael coefficiets. The process keeps util all the other odes except for these k i sources receive the packets from them. Note that as for each trasmissio from the k i sources to a specific Note that the size of packet i our cooperative scheme differs at differet layers. Each ode divide a packet ito i k i packets. Each of these packets is the used i the ext layer i. 6

7 ode, the process is a may-to-oe trasmissio ad this is equivalet to dividig the curret cluster ito smaller-size clusters ad the similar procedure executes i a smaller cluster. Note that our algorithm starts from the bottom layer, i.e., layer of the etwork ad cotiues to a higher layer util layer h. Figure 2: A example of a CT i multi-hop MIMO trasmissio. Assume oe destiatio has 4 sources. The red parts represet source clusters. /4 percet of the odes are allowed for trasmissio i the two source clusters o the left while /2 percet of the odes are active i the source cluster o the right.wheever several clusters flow to a commo cluster i the ext time slot, that cluster will be colored with the part several times larger tha all the trasmittig clusters to it. Oly the destiatio cluster (colored with yellow) will be etirely colored whe all the data fially flows to it. Step 2. Multi-hop MIMO Trasmissios: For the sake of eergy efficiecy, we costruct a multi-hop routig mode rather tha adoptig direct MIMO trasmissio betwee clusters i this step. Several source clusters start a series of MIMO trasmissios to reach their commo destiatio clusters i multi-hop maer. Sice each source cluster has i k i k i packets to sed i oe time slot, due to MIMO, several source clusters are allowed for cocurret trasmissio to oe cluster at the same time slot. To achieve asymptotically optimal coverge-cast capacity, we costruct a coverge-cast tree (CT) by coductig the three substeps preseted below, spaig from source clusters S ij s to their commo destiatio clusters D i. Here j k ci. Deote P i = {S ij, D i, j k ci }.. Costructig the Euclidea spaig tree E EST : Firstly we divide the uit square ito cells with side legth 2, where g = t, t = log g 4 k. For each cell that cotais s 2 clusters i P i, we radomly select a cluster p ij. For ay other p ik (k j) i the cell, let E EST E EST {p ix p iy } ad P P {p ik }. Subsequetly we coduct this process by lettig g = t 2,...,, Gettig the Mahatta routig tree E MRT : for each edge uv i E, assume that the coordiates of u ad v are (i u, j u ) ad (i v, j v ), respectively. We the fid a cluster w whose coordiate is (i u, j v ). Afterwards, E MRT E MRT {uw}, E MRT E MRT {wv}, E MRT E MRT {uv}. 3. Obtaiig the MT(P i ) for each edge uw i E MRT, we coect clusters crossed by uw i sequece to form a path, deoted as E(u, w). The E CT E CT E(u, w), E MRT E MRT {uw}. Fially, E CT is the set of edges of CT(P i ). 7

8 Figure 2 shows a simple example of the data flow o such coverge-cast tree (CT). Figure 3: 9-TDMA scheme where the whole etwork is divided ito clusters with equal area. Each 9 groups are categorized as a group. All the yellow cells i each group (umbered with ) ca trasmit simultaeously i a time slot. I the ext time slot all the cells umbered with 2 trasmit ad so o. Step 3. Cooperative Receptio: Give the total umber of coverge-cast sessios t i at layer i, cosider a particular ode i the cluster. It ca receive t ik i i packets from other odes, with each of them cotributig t ik i i i packets. Cosiderig i destiatios i each cluster, the traffic load are i k i i Θ t i Špackets. Sice the data exchages oly ivolve itra-cluster commuicatio, they ca work accordig to 9-TDMA scheme where the cells which are located 3 cells away from each other ca be active cocurretly, as is show i Figure Throughput ad Delay Aalysis uder May-to-oe Cooperative Scheme Now we focus o throughput ad delay that ca be achieved uder the scheme preseted i 3... We first the upper-boud of throughput ad our mai results as follows. Lemma Uder coverge-cast, with each of the odes i the etwork actig as destiatio ad X receivig packets from its distictive k sources, the aggregate capacity is upper-bouded by λ i C log (3) i= where C > 0 is a costat idepedet of. Proof Provided i Appedix A. Theorem I static wireless etworks, by adoptig our cooperative scheme, we ca achieve a aggregate throughput of Λ =ÜΘ 2h 2 2h k 2h (4) 8

9 with the delay of E[T ] = 8><>: ÜΘ 2h2 4h+3 2h k 2h2 2h 2h, if k = Ω( 2h 2 ) (5) ÜΘ h2 2h+2 2h k h2 4h+3 2h, if k = O( 2h 2 ) To prove Theorem, we will first itroduce the followig lemmas. Lemma 2 (Lemma i [5]) I a etwork with odes radomly ad uiformly distributed o a uit-square, the miimum distace betwee ay two odes is +δ w.h.p., for ay δ > 0. Lemma 3 (Lemma 4.3 i [5]) By 9-TDMA scheme, whe α > 2, oe ode i each cluster has a chace to operate data exchages at a costat trasmissio rate. Also whe α > 2, the iterferig power received by a ode from the simultaeously operatig clusters is upper-bouded by a costat. Lemma 4 Give k i idepedetly ad uiformly distributed source odes i the etwork at layer i, the umber of source clusters k ci =8< is give by :Θ (k i ), if k i = O( ci ) k ci. (6) Θ i i, if k i = Ω( ci ) Proof 2 The proof is similar to that of Lemma 4.5 i [5] ad we do ot preset the proof here. Lemma 5 Whe t i k i = O (( ci ) p 2 ) holds for all layer i, where 2 i h ad p 2 is a positive costat, if k i = Ω( i log ci ), the k i = Θk i ciw.h.p. if k i = O( i log ci ), the k i = Ok i ciw.h.p. Proof 3 The proof is similar to that of Lemma 4.6 i [5] ad we do ot preset the detailed proof here. Cosider the three steps i our scheme at layer i. Assume a aggregate coverge-cast throughput ÜΘ a i kb i Šis achievable at layer i w.h.p., where 0 a, b 0 ad a + b < 0. It is easy to obtai that the total time to complete iêk i t i traffic loads is k i t i ÜΘ a i k i k i t i!+o k i ki b i t Œ+Θ a i!. (7) k i i i i We say a evet occurs with high probability (w.h.p.) if its probability goes to as. 9

10 Hece, the throughput ca be expressed as T i = iq =ÜΘ ÜΘ k i t i a i +Ok i ki b i t +q k =ÜΘ =ÜΘ a i i k i t i k i i t i k i i i +Θ k a i i i i i k i + a i i ki b i i 2 a +qk i i i i k i + ki b 2 a i i i È i i k ci + 2 a i k b i I order to optimize the etwork divisio at layer i, we cosider two cases, i.e., ci = O(k i ) ad ci = Ω(k i ). Accordig to Lemma 4 ad Lemma 5, we have the followig two cases:. If ci = O(k i ), the k ci = O( ci ), k i =ÜΘk i ci; Ž. (8) 2. If ci = Ω(k i ), the k ci = Θ(k i ), k i =ÜΘ(); =ÜΘ I case, the throughput i Equatio (8) ca be writte as i i T i È i i k ci + 2 a =ÜΘ i + a b i k b i k b i i b i!. b i b i Ž (9) The result is optimized whe i =ÜΘk a a 2 ad k i = Ωi. b a+b 2 i b a+b 2 i. The, ci = i i = k b a+b 2 i a a+b 2 i = O(k i ) At the bottom layer, the aggregate throughput is k. If we divide the etwork i the optimal way at each layer, the relatioship betwee i, k i ad throughput at each layer is i = k b b ad T i = k b a+b 2 i b a+b 2 i, the recursio calculatio is listed as follows: i a b i T i k k k k h 4 2h 3 2h 2 2h 2 2h h 2h 3 kh h Note that h =, we obtai the aggregate throughput at layer h, i.e., 2h h k 2h 2 2h h i a+b 2 b i T =ÜΘ 2h 2 2h k 2h. (0) 0

11 I the optimized result, k = Ω a a 2=Ω 2h 2. ad Moreover, sice i = k 2i 2i 2 2i 2 i i k i i =ÜΘk i i ad k i =ÜΘk i ci=üθ k i i i Š, we have i Ad this yields to the followig derivatio: Hece, i = i k i = k = k = k = k = k = k = k 2i 2i 2i k i 2i 3 i 2i 3 2i i 2i 3 2i i k = 2i i. 2i+ 2i 2i+ 2i 3 2i k 2i + 2i 3 2i+ 2i + 2i 3 2i+3 +( h 2i 2i 2i+ i+œ2i 3 2i 3 2i+ i+ 2i+... 2i 3 2h 3) 2h 2h ( h 3 2h ) 2h 3 + 2( 3 2h ) 2h, from which we ca obtai = h 2h k h 2h. Now we tur to the aalysis o delay at layer i, deoted as D(i, k). Through Equatio (8), we Q ÜΘ h get D(h, k) = B T i=2 i k i Q, where B is miimum size of data trasmitted at layer h, i.e., B =ÜΘ h kœ=üθ kšh 2 k=() 2h 2 4h+2 2h k 2h2 2h 2h. Ad through recursio o D(i, k), the fial delay D(h, k) ca be obtaied, i.e, i= i k i+œ= D(h, k) =ÜΘ 2h2 4h+3 2h k 2h2 2h 2h. () The we focus o case 2, where =ÜΘ ci = Ω(k i ). Cosider the Žaggregate throughput at layer i, we have i i T i =ÜΘ È i i k i + 2 a i i i k i 2 i + a i Ž. (2)

12 Whe the result is optimized, i =ÜΘ( i k i ) 3 2a. Therefore, ci = i i = k 2a 3 i 2a 2 2a 3 i = Ω(k i ) a a 2 ad k i = Oi. Note that i this case, the aggregate throughput is at the bottom layer sice the traffic patter ca be treated as uicast at this layer. Dividig the etwork i the optimal way at each layer, the relatioship betwee i, k i ad throughput T i is i = ki 3 2a i calculatio is listed as follows: i a b i T i k k 5 3 2h 4 2h 3 2h 3 kh 2h 2h 3 h k 2h h T i = k 2h 2 2h h a 3 2a i 2 a 3 2a i. The recursive h Followig the same procedure i case, the aggregate throughput ad delay D(h, k) at layer h ca be obtaied, which are show as follows: T = T h =ÜΘ 2h 2 2h k 2h, (3) ad D(h, k) =ÜΘ h2 2h+2 2h k h2 4h+3 2h. (4) This completes our proof for THEOREM. 4 May-to-oe Scheme uder MANETs I Sectio 3, we aalyze the performace i static etworks. I this sectio, we tur to the mobile etworks. Due to the mobility characteristics of odes, the etwork performace may be quite differet from that i static oes. I the followig subsectios, we will itroduce the mobility model ad preset aother scheme that is suitable for mobile etworks. The, we will give our aalysis o throughput ad delay obtaied from the scheme. 4. Mobility Model We itroduce two-dimesioal i.i.d. mobility model ito the etwork, i.e., odes are uiformly distributed i the etwork. At the begiig of each time slot, each ode radomly chooses a poit i the uit square ad moves there. I this model, we assume that the odes move quickly so that the odes positios are idepedet from time slot to time slot. We also defie it as fast mobility model where the mobility of odes is at the same time-scale as that of data trasmissio. 4.2 May-to-oe Cooperative Scheme 2 uder MANETs Whe the positio of odes may be varyig with time, it is impossible to costruct a hierarchical scheme uder mobile etworks. Sice the relatioship determied i the curret time slot betwee 2

13 odes may be destroyed i the ext oe due to the radomess icurred by mobility. Hece, we eed to desig a ew scheme that ca take advatage of mobility of the odes. With appropriate schedulig, the etwork performace ca be improved May-to-oe Cooperative Scheme 2 We divide the whole etwork ito c cells such that there are M odes i each cell o average. To avoid the iterferece icurred to the etwork from the eighborig cells, we adopt the 9-TDMA strategy illustrated i sectio 3 agai. Accordig to our model defiitio, sice each ode acts as a destiatio, there are always some destiatios i each cell. Each cell becomes active oce every c 0 time slots. I a active cell, trasmissio occurs amog the odes withi the same cell. I a active cell, i each time slot, if there exist both a destiatio ad some of its sources, the we call there are sources-destiatio pair i the cell. If there are several such pairs i the cell, the we radomly choose oe pair, ad let all these sources i this pair form a atea array ad joitly sed their packets to their commo destiatio as well as all the other odes i that cell. All the other odes except the destiatio pick out a stored packet which is for the same destiatio as the ew oe, liearly combie the ew oe with it ad the replace the stored oe i the relayig buffer. If there are o sources-destiatio pairs i the cell, choose the maximum umber of sources that belog to the same destiatio i that cell. The, the chose sources joitly sed their packets to all the other odes i the same cell. Similarly, the odes will liearly combie the ew packet with the stored oe ad replace it i the relayig buffer. If there are either sources-destiatio pairs or sources that belog to the same destiatio i the cell, the choose the maximum umber of relays which hold the packets that are to be trasmitted to the same destiatio. Those chose relays the joitly sed their packets to all the other odes i the same cell. After receivig the packet, all the ode will update the packet as described above. A simple illustratio of our scheme is show i Figure Aalysis of Throughput ad Delay uder May-to-oe Scheme 2 I this subsectio, we will aalyze the achievable throughput ad delay uder our proposed scheme 2. First, we will first compute the boud of achievable delay ad the aalyze the correspodig throughput. The mai results obtaied uder scheme 2 is preseted i the followig theorem. Theorem 2 Suppose k = o(), the uder may-to-oe cooperative scheme 2, with the optimal etwork divisio M = Θ(), we ca achieve ideal performace o both the average delay required 3

14 8< Θ for a destiatio to receive packets from all its k correspodig sources ad the per-ode throughput, listed as follows: :λ = log(),e[d N ] = Θ(log()) if k = Θ(). (5) λ = Θ (), E[D N ] = Θ(k) if k = ω() The total square area is divided ito c cells where M odes are located i each cell o average. Figure 4: Illustratio of our may-to-oe cooperative scheme 2. The uit square is divided ito c cells with M odes located i each cell averagely. The cells colored with yellow ca be active cocurretly uder 9-TDMA scheme. For a active cell, what may be goig o is show i the subfigures (a) ad (b) o the right. (a) shows the case where there are at least oe sourcesdestiatio pairs i the etwork. (b) shows the case where there are o sources-destiatio pairs i the cell. To prove Theorem 2, we tur to the proof for delay i 4.3. first ad the prove the throughput i Aalysis o Delay Before the proof of delay, we fist itroduce the followig two lemmas. Lemma 6 Cosider odes uiformly distributed i the etwork area. The etwork is divided ito c idetical cells. The, the umber of odes i each cell is M = c w.h.p. if lim c log c =. Proof 4 Provided i Appedix B. Lemma 7 As for a destiatio ode, the coditio that it ca successfully decode the packets from all its k sources is that there should be at least k differet liear combiatios of these packets i 4

15 its receivig buffer ad the coefficiet vectors of these k combiatios are liearly idepedet of each other. Viewig from the perspective of etwork codig, the cetral problem arises: how log does it take for a destiatio ode to receive at least Θ(k) combiatio o average? If deotig the whole time as D N, the E[D N ] E[D ] + E[D 2 ], where E[D ] ad E[D 2 ] represet the time required for all odes i the etwork to have oe packet of the sources belog to that destiatio ad the time required for the destiatio to receive Θ(k) packets give that all the other odes already hold a packet, respectively. We focus o E[D ] first. Note that for each destiatio, it has k sources. That is to say, for oe sessio, iitially, there are oly k odes which hold the origial packets. Ad the process for lettig all the other odes i the etwork get these k packets is equivalet to the process for floodig k packets to all the other odes give that there are origially k distict odes holdig each of these packets. First, cosider a case where k distictive packets are stored i k odes iitially ad all the other odes i the etwork are empty. Now we first aalyze the delay required o the process for lettig all the odes i the etwork have these packets. Deote J t as the umber of odes holdig the packets from the k odes at time t. Note that J 0 = k. Ad let β t = J t /J t represet the growth factor after oe time slot. Obviously, we have β t+ = J t + a + a a Jt J t, (6) where a i represets the umber of ew odes to which the ith packet-holdig ode trasmits durig oe slot. Note that i each time cell, it is a multiple-trasmissio-multiple-receptio process. As the umber of packet-holdig odes grows, the growth factor β t will yield differet scale. Also otice that β t is also iflueced by differet etwork divisio. Joitly cosider these two factors, we discuss i the followig two cases :. If M = Ω(k), the k c = Θ (k). I this case, iitially, J t is much smaller tha M. Thus, there are o average oe such packet-holdig ode i each cell. This ode the trasmits the packet to all the other M odes durig oe time slot. As J t grows to M, the umber of cells where packet-holdig odes are located becomes the equal order of M, which ca guaratee that there are o average Θ J tm Šsuch packet-holdig odes per cell per time slot. 2. If M = O(k), the k c = Θ MŠ: I this case, the iitial umber of packet holdig odes k is already much larger tha the umber of cells M. Thus, the average umber of packet-holdig odes per cell is Θ km Š. I mobile etworks, to obtai the chael status iformatio (CSI) of other odes, a traiig sequece is cotaied i each packet. A destiatio ca the recover CSI through these traiig sequece cotaied i each packets. We assume it is of the equal size of a packet ad the packet size is sufficietly small compared to the total umber of odes i the etwork. Thus, the mobility of odes durig the acquiremet of a traiig sequece ca be eglected, if compared to data trasmissio time. For a ode who does ot have the packet iitially, it oly eeds to receive oe combiatio of some (or all) of these k origial packets. Ad for simplicity, i the followig part, whe we say a packet, we refer to the combiatio actually. 5

16 The, we have the followig lemmas: Lemma 8 I case, for each destiatio, kš the time required for all odes i the etwork to have oe packet from the sources correspodig to that destiatio is log E[D ] = Θ M ɛ + log( + M)!+Θ + log( M ) Œ, (7) M where ɛ is a arbitrarily small value greater tha zero. Proof 5 Provided i Appedix C. Lemma 9 I case 2, for each destiatio, the time required for all odes i the etwork to have oe packet from the sources correspodig to that destiatio is ( + log ( k)) E[D ] = Θ +. (8) M( k) Proof 6 For case 2, the procedure directly starts from the stage durig which the umber of odes grows from k to. Followig the same aalysis as that i case, we obtai ( + log ( k)) E[D ] = Θ +. (9) M( k) Now, we tur back to the real case where all the odes i the etwork are destiatios, each of which is required to receive packets from its correspodig k sources. Due to the fairess achieved i our scheme, each ode i a cell has equal chaces to receive packets. Sice there are o average M odes per cell, every ode ca be active oce every M time slots. Therefore, the average delay of the whole etwork durig this period is E[D ] = M E{D }. Now we give the followig lemma related to E[D ]. Lemma 0 The average delay for lettig all odes i the etwork to have oe packet of the sources belogig to the same destiatio =8< is bouded by :Θ(M) if M = ω(log()) E[D ]. (20) Θ() if M = o(log()) kš Proof 7 For case preseted i Lemma 8, we ca get log M E[D ] = Θ + + log( log( + M)!+Θ M M ). sice /M = Ω(k), we have /k = Ω(M). Thus, the term Θ log( k ) log(+m) is less tha ad is egligible compared to the term Θ M + + log( M )Š. Hece, we discuss o the term Θ M + + log( M )Š. Whe M = Θ(), log MŠapproaches to zero. Thus, M E[D ] = Θ(M + ) = Θ(). Whe M = ω(), the term log MŠ+Šca be omitted. Hece, M E[D ] = Θ(M). 6

17 For case 2 preseted i Lemma 9, we have M E[D ] = ΘM + (+log( k) Sice k = o(), k =8< discussio o M ad log(), we get :Θ(M) M E[D ] ( + log( k). k log(). Therefore, E[D ] = Θ(M + log()). With further if M = ω(log()). Θ(log()) if M = o(log()) =8< Joitly cosider case ad case 2, we have :Θ(M) if M = ω(log()) E[D ]. (2) Θ(log()) if M = o(log()) This completes our proof of Lemma 0. As for E{D 2 }, it is easy to kow that it takes a sigle destiatio k slots to receive k distictive ecoded packets give that all the odes i the etwork already hold oe of them. Ad cosider the fact that each destiatio i oe cell will have such chace oce every M time slots, we obtai the average delay required for all the destiatios to receive k distictive ecoded packets is M k. That is to say, E{D 2 } = Θ(Mk). Therefore, =8< we ca boud the total delay achieved uder our scheme as :Θ(Mk) if k = ω(log()) E[D N ] E[D ] + E[D 2 ]. (22) Θ(log()) if k = o(log()) Remark Durig the aalysis of E{D 2 }, we do ot take ito accout the case of decodig failure. That is, a destiatio still caot successfully decode the origial k packets after it receives k distictive ecoded packets. Such failure may be caused by the fact that the rak of the coefficiet matrix geerated from these k ecoded packets is less tha k. I this case, the destiatio has to receive more extra packets i order to successfully decode the k packets. However, based o the assumptio that k = o(), the probability of such decodig failure ca be upper-bouded by a costat factor, which does ot chage the order of our boud o E{D 2 }. If we remove the assumptio k = o(), the we caot simply treat decodig failure at the destiatio as a costat boud ad the average time spet for a sigle destiatio to successfully decode all the k packets must be larger tha Θ(k). It remais a iterestig future work that how decodig failure ca ifluece the delay if k = Θ() Aalysis o Throughput Lemma Uder our may-to-oe =8< :Θ cooperative scheme 2, we ca achieve a per-ode coverge-cast throughput of λ log() MŠ if k = o(log()) Θ if k = ω(log()). (23) i a MANET. 7

18 Proof 8 We calculate the throughput through the followig way: viewig from the perspective of a source ode, it belogs to k distictive destiatios o average. Thus, it has to trasmit at least k times. The, the umber of trasmissios per time slot is k/(e(d N )). As ca be see from the results o delay, the term Mk domiates the scale of the total delay. Ad we ca simply regard delay as Θ(M k). Thus, we obtai the per-ode throughput show i Equatio (). This completes our proof. Notice that both the throughput ad delay are optimized whe M = Θ(), which reders the results preseted i Theorem 2. 5 discussio I the previous sectios, we have derived all the performace metrics uder both static ad mobile etworks. I this sectio, we will further discuss these results. 5. The advatage of Our Cooperative Schemes I this sectio we focus o the effect of our proposed schemes brought to the etwork. I static etwork, our may-to-oe cooperative scheme allows for cocurret trasmissio, which coverts the iterferig sigals ito useful oes. This reduces the iterferece level to a extesive degree ad therefore udoubtedly leads to a improvemet o throughput. Whe the umber of layers h are sufficietly large i our scheme, the aggregate throughput ca reach Θ(). This is close to the upper-boud with differece of oly a log() factor. I MANETs, with further observatio o our scheme 2, we ca fid it is to some extet equivalet to a floodig algorithm but with more itelliget trasmissio. However, i previous floodig algorithm, packets are simply broadcasted arbitrarily to other odes i the cell, regardless of whether the receivers are destiatios of those packets. This udoubtedly leads to some uecessary waste o the umber of trasmissio, which icurs sacrifice o throughput. 5.2 Delay-throughput Tradeoff I this subsectio, we cosider delay-throughput tradeoff obtaied uder our schemes. 8>< Static etwork: By THEOREM, we obtai the delay/throughput tradeoff, as is show as follows: > : ÜΘ 2h2 4h+4 2h k 2h2 2h 2h, if k = Ω( 2h 2 ). (24) ÜΘ h2 2h+ 2h k h2 4h+2 2h, if k = O( 2h 2 ) Note that the tradeoff for k = O( 2h 2 ) is poor compared to that for k = Ω( 2h 2 ). I other words, a larger k helps to reduce delay. This is because the umber of clusters i our scheme is smaller tha that of sources whe k = Ω( 2h 2 ). This allows more simultaeous trasmittig odes to achieve largely reduced delay but at the cost of more extra eergy cosumptio. MANETs: The delay/throughput tradeoff obtaied uder mobile etwork is M 2 k. A couterituitive pheomeo ca be observed that a smaller umber of cells leads to poorer performace 8

19 o both throughput ad delay. However, from MIMO perspective, a icrease o the umber of odes per cell leads to the decrease o the umber of cocurret active cells, uder 9-TDMA scheme. Moreover, i each cell, as the umber of odes becomes large, each ode have to edure a loger waitig time before trasmissio. Both of the two factors reduces the efficiecy, which therefore leads to a larger delay for completig the whole process. Hece, the tradeoff is optimized whe M = Θ(), with per-ode throughput achieves Θ() ad the correspodig delay reduced to Θ(k). Because it ca guaratee the maximum umber of cocurret active cells as well as the shortest waitig time edured by each ode i the cell before trasmissio or receptio. 5.3 Coverig to Other Traffic Patters For MANETs, we choose the optimal etwork divisio, i.e., substitutig M = Θ() ito both our results o coverge-cast ad those applied to uicast, multicast ad broadcast Coverig to Uicast If we set the umber of sources per sessio k, uder coverge-cast as, the we get uicast traffic patter. Actually, uicast is a special type of coverge-cast. Therefore, all the results obtaied uder coverge-cast i both static ad mobile etworks ca be easily applied to uicast i this paper. To preset the exteded results more clearly, we discuss i the followig two cases:. Static etwork: i this case, whe we substitute k = ito Equatio (), we obtai the aggregate throughput for uicast, as is show i the followig equatio: Λ =ÜΘ 2h 2 2h. Ad the delay achieved is E[D] =ÜΘ h2 4h+3 2h. 2. MANETs: i this case, substitutig k = ito our resutls, we get the delay for uicast, E[D N ] = Θ(log()) Coverig to Multicast Static etwork: Cosider step ad step 2 i our may-to-oe cooperative scheme, sice the process i these two steps is may-to-oe ad coverget, it caot be simply reversed. Moreover, the data iformatio from the k sources may be differet, oe ca be treated as a duplicatio of those from other sources. Thus, the hierarchical scheme proposed i static etwork caot be applied to multicast traffic patter. MANETs: Due to the mobility of odes, spaig tree is ot eeded i routig establishmet. Thus, our may-to-oe scheme 2 ca easily be applied to multicast mode, oly with some mior modificatios i our the scheme. That is, i a cell, we figure out the maximum umber of odes that hold the packets from the same source. These odes the joitly broadcast these packets i the cell. Iitially, oly the source holds the packet ad whe there are several source-destiatios pairs 9

20 i the same cell, we radomly pick out oe such pair ad let the source broadcasts its packet to its destiatios as well as other odes i the cell. The algorithm cotiues util all the destiatios receives the packet they eed. Next we tur to the aalysis o throughput ad delay for multicast uder our modified scheme 2. Cosider the case where there are oly oe ode i the etwork holdig the packet at the begiig. Thus, the growth of the umber of packet-holdig odes is from to rather tha k to uder coverge-cast ad the total floodig time for a packet is bouded by Θ (log()). Moreover, sice the packet for each of the k destiatios is idetical, o extra time is eeded for each destiatio to wait for the additioal k packets. The destiatio ca immediately decode the packet after it receives oly oe copy. Ad this process is already cotaied i the floodig oe. Hece, we ca achieve a delay of E[D N ] = Θ (log()) (25) for multicast. As for the per-ode throughput, we cosider from the perspective of a source ode. It should trasmit oe packet but has to duplicate it times so that all the odes i the etwork ca get oe copy. Hece, the capacity yields to Coverig to Broadcast λ = E[D N ] =. (26) Sice broadcast ca be treated as a special type of multicast, we kow that the results obtaied uder static etwork caot cover broadcast case. Moreover, i mobile etwork, to make the result applicable to broadcast, we oly eed to refer to Equatio (26) ad Equatio (25) ad settig k as Θ(). Sice a source has oe packet to trasmit with copies, the results are the same as that of multicast listed i Compariso with Previous Work ad Geeralizatio For static etwork, whe applyig to uicast, our scheme still ca achieve a aggregate throughput of Ω ɛšfor ay ɛ > 0. This is idetical to that achieved i [6] while our delay is much larger. This is because the amout of data exchage i our scheme is much larger. Hece, if cocered with delay priority, our scheme is ot optimal for uicast to achieve a small delay. Next, cosider the extreme case where k = Θ(). The aggregate throughput is still close to Θ() with the delay reduced to Θ 4 2h 2h. There turs out to be a sigificat improvemet o capacity, compared with previous results i [7], [8] ad [9]. I [7], the aggregate capacity scales as Θ(log ) as goes to ifiity while i [8], the maximum rate for a collected etwork do ot exceed Θ log Š. I [9] where all the odes i the etwork flow their data a commo sik, the authors demostrate that total data aggregatio rates of Θ(log ) ad Θ() are optimal whe operatig i fadig eviromets with power path-loss expoets that satisfy 2 < α < 4 ad α > 4, respectively. Our result also achieves a gai of Θ(log ) compared with that i [20], where the capacity of data collectio is 20

21 Network Traffic Throughput Delay Uicast Ours:ÜΘ 2h 2h 2 ÜΘ h2 4h+3 2h [6]: - Θ 2h 4 2h Ours: Θ Θ() Š Static [7]: Θ(log ) - Network Coverge-cast [8]: Θ log - Θ Š [9]: Θ(log ad Θ() - [20]: log - Uicast Ours: log() Θ Š Θ(log()) [7]: Θ Θ(log ) Mobile Multicast log() Θ Š Θ(log()) Network [7]: Θ Θ(log ) Broadcast log() Š Θ(log()) [7]: Θ Θ(log()) Table 2: Throughput ad Delay of coverge-cast ad that exteded to uicast, multicast ad broadcast uder our schemes i both static ad mobile etworks. Compariso is also made betwee our Θ results ad previous oes. Θ Š log Šif there are siks with each oe collectig data from all the rest of the odes i the etwork. For MANETs,. Uicast: Our exteded results achieve the per-ode throughput of log with a Θ(log ) delay. A gai of is achieved o throughput compared with that obtaied i [7], where the per-ode throughput is Θ Šwhile the delay is also Θ(log ). The improvemet o throughput is due to our itelliget cooperatio betwee odes with the help of MIMO. Multiple odes ca trasmit simultaeously to other odes. Ad a ode ca successfully decode the origial packet oce it receives oly oe combiatio. Nevertheless, redudat trasmissio still has to be wasted i [7] eve with the adoptio of etwork codig sice a destiatio ca decode the origial packets oly whe it receives Θ(k) packets. 2. Multicast ad broadcast: We obtai the perode throughput of Θ Šad the delay of Θ(log ) uder both traffic patters. The result is the same as that of [7]. It is easy to uderstad sice i such traffic patters, a source seds idetical iformatio to several (or all) destiatios. I the scheme of both [7] ad ours, although all the destiatios ca receive the iformatio from their commo source withi log delay, a source has to edure several times duplicatio. Thus, a source has oly oe packet to share with all its destiatios alog with several data copies, which degrades the throughput performace. Through the compariso, it ca be see that all our results well cover ad uify those from the previous work, which also demostrates the robustess of our coverged-based aalysis. 2

22 6 Coclusio Through the compariso, it ca be see that all our results well cover ad uify those from the previous work, which also demostrates the robustess of our coverged-based aalysis.i this paper, with MIMO, we desig two differet cooperative schemes for static ad mobile ad hoc wireless etworks (MANETs), respectively. The hierarchical cooperatio scheme uder static etworks ca achieve a aggregate throughput of Ω( ɛ ) for ay ɛ > 0. The scheme uder MANETs features o joit multiple trasmissio ad receptio without hierarchical operatios. With optimal etwork divisio i the scheme, the achievable per-ode throughput ca be Θ() with the correspodig delay reduced to Θ(k). Moreover, we fid all the results derived from coverge-cast i this paper uder mobile etworks ca be easily applied to uicast, multicast ad broadcast. There are still may aspects for us to ivestigate i the future. For example, it remais a iterestig problem for desigig a scheme with MIMO uder other mobility models such as radom walk ad radom way poit models. The etwork performace may be quite differet from that obtaied uder i.i.d. mobility model. Furthermore, it also remais to be a future work o appropriate modificatio of our scheme uder MANET i order to make it also applicable to the sceario where all the odes trasmit their data to a sigle sik i the etwork, sice our curret scheme caot guaratee a small decodig failure probability for this extreme case. Appedix A. Proof of Lemma 2: For each destiatio ode i the etwork, there are k radomly chose sources belogig to it. If the sets of source odes for each destiatio do ot itersect with each other, k odes will serve as sources i total. However, there are oly odes i the whole etwork. Thus, by treatig the source-destiatio pair from a reverse view, for each ode s, there are o average k odes (deotig as d, d 2,..., d k ) which choose s as oe of its source odes. Assume each source ode trasmits data to d at data rate λ i. Sice d has to receive k distictive iformatio from these k sources, it acts as k differet odes durig each receptio. Thus, the total data rate to the destiatio d is upper-bouded by the capacity of a multiple-iput multiple-output chael betwee d ad the rest of the kx etwork. That is, X 2Ž λ i k log + PN0 G d α. (27) i= i=,s i d s i d Accordig to Lemma 2, the kx distace betwee s i ad d is larger tha w.h.p. Thus, +δ λ i k log + P G α(+δ)+ Ck log, (28) N 0 i= where C is a costat that does ot deped o ad k. If we assume that λ i, i k are idetical, the we have λ i C log, (29) 22

23 which implies that X λ i C log. (30) i= This completes our proof. B. Proof of Lemma 7: Cosider a particular cell. Let X i deote the 0- radom variable with X i = represetig v i i the cell ad X i = 0 represetig v i ot i the cell. Note that X i ad X j are idepedet with each other. Obviously, Pr[X i = ] = /c. Accordig to Cheroff bouds, we have ad i= Pr( Hece, with the assumptio that X lim c log c =, we have X i <δ θ c c c) e i= Pr(X X i > ( + δ) c) e δ2 3c (3) Pr(X i= X i < ( δ) c) e δ2 2c. (32) θ < e c log c 0 (33) as. Thus, M = c w.h.p. This completes our proof. C. Proof of Lemma 8: For case, it is obvious that E{a i } M. The we have E{β t+ J t } = J t + J t E{a i } J t + M. (34) Sice J t = β 0 β β 2... β t, E{J t } k( + M) t. (35) Note that the period eds whe J t = M. Deotig D as the time required for k packet-holdig odes to grow to M, the Moreover, we have Pr[T > t J t ] M tj t. (36) E{D } tpr[d > t] = te Jt {Pr[D > t J t ]} te Jt M tj t«(37) t M te Jt. The above iequality holds for all t > 0. We choose t to be a base ( + M) logarithm: t M log +M α( δš, k )M where α is chose as α (M + ) ad δ is ay costat umber less tha 23

24 . Usig this value of t i Iequality (37), we get E{D } log(α) + M δ log( k ) M log( + M) = log(α) + M δ log( k ) M log( + M) )MδŠ k α( M log( ) M k k M log(+m) ( M )MδŠ M log( ) k k log(+m) M)k α(. (38) )MδŠ Note that M M e Š whe goes to ifiity. Now we cosider the term k α( M log( k k ) log(+m). Obviously, k α( δš M k )M log( k ) < (/k)m ( δ) log(/k) 0 log( + M) /k as. Hece, it follows that This implies )MδŠ k α( M log( ) M k k M log(+m) lim E{D } kš. α log M δ log( + M). (39) kš To obtai a tight boud o Iequality (39), we choose δ to be a value very close to. Thus, it ca be iferred that log E{D } = Θ M ɛ log( + M)!, (40) where ɛ is a arbitrarily small value greater tha zero. Whe J t grows to M, ulike the previous process, where E[a i] almost remais uchaged per time slot, the duplicate rate E[a i ] starts to vary i differet time slots. Let m deote the umber of odes which do ot iitially have the packet (m M ) ad label these m odes with {x, x 2,..., x m }. Let X i represet the umber of time slots it takes for the o-packet holdig ode x i to reach a cell cotaiig a packet-holdig ode. Due to the multi-receptio, x i must receive a packet at this time. The probability that at least oe of the ew ode eters the same cell as packet-holdig ode x i is ϕ > M Š M e M. At all times X i are idepedet ad idetically distributed. Deotig D 2 as the time to expad the umber of packet-holdig odes from M to, the the radom variable D 2 is equal to the maximum value of at most m =š M i.i.d. variables. Hece, E[D 2 ] E{max{X, X 2,..., X m }}. Now we cosider ew radom variables {Y, Y 2,..., Y m } which are assumed to be i.i.d. distributed with rate ν = log(/( ϕ)). Note that + Y i is stochastically greater tha X i. Thus, E[D 2 ] + E{max{Y, Y 2,..., Y m }}. 24