Capacity of Large-scale CSMA Wireless Networks

Transcription

1 Capacity of Large-scale CSMA Wireless Networks Chi-Ki Chau, Member, IEEE, Mighua Che, Member, IEEE, ad Soug Ch Liew, Seior Member, IEEE Abstract I the literature, asymptotic studies of multi-hop wireless etwork capacity ofte cosider oly cetralized ad determiistic TDMA time-divisio multi-access coordiatio schemes. There have bee fewer studies of the asymptotic capacity of large-scale wireless etworks based o CSMA carriersesig multi-access, which schedules trasmissios i a distributed ad radom maer. With the rapid ad widespread adoptio of CSMA techology, a critical questio is that whether CSMA etworks ca be as scalable as TDMA etworks. To aswer this questio ad explore the capacity of CSMA etworks, we first formulate the models of CSMA protocols to take ito accout the uique CSMA characteristics ot captured by existig iterferece models i the literature. These CSMA models determie the feasible states, ad cosequetly the capacity of CSMA etworks. We the study the throughput efficiecy of CSMA schedulig as compared to TDMA. Fially, we tue the CSMA parameters so as to maximize the throughput to the optimal order. As a result, we show that CSMA ca achieve throughput as Ω, the same order as optimal cetralized TDMA, o uiform radom etworks. Our CSMA scheme makes use of a efficiet backboe-peripheral routig scheme ad a careful desig of dual carrier-sesig ad dual chael scheme. We also address the implemetatio issues of our CSMA scheme. Idex Terms Wireless Network Capacity, Achievable Throughput, Carrier-Sesig Multi-Access CSMA I. INTRODUCTION A importat characteristic that distiguishes wireless etworks from wired etworks is the presece of spatial iterferece, wherei the trasmissio betwee a pair of odes ca upset other trasmissios i its eighborhood. Spatial iterferece imposes a limit o the capacity of wireless etworks. The semial paper 4 by Gupta ad Kumar revealed that the capacity of wireless etworks costraied by spatial iterferece is upper bouded by O for umber of mutually commuicatig odes o a uiform radom etwork, regardless of the chose schedulig ad routig schemes. May similar upper bouds are derived for more sophisticated settigs e.g., with optimal source ad etwork codig schemes 24. I 6, Dai ad Lee derived the upper boud O for multi-hop radom access etworks usig a simple queuig aalytical argumet. They also showed that this upper boud is achievable oly if the maximum throughput of each local ode is a costat idepedet of. M. Che ad S. C. Liew are supported by Competitive Earmarked Research Grats Project # 4008, 4209, ad established uder the Uiversity Grat Committee of the Hog Kog, Chia. M. Che is also supported by a Direct Grat Project # of The Chiese Uiversity of Hog Kog CUHK. Major part of the work was doe whe C.-K. Chau visited CUHK, uder the support of the Direct Grat Project # of CUHK. C.-K. Chau is with the Istitute for Ifocomm Research, Sigapore, ad Uiversity of Cambridge, UK chi-ki.chau@cl.cam.ac.uk. M. Che ad S.-C. Liew are with Departmet of Iformatio Egieerig at the Chiese Uiversity of Hog Kog, Shati, Hog Kog, Chia {mighua,soug}@ie.cuhk.edu.hk. Sice the, a umber of solutios have bee proposed to achieve the upper bouds i various settigs. Particularly, 8 showed that by a efficiet backboe-peripheral routig scheme aalogously called highway system ad a twoste TDMA scheme, Ω is achievable o a uiform radom etwork with high probability. So far, the studies of achievable wireless capacity i the literature cosider oly cetralized cotrols ad a-priori schedulig schemes with TDMA. O the practical frot, carrier-sesig multi-access CSMA etworks e.g., Wi-Fi, which make use of distributed ad radomized medium-access protocols, are receivig wide adoptio across eterprises ad homes. It is ot clear whether the results related to cetrallyscheduled etworks are directly applicable to CSMA etworks. To bridge the gap betwee practice ad research, it will be iterestig to fid out to what extet the capacity of CSMA etworks ca be scaled. I particular, ca the simple distributed schedulig of CSMA scales etwork capacity as well as cetral schedulig ca? The aswer, accordig to our study, is yes. However, the way to go about achievig CSMA scalability is o-trivial ad several mechaisms must be i place before scalability ca be attaied. For example, the use of dual carrier-sesig res i two chaels will be eeded; ad oe must be able to assig differet back-off coutdow times to differet odes i a distributed maer. To establish our results, besides buildig o the past work of others, we fid it ecessary to clarify ad add rigor to the previous frameworks. It is well kow that spatial iterferece imposes a costrait o the liks that ca be active simultaeously. Give a iterferece model, i geeral there ca be a umber of subsets of liks that ca be active simultaeously. Each such subset of liks is called a feasible state. For a cetral scheduler, all feasible states are available for the desig of its schedule. For CSMA etworks, its distributed ature does ot allow us to dictate which particular feasible state will be active at what time. The problem becomes eve more challegig because if ot desiged properly, CSMA may allow a subset of liks that is ot iterferece-safe to trasmit simultaeously, leadig to the so-called hidde-ode problem. We defie the feasible states allowed by the CSMA protocol i a rigorous maer. We argue that the hidde-ode problem i CSMA etworks is caused by a mismatch betwee the feasible states allowed by CSMA ad the feasible states of a uderlyig iterferece model. We show how to resolve this mismatch to create hidde-ode free CSMA etworks. Most importatly, we show that hidde-ode free CSMA etworks ca achieve the same scalig of throughput as the cetral scheduler provided the aforemetioed dual carrier- A schedule is a sequece of feasible states that are active at differet times.

2 2 sesig ad dual chael scheme is i place. Our capacityoptimal CSMA scheme ot oly demostrates the theoretical achievable throughput of CSMA etworks, but also outlies a practical way to achieve it. II. BACKGROUND AND OVERVIEW The basic idea of CSMA is that before a trasmitter attempts its trasmissio, it eeds to ifer the chael coditio by sesig the chael. If it ifers that its trasmissio will upset or be upset by ay receiver s o-goig trasmissios icludig its ow receiver, the it defers its trasmissio. I additio, to prevet two trasmitters from begiig their trasmissios at the same time give that they both sese the chael to be safe for trasmissio, each trasmitter udergoes a radom backoff cout-dow period before trasmissio. The cout-dow will be froze whe chael is sesed to be ot iterferece-safe i.e., trasmissio is collisio-proe, ad will be resumed whe the chael is sesed to be iterferecesafe ai. A trasmissio will be cosidered successful, whe the trasmitter ca receive a ACK packet by the correspodig receiver, upo the completio of trasmissio. Compared to the cetralized TDMA scheme, the CSMA protocol has two distiguishig characteristics: i CSMA is a ACK-based protocol, i which the receivers are required to reply a ACK packet for each successful trasmissio. Thus, bi-directioal commuicatios eed to be explicitly cosidered whe formulatig the costraits o simultaeous trasmissio imposed by CSMA. The cetralized TDMA schemes i prior work 8, 9, 4, 7, 2, 24, however, did ot cosider bi-directioal commuicatios ad ACK packets. ii CSMA is a distributed radom access protocol. Each trasmitter chooses a radom time istace to iitiate its trasmissio, ad it ca oly rely o its limited local kowledge to ifer whether its trasmissio is compatible with other simultaeous trasmissio uder various iterferece settigs 2. Ulike the cetralized TDMA schemes, such a distributed cotrol requires limited a- priori coordiatio amog trasmitters ad receivers. Despite the popularity of CSMA protocols, capacity aalysis applicable to large-scale CSMA wireless etworks receive relatively little attetio i the literature. A likely reaso could be that CSMA protocols are geerally regarded as syoymous to the so-called protocol model i may TDMA based papers. The protocol model is, i fact, a simplified pairwise iterferece model that serves to model iterferece amog simultaeous liks, which either explicitly cosiders or precisely models the aforemetioed characteristics i-ii of CSMA 3. As such, it is ot clear whether the capacity results based o these iterferece models ca apply to CSMA etworks; ad 2 whether CSMA ca achieve the same throughput performace as cetralized TDMA. 2 Note that the iterferece is ot ecessarily symmetric a trasmissio could upset aother simultaeous trasmissio but ot the coverse. 3 Gupta ad Kumar s semial paper 4 appears to be the first to coi the phrase protocol model, but without specifyig ay distributed protocol that ca implemet the protocol model, other tha cetralized schemes by TDMA. There is a cosiderably large body of literature about siglehop CSMA etworks 5, 9. Here we study the more geeral multi-hop CSMA etworks, the results of which are quite limited i the literature 2, 20. We also ote that 3 has studied the capacity of multi-hop Aloha etworks. However, Aloha protocol is differet from CSMA protocol as it has o carrier-sesig operatios. Also, the defiitio of capacity i 3 appears to be differet from the covetioal Gupta-Kumar s oe 8, 3, 4, 7, 8, 24. I summary, our result is built upo two observatios. I CSMA etworks, there are two desig parameters that ca be used to adjust the behavior of a lik. The first parameter is the backoff coutdow rate which decides the legth of the idle period beig sesed by a lik before it ca iitiate a trasmissio. The secod parameter is the carrier-sesig power threshold related to carrier-sesig re which decides how sesitive the lik is to the surroudig trasmissios whe determiig the chael is busy or idle. Our first observatio is that the local CSMA schedulig algorithm with proper but differet cout-dow rates at differet liks ca achieve almost the same rate regio as ay cetralized TDMA schedulig scheme. Our secod observatio is that it is possible to tue the carrier-sesig thresholds to allow sufficiet spatial reuse i the etwork, so that the optimal Ω throughput ca be supported. Combiig these two observatios, we show that CSMA etworks with calibrated coutdow rates ad carriersesig thresholds ca achieve throughput of Ω, the same order as optimal TDMA schemes. A. Outlie of Our Results To explore the capacity of CSMA etworks, we first formulate the models of CSMA protocols to take ito accout characteristics i-ii. These models determie the upper ad lower boud o the capacity of CSMA etworks, ad are fuctios of various CSMA parameters. We the study the throughput efficiecy of CSMA relative to TDMA, followig the same procedure as i 9 ad 6. Fially we tue the CSMA parameters so that the capacity of a CSMA etwork is maximized to the optimal order. Our approach is divided ito four parts: Formulatio of Carrier-sesig Decisio Model Sec. III: Our models for CSMA protocol cosist of two compoets that capture two major fuctioalities of CSMA. The decisio model that formally formulates the costraits o simultaeously active liks imposed by CSMA carrier sesig operatios, such as distace-based carrier sesig. We explicitly distiguish the decisio model of CSMA protocols from the iterferece model. For istace, the fact that two simultaeously active liks are allowed by CSMA does ot ecessarily mea that they do ot iterfere with each other. This is the well-kow hidde ode problem. The radom access scheme that captures how CSMA access the wireless air time ad space. The key challege is to uderstad the throughput efficiecy of distributed ad radomized chael access mechaism of CSMA, as compared to cetralized TDMA scheme.

3 3 Ui-directioal feasible family Bi-directioal feasible family Carrier-sesig feasible family Pairwise Aggregate Pairwise Aggregate Pairwise iterferece iterferece iterferece iterferece carrier-sesig Radom Upper boud: O 4 Upper boud: O this paper Upper boud: O this paper etwork Achievable as: Ω Achievable as: Ω Achievable as: Ω capacity by TDMA 8 by TDMA this paper by dual carrier sesig this paper TABLE I CAPACITY OF UNIFORM RANDOM NETWORKS OVER VARIOUS FEASIBLE FAMILIES. We establish the relatioship betwee our CSMA models ad the iterferece models from the literature i Sec. III. 2 Hidde-ode-free Desig of CSMA Networks Sec. IV: There are various iterferece models i the literature icludig the so-called protocol model. They are iteded to capture ui-directioal trasmissios where ACK packets are ot required. I this paper, we exted these iterferece models to the settig of bi-directioal trasmissios, uder which CSMA protocols typically operate. It is well-kow that the distributed trasmissio schedulig i CSMA may ot be able to prevet spatial iterferece, as kow as the hidde ode problem 5. Utilizig our proposed carrier-sesig decisio models, we formally defie the hidde ode problem as due to a carrier-sesig decisio model violatig the feasibility of a bi-directioal iterferece model. Furthermore, we derive sufficiet coditios for pairwise carrier-sesig decisio model based o carrier-sesig re to elimiate the hidde ode problem uder various iterferece settigs Theorem. Our results iclude the prior oe i 5 as a special case. By elimiatig the hidde-ode problem, we ca apply elegat mathematical tools to aalyze the capacity ad throughput performace of multi-hop CSMA etworks. 3 Statioary State Aalysis of Radom Access Sec. V: To study the behavior of the radom access scheme, we cosider a idealized versio of IEEE 802. DCF based o a cotiuous-time Markov chai model i order to capture the essetial features of CSMA. This cotiuous Markov chai model has bee used i various aalyses 6, 9, 22. Based o the hidde-ode-free desig of CSMA etworks, the log-term throughput of CSMA with radom access is characterized by the statioary distributio of the cotiuoustime Markov chai model. Followig the same procedure as i 6, 9, 22, we preset the statioary distributio, ad hece, the log-term throughput of hidde-ode-free CSMA etworks uder various carrier-sesig decisio models i Sec. V. We also show that CSMA radom access schemes ca be tued to perform as well as TDMA schemes. 4 Desig of Dual Carrier-Sesig Secs. VI-VII: O hidde-ode-free CSMA etworks, we show that the curret CSMA settig with a sigle homogeeous carrier-sesig operatio fails to achieve the optimal capacity Ω o a uiform radom etwork. It ca at most achieve a capacity of O log with high probability show by Theorem 2. We the show that the desig of dual carrier-sesig operatios ca achieve the capacity of the same order as optimal cetralized TDMA. Our desig is draw from a efficiet backboe-peripheral routig scheme i 8, based o which we show that usig two differet carrier-sesig res are sufficiet to achieve optimal capacity of Ω o a uiform radom etwork with high probability show by Theorem 3. I this paper, we ot oly provide isights for the optimal asymptotic capacity of wireless etworks by our dual carrier-sesig scheme, but also address practical issues of implemetig our scheme. First, we address the scalability issue durig the dyamic switchig betwee the dual carriersesig operatios. We propose to use two frequecy chaels to distiguish the two carrier-sesig operatios. Secod, we address the issue of half-duplexity across two frequecy chaels, which eables low-cost implemetatio of our scheme. We summarize our results ad related work i Table I. III. FORMULATION AND MODELS First, ote that some key otatios are listed i Table II. TABLE II KEY NOTATIONS Notatio Defiitio N sd Set of source-sik pairs. λ k Data rate of source-sik pair k N sd. X Set of relayig liks iduced by the paths betwee all source-sik pairs i N sd. t i Coordiates of the trasmitter of lik i X. r i Coordiates of the receiver of lik i X. S Feasible state, a subset of liks that ca simultaeously trasmit. F, U, B, C Feasible family, a set of feasible states. P tx Fixed trasmissio power of all odes. N 0 Fixed oise power. α Power decayig factor i radio trasmissio. β Miimum Sigal-to-Iterferece-Noise ratio for successful receptios. Guard-zoe coefficiet, used i oise-absece pairwise SIR iterferece model. r xcl Iterferece re, used i fixed re iterferece models. r tx Commuicatio re, used i fixed re iterferece models. r cs Carrier sesig re, used i pairwise CSMA decisio models. t cs Carrier sesig power threshold, used i gregate CSMA decisio models. A cetral problem of multi-hop wireless commuicatios is defied as follows. Give a set of source-sik pairs N sd ad a set of data rate {λ k, k N sd }, we ask whether successful wireless commuicatios ca be established betwee all the sources ad siks i N sd to sustai the required rate {λ k, k N sd }, possibly usig other odes as relays, subject to a certai iterferece model of simultaeous wireless trasmissios. Specifically, we cosider the followig two degrees of freedom i establishig the wireless commuicatios:

4 4 Routig scheme that selects the appropriate relayig odes to coect the sources ad siks. 2 Schedulig scheme that assigs determiistically or radomly the slots of trasmissios at relayig odes. Furthermore, these wireless commuicatios should be established i a distributed maer with miimal global kowledge ad coordiatio amog the odes. Hece, we first preset several commo iterferece models of feasible simultaeous wireless trasmissios. The we exted these iterferece models to the settig of bi-directioal commuicatios. Next, we formulate carrier-sesig decisio models that capture distributed cotrol of trasmissios. A. Iterferece Models A iterferece model is defied by its iterferece-safe feasible family. Some commo iterferece-safe feasibility families i the literature are defied as follows. To simplify the defiitios, we implicitly assume i j. a.0 Pairwise fixed-re feasible family: S U fr X, rxcl, r tx, if ad oly if for all i, j S, t j r i r xcl ad t i r i r tx a. Pairwise oise-abset SIR feasible family: S U X,, if ad oly if for all i, j S, t j r i + t i r i 2 a.2 Pairwise SINR feasible family: S U, if ad oly if for all i, j S, β 3 N 0 + P tx t j r i a.3 Aggregate SINR feasible family: S U, if ad oly if for all i S, N 0 + β 4 P tx t j r i j S\{i} Also, we suppose r xcl > r tx, > 0, α > 2, β > 0, ad uiform power P tx at all odes. For a.2-a.3, P tx t i r i βn 0 for all i X. Otherwise, t i caot successfully trasmit packets to r i eve without iterferece. The otio of feasible family applies to both pairwise ad gregate iterferece models. Pairwise models a.0-a.2 ca be captured by the use of coflict graph, whereas the otio of feasible family is more geerally applicable to a.0-a.3. I 4, pairwise SIR iterferece model a. is called protocol model, whereas gregate SINR iterferece model a.3 is called physical model. The amig i this paper emphasizes the iterferece of trasmissios, ad avoids cofusio with CSMA protocol models 4. 4 We remark that also presets a geeralized protocol model with arbitrary iterferece footprit aroud the trasmitters that models more geeral pairwise iterferece settigs, ad a geeralized physical model that specifically applies to the Gaussia chael. B. Bi-directioal Iterferece Models The iterferece-safe costraits a.0-a.3 are uidirectioal, based o the assumptio that the receiver is ot required to reply a ACK packet to the trasmitter upo a successful trasmissio. For ACK-based trasmissios, iterferece ca occur betwee two trasmitters, betwee two receivers, ad betwee a trasmitter ad a receiver. See Fig. for a example of pairwise SIR iterferece model. Without the receptio of ACK packets, the trasmitter will cosider the trasmissio usuccessful ad retrasmit the DATA packet later o. Hece, we eed to esure that the trasmissios of DATA packets ad ACK packets of all simultaeous liks do ot iterfere with each other. D DATA DATA D ACK DATA r i t i t j r j a r i t i t j r j Fig.. I Fig. a the ormal DATA packet trasmissios from trasmitters will ot iterfere with each other, but i Fig. b there is iterferece whe trasmittig ACK packet. Let disti, j mi t j r i, r j t i, r j r i, t j t i. We cosider the bi-directioal versios of iterferece-safe costraits as follows. b.0 Bi-directioal pairwise fixed-re feasible family: S B fr X, rxcl, r tx, if ad oly if for all i, j S, disti, j r xcl ad t i r i r tx 5 b. Bi-directioal pairwise SIR feasible family: S B X,, if ad oly if for all i, j S, b disti, j + t i r i 6 b.2 Bi-directioal pairwise SINR feasible family: S B, if ad oly if for all i, j S, N 0 + P tx disti, j β 7 b.3 Bi-directioal gregate SINR feasible family 5 : S B, if ad oly if for all i S, N 0 + β 8 P tx disti, j j S\{i} Compared with the ui-directio iterferece-safe costraits, the bi-directioal couterparts cosider the iterferece from both the DATA ad ACK trasmissios. 5 A more precise defiitio should replace the deomiator of the LHS of Eq. 8 by N 0 + mi { j S\{i} Ptx mi{ tj -t i, r j -t i }, j S\{i} Ptx mi{ tj -r i, r j -r i } }. Here we choose the simpler ad more coservative form i Eq. 8, as it is sufficiet for our results.

5 5 C. Carrier-Sesig Decisio Models The iterferece-safe costraits a.0-a.3 ad b.0-b.3 capture the global spatial iterferece i the etwork. I CSMA, a trasmitter has oly local kowledge of its iterferece coditio, but ot the iterferece coditios at its targeted receiver or at the trasmittig ad receivig odes of other active liks. The decisio of a trasmitter whether to trasmit is oly determied by its carrier-sesig operatio, rather tha by the global kowledge of spatial iterferece. We defie carrier-sesig decisio models, i which a feasible family is a set of liks that may trasmit simultaeously uder a carrier sesig operatio. But this feasible family may or may ot be iterferece-safe uder the ui-/bi-directioal iterferece models. We preset two feasible families to capture carrier-sesig operatios, defied as follows. c. Pairwise carrier-sesig feasible family: S C X, r cs, if ad oly if for all i, j S, t j t i r cs 9 I pairwise carrier-sesig decisio model c., trasmissio decisio is based o the distace from other simultaeous trasmitters. c. is ofte used together with the pairwise iterferece model for aalysis i the literature. I fact, i aalysis ad i actual implemetatio, c. is also compatible with the gregate iterferece model. For istace, recetly itroduced a ovel ad practical approach to implemet pairwise carrier-sesig decisio model c. with respect to gregate iterferece model, usig Icremetal-Power Carrier Sesig IPCS. The basic idea of Icremetal-Power Carrier-Sesig IPCS is that a trasmitter t i ca estimate the distace to a idividual simultaeously active trasmitter t k by measurig the che of iterferece level. Suppose that iitially t i measures the gregate iterferece level as: N0 + j S\{k} P tx t j t i. The whe t k trasmits, the measured che of iterferece level at t i becomes P i = P tx t k t i, which reveals the distace to t k. This mechaism proceeds as follows. Hece, each trasmitter t i requires to maitai a couter ct i iitially set as 0. Whe t i detects ay che P i, if P i P tx r cs, the ct i ct i +. if P i P tx rcs, the ct i ct i. Trasmitter t i is allowed to trasmit oly if ct i = 0. Suppose that there is o trasmitters that will simultaeously start to trasmit at the same time 6, IPCS ca realize pairwise carriersesig decisio model c.. O the other had, the curret IEEE 802. etworks use a power-threshold based carrier sesig mechaism, such that a trasmitter decides its trasmissios based o the gregate iterferece level measured before the trasmissio: c.2 Aggregate carrier-sesig feasible family: S C X, t cs, if ad oly if there is a sequece i,..., i S, such that for each i k S N 0 + j {i,...,i k } P tx t j t ik t cs 0 6 This will be true, whe we use cotiuous expoetially radom coutdow as i the ext sectio. That is, whe each trasmitter i k sees the gregate iterferece level from other simultaeously active trasmitters that have started trasmissio before is below the power threshold t cs, i k decides that it is allowed to trasmit. Although gregate carrier-sesig decisio model c.2 is easier to implemet tha pairwise carrier-sesig decisio model c. which relies o IPCS, pairwise carrier-sesig does ot deped o the order of decisio sequece of trasmitters, which is more ameable to aalysis. IV. HIDDEN-NODE-FREE DESIGN Usig oly local iterferece coditios, the local decisios of trasmissios i CSMA caot completely prevet harmful spatial iterferece i.e., the hidde-ode problem, or may sometimes over-react to beig spatial iterferece i.e., the exposed-ode problem. While they are well recogized i the literature, lackig are formal defiitios that comprehesively cosider various iterferece ad carrier-sesig decisio models. Here, we provide formal defiitios to hiddeode ad exposed-ode problems based o the models i Sec.III. We the also provide sufficiet coditios to elimiate the hidde-ode problem. Because CSMA is a ACK-based protocol, we cosider a bi-directioal iterferece-safe feasible family B X from oe of b.0-b.3. Give a carrier-sesig feasible family C X from oe of c.-c.2, we defie Hidde-ode problem: if B X C X Exposed-ode problem: if C X B X Namely, hidde-ode problem refers to situatios where the carrier-sesig decisio violates the bi-directioal iterferece-safe costraits, whereas exposed-ode problem is where the carrier-sesig decisio is overly coservative i attemptig to coform to the bi-directioal iterferece-safe costraits. Our defiitios aturally geeralize the oes i 5, which cosiders oly pairwise iterferece ad carriersesig decisio models. For example, we illustrate a istace of hidde-ode problem for pairwise carrier-sesig decisio model ad pairwise SIR iterferece model i Fig. 2. r cs DATA DATA r cs DATA t i r i r j t j t i r i r j t j a Fig. 2. I Fig. a the carrier-sesig decisio model correctly permits the simultaeous liks for DATA packet trasmissio, but fails i the case of ACK packet trasmissio i Fig. b. Hece, B X C X As studied i 5, 22, hidde-ode problem causes ufairess i CSMA etworks. I this paper, we oly cosider CSMA etworks that are desiged to be hidde-ode free. Besides the beefit of better fairess, more importatly, the overall performace of a hidde-ode free CSMA etwork is b ACK

6 6 tractable aalytically. For example, the crucial Eq. 2 of CSMA statioary states to be preseted i Sec. V is valid oly for a CSMA etwork that is hidde-ode free. Oe of our cotributios is to establish the sufficiet coditios to elimiate hidde-ode problem i various iterferece models. We ote that it is more complicated to desig gregate carrier-sesig decisio model c.2 to prevet hidde odes. Hece, i the followig we oly cosider pairwise carrier-sesig decisio model c.. Lemma : If β α, the U X, U U Lemma 2: Let r tx = max i X t i r i. If r xcl P txkα Ptx β r α tx N 0 + r tx 2 where kα k= 4 π2k + 2 k, the U Ufr X, rxcl, r tx 3 Note that kα coverges rapidly to fiite costat 52, whe α > 2. See Fig. 3.a for a plot of the umerical values of kα. We remark that that 0 cosiders the simpler gregate oise-abset SIR model. Because of the absece of oise, usig a tighter packig lattice 0 yields a tighter costat kα. Lemma 3: If r xcl r xcl + 2r tx, the U fr X, rxcl, r tx B fr X, rxcl, r tx U fr X, r xcl, r tx 4 Lemma 4: If + 2, the U X, B X, U X, 5 Lemma 5: If β 2 + β α α, the U B U 6 Lemma 6: If β 2 + β α α, the U B U 7 Lemma 7: If r tx = max t i r i ad r cs r xcl + 2r tx, i X C X, r xcl B fr X, rxcl, r tx C X, r cs 8 Note that Lemma 4 ca be prove by applyig Lemma 5 ad lettig N 0 = 0, = β α ad = β α. Hece, 2 + β α α is a uiversal costat for both pairwise ad gregate iterferece models with/without oise. See Fig. 3.b for a plot of the umerical values of 2 + β α α. A. Hidde-ode-free Sufficiet Coditios Lemmas -7 establish a tree diram Fig. 4 of subsetrelatioships for the iterferece ad carrier-sesig decisio models, uder the respective sufficiet coditios. The tree diram Fig. 4 provides us a way to desig hidde-ode-free CSMA etworks. Give ay bi-directioal iterferece-safe feasible family B X from b.0-b.3, ad pairwise carrier-sesig feasible family C X, r cs, we start k Α Α a Fig. 3. Fig. a: Numerical values of kα, which coverges rapidly to fiite costat 52 whe α > 2. Fig. b: Numerical values of 2 + β α α. U Lemma 4 Lemma B Lemma 4 U U Lemma 5 B Lemma 5 U Lemma U Lemma 6 B Lemma 6 U Lemma 2 U Lemma 3 fr B Lemma 7 fr C Lemma Β B 6 fr b 8 0 Lemma 3 U fr Fig. 4. The tree diram represets the subset-relatioships for the iterferece ad pairwise carrier-sesig decisio model. at B X i the tree diram, ad follow the respective chais of lemmas to set the respective sufficiet coditios util reachig C X, r cs. The, we ca obtai a hidde-ode-free desig. Hece, it proves the followig theorem. Theorem : Suppose r tx = max i X t i r i. For ay bidirectioal iterferece-safe feasible family B X from b.0- b.3 ad pairwise carrier-sesig feasible family C X, r cs, there exists a suitable settig of r cs such that Hidde-ode-free Desig : B X C X, r cs 9 We summarize the sufficiet coditios for hidde-odefree CSMA etwork desig i Table III. We remark that although the virtual carrier sesig RTS/CTS i IEEE 802. is desiged to solve the hidde ode problem, usig RTS/CTS i multi-hop etworks does ot elimiate the hidde-ode problem 23, uless the carrier sesig re is large eough ad a umber of other coditios are met 5. The coditios for hidde-ode free operatio uder the RTS/CTS mode are much more complicated tha uder the basic mode, eve uder the pairwise iterferece model see 5 for details. To keep our focus i this paper, we will ot cosider the RTS/CTS mode. The extesio to icorporate RTS/CTS is certaily a iterestig subject for future studies, particularly for the hidde-ode free operatio uder the gregate iterferece model. V. STATIONARY THROUGHPUT ANALYSIS While Sec. III-IV address the distributed ad ACK-based ature of CSMA, this sectio addresses the characteristics of radom access i CSMA, ad study its achievable capacity as compared to TDMA schemes. 2 3 Α 4 5

7 7 Pairwise carrier-sesig feasible family C X, r cs Bi-directioal feasible family pairwise fixed re pairwise SIR pairwise SINR gregate SINR B fr X, rxcl, r tx B X, B B r tx = max i X t i r i r cs r xcl + 2r tx r tx = max i X t i r i r cs 3 + r tx 5 r tx = max t i r i i X r cs P tx P tx tx 2+β α α r N 0 α + 2r tx See 4 r tx = max t i r i i X r cs P txkα P tx 2+β α r α tx N 0 α + 3r tx TABLE III SUFFICIENT CONDITIONS FOR HIDDEN-NODE-FREE CSMA NETWORK DESIGN. RESULTS ARE DERIVED IN THIS PAPER UNLESS CITED OTHERWISE. A. Determiistic Schedulig Cosider a give routig scheme ad pairwise carriersesig decisio model c. implemeted by IPCS ad set to be hidde-ode-free by Theorem. For brevity, i the followig we let C X C X, r cs. If we assume slotted time, a determiistic schedulig scheme is defied as a sequece S t m t= where each S t C X, such that the trasmitters i each S t are allowed to trasmit oly at every timeslot t mod m. A TDMA scheme is simply a determiistic schedulig scheme. Such a TDMA scheme is oly a hypothetical scheme that ca serve as a referece scheme for the study of the radom access based CSMA etwork. Suppose the badwidth is ormalized to a uit costat. The for each lik i X, the throughput rate uder schedulig scheme S t m t= is: c det i St m t= m m i S t 20 t= Recall λ k is the data rate of source-sik pair k N sd. With the routig scheme, oe ca determie the feasible regio for λ k k N sd by solvig a multi-commodity flow problem. B. Multi-Backoff-Rate Radom Access More geerally, we cosider a radom access scheme e.g., IEEE 802. DCF, such that S t t= follows a radom sequece. We cosider a idealized versio CSMA radom access scheme as a cotiuous-time Markov process as i 7, 6, 9, 22, which is sufficiet to provide isights for the practical CSMA radom access scheme. We assume that the cout-dow time ad trasmissio time follow expoetial distributio 7. The avere cout-dow time ca be distict for differet liks. Thus, we call this multi-backoff-rate radom access. We formalize the radom access scheme by a Markov chai with its states beig C X. There is a possible trasitio betwee states S, S C X, if S = {i} S for some i X. Trasitio S {i} S represets that the trasmitter of lik i will start to trasmit, after some radom coutdow time. Trasitio {i} S S represets that the trasmitter of lik i will fiish trasmissio, after some radom trasmissio time. 7 The mai results of this paper is built upo the statioary probabilitya distributio i Eq showed that for geeral backoff ad trasmissio times that are ot expoetially distributed, Eq. 2 remais valid if the process is statioary. I particular, Eq. 2 has bee verified to be valid for may differet backoff time distributios, icludig the that of Wi-Fi. Thus, strictly speakig, the expoetial assumptio is ot eeded. Suppose the curret state of simultaeous trasmissios is S, ad trasmitter t i is coutig dow to trasmissio. Trasmitter t i will freeze cout-dow if it detects that the chael is busy i.e., S {j} S for some j i, ad {i, j} S / C X. It will resume cout-dow whe the state of simultaeous trasmissios becomes S such that {i} S C X. Let the rate of trasitio S {i} S be ν i, ad ormalize the rate of trasitio {i} S S as. Let ν ν i i X. The C X, ν deotes the cotiuous-time Markov process of idealized multi-backoff-rate CSMA radom access. Lemma 8: C X, ν is a reversible Markov process, with statioary distributio for each S C X as: exp i S P ν S = log ν i S C X exp j S log ν 2 j Lemma 8 is well-kow i the literature 9, 22. We preset it here for completeess. The log-term throughput is characterized by the statioary distributio of C X, ν. Therefore, for each lik i X, the throughput rate uder idealized multi-backoff-rate CSMA radom access is: c rad i C X, ν S C X:i S P ν S 22 We ca relate the throughput of a determiistic schedulig scheme with the log-term throughput of idealized multibackoff-rate CSMA radom access by the followig result. Lemma 9: Give a determiistic schedulig scheme S t m t=, let the fractio of time spet i S C X be P det S = m m t= S t = S. If P det S > 0 for all S C X, the there exists cout-dow rates ν, such that for each lik i X, it satisfies: St m t= c rad C X, ν 23 c det i Lemma 9 is a slightly modified versio of Propositio 2 i 6, which applies to the periodic TDMA schemes as cosidered i this paper. I the techical report 4, we give a simplified alterate proof, ispired by the set of Markov approximatio argumets elaborated i 5. A distributed algorithm is preseted i 6 to adapt the appropriate cout-dow rate ν to satisfy Lemma 9. The implicatio of Lemma 9 is that idealized multi-backoffrate CSMA radom access ca be adapted to perform at least as well as a class of TDMA schemes uder the same set i

8 8 of feasible states. Lemma 9 will be useful to explore the achievable capacity of multi-backoff-rate CSMA etworks, give the achievable capacity of the correspodig TDMA scheme o the same C X. VI. CAPACITY OF RANDOM NETWORK I this sectio, we apply the results from Sec. III-V to the capacity aalysis o a uiform radom etwork. The reaso for selectig a uiform radom etwork is to provide the simplest avere-case aalysis, without ivolvig other complicated radom etwork topologies. We cosider a Poisso poit process 8 of uit desity o a square plae 0, 0,. Every ode o the plae is a source or a sik that is selected uiform-radomly amog all the odes o the plae. We ext defie some otatios: deotes the radom set of source-sik pairs iduced by the Poisso poit process. R deotes a routig scheme that assigs each k N sd a path, such that each hop is withi the maximum trasmitter-receiver distace P tx /βn 0 α. X R deotes the radom set of liks iduced by routig scheme R over N sd N sd F X R deotes a feasible family from a.0-c.2 over the radom set of liks, X R. S F X R deotes the set of all possible determiistic schedulig schemes { S t F } X R m t=. λ F X R deotes the miimum data rate amog all the source-sik pairs i N sd, achieved by the optimal determiistic schedulig scheme:. λ F X R max S t m t= S FX R mi k N sd λ k 24 We ow defie the capacity over radom etworks. Sice λ F X R is a radom variable, we say that the capacity over N sd has a order as Θf with high probability w.h.p., if there exists fiite costats c > c > 0 such that lim P { λ F } X R = c f is feasible = lim if P { λ F X R = c f is feasible } < This is the covetioal defiitio of radom wireless etwork capacity 8, 4, 8, 24. A. Upper Boud for Sigle Carrier Sesig We first show that carrier sesig based o c.-c.2 caot achieve the optimal capacity Ω. Theorem 2: Cosider a carrier-sesig feasible family C X R from c.-c.2, for ay routig scheme R that coects all the source-sik pairs i N sd, λ C X R = O w.h.p. 25 log Proof: By Lemmas 3,7, there exists a suitable r xcl, such that C X R ca be cofigured as a subset of U fr X R, r xcl, r tx. It has bee show i 4 that λ U fr X R, r xcl, r tx = O w.h.p. 26 log 8 Oe ca cosider a alterative poit process where odes are placed o the plae by uiform distributio. But this poit process coverges to Poisso poit process asymptotically. for ay routig scheme R that coects all the source-sik pairs i N sd. Hece, it completes the proof. Noetheless, 8, 24 show that for ay iterferece-safe feasible family from a.-a.3, there exists a TDMA scheme to achieve throughput as Ω w.h.p.. We are thus motivated to adopt such a TDMA-based approach to CSMA etworks. B. Backboe-Peripheral Routig For the completeess of presetatio, we briefly revisit the efficiet routig scheme i 8 we call backboe-peripheral routig. Partitio the odes ito two classes: backboe odes ad peripheral odes. The backboe odes themselves are coected usig oly short-re liks, whereas every peripheral ode ca reach a backboe ode i oe-hop trasmissio. The basic idea is to use short-re backboe-backboe liks wheever possible. Sice short-re liks geerate miimal spatial iterferece, this icreases the umber of simultaeous active liks, ad hece the throughput. To implemet backboe-peripheral routig, we first partitio the square plae 0, 0, ito square cells with sidelegth s. Cosider the cells as vertices, a path ca be formed by coectig adjacet o-empty cells. Lemma 0: See 8 There exist costats c, c 2, c 3 idepedet of, such that whe we set s = c, the i every horizotal slab of /c c 2 log /c cells, there exist at least c 3 log disjoit paths betwee the vertical opposite sides of the plae w.h.p.. We build a backboe called highway system i 8 for routig o a uiform radom etwork as follows. Select a represetative ode i each o-empty cell. By Lemma 0, there is a coected sub-etwork that spas the plae w.h.p., formed by coectig the represetative odes i the adjacet cells. These coected represetative odes are the backboe odes, while the rest are the peripheral odes. Note that the distace betwee two adjacet backboe odes is at most 5c, while the distace betwee a peripheral ode to a earby backboe ode is at most c 2 log w.h.p.. Backboe-peripheral routig scheme operates as follows. The source first uses a oe-hop trasmissio to a backboe ode, if it is a peripheral ode. We cotrol the packet load from the peripheral odes such that each backboe ode is accessed by at most by some costat umber of peripheral odes. Next, the receivig backboe ode relays the packet followig multihop Mahatta-routig alog the adjacet backboe odes to the respective backboe ode that ca trasmit the packets to the sik i a sigle last hop. See Fig. 5 for a illustratio of backboe-peripheral routig. We defie a schedulig scheme uder backboe-peripheral routig cosistig of two stes: St P c4 log2 t= ad St B c5 t=, for some costats c 4, c 5. Backboe-peripheral Trasmissios: If i St P, the either t i or r i is a peripheral ode. Usig a spatial assigmet scheme, we divide the plae ito larger cells, each of which havig a area of Θlog 2 because the backboe-peripheral distace is Olog. It is show

9 9 /s cells /s cells Backboe ode Peripheral ode Backboe-backboe lik Backboe-peripheral lik Fig. 5. Backboe odes are a subset of coected odes by short-re liks, whereas peripheral odes relay all the packets to backboe odes. i 8 that we ca always pick a o-iterferig lik i each cell to trasmit i every timeslot t mod c 4 log 2 i the first ste, for some costat c 4. The throughput rate for each backboe-peripheral lik ca be show to be Θ log 2 Θ. 2 Backboe-backboe Trasmissios: If i S B t, the both t i ad r i are backboe odes. Sice the backboebackboe distace is O, we use a similar spatial assigmet scheme but cosiderig a cell with a area c 5, for some costat c 5. Sice each backboe ode is accessed by at most by some costat umber of peripheral odes, there are at most O peripheral odes that relays packets to each backboe ode. Thus, the throughput rate at each backboe-backboe lik divided by the umber of peripheral odes that relay packets to it is Θ. Overall, backboe-backboe liks are the bottleeck, ot backboe-peripheral liks. Hece, λ k = Ω is achievable w.h.p. o a uiform radom etwork based o backboeperipheral routig ad the above two-ste schedulig scheme. Note that sice the maximum backboe-peripheral distace may scale as Θlog, it is ecessary to decrease threshold β or icrease power P tx as icreases for these liks i the SINR models. If we opt to keep a fixed P tx ad decrease β, the data rate will decrease as icreases. However, the data rate does ot decrease as fast as the target per-flow throughput, which is O. Thus, the bottleeck will remai to be at backboe-backboe liks. VII. DUAL CARRIER-SENSING To adopt the TDMA scheme of backboe-peripheral routig i Sec. VI-B for CSMA etworks, i this sectio we employ dual carrier-sesig where multiple carrier-sesig res are allowed. Namely, smaller carrier-sesig res ca be used amog the short-re liks. This effectively eables more simultaeous liks ad improves the throughput. However, it is ot straightforward to implemet dual carriersesig i covetioal CSMA protocols e.g., IEEE 802., because the trasmitters may ot be aware if the other active liks are short-re or log-re. To address the above implemetatio issue of dual carrier-sesig, we are motivated to adopt a system with two frequecy chaels, i which the commuicatios o the backboe-backboe liks are carried out o oe frequecy chael, while the commuicatios o the peripheral liks are carried out o the other chael. I the followig, we provide a detailed study o the implemetatio of dual carrier-sesig o two frequecy chaels. First, Sec. VII-A cosiders a system that is full-duplex across the two frequecy chaels. The, Sec. VII-B cosiders a system that is half-duplex across the two frequecy chaels that is simpler to implemet, but whose coditios for hiddeode free operatio are more subtle. A. Full-duplexity across Two Frequecy Chaels Thus far, we have assumed that the commuicatio o a chael is half-duplex i that whe a ode trasmits, it caot receive. This is typically the case if oe strives for simple trasceiver desigs. We will cotiue to assume that a ode caot trasmit ad receive o the same chael simultaeously. However, we assume full-duplexity across differet frequecy chaels i that simultaeous trasmissio ad receptio o differet chaels are allowed. Specifically, whe a ode trasmits o frequecy, it could receive o frequecy 2; ad whe a ode trasmits o frequecy 2, it could receive o frequecy. Carrier-seig Mechaism: With such set-up, the peripheral odes will trasmit ad receive o oe of the frequecy chaels, referred to as the peripheral chael. The backboe odes will trasmit ad receive amog themselves o the backboe subet usig the other frequecy chael, referred to as the backboe chael. Whe trasmittig to or receivig from the peripheral odes, however, the backboes odes will use the peripheral chael. Thus, a backboe ode ca coceptually be thought of as cosistig of two virtual odes: a virtual peripheral ode for commuicatig with peripheral odes associated with it; ad a virtual backboe ode for relayig packets over the backboe etwork. This desig decouples the operatio of the peripheral access subet from that of the backboe highway. Formally, we partitio X ito two disjoit classes: X B for backboe-backboe liks, ad X P for backboe-peripheral liks. Assume rcs B < rcs. P The feasible family that captures the above carrier-sesig mechaism is defied as: d. Full-duplex pairwise dual carrier-sesig feasible family: S C ful X B, rcs, B X P, rcs P, if ad oly if for all i, j S, t j t i rcs c 27 such that i, j X c ad c {B, P}. That is, a peripheral ode will carrier-sese the peripheral chael oly. A backboe ode will carrier-sese the peripheral chael if it wishes to trasmit to a peripheral ode, ad will carrier-sese the backboe chael if it wishes to trasmit to a backboe ode. Throughput: We ow show that carrier-sesig model d. ca achieve throughput as Ω o two idepedet frequecy chaels.

10 0 Theorem 3: Cosider full-duplex pairwise dual carriersesig model d. o a uiform radom etwork based o backboe-peripheral routig. Let X B ad X P be the radom set of iduced backboe-backboe liks ad backboeperipheral liks, respectively. Usig multi-backoff-rate radom access scheme, there exists suitable rcs, B rcs, P such that λ C ful X B, rcs, B X P, rcs P = Ω w.h.p. 28 Proof: First, recall St P c4 log2 t= ad St B c5 t=, the two ste TDMA schemes i backboe-peripheral routig. Note that each of St P ad St B is a feasible state i some uidirectioal pairwise fixed-re feasible families a.0, where the trasmitter-receiver distace is Olog ad O respectively. By Lemma 3, we ca obtai respective families of bidirectioal pairwise fixed-re feasible families that iclude St P c4 log2 t= ad St B c5 t=, with larger res rp xcl ad rxcl B, idepedet of. We the select S P t c6 log2 t= ad S B t c7 t= that ca cover the schedules liks i St P c4 log2 t= ad St B c5 t=. Note that sice rxcl P ad rxcl B are set idepedet of, such selectios ca oly icur at most a costat multiple of the sizes by c 6 ad c 7. Hece, we ca use S P t c6 log2 t= ad S B t c7 t= i the backboe-peripheral routig scheme without alterig the order results o capacity. Sice the two frequecy chaels are idepedet, by Theorem, we ca obtai suitable settigs of rcs B ad rcs, P such that C X B, rcs B ad C X P, rcs P are hidde ode free i their respective chaels with respect to ay iterferece model b.0-b.2, ad S P t C X B, r B cs for all t =...c 6 log 2 29 S B t C X P, r P cs for all t =...c 7 30 Next, we employ Lemma 9 to establish a lower boud of the throughput of radom access o each of C X B, rcs B ad C X P, rcs, P by the throughput of a correspodig determiistic schedulig scheme as follows: For each S {S P t } c6 log2 t=, we set P det S = Θ log 2 For each S {S B t } c7 t=, we set Pdet S = Θ For feasible state S other tha S P t c6 log2 t= ad S B t c7 t=, we set P det S to be a small o-zero value, such that the sum of probabilities of these states is a costat, idepedet of. Hece, this satisfies the sufficiet coditio i Lemma 9 that P det S > 0. It is easy to see that C ful X B, rcs, B X P, rcs P is just a product of C X B, rcs B ad C X B, rcs. B Sice such a determiistic schedulig scheme ca achieve throughput as Ω o a uiform radom etwork w.h.p., it completes the proof by Lemma 9. B. Half-duplexity across Two Frequecy Chaels We ow cosider a system that is half-duplex across the two frequecy chaels to ease implemetatio further. A ode ca still receive o differet chaels simultaeously for the purpose of carrier-sesig both chaels simultaeously rather tha receivig data targeted for it. However, we place a restrictio o simultaeous trasmissio ad receptio o the same chael or differet chaels, as elaborated below. Half-duplexity Costraits: We itroduce the followig costraits to formulate half-duplexity: i a ode caot trasmit o chael i ad receive o chael j at the same time, whether i = j or i j. ii a ode ca oly trasmit o at most oe frequecy chael at ay time. Costrait i is maily to simplify implemetatio. Whe a ode trasmits, its ow trasmitted sigal power may overwhelm the received sigal. Although i priciple, the use of a frequecy filter may be able to isolate the sigals somewhat, the trasmit power may be very large compared with the receive power i.e., extreme ear-far problem, such that leake or crosstalk from the power at the trasmit bad may ot be egligible compared with the receive power. Referece 2 cotais a discussio o the eed for the assumptio of halfduplexity whe the trasmit ad receive frequecy chaels are the same, but the uderlyig ratioale ad priciples are the same whe the cross-frequecy leake is ot egligible. Costrait ii is maily due to the fact that i ACK-based CSMA schemes e.g., IEEE 802., there is a ACK packet i the reverse directio after the trasmissio of a DATA packet. If the odes trasmit o two frequecy chaels ad the DATA packets are of differet legths, oe of the DATA frames may fiish first ad the statio may ed up trasmittig DATA ad receivig ACK packets at the same time, thus violatig costrait i. Carrier-seig Mechaism: We ow describe the carrierseig mechaism uder costraits i ad ii as follows. The mechaism is illustrated i Fig. 6. The basic idea is that we allow a shorter carrier-sesig re to be used amog backboe-backboe liks, whereas a loger carrier-sesig re to be used i both chaels whe there is a active backboe-peripheral lik i the eighborhood. r B cs r P cs a Backboebackboe lik Backboeperipheral lik Fig. 6. There are two carrier sesig res as i Fig. a. I Fig. b shortre backboe-backboe liks will use a shorter carrier-sesig re amog themselves, while i Fig. c loger carrier-sesig re will used whe there is ay active backboe-peripheral lik. First, we cosider the case of a backboe-peripheral lik, where its carrier-sesig re is r P cs. I this case, either a peripheral ode dees to trasmit to a backboe ode, or a b c

11 backboe ode dees to trasmit to a peripheral ode. The trasmissio caot be allowed if there is ay simultaeous trasmitters withi the carrier-sesig re rcs P i either peripheral chael or backboe chael. Sice rcs P rcs, B this implies precludig trasmissio i backboe chael uder carrier-sesig re rcs. B The reaso for this requiremet is because of the followig cosideratio. Suppose that a peripheral ode wats to trasmit to its access backboe ode. It is possible that the backboe ode is i the midst of a commuicatio with aother backboe ode. To make sure that the peripheral ode does ot iitiate a trasmissio to the backboe ode i that situatio, the peripheral ode also has to perform carrier-sesig o the backboe chael. I practice, further implemetatio optimizatio is possible skipped here due to limited space. Next, we cosider the case of a backboe-backboe lik, where its carrier-sesig re is rcs. B I this case, a backboe ode wats to trasmit to aother backboe ode. The trasmissio is ot allowed if there is ay simultaeous trasmitters withi the carrier-sesig re rcs B i the backboe chael, or 2 there is ay simultaeous trasmitters withi the carrier-sesig re rcs P i the peripheral chael. The former coditio is obvious. The latter coditio is due to the fact that the target receiver backboe ode may be i the midst of a commuicatio with a peripheral ode. Agai, further optimizatio is possible with the latter case. Here, we simply set the carrier-sesig re i the later to be rcs, P sice the order results we wat to establish are ot compromised. The feasible family that captures the above carrier-sesig mechaism is defied as: d.2 Half-duplex pairwise dual carrier-sesig feasible family: S Chaf X B, rcs, B X P, rcs P, if ad oly if for all i, j S, t j t i max{rcs, c rcs}, c 3 where i X c, j X c ad c, c {B, P}. That is, there is a dyamic switchig process of carriersesig res, depedig o the presece of the classes of active liks. Throughput: Sice we are cosiderig half-duplexity across two frequecy chaels, the proof of throughput is differet tha Theorem 3. To show that carrier-sesig model d.2 i the presece of half-duplexity ca achieve the throughput as ad rcs. B We have to formally show carrier-sesig decisio model d.2 ca be implemeted practically, by cosiderig dual chael iterferece models that explicitly icorporate the costrait of halfduplexity across two frequecy chaels. Thus, we defie the gregate iterferece model i such case as follows: Ω, we first eed to determie r P cs e. Half-duplex bi-directioal dual chael gregate SINR feasible family: S Bhaf X B, β, X P, β, if ad oly if S = S c, where each S c B X c, β, c {B,P} 2 half-duplexity costrait for ay pair i, j S, {t i, r i } {t j, r j } =. Similarly, oe ca defie the respective dual chael iterferece models for a.0-a.3,b.0-b.2. Theorem 4: There exists a suitable settig of rcs, B rcs, P depedig o β ad the maximum trasmissio distace i X c, such that B haf X B, β, X P, β C haf X B, r B cs, X P, r P cs Theorem 4 establishes a hidde-ode-free desig for the dual carrier-sesig decisio model. The proof of Theorem 4 is to apply the sigle-chael hidde-ode-free desig Theorem o two idepedet frequecy chaels, ad the show the half duplexity costrait i e. will ot affect the settig of hidde-ode-free desig i d.2. Similar to Theorem 3, by Theorem 4 we ca immediately show that d.2 ca also achieve throughput as Ω. Theorem 5: Cosider half-duplex pairwise dual carriersesig model d.2 o a uiform radom etwork based o backboe-peripheral routig. Usig multi-backoff-rate radom access, there exists a suitable settig of rcs, B rcs, P such that λ C haf X B, r B cs, X P, r P cs = Ω w.h.p. 32 We remark that our CSMA capacity scalig-law results also hold for dese etworks where all odes are packed i a fixed area 0, 0,, because the costructio of backboeperipheral routig also applies to dese etworks 8. VIII. CONCLUSION This paper cotais a umber of ew results ad ideas that led isights ad solutios to maximize the achievable capacity i CSMA wireless etworks. We formulate a comprehesive set of CSMA models, cosiderig various distributed decisio cotrols ad commo iterferece settigs from the literature. We establish the relatioship betwee our CSMA models with the existig iterferece models from the literature. This ca characterize both the upper ad achievable bouds o the capacity of CSMA etworks to be Θ. We show that, based o a efficiet backboe-peripheral routig scheme ad a careful desig of dual carrier-sesig ad dual chael scheme, hidde-ode-free CSMA etworks ca achieve throughput as Ω, as optimal as TDMA schemes ca o a uiform radom etwork. Alog the jourey, we also show that ormal, sigle, ad homogeeous carrier sesig operatio is isufficiet to achieve the capacity as optimal as TDMA schemes ca o a uiform radom etwork. REFERENCES A. Agarwal ad P. R. Kumar. Capacity bouds for ad-hoc ad hybrid wireless etworks. ACM Computer Comm. Review, 343:7 8, J. G. Adrews, S. Shakkottai, R. Heath, N. Jidal, M. Haeggi, R. Berry, D. Guo, M. Neely, S. Weber, S. Jafar, ad A. Yeer. Rethikig iformatio theory for mobile ad hoc etworks. IEEE Commu. Mazie, 462:94 0, December F. Baccelli, B. Blaszczyszy, ad P. Muhlethaler. A Aloha protocol for multihop mobile wireless etworks. IEEE Tras. Iformatio Theory, 522:42 436, February 2006.

12 2 4 C.-K. Chau, M. Che, ad S. C. Liew. Capacity of large-scale CSMA wireless etworks. Techical report, Available o Arxiv: http: //arxiv.org/abs/ M. Che, S. Liew, Z. Shao, ad C. Kai. Markov approximatio for combiatorial etwork optimizatio. I Proc. IEEE INFOCOM, L. Dai ad T. T. Lee. Throughput ad delay aalysis of wireless radom access etworks. I Proc. CISS, The log versio available o Arxiv: 7 M. Durvy, O. Dousse, ad P. Thira. Border effects, fairess, ad phase trasitio i large wireless etworks. I Proc. IEEE INFOCOM, M. Fraceschetti, O. Dousse, D. N. C. Tse, ad P. Thira. Closig the gap i the capacity of wireless etworks via percolatio theory. IEEE Tras. Iformatio Theory, 533:009 08, March M. Fraceschetti, M. Migliore, ad P. Miero. The capacity of wireless etworks: iformatio-theoretic ad physical limits. IEEE Tras. Iformatio Theory, 558: , August L. Fu, S. C. Liew, ad J. Hu. Safe carrier sesig re i CSMA etwork uder physical iterferece model. arxiv.org/abs/ L. Fu, S. C. Liew, ad J. Hu. Effective carrier sesig i CSMA etworks uder cumulative iterferece. I Proc. INFOCOM, Y. Gao, D. M. Chiu, ad J. C. S. Lui. Determiig the ed-to-ed throughput capacity i multi-hop etworks: methodology ad applicatios. I Proc. ACM SIGMETRICS, pes 39 50, S. Guhu, C.-K. Chau, ad P. Basu. Gree Wave: Latecy ad capacityefficiet sleep schedulig for wireless etworks. I Proc. IEEE INFO- COM, P. Gupta ad P. R. Kumar. The capacity of wireless etworks. IEEE Tras. Iformatio Theory, 462: , L. B. Ji ad S. C. Liew. Improvig throughput ad fairess by reducig exposed ad hidde odes i 802. etworks. IEEE Tras. Mobile Computig, 7:34 49, Jauary L. B. Ji ad J. Walrad. A distributed CSMA algorithm for throughput ad utility maximizatio i wireless etworks. I Proc. Allerto Cof. o Comm., Cotrol, ad Computig, A. Josa, M. Liu, D. Neuhoff, ad S. Pradha. Throughput scalig i radom wireless etworks: A o-hierarchical multipath routig strategy. I Proc. Allerto, S. Li, Y. Liu, ad X.-Y. Li. Capacity of large scale wireless etworks uder Gaussia chael model. I Proc. ACM MobiCom, S. C. Liew, C. Kai, J. Leug, ad B. Wog. Back-of-the-evelope computatio of throughput distributios i CSMA wireless etworks. I Proc. IEEE ICC, To appear i IEEE Tras. Mobile Computig. 20 P. C. Ng ad S. C. Liew. Throughput aalysis of IEEE802. multi-hop ad hoc etworks. IEEE/ACM Tras. Networkig, 52: , A. Özgür, O. Lévêque, ad D. N. C. Tse. Hierarchical cooperatio achieves optimal capacity scalig i ad hoc etworks. IEEE Tras. Iformatio Theory, 533: , October X. W ad K. Kar. Throughput modellig ad fairess issues i CSMA/CA based ad-hoc etworks. I Proc. IEEE INFOCOM, K. Xu, M. Gerla, ad S. Bae. How effective is the IEEE 802. RTS/CTS hadshake i ad hoc etworks. I Proc. IEEE GLOBECOM, F. Xue ad P. R. Kumar. Scalig laws for ad hoc wireless etworks: A iformatio theoretic approach. Foudatios ad Treds i Networkig, 2:45 270, July IX. APPENDIX Lemma : For γ >, it is straightforward that A N 0 + γb < β A < γβ 33 N 0 + B Lemma 2: Let r tx = max i X t i r i. If there exists r xcl > r tx such that t j r i r xcl for all i, j S, the j S\{i} t j r i kαr xcl r tx 34 where kα k= 4 π2k + 2 k. Proof: It is adopted from 8 Lemma 3. See the techical report 4 for a complete proof. Corollary 3: By Lemma 2, if there exists r cs such that t j t i r cs for all i, j S, the j S\{i} t j t i kαr cs 35 Lemma : If β α, the U X, U U 36 Proof: U U is trivial. X, U follows from: U P tx t i r i N 0+P tx t j r i β Ptx ti ri P tx t j r i β t j r i β α t i r i 37 Lemma 2: Let r tx = max i X t i r i. If P tx r xcl P tx kα β r α tx N 0 + r tx 38 where kα k= 4 π2k + 2 k, the U Ufr X, rxcl, r tx 39 Proof: Suppose S U fr X, rxcl, r tx ad i S. By Lemma 2, we obtai: N 0 + P tx t j r i P tx rtx N 0 + P tx kαr xcl r tx j S\{i} Hece, r xcl Ptx P txkα β r tx α N 0 + r tx P txr tx N 0+P txkαr xcl r tx β S U Lemma 3: If r xcl r xcl + 2r tx, the U fr X, rxcl, r tx B fr X, rxcl, r tx U fr 40 4 X, r xcl, r tx 42 Proof: By Lemma 4 ad set = r xcl /r tx ad = r xcl /r tx Lemma 4: If + 2, the U U X, B X, U X, 43 Proof: By Lemma 5 ad set N 0 = 0, = β α ad = β α. Lemma 5: If β 2 + β α α, the B U 44 Proof: First, ote that P tx t i r i βn 0 for all i X. Otherwise, t i will be uable to trasmit to r i eve without iterferece. Secod, N 0 + P tx disti, j < β Ptx P tx β t i r i α N 0 > disti, j 45 The last iequality is due to the fact that α < 0 ad P tx t i r i βn 0. We eed to show that S {i} / B S {i} / U X, 2 + β α α 46 Suppose S {i} / B for some give lik i. The there are four cases as follows.

13 3 P tx t i r i N 0+P tx t j r i : Suppose < β for some j S. This is trivial that S {i} / U. 2: Suppose for some j S that < β 47 N 0 + P tx r j r i Without loss of geerality, we also assume t i r i t j r j 48 Otherwise, if t j r j > t i r i, the r j r i < Ptx P tx β t i r i α N 0 Ptx P tx β t j r j α N 0 Therefore, we ca equivaletly assume 49 P tx t j r j N 0 + P tx r j r i < β ad t j r j t i r i, 50 by iter-chig i ad j. Next, we obtai: t j r i r j r i + t j r j < Ptx P tx β t i-r i α N 0 + t j -r j by Eqs. 47,45 Ptx P tx β t i-r i α N 0 + t i -r i by Eq β α Ptx P tx β t i-r i α N 0 5 The last iequality is due to the fact that whe N 0 0, β P tx Ptx β t i r i N 0 ti r i 52 By Lemma, settig γ = + β α α, we obtai: N 0 + P tx t j r i < +β α α β = +β α α 53 P tx t i r i 3: Suppose N 0+P tx t j t i < β for some j S. This is show i a similar way as Case 2. 4: Suppose for some j S that I additio, we assume < β 54 N 0 + P tx t i r j P tx t j r j β 55 N 0 + P tx t i r j Otherwise, it reduces to Case by iter-chig i ad j. Hece, we obtai: P tx Ptx t i t j < β t j r j α N 0 t j r j < t i r i Next, we obtai: Ptx P tx β t i r i α N 0 56 t j r i t j r j + r j t i + t i r i < t j -r j + Ptx P tx β t - j-r j -α α N 0 + t i -r i by Eq. 55 < + 2β α Ptx P tx β t i-r i α N 0 by Eqs. 56,52 57 By Lemma, settig γ = 2 + β α α, we obtai: N 0 + P tx t j r i < β + 2β α α = 2 + β α α Therefore, S {i} / U. This proves Eq. 46. Lemma 6: If β 2 + β α α, the U B 58 U 59 Proof: Suppose S B. We eed to show S {i} / B S {i} / U X, 2 + β α α 60 First, we assume ad P tx t j r j N 0+P tx distj,i Ptx tj rj P tx distj,i P tx t i r i N 0+P tx disti,j Ptx ti ri P tx disti,j β 6 β t j r j β α distj, i β 62 β t i r i β α disti, j Otherwise, we complete the proof by Lemma 5, such that S {i} / U X, 2 + β α α 63 Next, we obtai: t j r i t j r j + r j r i β α r j r i + r j r i by Eq. 62 t j r i + β α r j r i t j r i t i r i + t j t i t j r i + β α t j t i by Eq. 6 t j r i t j r j + r j t i + t i r i t j r i + 2β α r j t i by Eqs. 6,62 64 Therefore, t j r i < + 2β α disti, j 65 Also, sice S {i} / B, we obtai: P tx t i r i N 0+ P tx disti,j j S\{i} j S\{i} j S\{i} < β disti, j -α > +2β - α α t j -r i -α > P tx Ptx P tx Ptx β t i-r i -α -N 0 β t i-r i -α -N 0 by Eq. 65 By Lemma, settig γ = 2 + β α α, we obtai: 66 N 0 + j i P tx t j r i α < β + 2β α α = 2 + β α α 67 Fially, we complete the proof by combiig with Lemma 5. Lemma 7: If r tx = max t i r i ad r cs r xcl + 2r tx, i X C X, r xcl B fr X, rxcl, r tx C X, r cs 68

14 4 Proof: It ca be prove i a similar fashio as Lemma 3, where we replace costrait t j r i r xcl by t j t i r cs. Lemma 8: C X, ν is a reversible Markov process. The statioary distributio for each S C X is: exp i S P ν S = log ν i S C X exp j S log ν 69 j Proof: Eq. 69 satisfies the detailed balaced eq: exp j {i} S log ν j = exp j S log ν j exp log νi P ν {i} S= P ν S ν i Hece, C X, ν is a reversible Markov process, Eq. 69 is the statioary distributio. Theorem 4: There exists a suitable settig of rcs, B rcs, P depedig o β ad the maximum trasmissio distace i X c, such that B haf X B, β, X P, β C haf X B, r B cs, X P, r P cs 70 Proof: First, we ote that B haf X B, β, X P, β { = c {B,P} Sc S c B X c, β } { } 7 S i, j S, {t i, r i } {t j, r j } = By Theorem, for each c {B, P}, there exists a suitable rcs, c depedig o β ad the maximum trasmissio distace i X c such that B X c, β C X c, rcs c 72 Also, it follows that { c {B,P} Sc S c C X c, r c cs } C haf X B, rcs, B X P, rcs 73 P Next, we eed to show there exists suitable r c cs, such that { } S i, j S, {t i, r i } {t j, r j } = C haf X B, r B cs, X P, r P cs 74 If i X c ad j X c, the Eq. 74 follows from the tree diram Fig. 4 that there exists a suitable r c cs, such that X c, c C X c, r c cs 75 B B for ay c > 0. Else if i X c ad j / X c, the without loss of geerality, we cosider i X c ad j X c, ad r c cs > r c cs. The, Eq. 74 follows from the fact that there exists a suitable r c cs, such that X c {j}, c C X c {j}, r c cs C X c, r c cs 76 for ay c > 0. Fially, we take the maximum carrier-sesig re amog rcs c ad r c cs, for each c {P, B}. Hece, we complete the proof. Chi-Ki Chau is a Seior Research Fellow at the Istitute for Ifocomm Research I 2 R, Sigapore. He is also affiliated with Uiversity of Cambridge as a primary researcher for the Iteratioal Techology Alliace i Network ad Iformatio Sciece ITA program. He was awarded a Croucher Foudatio Research Fellowship at the EE Dept., Uiversity College Lodo. He has bee a regular visitig scholar ivited to IBM T. J. Watso Research Ceter Hawthore, US, BBN Techologies Bosto, US, Uiversity of Massachusetts Amherst, US, Shhai Research Ceter for Wireless Commuicatios Chiese Academy of Scieces, ad the Chiese Uiversity of Hog Kog. He received the Ph.D. from Uiversity of Cambridge supported by a Croucher Foudatio scholarship, ad the B. Eg. First-class Hoors from the Dept. of Iformatio Egieerig, the Chiese Uiversity of Hog Kog. His research iterests cocer diverse areas of etworkig, commuicatios, ad radomized algorithms. Mighua Che received his B.Eg. ad M.S. degrees from the EE dept. at Tsighua Uiversity i 999 ad 200, respectively. He received his Ph.D. degree from the EECS dept. at UC Berkeley i He spet oe year visitig Microsoft Research Redmod as a Postdoc Researcher. He joied the Dept. of Iformatio Egieerig, the Chiese Uiversity of Hog Kog, i 2007 as a Assistat Professor. He received the Eli Jury award from UC Berkeley i 2007, the ICME Best Paper Award i 2009, ad the IEEE Trasactios o Multimedia Prize Paper Award i His curret research iterests iclude complex systems, distributed ad stochastic etwork optimizatio ad cotrol, peer-to-peer etworkig, wireless etworkig, ad etwork codig. Soug Ch Liew received his S.B., S.M., E.E., ad Ph.D. degrees from MIT. From March 988 to July 993, Soug was at Bellcore ow Telcordia, New Jersey, where he eged i Broadbad Network Research. Soug has bee Professor at the Dept. of Iformatio Egieerig, the Chiese Uiversity of Hog Kog, sice 993. Soug s primary research area is wireless etworkig. Soug ad his studet wo the best paper awards i IEEE MASS 2004 ad IEEE WLN Separately, TCP Veo, a versio of TCP to improve its performace over wireless etworks proposed by Soug ad his studet, has bee icorporated ito a recet release of Liux OS. I additio, Soug iitiated ad built the first iter-uiversity ATM etwork testbed i Hog Kog i 993. More recetly, Soug s research group pioeers the cocept of Physicallayer Network Codig PNC for applicatio i wireless etworks. Besides academic activities, Soug is also active i the idustry. He co-fouded two techology start-ups i Iteret Software ad has bee servig as cosultat to may compaies ad idustrial orgaizatios. He is curretly cosultat for the Hog Kog Applied Sciece ad Techology Research Istitute ASTRI, providig techical advice as well as helpig to formulate R&D directios ad strategies i the areas of Wireless Iteretworkig, Applicatios, ad Services. Soug is the holder of four U.S. patets ad Fellow of IEE ad HKIE. Publicatios of Soug ca be foud i