Capacity of Large-scale CSMA Wireless Networks

Transcription

1 Capacity of Large-scale CSMA Wireless Networks Chi-Ki Chau Computer Laboratory, Uiversity of Cambridge; Dept. of E. & E. Egg., Uiversity College Lodo Mighua Che Dept. of Iformatio Egg., The Chiese Uiversity of Hog Kog Soug Chag Liew Dept. of Iformatio Egg., The Chiese Uiversity of Hog Kog 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 (carrier-sesig 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 practical implemetatio issues of our capacity-optimal CSMA scheme. Categories ad Subjects: C.2. [Computer-Commuicatio Networks: Wireless commuicatio, G.3 [Probability Ad Statistics: Stochastic processes Geeral Terms: Theory, Desig, Performace Keywords: Network Capacity, Achievable Throughput, Carrier-Sesig Multi-Access (CSMA), Radom Networks. INTRODUCTION A importat characteristic that distiguishes wireless etworks from wired etworks is the presece of spatial i- Permissio to make digital or hard copies of all or part of this work for persoal or classroom use is grated without fee provided that copies are ot made or distributed for profit or commercial advatage ad that copies bear this otice ad the full citatio o the first page. To copy otherwise, to republish, to post o servers or to redistribute to lists, requires prior specific permissio ad/or a fee. MobiCom 09, September 20 25, 2009, Beijig, Chia. Copyright 2009 ACM /09/09...$5.00. terferece, wherei the trasmissio betwee a pair of odes ca upset other trasmissios i its eighborhood. Such spatial iterferece imposes a limit o the capacity of wireless etworks. The semial paper [0 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, o matter how optimal schedulig ad routig schemes are chose. May similar upper bouds are derived for more sophisticated settigs (e.g., with optimal source ad etwork codig schemes [7). I [5, Dai ad Lee derive the upper boud O( ) for multi-hop radom access etworks by usig a simple queuig aalytical argumet. They also show that this upper boud is achievable oly if the maximum throughput of each local ode is a costat idepedet of. Sice the, a umber of solutios have bee proposed to achieve the upper bouds i various settigs. Particularly, [7 showed that by a efficiet backboe-peripheral routig scheme (aalogously called highway system ) ad a two-stage 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. Meawhile, o the practical frot, carrier-sesig multi-access (CSMA) etworks (e.g., Wi-Fi), which make use of distributed radom-access medium-access protocols, are receivig wide adoptio across eterprises ad homes. It is ot clear, however, whether the results related to cetrally-scheduled 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 otrivial ad several mechaisms must be i place before scalability ca be attaied. For example, the use of dual carriersesig power thresholds 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 si-

2 multaeously. 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-sesig ad dual chael scheme is i place. Our capacity-optimal CSMA scheme ot oly demostrates the theoretical achievable throughput of CSMA etworks, but also provides a implemetable solutio by more practical distributed CSMA protocols. Because of limited space, the proofs of lemmas are deferred to the full techical report [4. 2. 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. Eve if the chael is sesed to be suitable for trasmissio, the trasmitter eeds to rely o a radom collisio avoidace mechaism, i which the trasmitter iitializes a radom cout-dow period before trasmissio. The cout-dow will be froze whe chael is sesed to be ot iterferece-safe (i.e., trasmissio is collisioproe), ad will be resumed whe the chael is sesed to be iterferece-safe agai. 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, such that 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 [7, 0, 3, 7, however, did ot cosider bidirectioal 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 com- A schedule is a sequece of feasible states that are active at differet times. patible with other simultaeous trasmissio uder various iterferece settigs 2. Such a distributed cotrol requires oly limited a-priori coordiatio amog trasmitters ad receivers (ulike the cetralized TDMA schemes). Despite the popularity of CSMA protocols, capacity aalyses 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. There are other variats of protocol models i the literature [ that also model iterferece amog simultaeous liks, rather tha distributed multi-access protocols. 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. These are the key questios to be addressed i this paper. There is cosiderably large body of literature about siglehop CSMA etworks [, 4, 8, whereas here we study the more geeral multi-hop CSMA etworks, the results of which are quite limited i the literature [9, 5. 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 differet from the covetioal Gupta-Kumar s oe [7, 0, 3, 7, because [3 does ot require full coectivity betwee every pair of source ad destiatio. 2. Outlie of Our Results To explore the capacity of CSMA etworks, we first formulate the models of CSMA protocols that 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 [4 ad [2. Fially we tue the CSMA parameters so that the capacity of a CSMA etwork is maximized. More specifically, our approach is divided ito four parts:. Formulatio of Carrier-sesig Decisio Model (Sec. 3): 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 simultaeous liks imposed by CSMA carrier sesig operatios, such as power-thresholdbased carrier sesig. We explicitly distiguish the decisio model of CSMA protocols from the 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 [0 appears to be the first paper to coi the phrase protocol model, but without specifyig ay distributed protocol that actually implemets the protocol model, other tha cetralized schemes by TDMA.

3 Ui-directioal feasible family Bi-directioal feasible family Carrier-sesig feasible family Pairwise Aggregate Pairwise Aggregate Pairwise Aggregate iterferece iterferece iterferece iterferece carrier-sesig carrier-sesig Radom Upper boud: O( ) [0 Upper boud: O( ) (this paper) Upper boud: O( ) (this paper) etwork Achievable as: Ω( ) Achievable as: Ω( ) Achievable as: Ω( ) capacity by TDMA [7 by TDMA (this paper) by dual carrier sesig (this paper) Table : Capacity of uiform radom etworks over various feasible families. iterferece model. For istace, two simultaeous liks are allowed by CSMA does ot ecessarily mea 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 chael access mechaism of CSMA, as compared to cetralized TDMA scheme. We establish the relatioship betwee our CSMA models ad the existig iterferece models from the literature i Sec Hidde-ode-free Desig of CSMA Networks (Sec. 4): 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 local distributed cotrols of trasmissios i CSMA may ot be able to prevet spatial iterferece, as kow as the hidde ode problem [. 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 our carrier-sesig decisio models to elimiate the hidde ode problem uder various iterferece settigs (Theorem ). Our defiitio ad sufficiet coditios iclude the prior result i [ 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. 5): 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 already bee used i various aalyses [2, 4. 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 cotiuous-time Markov chai model. Followig the same procedure as i [2, 4, 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. 5. 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. 6-7): 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 [7, based o which we show two differet carriersesig power thresholds are sufficiet to achieve optimal capacity of Ω( ) o a uiform radom etwork with high probability (show by Theorem 3). Not oly have we provided isights i this paper for the optimal asymptotic capacity of wireless etworks by our dual carrier-sesig scheme, but also addressed practical issues of implemetig our scheme. First, we address the scalability issue durig the dyamic switchig betwee the dual carrier-sesig 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 dual chael scheme. We summarize our results ad related work i Table. 3. FORMULATION AND MODELS First, ote that some key otatios are listed i Table 2. A cetral problem of multi-hop wireless commuicatios is defied as follows. Give a set of source-destiatio 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 destiatios 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:. Routig scheme that selects the appropriate relayig odes to coect the sources ad destiatios. 2. Schedulig scheme that assigs (determiistically or radomly) the opportuities of trasmissios at relayig odes.

4 Table 2: Key Notatios Notatio Defiitio N sd set of source-destiatio pairs. λ k data rate of source-destiatio pair k N sd. X set of relayig liks iduced by the paths betwee all source-destiatio 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 rage, used i fixed rage iterferece models. r tx commuicatio rage, used i fixed rage iterferece models. r cs carrier sesig rage, used i pairwise CSMA decisio models. t cx carrier sesig power threshold, used i aggregate CSMA decisio models. Furthermore, these wireless commuicatios should be established i a distributed maer with miimal global kowledge ad coordiatio amog the odes. To shed light o this problem, 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. 3. 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. a.0) Pairwise fixed-rage feasible family: S U pw [ 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 pw [ sir 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 pw sir[ X, β, if ad oly if for all i, j S, P tx t i r i α β (3) N 0 + P tx t j r i α a.3) Aggregate SINR feasible family: S U ag sir[ X, β, if ad oly if for all i S, P tx t i r i α N 0 + β (4) P tx t j r i α j S\{i} We assume r xcl > r tx, > 0, α > 2, β > 0, ad uiform power P tx at all odes. For a.2)-a.3), we assume P tx t i r i α βn 0 for all i X. Otherwise, t i caot successfully trasmit packets to r i eve without iterferece from other odes. The otio of feasible family geeralizes the otio of coflict graph implicit i a.0)-a.2), i which a feasible state is a idepedet set of the coflict graph arisig from the model. a.0)-a.2) ca serve as approximatios to the more realistic iterferece-safe costrait a.3), whe path-loss expoet α is large ad backgroud oise N 0 is small. I [0, pairwise SIR iterferece model a.) is called protocol model, whereas aggregate 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 Bi-directioal Iterferece Models The iterferece-safe costraits a.0)-a.3) are ui-directioal, 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. DATA DATA DATA t i r i (a) r j t j Figure : I Fig. (a) the ormal DATA packet trasmissios from trasmitters will ot iterfere with each other, but i Fig. (b) the ACK packet trasmissios from receivers will iterfere. Let dist(i, 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-rage feasible family: S B pw [ fr X, rxcl, r tx, if ad oly if for all i, j S, ACK (b) DATA dist(i, j) r xcl ad t i r i r tx (5) b.) Bi-directioal pairwise SIR feasible family: S B pw sir[ X,, if ad oly if for all i, j S, dist(i, j) ( + ) t i r i (6) b.2) Bi-directioal pairwise SINR feasible family: 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. ACK

5 S B pw sir[ X, β, if ad oly if for all i, j S, P tx t i r i α N 0 + P tx ( dist(i, j) ) α β (7) b.3) Bi-directioal aggregate SINR feasible family: S B ag sir[ X, β, if ad oly if for all i S, N 0 + P tx t i r i α ( ) α β (8) P tx dist(i, j) j S\{i} Compared with the ui-directio iterferece-safe costraits, the bi-directioal couterparts cosider the iterferece effect from both trasmitters ad receivers. 3.3 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. Two useful feasible families to capture commo carrier-sesig operatios are as follows. c.) Pairwise carrier-sesig feasible family: S C pw[ X, r cs, if ad oly if for all i, j S, t j t i r cs (9) c.2) Aggregate carrier-sesig feasible family: S C ag[ X, t cs, if ad oly if for all i S, N 0 + j SP tx t j t i α t cs (0) Aggregate carrier-sesig decisio model c.2) captures powerthreshold based carrier sesig, where a trasmitter decides its trasmissios based o the chael sesig result. Pairwise carrier-sesig decisio model c.) ca serve as a approximatio to c.2), whe path-loss expoet α is large ad backgroud oise N 0 is small. Alteratively, c.) ca be carrier sesig based o had-shakig messages, i which r cs ca be iterpreted as the coverage area of a iteded trasmitter, such that withi the distace r cs the had-shakig messages (e.g., RTS, DATA) trasmissios from this trasmitter ca be successfully sesed by other eighborig trasmitters i its coverage, deterrig them from trasmissios. 4. 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 carriersesig decisio models. Here, we provide formal defiitios to hidde-ode ad exposed-ode problems based o the models i Sec.3. 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 iterferecesafe 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 [, which cosiders oly pairwise iterferece ad carrier-sesig 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. DATA DATA DATA t i r i (a) r j t j Figure 2: I Fig. (a) the carrier-sesig decisio model correctly permits the simultaeous liks for DATA packets, but fails i the case of ACK trasmissio i Fig. (b). Hece, B pw [ [ sir X C pw X As studied i [, 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 tractable aalytically. For example, the crucial Eq. (23) of CSMA statioary states to be preseted i Sec. 5 is valid oly for a CSMA etwork that is hidde-ode free. Oe of our cotributios is to establish a comprehesive set of sufficiet coditios to elimiate hidde-ode problem i various iterferece ad carrier-sesig decisio models. The basic idea is to establish a set of subset-relatioships amog all the feasible families of iterferece ad carriersesig decisio models i the followig. Hece, uder some suitable settigs, a carrier-sesig feasible family ca be cofigured as a subset to a give iterferece-safe feasible family, thus elimiatig the hidde-ode problem altogether. Lemma. If β α, the U pw pw ag sir [X, Usir [X, β Usir [X, β () (b) ACK

6 Lemma 2. Let r tx = max i X t i r i. If ( r xcl P txk(α) ( Ptx β r α tx N 0 ) ) α + r tx (2) where k(α) i= 2 π(2i + 2) i α, the pw[ [X, β U X, rxcl, r tx U ag sir fr (3) Note that k(α) coverges rapidly to fiite costat 26, whe α > 2. See Fig. 3.(a) for a plot of the umerical values of k(α). We remark that that [8 cosiders the simpler aggregate oise-abset SIR model. Because of the absece of oise, usig a tighter packig lattice [8 yields a tighter costat k(α). Lemma 3. If r xcl r xcl + 2r tx, the U pw [ fr X, rxcl, r tx B pw[ fr X, rxcl, r tx U pw[ fr X, r xcl, r tx (4) Lemma 4. If + 2, the U pw [ sir X, B pw[ sir X, U pw[ sir X, (5) Lemma 5. If β (2 + β α ) α, the U pw [ sir X, β B pw [ sir X, β U pw sir[ X, β (6) Lemma 6. If β (2 + β α ) α, the U ag [ sir X, β B ag [ sir X, β U ag sir[ X, β (7) Lemma 7. If r tx = max i X ti ri ad rcs r xcl + 2r tx, C pw[ X, r xcl B pw[ fr X, rxcl, r tx C pw [ X, r cs Lemma 8. If t cs N 0 + P tx rcs α, the C pw[ X, r cs C ag [ X, t cs Lemma 9. If r cs ( P tx k(α) ( tcs N 0 )) α, the C ag[ X, t cs C pw [ X, r cs (8) (9) (20) 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 aggregate iterferece models with/without oise. See Fig. 3.(b) for a plot of the umerical values of (2 + β α ) α Α k Α (a) Β 6 8 (b) Figure 3: Fig. (a): Numerical values of k(α), which coverges rapidly to fiite costat 26 whe α > 2. Fig. (b): Numerical values of (2 + β α ) α Α 4 5 U pw Lemma 4 sir Lemma B pw Lemma 4 sir U pw sir U pw Lemma 5 sir B pw Lemma 5 sir U pw sir Lemma U ag Lemma 6 sir B ag Lemma 6 sir U ag sir Lemma 2 U pw Lemma 3 fr B pw Lemma 7 fr C pwlemma 8 C aglemma 9 C pwlemma 7 B pw fr Figure 4: The tree diagram represets the subsetrelatioships for the iterferece ad carrier-sesig decisio models. 4. Hidde-ode-free Sufficiet Coditios Lemmas -8 establish a tree diagram Fig. 4 of subsetrelatioships for the iterferece ad carrier-sesig decisio models, uder the respective sufficiet coditios. The tree diagram 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 ay carrier-sesig feasible family C [ X from c.)-c.2), we start at B [ X i the tree diagram, ad follow the respective chais of lemmas to set the respective sufficiet coditios util reachig C [ X. The, we ca obtai a hidde-odefree 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 ay carrier-sesig feasible family C [ X from c.)- c.2), there exists a suitable settig of r cs or t cs such that (Hidde-ode-free Desig) : B [ X C [ X (2) We summarize the sufficiet coditios for hidde-odefree CSMA etwork desig i Table 3. 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 [6, uless the carrier sesig rage is large eough ad a umber of other coditios are met [. 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 [ 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 aggregate iterferece model. Moreover, Theorem covers hidde-ode free operatio of the basic mode uder the aggregate iterferece model, which was ot treated i [. Lemma 3 U pw fr 5. STATIONARY THROUGHPUT ANALYSIS While Sec. 3-4 address the distributed ad ACK-based ature of CSMA, this sectio addresses the characteristics of

7 Pairwise carrier-sesig feasible family C pw[ X, r cs Aggregate carrier-sesig feasible family C ag[ X, t cs Bi-directioal feasible family pairwise [ fixed rage pairwise [ SIR pairwise [ SINR aggregate [ SINR X, rxcl, r tx X, X, β X, β B pw fr r tx = max ti ri i X r cs r xcl + 2r tx B pw sir r tx = max ti ri i X r cs (3 + )r tx [ B pw sir r tx = max t i r i ( i X P r cs tx ( P tx r α (2+β α ) α tx ) ) α N 0 + 3r tx B ag sir r tx = max t i r i ( i X r cs P tx k(α) ( P tx r α (2+β α ) α tx ) ) α N 0 + 3r tx Let r cs be a carrier-sesig rage satisfyig the above correspodig coditios, t cs N 0 + P tx rcs α Table 3: Sufficiet coditios for hidde-ode-free CSMA etwork desig. Results are derived i this paper uless cited otherwise. radom access i CSMA, ad study its achievable capacity as compared to TDMA schemes. 5. Determiistic Schedulig Cosider a give routig scheme ad a carrier-sesig feasible family C [ X (that is set to be hidde-ode-free by Theorem ). 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) (22) Recall that λ k is the date rate of source-destiatio pair k N sd. Together with the routig scheme, oe ca determie the feasible regio for (λ k ) k N sd by solvig a multicommodity flow problem. 5.2 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 [6,2,4, 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 5. The average 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 5 A mai result upo which the results of this paper is built is the statioary probability distributio i Eq. (23). It turs out that Eq. (23) is isesitive to the distributios of the cout-dow ad trasmissio times. That is, Eq. (23) is still the statioary probability distributio uder geeral trasmissio-time distributio ad geeral cout-dow time distributio with memory. See the full techical report [4 for further discussio. t= for some i X. Suppose S C [ X ad ({i} S C [ X or {i} S C [ X ). Trasitio S {i} S represets that the trasmitter of lik i will start to trasmit, after some radom cout-dow time. Trasitio {i} S S represets that the trasmitter of lik i will fiish trasmissio, after some radom trasmissio time. Suppose the curret state of simultaeous trasmissios is S, ad trasmitter i is coutig dow to trasmissio. Because of carrier-sesig, 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 ). i will resume cout-dow util 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 0. C [ X, ν is a reversible Markov process, with statioary distributio for each S C [ X as: exp ( i S P ν(s) = log ) νi S C [X exp ( ) (23) j S log ν j Lemma 0 is well-kow i prior work [6, 2, 4. 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-backoffrate CSMA radom access is: c rad i [ C [ X, ν S C [X:i S P ν(s) (24) We ca relate the throughput of a TDMA scheme with the log-term throughput of idealized multi-backoff-rate CSMA radom access by the followig result. Lemma. 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= (St = S). If Pdet (S) > 0 for all S C [X, the there exists cout-dow rates ν, such that for each lik i X, it satisfies: c det i [ (St ) m t= c rad[ [ C X, ν i (25)

8 Lemma was origially prove i [2. I the full techical report [4, we also give a simplified alterate proof. The implicatio of Lemma 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 of feasible states. Lemma will be useful to show the existece of the achievable capacity of multi-backoff-rate CSMA etworks, if we kow the achievable capacity of the correspodig TDMA scheme with the same feasible family C [X. 6. CAPACITY OF RANDOM NETWORK I this sectio, we apply the results from Sec. 3-5 to the capacity aalysis o a uiform radom etwork. The ramificatio of selectig a uiform radom etwork is to provide the simplest average-case aalysis, without ivolvig other complicated radom etwork topology. We cosider a Poisso poit process 6 of uit desity o a square plae [0, [0,. Every ode o the plae is a source or a destiatio that is selected uiform-radomly amog all the odes o the plae. We ext defie some otatios: N sd deotes the radom set of source-destiatio 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. 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-destiatio pairs i N sd, achieved by the most optimal determiistic schedulig scheme: λ ( F [ X R ) ) max mi λ k (26) (S t ) m t= S (F[X R ) ( k N sd 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 Θ(f()) 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 } < (27) This is the covetioal defiitio of radom wireless etwork capacity [7, 0, 3, Upper Boud for Sigle Carrier Sesig We first show that carrier-sesig based o c.)-c.2) caot achieve the optimal capacity Ω ( ). 6 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. 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-destiatio pairs i N sd, λ ( C [ X R ) ( ) = O (w.h.p.) (28) log Proof. By Lemmas 3,7-8, there exists a suitable r xcl, such that C [ X R ca be cofigured as a subset of U pw[ fr X R, r xcl, r tx. It has bee show i [0 that λ ( U pw [ fr X R ) ( ), r xcl, r tx = O (w.h.p.) (29) log for ay routig scheme R that coects all the source-destiatio pairs i N sd. Hece, it completes the proof. Noetheless, [7, 7 shows 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. 6.2 Backboe-Peripheral Routig We briefly revisit the efficiet routig scheme i [7 (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-rage liks, whereas every peripheral ode ca reach a backboe ode i oe-hop trasmissio. The basic idea is to use shortrage backboe-backboe liks wheever possible. Sice short-rage 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. Cosiderig the cells as vertices, a path ca be formed by coectig adjacet o-empty cells. Lemma 2. (See [7) 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 [7) for routig o a uiform radom etwork as follows. Select a represetative ode i each o-empty cell. By Lemma 2, 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 multi-hop Mahatta-routig alog the adjacet backboe odes to the respective backboe ode that ca trasmit the packets to the destiatio i a sigle last hop. See Fig. 5 for a illustratio of backboe-peripheral routig.

9 /s chaels that oly icurs small overhead at low implemetatio cost. 7. Simple Desigs of Dual Carrier-Sesig To illustrate simple desigs of dual carrier-sesig, we cosider pairwise carrier-sesig decisio model i Fig. 6. The basic idea is that we allow a shorter carrier-sesig rage to be used amog backboe-backboe liks. A loger carrier-sesig rage will be used whe there is a backboeperipheral lik i the eighborhood. /s Figure 5: Backboe odes are a subset of coected odes by short-rage liks, whereas peripheral odes relay all the packets to backboe odes. (b) We defie a schedulig scheme uder backboe-peripheral routig cosistig of two stages: (St P ) c 4 log 2 t= ad (St B ) c 5 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 Θ(log 2 ) (because the backboe-peripheral distace is O(log )). It is show i [7 that we ca always pick a oiterferig lik i each cell to trasmit i every timeslot (t mod c 4 log 2 ) i the first stage, for some costat c 4. The throughput rate for each backboeperipheral lik ca be show to be Θ( ) Θ( log 2 ). 2. (Backboe-backboe Trasmissios): If i St B, 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-stage schedulig scheme. 7. DUAL CARRIER-SENSING To adopt the TDMA scheme of backboe-peripheral routig i Sec. 6.2 for CSMA etworks, i Sec 7. we employ dual carrier-sesig where multiple carrier-sesig rages (or power thresholds) are allowed. Namely, smaller carriersesig rages (or larger power thresholds) ca be used amog the short-rage liks. This effectively eables more simultaeous liks ad improves the throughput. However, simple desigs of multi-carrier sesig may ot be scalable ad eve implemetable. I Sec 7.2, we discuss some practical issues of implemetig multi-carrier sesig. We preset a careful desig of dual carrier-sesig usig two (a) Figure 6: There are two carrier sesig rages as i Fig. (a). I Fig. (b) short-rage backboebackboe liks will use a shorter carrier-sesig rage amog themselves, while i Fig. (c) loger carrier-sesig rage will used whe there is ay backboe-peripheral lik. Formally, we partitio the set of liks X ito two disjoit classes: X B for backboe-backboe liks, ad X P for backboe-peripheral liks. Assume r B cs < r P cs ad t B cs > t P cs. Two simple dual carrier-sesig desigs by extedig c.)- c.2) are: (c) d.) Pairwise dual carrier-sesig feasible family: S C pw mcs[ (X B, r B cs), (X P, r P cs), if ad oly if for all i, j S, t j t i max{r c cs, r c cs}, (30) where i X c, j X c ad c, c {B, P}. d.2) Aggregate dual carrier-sesig feasible family: S Cmcs[ ag (X B, rcs), B (X P, rcs) P, if ad oly if for all i S, N 0 + j SP tx t j t i α { mi t c } cs (3) c {B,P}:X c S That is, there will be a dyamic switchig process of carriersesig rages (or power thresholds), depedet o the presece of classes of active liks. Theorem 3. Cosider dual carrier-sesig decisio 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 backboe-peripheral liks, respectively. Usig multi-backoff-rate radom access scheme, there exists a suitable settig of (rcs, B rcs), P such that λ ( Cmcs[ pw (X B, rcs), B (X P, rcs) P ) ( ) = Ω (w.h.p.) (32)

10 Proof. Recall that (S P t ) c 4 log 2 t= ad (S B t ) c 5 t= are the two stage TDMA schemes i backboe-peripheral routig. Note that each S P t ad S B t is a feasible state i a ui-directioal pairwise SIR iterferece model. We set r B cs = c 6 ad r P cs = c 7 log for some costats c 6, c 7. The accordig to Theorem, we ca obtai some costats c 6, c 7, such that S P t C pw mcs[ (X B, r B cs), (X P, r P cs) for all t =...c 4 log 2 (33) S B t C pw mcs[ (X B, r B cs), (X P, r P cs) for all t =...c 5 (34) Next, we employ Lemma to establish a lower boud of the throughput of radom access o Cmcs[ pw (X B, rcs), B (X P, rcs) P, by the throughput of a correspodig determiistic schedulig scheme as follows: For each S {St P } c 4 log 2 t=, we set P det (S) = Θ( For each S {S B t } c 5 t=, we set P det (S) = Θ() ) log 2 For other S, we equally divide the time amog them, such that P det (S) = Θ() S Cmcs [(X pw B,rcs B ),(XP,rcs P ) \{St P}c 4 log2 t= {St B}c 5 t= (35) Therefore, this satisfies the sufficiet coditio i Lemma that P det (S) > 0 for all S Cmcs[ pw (X B, rcs), B (X P, rcs) P. 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. [ Note that the aggregate model Cmcs ag (X B, rcs), B (X P, rcs) P ca be show i a similar fashio. 7.2 Dual Chael Dual Carrier-Sesig Although the simple dual carrier-sesig decisio models d.)-d.2) ca achieve optimal capacity, they suffer from some implemetatio issues. First, trasmitters are required to kow the classes of all active liks, durig the dyamic switchig betwee the dual carrier-sesig operatios. I pairwise model d.), this ca be achieved by relyig o overhearig the physical preambles like MAC addresses of trasmitters, ad resolvig the respective classes. However, this icurs cosiderably high overhead i dese etworks. Particularly, o a uiform radom etwork, the distace for backboe-peripheral lik is Θ(log ). Thus, the iduced overhead ca scale as large as O(log 2 ). Secod, i aggregate model d.2), such kowledge of all active liks is ot available, because a trasmitter ca oly perceive a aggregate power level from all the active liks, ad is uable to resolve the power levels of idividual class. To address the above implemetatio issues of dual carriersesig, we are motivated to adopt a system with two frequecy chaels, i which the commuicatios o the backboebackboe liks are carried out o oe frequecy chael, while the commuicatios o the peripheral liks are carried out o the other chael. That is, the liks i class X B will use oe frequecy chael with carrier-sesig power threshold as t B cs, while the liks i class X P will use the other chael with a threshold t P cs. Sice covetioal CSMA protocols (e.g., IEEE 802.) ofte support more tha two chaels, our scheme ca be coveietly implemeted o these CSMA protocols. First, Sec cosiders a system that is full-duplex across the two frequecy chaels. The, Sec cosiders a system that is half-duplex across the two frequecy chaels that is simpler to implemet, but whose coditios for hidde-ode free operatio are more subtle 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. With this 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. (I) Carrier-Sesig: The followig is the correspodig carrier-sesig decisio model: d.3) Full-duplex dual chael dual aggregate carrier-sesig feasible family: S Dful[ ag (X B, t B cs), (X P, t P cs), if ad oly if for all i S, N 0 + P tx t j t i α t c cs (36) j X c S where i X c ad c {B, P}. I essece, 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. (II) Throughput: Oe ca easily show that carrier-sesig model d.3) ca achieve throughput as Ω ( ) by relaxig Theorem 3 o two idepedet frequecy chaels 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. However, we disallow simultaeous trasmissio ad receptio, whether o the same chael or differet chaels. Specifically, we itroduce the followig costraits: (i) a ode caot trasmit ad receive (eve o differet frequecy chaels) simultaeously; ad (ii) a ode ca oly trasmit o at most oe frequecy chael at ay time.

11 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 leakage 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 half-duplexity whe the trasmit ad receive frequecy chaels are the same, but the uderlyig ratioale ad priciples are the same whe the crossfrequecy leakage is ot egligible. Costrait (ii) is maily due to the fact that i ACKbased 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). (I) Carrier-Sesig: Let us ow cosider the implicatio of costraits (i) ad (ii) for the carrier sesig operatio. The mathematical descriptio of a carrier-sesig decisio model that takes care of the costraits is as follows: d.4) Dual chael dual aggregate carrier-sesig feasible family: S Dhaf[ ag (X B, t B cs), (X P, t P cs), if ad oly if for all i S ad all c {B, P}, N 0 + P tx t j t i α mi{t c cs, t c cs} (37) j X c S where i X c ad c {B, P}. To uderstad the above, let us first cosider the case of a backboe-peripheral lik (i.e., c = P). I this case, either a peripheral ode desires to trasmit to a backboe ode, or a backboe ode desires to trasmit to a peripheral ode. The trasmissio caot be allowed if the power sesed is larger tha the carrier-sesig threshold of the peripheral chael. It is obvious as to why this should apply to cumulative power sesed o the peripheral chael. But accordig to the above iequality, the cumulative power sesed o the backboe chael (i.e., for the case where c = B o the lefthad side of the above) should ot exceed the peripheralchael threshold either (Note: o the right-had side of the above, the threshold of the peripheral chael, t c cs, is smaller tha the threshold of the backboe chael, t c cs ). The reaso for this requiremet is costraits (i) ad (ii). Cosider the followig, 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 the above iequality, to make our aalysis simpler ad cleaer, we simply set the threshold to that of the peripheral chael. This does ot chage the order of results. I practice, further implemetatio optimizatio is possible (skipped here due to limited space). Next, we cosider the case of a backboe-backboe lik (i.e., c = B). I this case, a backboe ode wats to trasmit to aother backboe ode. The trasmissio is ot allowed if the power sesed o the backboe chael is larger tha the backboe threshold, or if the power sesed o the peripheral chael is larger tha the peripheral threshold. The former is obvious. The latter 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 threshold to the peripheral threshold, sice the order results we wat to establish are ot compromised. (II) Throughput: To show that carrier-sesig model d.4) ca achieve throughput as Ω ( ), we first eed to properly determie t P cs ad t B cs. We have to formally show carriersesig decisio model d.4) ca be implemeted practically, by cosiderig dual chael iterferece models that explicitly icorporate the costrait of half-duplexity across two frequecy chaels. e.) Bi-directioal dual chael aggregate SINR feasible family: S Bhaf[ ag (X B, β B ), (X P, β P ), if ad oly if ) S = S c, where each S c Bsir[ ag X c, β c, 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 (t B cs, t P cs), depedig o (β B, β P ), such that B ag [ haf (X B, β B ), (X P, β P ) Dhaf[ ag (X B, t B cs), (X P, t P cs) (38) Theorem 4 establishes a hidde-ode-free desig for the dual chael 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.4). Lemma 3. For ay c {B, P}, if ( rcs c ( t c ) ) α cs N 0, (39) P tx k(α) the D ag haf [ (X B, t B cs), (X P, t P cs) C pw mcs[ (X B, r B cs), (X P, r P cs) (40) Lemma 3 shows that the feasible states of the dual chael dual aggregate carrier-sesig decisio model d.4) ecompass the feasible states of the pairwise dual carrier carriersesig decisio model d.), with suitably chose carriersesig rage. Therefore, by Lemma 3 ad Theorem 3, we ca show that the implemetable dual chael dual aggregate carrier-sesig decisio model d.4) ca achieve throughput as as Ω ( ) o a uiform radom etwork. Theorem 5. Cosider dual chael dual carrier-sesig decisio model d.4) o a uiform radom etwork based o backboe-peripheral routig. Usig multi-backoff-rate radom access scheme, there exists a suitable settig of (t B cs, t P cs), such that λ ( D ag haf[ (X B, t B cs), (X P, t P cs) ) = Ω ( ) (w.h.p.) (4)