On the Capacity of k-mpr Wireless Networks

3878 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 O the Capacity of -MPR Wireless Networs Mig-Fei Guo, Member, IEEE, Xibig Wag, Member, IEEE, Mi-You Wu, Seior Member, IEEE Abstract The capacity of wireless ad hoc etwors is maily restricted by the umber of cocurret trasmissios. Recet studies foud that multi-pacet receptio (MPR) ca icrease the umber of cocurret trasmissios ad improve etwor capacity. This paper studies the capacity of 2-D wireless etwors wherei each ode ca decode at most simultaeous trasmissios withi its receivig rage. We call such etwors -MPR wireless etwors. For compariso, we call traditioal etwors -MPR wireless etwors. Suppose that the umber of odes i a wireless etwor is ad each ode ca trasmit at W For arbitrary -MPR wireless etwors, we show that whe = O(), the capacity gai over -MPR etwors is Θ( ). Whe =Ω(), the capacity is Θ(W) bit-meters/sec ad the etwor is scalable. For radom -MPR wireless etwors, we show that whe = O( log ), the capacity upper boud ad lower boud match ad the capacity gai over -MPR etwors is Θ(). Whe =Ω( log ), eve the lower boud has a capacity gai of Θ( log ) over -MPR etwors. From these results, we coclude that the mai costraits for -MPR wireless etwors to utilize MPR ability are the limited umber of trasmitters ad the limited umber of flows served by each ode. Idex Terms Capacity, scalig law, multi-pacet receptio. I. INTRODUCTION THE semial wor of Gupta ad Kumar [] derived that the wireless etwor capacity scales as Θ(W ) bitmeters/sec i arbitrary etwors while scales as Θ(W log ) bits/sec i radom etwors where is the umber of odes. For ifrastructure wireless mesh etwors, P. Zhou et al. [2] derived that the per-cliet throughput decreases as the umber of cliets icreases. The mai reaso for these throughput degradatios is that all wireless odes share the same wireless medium ad the umber of cocurret trasmissios is limited. As the successive iterferece cacellatio (SIC) circuits with simple implemetatio ad low complexity have bee itroduced, multi-pacet receptio (MPR) becomes a reality [8], which provides potetial to icrease the umber of cocurret trasmissios ad improve the etwor capacity. Past researches o the capacity of MPR-based wireless etwors derived their results assumig that all the trasmissios withi the receivig rage of a ode ca be decoded, however, as the umber of trasmissios withi the receivig rage icreases, the receiver caot decode all of them. The reaso is as follows. As show i [8], the mai idea of SIC is to cacel Mauscript received August 4, 2008; revised Jauary 4, 2009; accepted March 25, 2009. The associate editor coordiatig the review of this paper ad approvig it for publicatio was A. Yeer. M. F. Guo ad M. Y. Wu are with the Departmet of Computer Sciece, Shaghai Jiao Tog Uiversity, Shaghai, 200240 Chia (e-mail: {mfguo, mwu}@sjtu.edu.c). X. Wag is with the Departmet of Electroic Egieerig, Shaghai Jiao Tog Uiversity, Shaghai, 200240 Chia (e-mail: xwag8@sjtu.edu.c). Digital Object Idetifier 0.09/TWC.2009.090265 536-276/09$25.00 2009 IEEE each received sigal oe by oe i the decreasig order of the sigal stregth ad the sigal cacellatio process delay is restricted by the speed of performig Walsh-Hadamard Trasform (WHT), so the possible umber of cacellatios is limited, which leads to the limited umber of decoded trasmissios. Usig a more practical iterferece model compared with [5], [6], this paper studies the capacity of 2-D MPR-based wireless ad hoc etwors assumig that a wireless iterface ca decode at most, trasmissios withi its receivig rage. We call such etwors -MPR wireless etwors. For compariso, we call traditioal etwors -MPR wireless etwors. The MPR ability,, depeds o the hardware implemetatio. We study how the capacity of 2-D -MPR wireless etwors scales with ad the umber of wireless odes,, i both arbitrary ad radom scearios. The remaider of the paper is orgaized as follows. We summarize the related wor i Sectio II. Sectio III describes the etwor model ad mai results. I Sectio IV, we prove the results for arbitrary etwors. The proofs for the results of radom etwors are preseted i Sectio V. We discuss the derived results i Sectio VI. Sectio VII cocludes our wor. II. RELATED WORK Assumig that a ode ca cocurretly sed to ad receive from may odes whe usig FDMA ad CDMA, R. M. D. Moraes et al. [3] studied the upper boud ad lower boud of li s Shao capacity ad per source-destiatio throughput. G. D. Celi et al. [4] preseted ew bacoff mechaisms for MPR-based wireless etwors to deal with ufairess ad improve etwor throughput. Assumig that all the trasmissios withi the receivig rage of a receiver ca be decoded, J. J. Garcia-Lua-Aceves et al. [5] have show that 3-D radom MPR-based wireless etwor has a capacity gai of Θ(log ). Also assumig that all the trasmissios withi receivig rage ca be decoded, J. J. Garcia-Lua- Aceves et al. [6] too SINR ad Shao li capacity ito the capacity aalysis ad gave their results. Usig more realistic protocol model compared with [5], [6], X. Wag et al. [7] assumed that at most M simultaeous trasmissios withi the receivig rage of a receiver ca be decoded ad maximized the aggregate etwor throughput by formulatig a optimizatio problem. Accordig to the survey of the related wor, we ca see that this paper is the first wor studyig the capacity of 2-D MPR-based wireless etwors usig a practical model, which assumes that a iterface ca decode at most trasmissios withi its receivig rage. Our cotributios ca be summarized as follows:

GUO et al.: ON THE CAPACITY OF K-MPR WIRELESS NETWORKS 3879 Networ capacity (bit-meters/sec) W W (-MPR) -MPR -MPR The maximum umber of cocurret receptios, Fig.. The capacity of arbitrary -MPR wireless etwors (figure is ot to scale). The upper boud ad lower boud match exactly. To the best owledge of us, this paper is the first wor studyig the capacity of such etwors i both arbitrary ad radom scearios. From the derived results, we get some valuable desig implicatios for -MPR wirelesss etwors. III. NETWORK MODEL AND MAIN RESULTS A. Networ Model I this wor, we suppose that each ode is equipped with oe -MPR wireless iterface, which ca decode at most trasmissios withi its receivig rage. We call such ability -MPR ability. Each ode ca trasmit at W bits/sec over a commo wireless chael. We cosider a more geeral sceario where the chael is divided ito M subchaels, each of which has a capacity of W m bits/sec, m M ad M m= W m = W. Trasmissios are slotted ito sychroized slots of the same legth τ. Pacets are set from source to destiatio i multi-hop.. Arbitrary Networs I a arbitrary -MPR wireless etwor, odes are arbitrarily located i a dis of uit area i the plae. Each ode has a arbitrarily chose destiatio to which it could sed traffic at a arbitrary rate. Each ode ca choose a arbitrary power level for each trasmissio. Uder these assumptios, we give the protocol iterferece model for arbitrary -MPR wireless etwors as follows. Suppose odes, {X ip p }, trasmit to ode X j simultaeously. These trasmissios are successfully decoded by ode X j if X q X j max ( + Δ) X ip X j p for ay other ode X q simultaeously trasmittig over the same subchael. Similar as [], the quatity Δ > 0 models a guard zoe that prevets a eighborig ode from trasmittig o the same subchael at the same time. Networ capacity (bits/sec) W log log W log log log W W log (-MPR) log log log log Upper Boud Lower Boud -MPR The maximum umber of cocurret receptios, Fig. 2. Upper boud ad lower boud for the capacity of radom -MPR wireless etwors (figure is ot to scale). 2. Radom Networs I a radom -MPR wireless etwor, odes are radomly located, i.e., idepedetly ad uiformly distributed, o the surface of a torus of uit area. Each ode has oe flow to a radomly chose destiatio to which it wishes to sed at λ() Each ode employs the same receivig rage r() for each receptio. Uder these assumptios, we give the protocol iterferece model for radom -MPR wireless etwors as follows. Suppose odes, {X ip p }, trasmit to ode X j simultaeously. These trasmissios are successfully decoded by ode X j if () The distace betwee X ip ad X j is o more tha r(). max X ip X j r() p ad (2) For ay other ode X q simultaeously trasmittig over the same subchael, the followig iequatio holds. X q X j ( + Δ)r() B. Mai Results The results of this paper are preseted as follows.. Arbitrary Networs I a arbitrary etwor, the etwor capacity is measured i terms of bit-meters/sec (itroduced by []). As show i Fig., the capacity of arbitrary -MPR wireless etwors is as follows: () Whe = O(), the etwor capacity is Θ(W ) bit-meters/sec. (2) Whe =Ω(), the etwor capacity is Θ(W) bitmeters/sec. Note that Fig. is ot to scale. To simplify the illustratio, we use piecewise liear curve to represet the capacity scalig law, although the scalig fuctio is ot piecewise liear. This figure covetio also applys i Fig. 2.

3880 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 S S D 4 S 2 N flow Q 2 3 W flow S flow D 3 W N C S E S 3 D E flow D D 2 S 4 Fig. 3. Routig through cells i a radom -MPR wireless etwor Fig. 4. The flows served by cell C are classified ito 4 categories (E, S, W, N flows). 2. Radom Networs I radom etwors, the etwor capacity is measured i terms of bits/sec. We give a lower boud ad a upper boud for the capacity of radom -MPR wireless etwors. As show i Fig. 2, the etwor capacity exhibits differet bouds as the order of varies. Whe = O( log ), the lower boud ad upper boud match exactly ad the etwor capacity is Θ(W log ) Whe =Ω( log ), the etwor capacity is Ω(W ) The capacity upper boud is preseted as follows: () Whe =Ω( log ) ad = O( log log log ),etwor capacity is O(W log ) (2) Whe =Ω( log log log ) ad = O(), etwor capacity is O(W log log ) W (3) Whe = Ω(), etwor capacity is O( log log ) IV. CAPACITY OF ARBITRARY NETWORKS Firstly, we derive the upper boud for the capacity of arbitrary -MPR wireless etwors i Sectio IV-A. Secodly, i order to illustrate that the upper boud is tight, we give a costructive lower boud, which matches the upper boud exactly, i Sectio IV-B. A. Upper Boud I a arbitrary -MPR wireless etwor, we suppose the whole etwor trasports λt bits over T secods. The average distace betwee the source ad the destiatio of a bit is L meters. Cosequetly a capacity of λl bit-meters/sec is achieved. Uder the etwor model we described i Sectio III-A, we have the followig theorem. Theorem : Uder the protocol model, the upper boud for the capacity λl of arbitrary -MPR wireless etwors is preseted as follows:. Whe = O(), λl = O(W ) bit-meters/sec. 2. Whe = Ω(), λl = O(W) bit-meters/sec. Proof: Whe is O(), we use similar techiques used i [] to prove the result. There are two mai differeces i our proof. Firstly, differet from [], the odes should be grouped to utilize the -MPR ability. Secodly, i [], the diss cetered at receivers should be disjoit ad at oe time slot there is at most oe trasmissio i each dis. I our proof, these diss are also disjoit but there are at most trasmissios i each dis. Whe =Ω(), sice there are ot eough trasmitters to further improve the throughput, the etwor capacity is at most the capacity whe =Θ(), which has bee give. Due to space costrait, we omit the full proof. B. A Costructive Lower Boud I this sectio, we will show that the upper boud i the previous sectio is tight by achievig it. For derivig the achieved lower boud, we preset the followig lemma. Lemma : Suppose = O( 2 ), C is the capacity of -MPR wireless etwors, C 2 is the capacity of 2 -MPR wireless etwors, the C = O(C 2 ). Proof: The 2 -MPR wireless etwors ca imitate -MPR wireless etwors by restrictig 2 -MPR ability to -MPR ability. Hece the capacity of 2 -MPR wireless etwors is at least the capacity of -MPR wireless etwors. Hece we get the lemma. Note that Lemma applys i both arbitrary etwors ad radom etwors. Theorem 2: The lower boud for the capacity of arbitrary -MPR wireless etwors is preseted as follows:. Whe = O(), λl =Ω(W ) bit-meters/sec. 2. Whe =Ω(), λl =Ω(W) bit-meters/sec. Proof: Whe = o(), we use the similar ode deploymet used i [] to prove the result. The oly differece is that at each trasmittig locatio we place odes rather tha oly deploy oe ode. A receiver ad its earest trasmitters form a trasmittig group. There are trasmissios for each group. Whe = Θ(), for costructio we suppose that = + ad there is oly oe trasmittig group. The result whe =Ω() ca be deduced from the result whe =Θ() by Lemma. Due to space costrait, we omit the full proof. V. CAPACITY OF RANDOM NETWORKS I a radom -MPR wireless etwor, we suppose that each ode seds at λ() bits/sec to its destiatio. The highest value

GUO et al.: ON THE CAPACITY OF K-MPR WIRELESS NETWORKS 388 2 T T T R T Fig. 5. Structure ad its R-Cell ad T-Cells i schedulig scheme Fig. 6. -Coflict Structures Fig. 7. 2-Coflict Structures of λ() that ca be supported by every source-destiatio pair with high probability is defied as the per-ode throughput of the etwor. The traffic betwee a source-destiatio pair is referred to as flow. Sice there are flows i total, the etwor capacity is defied to be λ(). I Sectio V-A, we give a costructive lower boud for the capacity of radom -MPR wireless etwors. I Sectio V-B, we give a loose upper boud. A. A Costructive Lower Boud I order to establish a lower boud, we costruct a routig scheme ad a trasmissio schedulig scheme for ay radom -MPR wireless etwor.. Cell Costructio Utilizig the approach used i [9], we divide the surface of the uit torus ito square cells, each of which has a area of a(). We choose the receivig rage r() of each ode to be 8a(). With this rage, a ode i oe cell ca commuicate with ay ode i its eight eighborig cells. The area of each cell, a(), should be carefully chose to satisfy multiple costraits, which will be described later. 00 log We set a() =mi(max(, log log ), ( log )2 ).Thefirst costrait is coectivity costrait. The cell size a() should 00 log be large eough, say, a(), to guratee etwor coectivity. The secod costrait is used to guratee that each odehas at least eighborsto utilize the -MPR ability. The cell size should be large eough, say, a(), to esure that each ode has at least eighbors. The third costrait is to esure that the etwor ca at least accommodate the termiatig flows of each ode. The cell size should be chose log log small eough, say, a() ( log )2 to satisfy this costrait. We will discuss the third costrait i detail i the discussio o the routig scheme. For costructio, we use the followig lemma to boud the umber of odes i each cell. This lemma has bee proved i []. 50 log Lemma 2: If a() >, the each cell has Θ(a()) odes, with high probability. 00 log By costructio, we guratee that a() for large 00 log by settig a() = mi(max(, log log ), ( log )2 ). Cosequetly, Lemma 2 holds ad each cell has Θ(a()) odes. 2. Routig Scheme The routig scheme cosists of two steps, cell assigmet, ad the ode assigmet. Cell Assigmet Cells are assiged to serve each flow of the etwor. As show i Fig. 3, pacets of a flow are routed through the cells that lie alog the straight lie joiig the source ad the destiatio ode. For each itersected cell of a flow (source-destiatio lie), we choose a ode to relay the traffic ofthisflow (we will describe the ode assigmet scheme later). We should cosider a special case wherei the lie passes a grid poit exactly, say, Q, ifig.3.ithiscase, we require the cell o the right side of Q to serve this flow. Hece i Fig. 3 the pacets of flow S-D should be firstly relayed from cell to cell 2 ad the be relayed from cell 2 to cell 3. Node Assigmet For each flow served by a cell, we select a ode from the cell to serve this flow. Accordig to the cell assigmet scheme, the ext hop for ay flow must be withi oe of the four eighborig cells: the orther, souther, easter ad wester eighborig cell. Each flow served by a cell ca be classified by directio ito oe of four cardial categories. A served flow whose ext hop is withi the orther eighborig cell is called a N flow served by this cell. Similarly, we ca defie S, E, W flows served by a cell. The examples of N, S, E ad W flows served by cell C ca be foud i Fig.4. The ode assigmet scheme has two steps. I step oe, the source ode ad destiatio ode of a flow are assiged to serve the flow. I step two, we assig odes to serve those flows that pass through a cell. For each passig flow of a cell, the ode, which has bee assiged to the least umber of flows of the same category so far, is assiged to serve the flow. This step evely distributes the flows of each category amog the odes of a cell. Hece for each flow category, all odes serve early the same umber of flows. The ode assiged to a flow will receive pacets from the assiged ode i the previous cell ad sed the received pacets to the assiged ode i the ext cell. We preset the followig lemma to boud the umber of source-destiatio lies that pass through ay cell. This lemma is proved i [9]. Lemma 3: The umber of source-destiatio lies passig through ay cell (icludig lies origiatig ad termiatig i the cell) is O( a()) with high probability. From Lemma 2 ad Lemma 3, we coclude that each ode serves O( ) flows. a() After itroducig the routig scheme, we discuss the third costrait for cell size. Because each ode pics a destiatio ode radomly, a ode may be the destiatio of multiple

3882 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 flows. Let D() be the maximum umber of flows that have the same destiatio ode. We use the followig lemma to boud D(). This lemma is proved i []. Lemma 4: The maximum umber of flows for which a ode i the etwor is a destiatio, D(), isθ( log log log ), with high probability. Because each ode should at least accommodate the flows that have the ode as destiatio, we should esure = Ω(D()). Hece the cell size a() should ot a() log log exceed Θ(( log )2 ). 3. Schedulig Trasmissios The schedulig scheme aims to schedule each flow with equal opportuity (i.e., all flows i a local regio are served with same time slots) while satisfyig the followig costraits. () A ode caot trasmit ad receive simultaeously, sice each ode has oly oe -MPR wireless iterface. (2) A ode caot trasmit to more tha oe receivers simultaeously. The reaso is the same as that of costrait (). (3) Ay two simultaeous trasmissios should ot iterfere with each other uder the protocol model. (4) I order to utilize -MPR ability, the trasmissios for the same receiver should be cocurret. For schedulig trasmissios, we defie a ew schedulig uit Structure. I Fig. 5, the five cells surrouded by the red lie compose a structure. We call the ceterig cell receivig cell (R-Cell). Each of the four eighborig cells of a R-Cell is called trasmittig cell (T -Cell). For calculatig the achieved etwor capacity, we cosider the trasmissio schedulig i oe secod. We divide oe secod ito several big slots, Structure Slots. The we further divide each structure slot ito smaller slots, Flow Slots. We build a schedule usig a two-layer process. The first layer is Structure Layer ad the secod layer is Flow Layer. O the structure layer, we schedule the structures with structure slots to avoid coflicts ad iterfereces betwee ay two structures. O the flow layer, we schedule the idividual flows with flow slots. After illustratig the two layer schedulig scheme, we will prove that the schedulig scheme schedules each flow with equal opportuity while satisfyig the four costraits. Hece this scheme is feasible. Structure Layer O the structure layer, we schedule the structures satisfyig the followig requiremets () I ay structure slot, a cell caot act as both a T -Cell ad a R-Cell. (2) I ay structure slot, there is oly oe R-Cell for each T -Cell. (3) I ay structure slot, the trasmissios of oe structure should ot iterfere with the trasmissios of ay other structure uder the protocol model. Based o the three requiremets, we itroduce the defiitio of i-coflict structure, i 3. If structure A ad B caot be scheduled i same structure slot due to requiremet (i), i 3, we say that structure A is a i-coflict structure of structure B, ad vice versa. Structure A ad B are i-coflict structures with each other. Theorem 3: Uder the protocol model, there is a schedule such that i every c structure slots, each structure i the tessellatio gets oe structure slot such that the three requiremets are satisfied where c depeds oly o Δ. Proof: To prove the theorem, we should satisfy each of the three requiremets. Firstly, to satisfy the requiremet (), we should esure that the -coflict structures are scheduled i differet structure slots. As show i Fig. 6, cell acts as a R-Cell i the red structure (the left structure) while it acts as a T -Cell i the gree structure (the right structure). Hece red ad gree structures are -coflict structures with each other. I our grid tessellatio, the umber of -coflict structures for a structure is a costat, say, c (i fact, it is four). Secodly, as show i Fig. 7, cell 2 acts as a T -Cell i the red structure (the lower left structure) while it acts as a T -Cell i the blue structure (the upper right structure). Hece the red ad blue structures are 2-coflict structures with each other. I our grid tessellatio, the umber of 2-coflict structures for a structure is costat, say, c 2 (i fact, it is eight). Thirdly, we will satisfy the requiremet (3). Let N deote the umber of iterferig cells of each cell i a radom - MPR etwor (the cell tessellatio is the same with that we described above). Let N 2 deote the umber of iterferig cells for each cell i a radom -MPR etwor. Let N 3 deote the umber of iterferig structures of each structure i a radom -MPR etwor. Sice oly the R-Cell of a structure ca be iterfered by other cells, we have N 3 N 2. Sice the oly differece betwee the protocol models of -MPR ad - MPR etwors is cocurret receptio, we have N 2 N. HecewehaveN 3 N. [] has show that N is upper bouded by a umber, say, c 3, which depeds oly o Δ. Hece the umber of 3-coflict structures for a structure is less tha c 3. Lettig each structure deote a ode, we build a iterferece graph. There is a edge betwee two odes, if the correspodig two structures of the two odes are i-coflict structures, i 3, with each other. From the aalysis above, we coclude that the maximum vertex degree of this iterferece graph is at most c +c 2 +c 3.Letc = c +c 2 +c 3. Accordig to [2], the iterferece graph ca be vertexcolored with at most (c +)colors. Lettig c = c +,we get the theorem. Flow Layer O the flow layer, we build a two-phase scheme to schedule each flow served by R-Cell usig the -MPR ability whe a structure is scheduled for a structure slot. To achieve the schedulig objective, flow layer schedulig should esure that each flow is served with equal umber of flow slots i a structure slot. To achieve this goal ad to be feasible, flow layer schedulig should satisfy the followig requiremets. () I both phases, the umber of cocurret receptios caot exceed the umber of trasmitters. (2) I the secod phase, the umber of cocurret receptios caot exceed the umber of flows each ode serves. We will show that these requiremets are ecessary codi-

GUO et al.: ON THE CAPACITY OF K-MPR WIRELESS NETWORKS 3883 tios for feasible flow layer schedulig later. Because of these requiremets, we caot always fully utilize the -MPR ability ad the feasible umber of cocurret receptios caot always be. Hece for flow layer schedulig, we defie g as the feasible umber of cocurret receptios where g. First Phase I this phase, each flow gets a flow slot to be trasmitted from the T -Cell to the R-Cell. For each ode i the T -Cells, each of the served flows whose ext hops are i the R-Cell is trasmitted for oe flow slot. Sice we use multi-pacet receptio, the umber of receivers is less tha the umber of trasmitters ( ideally, the ratio betwee trasmitters ad receivers is, whe-mpr ability is fully utilized). The other odes i R-Cell do othig i first phase. By costructio, we arbitrarily select a portio (this portio, which is depedet o g, will be discussed later) of odes from each cell ad use them as receivers whe the cell acts as a R-Cell i the first phase. We call these receivers of a R-Cell i first phase Busy Nodes ad call all of the other odes Free Nodes. I the first phase, busy odes receive trasmissios o behalf of a R-Cell. Additioally, the flows served by free odes but received by busy odes i first phase should be distributed uiformly amog the busy odes for utilizig multi-pacet receptio i secod phase. Secod Phase I this phase, each flow served by free odes is assiged a flow slot to be trasmitted from busy odes to free odes usig the -MPR ability. The trasmitters i this phase are busy odes. Theorem 4: Usig the two-phase flow layer schedulig, for a scheduled structure, each flow served by the R-Cell ca be assiged oe effective flow slot to be trasmitted from the servig ode i the T -Cell to the servig ode i the R-Cell. Proof: The flowsservedbyther-cell ca be classified ito two categories. The first category icludes all the flows served by the busy odes of the R-Cell. The secod category icludes all the flows served by free odes of R-Cell. For the first category, each of the flows is assiged oly oe flow slot i the first phase. Hece each flow is trasmitted from the servig ode i the T -Cell to the servig ode i the R-Cell for oe effective flow slot. For the secod category, each of the flowsisassigedtwoflow slots (oe i the first phase ad oe i the secod phase). Sice i the first assiged flow slot, the flow is trasmitted from T -Cell to a temporary receiver, the umber of effective assiged flow slots is oly oe (the oe assiged i secod phase). Hece i flow layer schedulig, each of the flows served by the R-Cell gets oe effective flow slot to be trasmitted from the servig ode i the T -Cell to the servig ode i the R-Cell. After itroducig the two-phase process, we preset the followig theorem to show that the two requiremets are ecessary for feasible flow layer schedulig. Theorem 5: If flow layer schedulig with g cocurret receptios is feasible, the requiremet () ad (2) are satisfied. Proof: Firstly, we prove that requiremet () is a ecessary coditio for feasible flow layer schedulig with g cocurret receptios. If requiremet () is ot satisfied, g is larger tha the umber of trasmitters i the first or secod phase. Hece we have ot eough trasmitters to fully use g cocurret receptios ad flow layer schedulig with g cocurret receptios is ifeasible, which cotradicts the assumptio. Secodly, we prove that requiremet (2) is a ecessary coditio for feasible flow layer schedulig with g cocurret receptios. If requiremet (2) is ot satisfied, g is larger tha the umber of flows each ode serves i secod phase. As a result, to mae g cocurret receptios feasible, some or all of the flows served by free odes are assiged more tha oe flow slot because there are ot eough flows for assigig each flow oe flow slot to fully use g cocurret receptios. However, each of these flows is assiged oly oe flow slot i first phase ad i each flow slot, a ode seds at a data rate which is at most W bits/sec, so the maximum umber of bits of each flow to be trasmitted i secod phase is at most W (legth of flow slot). Hece i order to eep flow coservatio, i secod phase, i all the assiged flow slots for each flow (icludig the added flow slots), the total umber of trasmitted bits of the flow caot exceed W ( legth of flow slot). Cosequetly the added flow slots due to g cocurret receptios cotribute othig to the capacity. Hece g cocurret receptios is ifeasible, which cotradicts the assumptio. We give the followig theorem to justify the proposed twolayer schedulig scheme. Theorem 6: The proposed two-layer schedulig scheme is feasible ad schedules each flow with equal opportuity. Proof: As each cell ca act either as a R-Cell or a T -Cell ad accordig to the routig scheme all the flowsservedbya cell are from the four T -Cells, each flow i the etwor ca be served. From Theorem 4, we coclude that each flow is served with equal opportuity. Next, we prove that the scheme satisfies the four costraits to coclude that the scheme is feasible. Firstly, o the structure layer, a cell caot act both as a R-Cell ad act as a T -Cell i ay structure slot ad o the flow layer, a ode caot both trasmit ad receive i ay flow slot, so costrait () is satisfied. Secodly, o the structure layer, there is oly oe R-Cell for each T -Cell i ay structure slot ad o the flow layer, a trasmitter has oly oe receiver i ay flow slot, so costrait (2) is satisfied. Thirdly, o the structure layer, the trasmissios of a structure do ot iterfere with trasmissios of ay other structure uder the protocol model i ay structure slot ad o the flow layer, at most flows are scheduled i ay flow slot, so there is o iterferece betwee ay two trasmissios uder the protocol model ad costrait (3) is satisfied. Fourthly, o the flow layer, g receptios of a receiver is scheduled i same flow slot, so the costrait (4) is satisfied. Cosequetly we get the theorem. 4. The achieved capacity lower boud After itroducig the whole schedulig scheme, we discuss the capacity lower boud for radom -MPR wireless etwors. Recallig that each ode serves O( ) flows, for a() requiremet (2) of flow layer schedulig, we should esure that the umber of cocurret receptios is O( ) i the a() secod phase.

3884 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 a. Whe = O( log ), we have the followig lemmas for the capacity lower boud. Lemma 5: Whe = O( log ), we ca fully utilize - MPR ability ad the umber of cocurret receptios is. Proof: To prove the lemma, we eed to show that with cocurret receptios, the requiremet () ad (2) are satisfied. Note that whe = O( log ), the cell size 00 log a() is. I first phase, it ca be cocluded from the ode assigmet scheme that the umber of trasmitters is Θ(a()) = Θ(log ) = Ω(). I secod phase, the umber of trasmitters (busy odes), Θ( a() ), is Ω(). Hece requiremet () is satisfied. Sice = O( log ), is O( ). Hece requiremet (2) is satisfied. a() Based o Lemma 5, we preset the followig lemma for the capacity lower boud whe = O( log ). Lemma 6: Whe = O( log ), λ() =Ω(W log ) Proof: I the first phase, sice the umber of flows is O( a()) ad the umber of cocurret receptios is, we eed O( a() ) flow slots to assig each flow oe flow slot. I the secod phase, the umber of flows to be redistributed from busy odes to free odes is O(( ) a()) = O( a()), so we also eed O( a() ) flow slots to assig each flow oe flow slot. Cosequetly, for flow layer schedulig, we eed O( a() ) flow slots to schedule the flows served by R-Cell such that each flow is assiged oe effective flow slot to be trasmitted from the servig ode i T -Cell to the servig ode i R-Cell. We divide oe secod ito c structure slots ad divide each structure slot ito O( a() ) flow slots, each of which has a legth of Ω( ) secods. Sice each ode ca c a() trasmit at the rate of W bits/sec, i each flow slot, λ() = W Ω( ) bits ca be trasported. Hece Ω( W ) bits c a() c a() of each flow ca be trasmitted i oe secod. Accordig to Theorem 3, c is a umber depedet oly o Δ. Sice a() =,wegetthelemma. 00 log b. Whe =Ω( log ) ad = O( ), wehavethe followig lemmas for the capacity lower boud. Lemma 7: Whe =Ω( log ) ad = O( ), we ca partially use the -MPR ability ad the umber of cocurret receptios is Θ( a()). Proof: Note that whe =Ω( log ) ad = O( ), the cell size a() is. Firstly, we prove that whe =Ω( log ) ad = O( ), we caot fully use the -MPR ability ad the umber of cocurret receptios, g, caot be. To see this, if the umber of cocurret receptios is, i secod phase, the umber of trasmitters (busy odes) is )=Θ()=O(). Hece the umber of cocurret receptios exceeds the umber of trasmitters, which cotradicts requiremet (). Hece we caot fully use the - MPR ability. Secodly, we prove that we ca partially use the -MPR ability ad the umber of cocurret receptios is Θ( a()). To see this, if the umber of cocurret Θ( a() receptios, g, isθ( a()), ithefirst phase, it s easy to see that the umber of cocurret trasmissios does ot exceed the umber of trasmitters. I secod phase, the umber of trasmitters is Θ( a() )=Θ( a()), which has the same g order compared with the umber of cocurret receptios, g. Hece requiremet () is satisfied. Sice = O( ), wehave g =Θ( a()) = O( ) i the secod phase. Hece a() requiremet (2) is satisfied. Based o Lemma 7, we preset the followig lemma for the capacity lower boud whe =Ω( log ) ad = O( ). Lemma 8: Whe = Ω( log ) ad = O( ), λ() =Ω(W ) Proof: Whe =Ω( log ) ad = O( ), the umber of cocurret receptios, g, isθ( a()). Substitutig with Θ( a()), usig the similar techiques used i proof for Lemma 6, we get the lemma. c. Whe =Ω( ), we have the followig lemma for the capacity lower boud. Lemma 9: Whe =Ω( ), λ() =Ω(W ) Proof: Accordig to Lemma, whe = Ω( ), the achieved capacity is at least the achieved capacity whe =Ω( log ) ad = O( ). Siceλ() =Ω(W ) bits/sec whe = Ω( log ) ad = O( ), whe =Ω( ), λ() =Ω(W ) Based o Lemmas 6, 8 ad 9, the capacity lower boud ca be preseted as follows. Theorem 7: The lower boud for the capacity of radom -MPR wireless etwors is as follows:. Whe = O( log ), λ() = Ω(W log ) 2. Whe =Ω( log ), λ() =Ω(W ) B. Upper Boud I this sectio, we give the upper boud for the capacity of radom -MPR wireless etwors. The capacity of radom -MPR wireless etwors is costraied by three costraits, destiatio bottleec costrait, geeral costrait ad iterferece costrait respectively. For each of these costraits, there is a upper boud for the etwor capacity. These upper bouds together defie the upper boud for the capacity of radom -MPR wireless etwors. Destiatio Bottleec Costrait The capacity of radom -MPR wireless etwors is costraied by the data that ca be received by a destiatio ode. Usig similar techiques used i [], we ca get followig upper boud for this costrait.. Whe = O(), λ() =O( Wlog log log ) ) 2. Whe =Ω(), λ() =O( W2 log log log Geeral Costrait Sice a radom -MPR wireless etwor is a special id of arbitrary -MPR wireless etwor, the capacity upper boud for arbitrary etwors is also applicable i radom -MPR wireless etwors. Because the distace betwee the source ad destiatio of a flow is Θ(), we have the followig capacity upper boud.

GUO et al.: ON THE CAPACITY OF K-MPR WIRELESS NETWORKS 3885. Whe = O(), λ() =O(W ) 2. Whe =Ω(), λ() =O(W) Iterferece Costrait Sice we eed to avoid the iterferece, we preset the upper boud for this costrait below. Theorem 8: To satisfy the iterferece costrait, the capacity of radom -MPR wireless etwors is bouded as follows. Whe = O( log log log ), λ() = O(W log ) 2. Whe = Ω( log log log ) ad = O(), λ() = O(W log log ) W 3. Whe =Ω(), λ() =O( log log ) Proof: To prove the theorem, we use similar techiques used i []. Firstly, based o the area cosumig observatio, we ca boud the total data rate of the whole etwor, D. Secodly, by lower boudig the umber of hops for a flow, we ca lower boud the data rate served by the whole etwor, D 2, which is depedet o the receivig rage, r(). Obviously, we have D 2 D. To mae the upper boud tight, we should esure that the etwor is coected ad each ode has at least eighbors to utilize the -MPR ability. Accordig to the results of [0], we ca tae the log +2 log log π receivig rage, r() to satisfy these two requiremets. Discussig the order of, we ca get the theorem based o the iequatio D 2 D. Due to the space costrait, we omit the full proof. Because the first two upper bouds are loose compared with the third oe, the third upper boud is the upper boud for the capacity of radom -MPR wirelesss etwors. VI. DISCUSSIONS We aalyze the derived results i this sectio. As show i Sectio IV, the capacity of arbitrary -MPR wireless etwors becomes Θ(W) bit-meters/sec whe = Ω(), which meas the etwor is truly scalable. The mai reasos for this scalable etwor are the arbitrary deploymet of odes ad arbitrary traffic patter, which allow us to deploy odes ad pla the traffic patter properly i order to fully utilize the -MPR ability. O the other had, i radom -MPR wireless etwors, both the deploymet of odes ad the traffic patter are radom, so we caot itetioally deploy the odes ad pla the traffic patter to fully utilize the -MPR ability. Receivers should redistribute the received flows to more odes i order to allow more trasmitters to trasmit the flows, which utilizes the -MPR ability better. Recallig our costructive procedure for the lower boud, whe is small eough, say, = O( log ), the radom etwor has eough trasmitters ad flows to fully utilize the -MPR ability so the capacity gai over -MPR etwors is exactly Θ(). Whe =Ω( log ), sice is too large, there are ot eough trasmitters or eough flows to fully utilize the -MPR ability. Cosequetly we ca oly employ part of the -MPR ability ad the capacity gai of the lower boud over -MPR etwors is Θ( log ). Satisfyig the destiatio bottleec costrait, the geeral costrait ad the iterferece costrait, we give a loose upper boud. We see that this upper boud is loose by otig that i the derivig process of this upper boud, the receivers do ot redistribute flows to more odes for future trasmissios usig the -MPR ability. Cosequetly, for later trasmissios, there are ot eough trasmitters to utilize the -MPR ability. Hece this upper boud caot be achieved by ay radom -MPR wireless etwor i fact. I summary, the mai costraits for radom -MPR wireless etwors to utilize the -MPR ability are the limited umber of trasmitters ad the limited umber of flows each ode serves. To address the first costrait, the receivers eed to distribute the received flows to more odes for future trasmissios. To address the secod costrait, we should assig more flows to each ode (i.e., assig the same flow to more odes to balace the traffic load ad improve the robustess of the etwor). VII. CONCLUSIONS I this wor, we study the capacities of both arbitrary - MPR wireless etwors ad radom etwors. We equip each ode i the etwor with oe -MPR wireless iterface. Each iterface is able to decode at most cocurret trasmissios withi its receivig rage. Uder these assumptios, we have show that for arbitrary etwors, whe = O(), thereisa Θ( ) capacity gai over -MPR wireless etwors. Whe =Ω(), the capacity of arbitrary -MPR wireless etwors is Θ(W) bit-meters/sec ad the etwor is scalable. For radom etwors, we give a costructive lower boud ad a upper boud for the etwor capacity. We have show that eve the lower boud has a capacity gai of Θ( log ) over - MPR radom wireless etwors whe is large eough. From these results, we coclude that the mai costraits for the capacity of radom -MPR wireless etwors are the limited umber of trasmitters ad the limited umber of flows served by each ode. ACKNOWLEDGMENT This wor is supported by NSF Chia (No. 6057338, 6077309, 60702046, 60832005); Natioal Grad Fudametal Research 973 Program of Chia uder Grat No.2006CB303000; Chia Miistry of Educatio (No. 20070248095); Shaghai Jiaotog Uiversity Youg Faculty Fudig (No. 06ZBX800050); Qualcomm Research Grat; Chia Iteratioal Sciece ad Techology Cooperatio Programm (No. 2008DFA630); PUJIANG Talets (08PJ4067); Shaghai Iovatio Key Project (085500400). REFERENCES [] P. Gupta ad P. R. Kumar, The capacity of wireless etwors," IEEE Tras. Iform. Theory, vol. 46, o. 2, pp. 388-404, Mar. 2000. [2] P. Zhou, X. Wag, ad R. R. Rao, Asymptotic capacity of ifrastructure wireless mesh etwors," IEEE Tras. Mobile Computig, vol. 7, o. 8, pp. 0-024, Aug. 2008. [3] R. M. D. Moraes, H. R. Sadjadpour, ad J. J. Garcia-Lua-Aceves, May-to-may commuicatio: a ew approach for collaboratio i MANETs," IEEE INFOCOM, 2007. [4] G. D. Celi, G. Zussma, W. F. Kha, ad E. Modiao, MAC for etwors with multipacet receptio capability ad spatially distributed odes," IEEE INFOCOM, 2008.

3886 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 7, JULY 2009 [5] J. J. Garcia-Lua-Aceves, H. R. Sadjadpour, ad Z. Wag, Challeges: towards truly scalable ad hoc etwors," ACM MobiCom, 2007. [6] J. J. Garcia-Lua-Aceves, H. Sadjadpour, ad Z. Wag, Extedig the capacity of ad hoc etwors beyod etwor codig," ACM IWCMC, 2007. [7] X. Wag ad J. J. Garcia-Lua-Aceves, Embracig iterferece i ad hoc etwors usig joit routig ad schedulig with multiple pacet receptio," IEEE INFOCOM, 2008. [8] P. Patel ad J. Holtzma, Aalysis of a simple successive iterferece cacellatio scheme i a DS/CDMA system," IEEE J. Select. Areas Commu., vol. 2, o. 5, pp. 796-807, Jue 994. [9] A. E. Gamal, J. Mame, B. Prabhaar, ad D. Shah, Throughput-delay trade-off i wireless etwors," IEEE INFOCOM, 2004. [0] P. J. Wa ad C. W. Yi, Asymptotic critical trasmissio radius ad critical eighbor umber for -coectivity i wireless ad hoc etwors," ACM MobiHoc, 2006. [] P. Kyasaur ad N. H. Vaidya, Capacity of multi-chael wireless etwors: impact of umber of chaels ad iterfaces," ACM MobiCom, 2005. [2] D. B. West, Itroductio to Graph Theory. Pretice Hall, 2d ed., 200. Mig-Fei Guo (M 08) Received the B.S. degree i Electroic Egieerig from Xidia Uiversity, Xi a, Chia, i 2004, ad the M.S. degree i Computer Sciece from Shaghai Jiao Tog Uiversity, Shaghai, Chia, i 2007. Sice 2006, he has bee worig towards the Ph.D degree i Computer Sciece at Shaghai Jiao Tog Uiversity, Shaghai, Chia. His research lies i the fields of iformatio theory ad wireless commuicatios, especially of wireless Ad-hoc ad Sesor etwors. Xibig Wag received the B.S. degree (with hos.) from the Departmet of Automatio, Shaghai Jiaotog Uiversity, Shaghai, Chia, i 998, ad the M.S. degree from the Departmet of Computer Sciece ad Techology, Tsighua Uiversity, Beijig, Chia, i 200. He received the Ph.D. degree, major i the Departmet of electrical ad Computer Egieerig, mior i the Departmet of Mathematics, North Carolia State Uiversity, Raleigh, i 2006. Curretly, he is a faculty member i the Departmet of Electroic Egieerig, Shaghai Jiaotog Uiversity, Shaghai, Chia. His research iterests iclude resource allocatio ad maagemet i mobile ad wireless etwors, TCP asymptotics aalysis, wireless capacity, cross layer call admissio cotrol, asymptotics aalysis of hybrid systems, ad cogestio cotrol over wireless ad hoc ad sesor etwors. Dr.Wag has bee a member of the Techical Program Committees of several cofereces icludig IEEE INFOCOM 2009-200, IEEE ICC 2007-200, IEEE Globecom 2007-200. Mi-You Wu (S 84 M 85 SM 96) received the M.S. degree from the Graduate School of Academia Siica, Beijig, Chia, ad the Ph.D. degree from Sata Clara Uiversity, Sata Clara, CA. He is a IBM Chair Professor with the Departmet of Computer Sciece ad Egieerig, Shaghai Jiao Tog Uiversity, Shaghai, Chia. He serves as the Chief Scietist at the Grid Ceter, Shaghai Jiao Tog Uiversity. He is also a Research Professor with the Uiversity of New Mexico, Albuquerque. His research iterests iclude grid computig, wireless etwors, sesor etwors, overlay etwors, multimedia etworig, parallel ad distributed systems, ad compilers for parallel computers. He has published over 50 joural ad coferece papers i the aforemetioed areas.