Transport Capacity of Distributed Wireless CSMA Networks

Transcription

1 Trasport Capaity of Distributed Wireless CSMA Networks Tao Yag, Guoqiag Mao, Seior Member, IEEE, Wei Zhag, Seior Member IEEE ad Xiaofeg Tao, Seior Member IEEE arxiv:45.48v [s.ni] 7 May 4 Abstrat I this paper, we study the trasport apaity of large multi-hop wireless CSMA etworks. Differet from previous studies whih rely o the use of etralized shedulig algorithm ad/or etralized routig algorithm to ahieve the optimal apaity salig law, we show that the optimal apaity salig law a be ahieved usig etirely distributed routig ad shedulig algorithms. Speifially, we osider a etwork with odes oissoly distributed with uit itesity o a square B R. Furthermore, eah ode hooses its destiatio radomly ad idepedetly ad trasmits followig a CSMA protool. By resortig to the perolatio theory ad by arefully tuig the three otrollable parameters i CSMA protools, i.e. trasmissio power, arrier-sesig threshold ad out-dow timer, we show that a throughput of Θ is ahievable i distributed CSMA etworks. Furthermore, we derive the preostat preedig the order of the trasport apaity by givig a upper ad a lower boud of the trasport apaity. The tightess of the bouds is validated usig simulatios. Idex Terms Capaity, per-ode throughput, CSMA, wireless etworks I. INTRODUCTION Wireless multi-hop etworks have bee ireasigly used i ivilia ad military appliatios. I a wireless multi-hop etwork, odes ommuiate with eah other via wireless multi-hop paths, ad pakets are forwarded ollaboratively hop-by-hop by itermediate relay odes from soures to their respetive destiatios. Studyig the apaity of these etworks is a importat problem. Capaity of large wireless etworks has bee extesively ivestigated with a partiular fous o the throughput salig laws whe the etwork beomes suffiietly large [] []. Two metris are widely used i the study of etwork apaity: trasport apaity ad trasmissio apaity. The trasport apaity quatifies the ed-to-ed throughput that a be ahieved betwee soure-destiatio pairs whereas the trasmissio apaity, ofte used together with aother metri outage probability, quatifies the ahievable sigle-hop rates i large wireless etworks. The trasport apaity is useful to apture the impat of etwork topology, routig ad T. Yag is with the Shool of Eletrial ad Iformatio Egieerig, The Uiversity of Sydey. tao.yag@sydey.edu.au. G. Mao is with the Shool of Computig ad Commuiatios ad Ceter for Real-time Iformatio Networks, The Uiversity of Tehology, Sydey, ad Natioal ICT Australia. guoqiag.mao@uts.edu.au. W. Zhag is with the Shool of Eletrial Egieerig & Teleommuiatios, The Uiversity of New South Wales. wzhag@ee.usw.edu.au. X. Tao is with Natioal Egieerig Lab. for Mobile Network Seurity, Beijig Uiversity of osts ad Teleommuiatios. taoxf@bupt.edu.. This researh is fuded by ARC Disovery projets: D58 ad D. shedulig algorithms o etwork apaity [] [6], [8] [], [] [4]. Comparatively, the trasmissio apaity is more useful whe the fous is o the impat of physial layer details, e.g., fadig, iterferee ad sigal propagatio model, o the apaity of large etworks [5] [9]. I this paper, we fous o the study of the trasport apaity. I the groud-breakig work [6] by Gupta ad Kumar, it was show that i a stati etwork of odes uiformly ad i.i.d. o a area of uit size ad eah ode is apable of trasmittig at W bits/seod ad usig a fixed ad idetial trasmissio W rage, the ahievable per-ode throughput is Θ log whe eah ode hooses its destiatio radomly ad idepedetly. If odes are optimally ad determiistially plaed to maximize apaity, the ahievable per-ode throughput W beomes Θ. I a more geeral settig, assumig oly that power atteuates with distae followig a power-law relatioship, Xie ad Kumar [] showed that Θ is a upper boud o the per-ode throughput of wireless etworks, regardless of the shedulig ad routig algorithm beig employed. Sie the, a umber of solutios have bee proposed to ahieve the above upper bouds uder various etwork settigs ad usig various routig ad shedulig algorithms [] [6], [8] [], [] [4]. I [4], Fraeshetti et al. osidered the same etwork as that i [6] exept that odes are allowed to use two differet trasmissio rages. They showed that by usig a routig sheme based o the so-alled highway system ad a etralized/determiisti time divisio multiple aess TDMA sheme, the per-ode throughput a reah Θ eve whe odes are radomly loated. Speifially, the highway system is formed by odes usig the smaller trasmissio rage, whereas the larger trasmissio rage is used for the last mile, i.e., betwee the soure or destiatio ad its earest highway ode. The existee of highway system was established usig the perolatio theory. Other work i the field iludes [5] i whih Grossglauser ad Tse showed that i mobile etworks, by leveragig o the odes mobility, a per-ode throughput of Θ a be ahieved at the expese of large delay. Their work [5] has sparked huge iterest i studyig the apaity-delay tradeoffs i mobile etworks assumig various mobility models ad the obtaied results ofte vary greatly with the differet mobility models beig osidered, see [] [5] ad referees therei for examples. I [6], Che et al. studied the apaity of wireless etworks uder a differet traffi distributio. I partiular, they osidered a set of radomly deployed odes trasmittig to a sigle sik or multiple siks where the siks a be either regularly deployed or radomly deployed.

2 They showed that with sigle sik, the trasport apaity is give by Θ W ; with k siks, the trasport apaity is ireased to Θ kw whe k O log or Θ log W whe k Ω log. Furthermore, there is also sigifiat amout of work studyig the impat of ifrastruture odes [7] ad multiple-aess protools [8], [9] o the apaity ad the multiast apaity []. We refer readers to [] for a omprehesive review of related work. The above work of Fraeshetti et al. [4] ad Gupta ad Kumar [6], [], ad most other work i the field [], [], [8] [], [], [], established the apaity of wireless multi-hop etworks usig etralized shedulig ad routig shemes, whih may ot be appropriate for large-sale etworks beig ivestigated i [4], [6], []. I a reet work [], Chau et al. took the lead i studyig the throughput of CSMA etworks. They showed that CSMA etworks a ahieve the per-ode throughput Θ, the same order as etworks usig optimal etralized TDMA, if multiple bak off outdow rates are used i the distributed CSMA protool ad pakets are routed usig the highway system proposed i [4]. While the use of distributed CSMA for shedulig i [] ostitutes a sigifiat advae ompared with the etralized TDMA osidered i previous work, the routig sheme i [] still relies o the highway system, whih eeds etralized oordiatio to idetify the highway odes ad to establish the highway. The etralized routig sheme used i [] is ot ompatible with the distributed CSMA shedulig sheme. I this sese, the routig ad shedulig sheme i [] is ot etirely distributed ad may ot be suitable for large-sale etworks. Furthermore, the deploymet of the highway system i CSMA etworks requires two differet arrier-sesig rages to be used: a smaller arrier-sesig rage used by the highway odes ad a larger arrier-sesig rage used by the remaiig odes to aess the highway. The use of two differet arrier-sesig rages may exaerbate the hidde ode problem i CSMA etworks, whih will be explaied i detail i Setio V. To oquer the potetial hidde problem brought by the use of two differet arriersesig rages, the etire frequey badwidth is divided ito two sub-bads for use by the two types of odes employig differet arrier-sesig rages respetively. This imposes additioal hardware requiremets o the odes ad also auses spetrum waste. Based o the above observatio, we are motivated to develop a distributed shedulig ad routig algorithm to ahieve the order-optimal throughput i CSMA etworks i this paper. Speifially, by resortig to the perolatio theory ad by arefully tuig the three otrollable parameters i CSMA protools, i.e., trasmissio power, arrier-sesig threshold ad out-dow timer, we show that a throughput of Θ is ahievable i distributed CSMA etworks. Furthermore, we aalyze the pre-ostat preedig the order of the trasport apaity by givig a upper ad a lower boud of the trasport apaity. The tightess of the bouds is established usig simulatios. The followig is a detailed summary of our otributios: We develop a distributed routig ad shedulig algorithm that is able to ahieve the order-optimal throughput i CSMA etworks. More speifially, the routig deisio relies o the use of loal eighborhood kowledge oly ad eah ode ompetes for hael aess i a distributed ad radomized maer usig CSMA protools. We demostrate that by joitly tuig the arrier-sesig threshold ad the trasmissio power, the hidde ode problem a be elimiated eve for odes usig differet arrier-sesig thresholds, differet trasmissio powers ad a ommo frequey bad. This is differet from the tehiques used i the previous work [] where odes usig differet arrier-sesig rages have to use differet frequey bad for trasmissio. The tehique developed provides guidae o settig the arrier-sesig threshold ad the trasmissio power to avoid the hidde ode problem i CSMA etworks i a more geeral settig. We aalyze the pre-ostat preedig the order of the trasport apaity by givig a upper ad a lower boud of the trasport apaity. As poited out i [], the preostat is importat to fully uderstad the impat of various parameters o etwork apaity. Extesive simulatios are arried out whih validate the tightess of our aalytial results. The rest of this paper is orgaized as follows. Setio II reviews related work; Setio III presets the etwork model ad defies otatios ad oepts used i the later aalysis; Setio IV desribes the routig algorithm ad aalyzes the traffi load of eah ode; Setio V presets the solutio for obtaiig a hidde ode free CSMA etwork; Setio VI optimizes the medium aess probability for eah ode by tuig the bak off timer ad aalyzes the per-ode throughput uder our proposed ommuiatio strategy; Fially, Setio VII oludes the paper. II. RELATED WORK I additio to the work metioed i Setio I o geeral studies of etwork apaity, i this setio we further review work losely related to researh ad theoretial aalysis i this paper. Limited work exists o aalyzig apaity of large etworks ruig distributed routig ad shedulig algorithms, despite their extesive deploymet i real etworks. Referee [] disussed i Setio I was amog the first work studyig the apaity of etworks employig distributed ad radomized CSMA protools ad showed that these etworks a ahieve the same order-optimal throughput of Θ as etworks employig etralized TDMA shemes. Yag et al [4] studied the ahievable throughput of three dimesioal CSMA etworks. Ko et al [] showed that i CSMA etworks, by joitly optimizig the trasmissio rage ad paket geeratio rate, the ed-to-ed throughput ad ed-to-ed delay a sale as Θ log ad Θ log, respetively. Byu et al [] showed that distributed slotted ALOHA protools a have order-optimal throughput. Ulike i ALOHA, where eah ode aess the medium idepedetly with a presribed probability, odes of CSMA etworks suffer from a spatial

3 orrelatio problem, whih meas that the ativity of a ode is depedet o the ativities of other odes due to the arrier-sesig operatio. This orrelatio problem makes the aalysis of iterferee ad apaity of CSMA etworks more hallegig tha that of ALOHA etworks. Therefore, although both ALOHA ad CSMA are distributed medium aess otrol protools, the results obtaied for ALOHA etworks are ot diretly appliable to CSMA etworks. Some researh efforts were also devoted to modelig the spatial distributio of ourret trasmitters obeyig arriersesig ostraits ad the distributio of iterferee resultig from these trasmitters. The spatial distributio of ourret trasmitters followig CSMA protools are ofte modeled by the Matï œr hard-ore poit proess p.p. ad sometimes approximated by the oisso poit proess [9], [4] [7]. I more reet studies [9], [5], [7], the Radom Sequetial Absorptio RSA p.p. was proposed as a more atural model for represetig the spatial distributio of ourret CSMA trasmitters. Nguye ad Baelli [4] studied the RSA p.p. by haraterizig its geeratig futioal ad derived upper ad lower bouds for the geeratig futioal. Furthermore, the authors of [4] derived the etwork performae metris, viz., average medium aess probability ad average trasmissio suess probability two ommoly used metris i the study of trasmissio apaity, i terms of the geeratig futioal. Alfao et al. i [9] obtaied approximately the trasmissio apaity distributio. The above work [9], [4] studied the trasmissio apaity by ivestigatig the trasmissio suess probability ad the medium aess probability of a typial ode, whih quatifies the spatial average performae of the etwork. I ompariso, the trasport apaity ofte quatifies the throughput that a be ahieved by every soure-destiatio pair asymptotially almost surely, whih is ofte assoiated with the worst ase performae. Improvig spatial frequey reuse of CSMA etworks is a importat problem that has also bee extesively ivestigated, see [8] [4] for the relevat work. However, high level of spatial frequey reuse does ot diretly lead to ireased edto-ed throughput beause the latter performae metri also ritially relies o the ommuiatio strategies, i.e., routig algorithm ad shedulig sheme, used i the etwork. I this paper we fous o the study of ahievable ed-to-ed throughput. III. NETWORK MODEL AND SETTINGS I this setio, we itrodue the etwork model, the sigal propagatio model, the SINR model ad defie otatios ad oepts that are used i later aalysis. Two etwork models are widely used i the study of asymptoti etwork apaity: the dese etwork model ad the exteded etwork model. By appropriate salig of the distae, the results obtaied uder oe model a ofte be exteded to the other oe [4]. I this paper, we osider the exteded etwork model. artiularly we osider a etwork with odes deployed o a box B R aordig to a oisso poit proess with uit itesity. Eah ode hooses its destiatio radomly ad idepedetly of other odes. We study the apaity of the above etwork as. It is assumed that all data trasmissios are oduted over a ommo wireless hael. We are maily oered with the evets that our iside B asymptotially almost surely a.a.s. as. A evet ξ depedig o is said to our a.a.s. if ad oly if iff its probability approahes as. The followig otatios are used throughout the paper oerig the asymptoti behavior of positive futios: f O g if that there exist a positive ostat ad a iteger suh that f g for ay > ; f Ω g if g O f ; f Θ g if that there exist two ostats, ad a iteger suh that g f g for ay > ; f f o g if lim g. A. Iterferee model Let x k, k Γ, be the loatio of ode k, where Γ represets the set of idies of all odes. Whe ode i is trasmittig with power i, the reeived power at ode j loated at x j from ode i is give by i x i x j where x i x j represets the path-loss from ode i to ode j, is the path-loss expoet ad x i x j is the Eulidea distae betwee the two odes. I the paper we assume that >. This hael model is widely used i the literature [], [4], [6], [6]. A trasmissio from ode i to ode j is suessful iff the SINR at ode j is above a predetermied threshold β, i.e., SINR x i x j i x i x j N + β k x k x j k T i where T i Γ deotes the set of simultaeously ative trasmitters as ode i ad N represets bakgroud oise. I this paper we osider a iterferee-limited etwork ad assume that the impat of N is egligible. Despite the ommo kowledge that a higher SINR a lead to a ireased lik apaity, i reality trasmissio from a trasmitter to a reeiver a oly our at oe of a set of preset data rates after the SINR threshold is met [9], [4]. Therefore for a trasmitterreeiver pair, whe its assoiated SINR is above β, it is osidered that the trasmitter a trasmit to the reeiver at a fixed rate of W log + β b/s B. Defiitio of throughput Eah ode seds pakets to a idepedetly ad radomly hose destiatio ode via multiple hops. A ode a be a soure ode, a destiatio ode for aother soure ode, a relay ode or a mixture. The per-ode throughput or equivaletly the trasport apaity of the etwork, deoted by λ, is defied as the maximum rate that ould be

4 4 ahieved a.a.s. by all soure-destiatio pairs simultaeously. Similar as that i [6], we say that a perode throughput of λ is feasible if there is a temporal ad spatial routig ad shedulig sheme suh that every ode a sed λ bits/se o time average to its destiatio a.a.s., i.e., there exists a suffiietly large positive umber τ suh that i every fiite time iterval [j τ, jτ] every ode a sed τλ bits to its destiatio a.a.s.. C. CSMA protool The geeral idea of CSMA protools is that before trasmissio, a ode will sese other ative trasmissios i its viiity suh that earby odes will ot trasmit simultaeously. More speifially, a ode j is said to be i otetio with ode i if the reeived power by ode i from ode j is above the arrier-sesig threshold T i of ode i, i.e., j x i x j > T i. Node i a oly trasmit if it seses o other ative trasmissios i otetio, or i other words the ode seses the hael idle. To prevet the situatio where several earby odes simultaeously start trasmittig whe their ommo eighbor stops its trasmissio, hee ausig a ollisio, a bak off mehaism is ofte employed suh that a ode sesig the hael idle will wait a radom amout of time before startig its trasmissio. The followig bak off mehaism is osidered i this paper. Eah ode seses the hael otiuously ad maitais a outdow timer, whih is iitialized to a o-egative radom value. The timer of a ode outs dow whe it seses the hael idle; whe the hael is sesed as busy, the ode freezes its timer. A ode iitiates its trasmissio whe its outdow timer reahes zero ad the hael is sesed as idle. After fiishig its trasmissio, the ode resets its outdow timer to a ew radom value for the ext trasmissio. The distributio of the radom iitial outdow timer will be speified later i the paper. IV. ROUTING ALGORITHM AND TRAFFIC LOAD I this setio we desribe the routig algorithm to be used ad aalyze the traffi load for eah ode uder the algorithm. The routig algorithm hooses the sequee of odes to deliver a paket from its soure to its destiatio without osiderig physial layer implemetatio details. To begi the ostrutio of our routig algorithm, we partitio the box B of size ito squares of side legth log where is a positive ostat. Eah of these squares is the further subdivided ito smaller ells of ostat side legth. The values of ad will be speified later. See Fig. for a illustratio. Followig ommo termiology used i the perolatio theory, we also refer to these ells as sites ad use the two terms ells ad sites exhageably. We all a site ope if it otais at least oe ode, ad losed otherwise. Due to the oisso distributio of odes with uit itesity, it a be easily obtaied that a site is ope with probability p e. Furthermore, the evet that a site is ope or losed is idepedet of the evet that aother distit site is ope or losed. The total umber of sites i a square is log, the total umber of sites i B is. ad the total umber of squares i B is log The tehiques to hadle the situatio that log, ad log are ot itegers are well-kow [4]. Therefore i this paper we igore some trivial disussios ivolvig the situatios that log, ad log are ot itegers ad osider them to be itegers. Before we a further explai our routig algorithm, we eed to first establish some prelimiary results. The etwork area B a be slied ito horizotal retagles of size log, where eah horizotal retagle osists of squares. Deote by H i the i-th horizotal retagle log where i log. We all two sites adjaet if they share a ommo edge. We defie a left to right ope path i H i as a sequee of distit ad adjaet ope sites that starts from a ope site o the left border of H i ad eds at a ope site o the right border of H i. The followig theorem, due to [4, Theorem 4..9], gives a lower boud o the umber of ope paths i H i. Theorem. [4, Theorem 4..9]Cosider site perolatio with parameter p e. For suffiietly large, there exist ostats ad ω idepedet of, satisfyig ad ω log 5 < p <, log 6 p < 5 p p + log 6 p + <, 6 suh that a.a.s. there exist at least ω log left to right disjoit ope paths i every horizotal retagle. log By symmetry, if we partitio B ito vertial retagles. Eah oe is of size log ad osists of log squares. Deote by V j the j-th, j log vertial retagle. It a also be established that a.a.s. there are at least ω log top to bottom disjoit ope paths i every V j, j log. The followig result a be readily established [4]: Corollary. There are a.a.s at least ω log left-to-right ope paths ad ω log top-to-bottom ope paths i every square. A. Desriptio of the distributed routig algorithm We are ow ready to explai our routig algorithm. Deote by SD i the lie segmet oetig ode i to its destiatio. The pakets geerated by soure ode i are routed alog the squares itersetig SD i. A square will oly serve the traffi of a soure-destiatio pair if the assoiated SD lie itersets the square. Note that it is trivial to establish that a.a.s. every square has at least oe ode.

5 5.5 Number of ope paths i a retagle / log.5.5 Simulatio Lower boud o δ Network size x 4 Figure. A illustratio of the umber of left-to-right ope paths i a horizotal retagle as the etwork size varies. Vertial axis shows the ratio of the umber of ope paths to log. Figure. A illustratio of partitio of B ad the routig algorithm. Blak square represets a losed site ad white square represets a ope site. Grey square represets a ope site that forms a ope path. S ad D, idiated by two small hollow irles, are a pair of soure ad destiatio odes. H ad H, idiated by two small blak squares, are two odes loated i ope sites that form ope paths. First S trasmits its pakets to H usig a trasmissio rage of up to log. The the pakets will be routed alog the ope paths to H, usig a trasmissio rage of up to 5. Fially, H trasmits the pakets to the destiatio D. If H itself is a soure ode, the it trasmits its paket diretly to the ext-hop ode alog the ope path, usig a trasmissio rage of up to 5. The routig a be divided ito three stages: I the first stage, a soure ode S, if it is ot a ode loated i a ope site that forms oe of the ope paths, will trasmit its paket to a ode i a radomly hose ope site that forms a ope path. If there are multiple odes i a ope site, a ode will be desigated radomly to relay all traffi passig through the site. If the soure ode is already i a site that forms a ope path, this stage of routig a be omitted ad the routig proeeds diretly to the ext stage. The maximum distae betwee the soure ode ad its ext-hop ode i this stage is bouded by log beause the distae betwee ay two odes loated i a square is at most log. I the seod stage, the paket will be routed to the adjaet square itersetig the SD lie alog oe of these left-to-right ope path or top-to-bottom ope paths util the paket reahes a ode i the ext square. Depedig o the loatio of the ope path otaiig the relay ode ad the loatio of the adjaet square, the paket may be routed alog a left-to-right ope path whe the adjaet square is o the left or o the right of the urret square or alog a top-to-bottom ope path whe the adjaet square is o the top or o the bottom of the urret square. If the paket eeds to be swithed from a leftto-right ope path to a top-to-bottom ope path e.g., whe the previous square is o the left of the urret square but the ext square is o the bottom of the urret square, a top-to-bottom ope path is hose radomly from the at least ω log ope path available. The above proess otiues util the paket reahes the square that otais the destiatio ode. I this stage, the maximum distae betwee a ode ad its ext-hop ode is bouded by 5 beause the distae betwee ay two odes loated i two adjaet ells is at most 5. I the third stage, after reahig the square otaiig the destiatio ode, if the destiatio ode is loated o oe of the ope paths, the paket will be routed alog a multihop path to the destiatio via ope paths; if the destiatio is ot loated o oe of the ope paths, the paket will be trasmitted to the destiatio diretly ad the maximum trasmissio distae is bouded by log. The same route is used for all pakets belogig to the same soure-destiatio pair. The feasibility of the above routig algorithm is guarateed by Corollary. A ode oly eeds eighborhood iformatio of odes o more tha 5 log away to make a routig deisio. The required iformatio for makig a proper routig deisio is vaishigly small ompared with that i the highway algorithm. Furthermore, ompared with the etwork size, the required iformatio is also vaishigly small as. Therefore the routig algorithm a be exeuted i a distributed maer. O the other had, we readily akowledge that eighborhood iformatio of odes up to 5 log away may be required by the routig algorithm. The required eighborhood iformatio grows logarithmially with the size of the etwork. Corollary is a ready osequee of Theorem : Corollary. Let.78 ad, a.a.s. there are at least.5474 log left-to-right ope paths i every horizotal retagle. I the rest of this paper, we arry out aalysis assumig that ad take values speified i Corollary ad ω Fig. shows simulatio results of the umber of ope paths i a horizotal retagle as the etwork size varies. Eah radom simulatio is repeated a large umber of times ad the average result is show. The ofidee iterval is very small ad egligible, ad thus ot plotted i the figure. The lower boud o the umber of ope paths suggested i Corollary is also plotted for ompariso. As show i Fig., the lower boud is reasoably tight. Fig., draw from a simulatio, further gives a ituitive illustratio of the ope paths i a horizotal retagle. After establishig the routig algorithm, ext we aalyze the traffi load for eah ode uder the algorithm, whih forms a key step i aalyzig the etwork apaity.

6 6 Figure. A illustratio of left-to-right ope paths i a retagle obtaied by omputer simulatios. Blak ells represet losed sites while white ells represet ope sites. Number of SD lies passig through a square / log Simulatio Upper boud Network size x 4 Figure 4. The umber of SD lies passig through a square versus the upper boud i Lemma 4. Vertial axis shows the ratio of the umber of SD lies passig through a square to log. Lemma 4 shows that the radom umber of SD lies passig through a arbitrarily hose square, iludig the SD lies origiatig from ad edig at the square, is upper bouded. Lemma 4. For a arbitrary square i B, the radom umber of SD lies passig through it, deoted by Y, satisfies that lim r Y ω log 7 where ω. + ɛ + δ, ɛ ad δ are arbitrarily small positive ostats. roof: See Appedix I. As a way of establishig the tightess of the boud i Lemma 4, Fig. 4 shows simulatio results of the umber of SD lies passig a square i ompariso with the upper boud i Lemma 4. Usig Corollary ad Lemma 4, the followig result a be readily established: Lemma 5. Eah relay ode eeds to arry the traffi of at most soure-destiatio pairs a.a.s. ω.5474 Note that a ode ot o a ope path does ot eed to arry the traffi of other soure-destiatio pairs. V. A SOLUTION TO THE HIDDEN-NODE ROBLEM Our routig algorithm desribed i the last setio eeds to use two differet trasmissio rages of legths Θ ad Θ log respetively. The use of two differet trasmissio rages i CSMA etworks will exaerbate the soalled hidde ode problem. See Fig. 5 for a illustratio. Figure 5. A illustratio of the hidde ode problem whe odes use differet trasmissio power. Assume that the same arrier-sesig threshold is used by ode A ad B. The trasmissio of A usig a lager trasmissio power ode B usig a smaller trasmissio power, respetively a be deteted by odes loated withi a distae R A R B, respetively, ad R A > R B. Cosequetly B a detet A s trasmissio but ode A aot detet ode B s. Therefore eve whe ode B is trasmittig, ode A still a start its ow trasmissio, thereby resultig i a ollisio ad ausig the hidde ode problem. I [], the problem was addressed by lettig odes operate o two frequey bads, amely, short-rage trasmissios operate o oe frequey bad while log-rage trasmissios operate o the other. Their solutio may result i lower spetrum usage beause log-rage trasmissio is used less frequetly ad also poses additioal hardware requiremets o odes. Therefore, we preset a solutio by joitly tuig the trasmissio power ad the arrier-sesig threshold. A. A formal defiitio of the hidde ode problem Uder the SINR model, a set of ourret trasmissios or liks are said to form a idepedet set if the SINRs are all above the SINR threshold β. Let F be the set of all idepedet sets. Beause of the radom ad distributed ature of the arrier-sesig operatios by idividual odes, the set of simultaeous trasmissios observig the arriersesig ostrait, deoted by S CS, may or may ot belog to F, i.e., some trasmissios observig the arrier-sesig ostraits may still ause the SINRs at some reeivers to be above β. Let F CS be the set of all S CS s. Let Ψ be the set of ourret trasmissios i a CSMA etwork. More formally, a hidde ode problem is said to our if Ψ F CS but Ψ / F. A CSMA etwork is said to be hidde ode free if its arrier-sesig operatios ad trasmissio powers are arefully desiged suh that all Ψ F CS also meets the oditio that Ψ F. For a CSMA etwork i whih uiform trasmissio power is i use, by settig the arrier-sesig rage to be a ostat multiple of the trasmissio rage, the hidde ode problem a be effetively elimiated [], [4]. For our routig algorithm usig two trasmissio rages of legths Θ ad Θ log, if the arrier-sesig rage is set to be Θ log, although the hidde ode problem a be elimiated, the umber of ourret trasmissios hee the spatial frequey reuse will be redued ompared with a arrier-sesig rage of Θ, whih i tur auses a redued apaity. Therefore we maage to have trasmissios with differet rages to oexist ourretly istead. I this way, the apaity will be maximized while elimiatig the hidde ode problem. More speifially, let i be the trasmissio power used for the i th trasmissio where the same trasmitter may

7 7 use differet power whe trasmittig to differet reeiver. The trasmitter also uses differet arrier-sesig threshold whe differet trasmissio power is used. Deote by T i the arrier-sesig threshold used for i. Furthermore, let the trasmissio power of a trasmitter be suh that the power reeived at its iteded reeiver is at least is a ostat ot depedig o ad the value of will be speified shortly later i this setio. The followig lemma speifies the relatio betwee i ad T i required for two trasmitters to be able to sese eah other s trasmissio. Lemma 6. Let the values of i ad T i be hose suh that the followig oditio is met i T i. 8 For two arbitrary trasmitters loated at x i ad x j respetively, they a sese eah other s trasmissio iff i j x i x j < 9 T i T j roof: Whe ode i loated at x i trasmits usig power i, the power reeived at ode j at loatio x j is give by i x i x j. Let T j be the arrier-sesig threshold of ode j. The trasmissio of ode i a be deteted iff i x i x j > T j. Usig 8, ode j a detet ode i s trasmissio iff T i x i x j > T j or equivaletly x i x j < i T it j j. Usig a similar argumet, ode i a detet ode j s trasmissio iff 9 is met. Lemma 6 shows that by arefully hoosig the arriersesig threshold aordig to the trasmissio power for eah trasmitter, a major ause of the hidde ode problem: a ode A seses aother ode B s trasmissio but ode B aot sese ode A s trasmissio a be elimiated. I the ext several paragraphs, we shall demostrate how to hoose, whih determies the miimum power reeived at a reeiver, suh that the SINR requiremet a also be met. I the first ad third stages of our routig algorithm, the maximum distae betwee a trasmitter ad a reeiver is log while the the maximum distae betwee a trasmitter ad a reeiver i the seod stage is 5. Aordigly, for the first ad third stages, we let the trasmissio power be h log while for the seod stage, the trasmissio power is set at l 5 It is trivial to show that the reeived sigal power of all trasmissios is at least. Furthermore, the followig theorem established i our previous work [4, Theorem ] helps to obtai a upper boud o the iterferee experieed by ay reeiver i the etwork. Theorem 7. Cosider a CSMA etwork with odes distributed arbitrarily o a fiite area i R where all odes trasmit at the same power ad use the same arrier-sesig threshold T. Furthermore, the power reeived by a ode at x j from a trasmitter at x i is give by x i x j. Let r be the distae betwee a reeiver ad its trasmitter. The maximum iterferee experieed by the reeiver is smaller tha or equal to N d, r + N d where 4 N d, r ad d T. 5 4 d r 4 d r d + d r + + d r d r d N d d + d + d d Notig that N d, r is a mootoially ireasig futio of r, it a be readily established usig Theorem 7 that i the CSMA etwork aalyzed i this paper i whih two sets of trasmissio powers, arrier-sesig threshold ad the maximum trasmissio rage are employed, the maximum iterferee for ay value of is bouded by h N T h, h log + N T h l +N, l 5 + N 4 T l where T l ad T h are the arrier-sesig threshold hose for l ad h respetively aordig to 8. Remark 8. At the expese of more aalytial efforts, a tighter boud o iterferee a be established that the maximum iterferee i the CSMA etwork osidered i this paper is bouded by N l, 5 + N l T l for ay T l value of. Beause for a suffiietly large etwork, whih is the fous of this paper, the differee betwee this boud ad the upper boud i 4 is egligibly small, we hoose to omit the aalysis due to spae limitatio. Notig that h T h log, l T l 5 ad d T, it is easy to olude usig ad that whe >, the otributio of the first two terms N h, T h log + N h, T h attributable to trasmissios usig a larger trasmissio power, beome vaishigly small ompared with the last two terms as. The followig theorem provides guidae o how to hoose to meet the SINR requiremets for all ourret trasmissios i a large CSMA etwork. Theorem 9. For a arbitrarily high SINR requiremet β, there exists a value of for suffiietly large suh that the SINR of all trasmissios i a CSMA etwork, i whih eah T l

8 8 trasmitter sets its trasmissio power ad arrier sesig threshold aordig to the relatioship i Lemma 6, is greater tha or equal to β. Furthermore, the value of is give impliitly by the followig equatio N l, 5 T l β 5 + N l T l l T l roof: Notig that the miimum reeived power is, the theorem beomes a easy osequee of the iterferee upper boud established earlier i the setio. As a brief summary of the results of this setio, Theorem 9 gives guidae o how to hoose to meet the SINR requiremet. More speifially, otig that 5 ad usig equatios ad, equatio 5 beomes a impliit equatio of. Solvig the equatio, the value of that meets the SINR requiremet a be obtaied, whih is idepedet of. Give the value of, the other parameters i the CSMA etwork, i.e. h, l ad the arrier sesig thresholds, a all be determied usig equatios 8, ad respetively. It a be readily established usig the aalysis preseted i this setio that the CSMA etwork whose trasmissio power ad arrier sesig threshold are hose followig the above steps are immue from the hidde ode problem. VI. BACK OFF TIMER SETTING AND CAACITY ANALYSIS I the last setio, we demostrated how to hoose the trasmissio power ad the arrier sesig threshold to solve the hidde ode problem. I the CSMA etwork i whih odes may use two differet trasmissio powers, a potetial problem that may arise is that odes usig the larger trasmissio power may potetially oted with more odes for trasmissio opportuities. Therefore odes usig the larger trasmissio power may ot get a fair trasmissio opportuity ompared with odes usig the smaller trasmissio power. This may potetially auses odes usig the larger trasmissio power to beome a bottleek i throughput whih redues the overall etwork apaity. I this setio, we demostrate how to hoose aother otrollable parameter i CSMA protools, i.e., bak off timer, to oquer the diffiulty. Same as that i referees [] ad [44], we osider a CSMA protool i whih the iitial bak off timer is a radom variable followig a expoetial distributio. Nodes usig differet trasmissio power may however hoose differet mea value to use i the expoetial distributio goverig their respetive radom iitial bak off timer. The followig theorem provides the basis for hoosig these mea values. Theorem. Let δ ad δ be two small positive ostats. If trasmissios usig a low trasmit power l set their iitial bak off time to be expoetially distributed with mea λ l ad trasmissios usig a high trasmissio power h set their iitial bak off time to be expoetially distributed with mea λ h log, the Medium aess probability of a low power trasmissio x Simulatio Lower boud Network size x 4 Figure 6. A ompariso betwee the simulatio result o the medium aess probability of a ode usig the low power trasmissio with the lower boud i Theorem where β ad 4. i a.a.s. eah low power trasmissio a be ative with a ostat probability greater tha or equal to ω π δ π + 6 ii a.a.s. eah high power trasmissio a be ative with a probability greater tha or equal to ω 4 π log δ 4π 4 log 4 + roof: See Appedix II. Fig. 6 shows the trasmissio opportuity or the medium aess probability of a ode usig l versus the lower boud i Theorem for differet values of. O the basis of the results established i this setio ad i the earlier setios, we preset the followig theorem whih forms the major result of this paper. Theorem. The ahievable per-ode throughput i the CSMA etwork is greater tha or equal to ad is smaller tha or equal to.5474ω ω W ; 7.5 5π W + a.a.s. as, where ω is give i Lemma 4 ad ω is give by 6. roof: We first show that the ahievable per-ode throughput is lower bouded by.5474ω ω W. Let λ λ, respetively be the per-ode throughput that a be ahieved i the first ad the third the seod, respetively stages of our routig algorithm. Obviously the fial per-ode throughput λ satisfies λ mi {λ, λ }. I the followig, we aalyze λ ad λ separately. As a easy osequee of Lemma 5, a.a.s. eah relay ω ode arries the traffi of at most.5474 soure-destiatio pairs. Aordig to the first statemet of Theorem, a.a.s. eah relay ode o a ope path a aess the hael

9 9 with a probability of at least ω, whih is a ostat idepedet of. The olusio the readily follows that lim r λ.5474ω ω W. For the seod stage of the routig, ote that a soure or a destiatio ode ot o a ope path does ot eed to arry traffi for other soure-destiatio pairs. Usig the seod statemet of Theorem, olusio follows that log 4 λ Ω. Combiig the above two results o λ ad λ ad otig that the apaity bottleek lies i the seod stage, the first statemet i this theorem is proved. We ow further show that the ahievable per-ode throughput is upper bouded by W.5 5π +. The upper boud is to be established usig a result proved i our previous work [45, Corollary 6], whih shows that the perode throughput is equal to the produt of the average umber of simultaeous trasmissios ad the lik apaity divided by the produt of the average umber of trasmissios required to deliver a paket to its destiatio ad the umber of souredestiatio pairs. We first aalyze the average umber of trasmissios required for a paket to reah its destiatio. The average distae betwee a radomly hose souredestiatio pair is.5 [46]. A paket moves by oe ell i eah hop o a ope path where the otributio of the last mile trasmissio betwee a soure a destiatio ad a ope-path ode is vaishigly small ompared with.5. Thus a.a.s. the average umber of hops traversed by a paket is at least.5. Next we aalyze the average umber of simultaeous trasmissios. Sie there is at most oe ode i a ell atig as a ope path ode, there are at most ope path odes i the etwork. Followig the same proedure i obtaiig, 4 ad 5, we have that r [ ηi] l. Therefore, the average umber 5π + of simultaeous trasmissios is at most 5π + Note that whe a o-ope-path ode trasmits usig h, the umber of simultaeous trasmissios will oly redue. As a ready osequee of the above aalysis ad [45, Corollary 6], a upper boud o the per-ode throughput results. The lower boud o the per-ode throughput provided i Theorem is order optimal i the sese that the throughput is of the same order as the kow result o the optimum per-ode throughput [4] of etworks uder the same settigs. Furthermore, Theorem gives the pre-ostat preedig the order of the per-ode throughput:.5474ω ω. A detail examiatio of the pre-ostat reveals that the pre-ostat a be separated ito the produt of two terms:.5474 ω ad ω. The first term.5474 ω is etirely determied by the routig algorithm, more speifially determied by how the routig algorithm distribute traffi load amog relay odes ad amog soure-destiatio pairs. The seod term ω is etirely determied by the shedulig algorithm ad some physial layer details, i.e., the SINR requiremet, iterferee ad propagatio model. The above observatio appears to suggest that impat of the routig algorithm ad the shedulig algorithm a be deoupled ad studied separately, ad the two algorithms that determie the overall etwork apaity a be optimized separately. er ode throughput/ x Simulatio Lower boud Upper boud Network size x 4 Figure 7. A simulatio of per-ode throughput with 4 ad β. For ompariso, the upper ad the lower boud obtaied i the paper is also show Simulatio / lower boud Upper boud / lower boud Network size x 4 Figure 8. A simulatio of per-ode throughput with 4 ad β. For ompariso, the upper ad the lower boud obtaied i the paper are also show. To failitate ompariso, both the per-ode throughput obtaied from simulatios ad the per-ode throughput upper boud are ormalized by the per-ode throughput lower boud. Fig. 7 shows a ompariso of the per-ode throughput obtaied from simulatios, the upper lower bouds obtaied i Theorem for differet values of. To failitate ompariso, Fig. 8 further shows the ratio of the per-ode throughput obtaied from simulatios to the throughput lower boud ad the ratio of the throughput upper boud to the throughput lower boud. As show i the figures, the lower boud is fairly tight ad the upper boud is also withi a fator of of the simulatio result. The simulatio results demostrate that the pre-ostat obtaied i our study provides a pretty aurate haraterizatio of the per-ode throughput. VII. CONCLUSION I this paper, we studied the trasport apaity of large wireless multi-hop CSMA etworks. We showed that by arefully hoosig the otrollable parameters i the CSMA protool ad desigig the routig algorithm, a etwork ruig distributed CSMA shedulig algorithm ad eah ode makig routig deisios based o loal iformatio oly a also ahieve a order-optimal throughput of Θ, whih is the same as that of large etworks employig etralized routig ad shedulig algorithms. Furthermore, we ot oly gave the order of the throughput but also derived the preostat preedig the order by givig a upper ad a lower

10 Figure 9. A illustratio of a SD lie itersetig the irumsribed irle boud of the trasport apaity. The tightess of the bouds was validated usig simulatios. Theoretial aalysis was preseted o tuig the arrier-sesig threshold ad the trasmissio power to avoid hidde ode problems ad o tuig the bak off timer distributio to esure eah ode gai a fair aess to the hael i CSMA etworks usig o-uiform trasmissio powers. The priiple developed through the aalysis was expeted to be also helpful to set the orrespodig parameters of CSMA etworks i a more realisti settig. AENDIX I ROOF OF LEMMA 4 I the proof of Lemma 4, we will make use of a result established i the stohasti orderig theory [47]. For two real valued radom variables X ad X, we say X st X iff for all x,, r X > x r X > x. Theorem. [47, Theorem a]suppose X i follows a Biomial distributio with parameters i N ad p i,, deote the distributio of X i by B i, p i, i,, i.e., X i B i, p i. We have X st X iff p p ad. As a easy osequee of the above theorem, for three idepedet Biomial radom variables X B, p, X B, p ad X B, p with ad p p, it a be oluded that X st X st X. Now we are ready to prove Lemma 4. Let Y j i be the idiator radom variable for the evet that the SD i passes through the j th square: Y j i { if SD i passes through the jth square otherwise. We shall derive a upper boud o r [, log [ ] Y j i for ay j ]. Cirumsribe the j th square with a small irle of radius log, as show i Fig. 9. For a soure S loated outside the square ad at a distae x from the eter of the square, the agle θ x subteded by the irle at S is θ x arsi x. Usig the fat that arsi x.6x whe x, we have log θ x.6 arsi log. log x x 8 Notig that B is of size, the area of the setor formed by the two dashed tagets Fig. 9 ad the boarder of B is at most θx π. If the destiatio of S, deoted by D, does ot lie i this setor, the the assoiated SD lie does ot pass through the irle. Therefore, the probability that the SD lie itersetig the irle is at most θx π. Cosiderig that the irle is loated i a box B, the probability desity that S is at a distae x from the irle a be show to be upper bouded by πx. It follows from the above aalysis ad 8 that [ ] r Y j i. log πx πx dx. log 9 Reall that Γ represets the set of idies of all odes i the etwork. For a fixed square j, the total umber of SD lies passig through it is give by Y j Γ i Y j i, whih is the sum of i.i.d. Beroulli radom variables sie the loatios of odes are idepedet ad Y j i depeds oly o the loatios of soure ad destiatio odes of the i th soure-destiatio pair. Therefore Y j [ follows] the Biomial distributio, i.e., Y j B Γ, r Y j i. As a easy osequee of the oisso distributio of odes, a.a.s. the total umber of odes Γ + ɛ, where ɛ is a arbitrarily small positive ostat. Defie aother Biomial radom variable Ỹ j B 4 log + ɛ,. It follows from Theorem that Y j st Ỹ j It a be further show that for ay < δ <, [ r Y j > + δ + ɛ. ] log [ r Ỹ j > + δ + ɛ. ] log r [Ỹ j j > + δ E [Ỹ ]] exp δ [Ỹ ] E j exp. + ɛ δ log where results from the Cheroff boud. Usig the uio boud ad the above result, we have log r Y j >. + ɛ + δ log j log exp. + ɛ δ log Notig that log exp.+ɛδ log as, therefore a.a.s. Y j [ ]. + ɛ + δ log for ay j, whih ompletes the proof of Lemma 4. log

11 AENDIX II ROOF OF THEOREM Cosider a ode i o a ope path loated at x i trasmittig with power l 5. Sie the highest trasmissio power used i the etwork is h log, by 9, the furtherest trasmitter that ode i a sese is withi a distae of log. Deote by D x, r a disk etered at x ad with a radius of r. All odes that are possibly ompetig with ode i for trasmissio opportuities are loated withi D x i, log. Deote by A x, r, r a aulus area etered at x with a ier radius r ad a outer radius r. A little refletio shows that all odes usig the low trasmissio power l ad ompetig with ode i must be loated i D x i, 5, ad the odes i A x i, 5, log that ompete with ode x i must use the high trasmit power h. Note that i eah ope site that forms the ope path, oly oe ode serves as the relay π 5 π 5 ode. Hee, there are at most ope path odes i D x i, 5 that use l. Let N x, r be the radom umber of odes loated i D x, r. Next we provide a asymptoti upper boud o the umber of odes i D x i, log for ay ode i o a ope path. Deotig by H the set of idies of odes o ope paths, learly H <. By Cheroff boud ad the uio boud, we have for a arbitrarily small positive ostat δ, [ r N i H x i, log + δ π ] log [ r N x i, log i H + δ E [ N x i, ]] log δ e [N x E ] i, log where E deotes the { expetatio [ operator. It a be readily show that exp δ E N x i, ]} log approahes as. Therefore a.a.s. the umber of odes withi a distae log of a ope path ode is bouded above by + δ π log. Next we aalyze the trasmissio opportuity of a ope path ode. Deote by t i the bak off timer of ode i at a partiular time istat whe the hael is idle. Deote by C i the set of idies of odes that ompete with ode i for trasmissio. Followig the CSMA protool, ode i a beome a ative trasmitter i the ompetitio if t i < mi t j. j C i\{i} Let ηi l be the evet that a trasmissio of ode i usig the low trasmit power is ative. Usig the memoryless property of a expoetial distributio that for a timer followig a expoetial distributio, the amout of lapsed time does ot alter the distributio of the remaiig value of the timer, it a be show that for ay i H r [ ηi l ] [ ] r t i < mi t j j C i\{i} e λ l t π 5 e λ h t +δ π log λ l e λ lt dt 4 where i the above equatio the term e λ lt π 5 represets the probability that at a radomly hose time istat whe the hael is idle, all π 5 ope path x i, 5, whih are ompetig for tras- odes i D missio opportuities with ode i, have their respetive bak off timer larger tha a partiular value t; the term e λ h t +δ π log represets the probability that all odes usig h i D x i, log, whih are ompetig for trasmissio opportuities with ode i, have their respetive bak off timer larger tha t; the term λ l e λ lt is the pdf of the bak off timer of ode i. It a be further show from 4 that for ay i H, r [ ηi l ] λ l e π 5 λl +λ h +δ π log +λ l t dt λ l π 5 λl + λ h + δ π log + λ l π 5 + λ h λ l + δ π log + π δ π. 5 + Now we otiue to prove the seod part of Theorem. Cosider that a ode j trasmits usig the high power h log. By 9, all odes that are possibly ompetig with ode j are loated withi D x j, log. Furthermore, amog the odes ompetig with ode j, those ope path odes usig the lower trasmissio power l must be loated i D x j, log, ad the umber π log of these ope path odes is at most. π log Next we derive a upper boud o the umber of odes i D x j, log ompetig with ode j for ay j O where O is the set of idies of odes usig the high power. It a be easily show that lim r O <. Usig the uio boud ad the Cheroff boud, we have for ay small positive ostat δ, r N x j, log j O

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13 [4] M. Fraeshetti ad R. Meester, Radom Networks for Commuiatio from Statistial hysis to Iformatio Systems. Cambridge Uiversity ress, 7. [4] L. Fu, S. Liew, ad J. Huag, Effetive arrier sesig i CSMA etworks uder umulative iterferee, IEEE Tras. Mobile Comput., vol., o. 99, pp.,. [4] T. Yag, G. Mao, ad W. Zhag, Coetivity of large-sale sma etworks, IEEE Tras. Wireless Commu., vol., o. 6, pp ,. [44] L. Jiag ad J. Walrad, A distributed sma algorithm for throughput ad utility maximizatio i wireless etworks, IEEE/ACM Tras. Netw., vol. 8, o., pp ,. [45] G. Mao, Z. Li, X. Ge, ad Y. Yag, Towards a simple relatioship to estimate the apaity of stati ad mobile wireless etworks, IEEE Tras. Wireless Commu., vol., o. 8,pp ,. [46] J. hilip, The probability distributio of the distae betwee two radom poits i a box. KTH mathematis, Royal Istitute of Tehology, 7. [47] A. Kleke ad L. Matter, Stohasti orderig of lassial disrete distributios, Advaes i Applied robability, vol. 4, o., pp. 9 4,. Wei Zhag S -M 6-SM reeived the h.d. degree i eletroi egieerig from The Chiese Uiversity of Hog Kog i 5. He was a Researh Fellow with the Departmet of Eletroi ad Computer Egieerig, The Hog Kog Uiversity of Siee ad Tehology, durig 6-7. From 8, he has bee with the Shool of Eletrial Egieerig ad Teleommuiatios, The Uiversity of New South Wales, Sydey, Australia, where he is a Assoiate rofessor. His urret researh iterests ilude ogitive radio, ooperative ommuiatios, spae-time odig, ad multiuser MIMO. He reeived the best paper award at the 5th IEEE Global Commuiatios Coferee GLOBECOM, Washigto DC, USA, i 7 ad the IEEE Commuiatios Soiety Asia-aifi Outstadig Youg Researher Award i 9. He was TC Co-Chair of Commuiatios Theory Symposium of the IEEE Iteratioal Coferee o Commuiatios ICC, Kyoto, Japa, i. He serves TC Chair of the Symposium o Sigal roessig for Cogitive Radios ad Networks i the d IEEE Global Coferee o Sigal ad Iformatio roessig GlobalSI - Atlata, USA, i 4. He is a Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS ad a Editor of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS Cogitive Radio Series. Tao Yag reeived her BEg degree ad MS degree i Eletrial Egieerig from Southwest Jiaotog Uiversity, Chia. She is urretly workig towards the hd degree i Egieerig at The Uiversity of Sydey. Her researh iterests ilude wireless multihop etworks ad etwork performae aalysis. Xiaofeg Tao FIET, SMIEEE reeived his B.S degree i eletrial egieerig from Xi a Jiaotog Uiversity, Chia, i 99, ad M.S.E.E. ad h.d. degrees i teleommuiatio egieerig from Beijig Uiversity of osts ad Teleommuiatios BUT i 999 ad, respetively. He was a visitig rofessor at Staford Uiversity from to, hief arhitet of Chiese Natioal FuTURE 4G TDD workig group from to 6 ad established 4G TDD CoM trial etwork i 6. He is urretly a rofessor at BUT, the ivetor or o-ivetor of 5 patets, the author or o-author of papers, i 4G ad beyod 4G. Guoqiag Mao S 98-M -SM 8 reeived hd i teleommuiatios egieerig i from Edith Cowa Uiversity. Betwee ad, he was a Leturer, a Seior Leturer ad a Assoiate rofessor at the Shool of Eletrial ad Iformatio Egieerig, the Uiversity of Sydey, all i teured positios. He urretly holds the positio of rofessor of Wireless Networkig, Diretor of Ceter for Real-time Iformatio Networks at the Uiversity of Tehology, Sydey. He has published more tha papers i iteratioal oferees ad jourals, whih have bee ited more tha times. His researh iterest iludes itelliget trasport systems, applied graph theory ad its appliatios i etworkig, wireless multihop etworks, wireless loalizatio tehiques ad etwork performae aalysis. He is a Seior Member of IEEE, a Editor of IEEE Trasatios o Vehiular Tehology ad IEEE Trasatios o Wireless Commuiatios ad a o-hair of IEEE Itelliget Trasport Systems Soiety Tehial Committee o Commuiatio Networks.