This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. Coetivity ad Capaity o Multi-Chael Wireless Networks with Chael Swithig Costraits Vartika Bhadari Dept. o Computer Siee, ad Coordiated Siee Laboratory Uiversity o Illiois at Urbaa-Champaig vbhadar@uiu.edu Niti H. Vaidya Dept. o Eletrial ad Computer Eg., ad Coordiated Siee Laboratory Uiversity o Illiois at Urbaa-Champaig hv@uiu.edu Abstrat This paper argues or the eed to address the issue o multi-hael etwork perormae uder ostraits o hael swithig. We preset examples rom emerget diretios i wireless etworkig to motivate the eed or suh a study, ad itrodue some models to apture hael swithig ostraits. For some o these models, we study oetivity ad apaity o a wireless etwork omprisig radomly deployed odes, equipped with a sigle iterae eah, whe there are = O(log) haels o equal badwidth W available. We osider a adjaet (, ) hael assigmet where a ode may swith betwee adjaet haels, but the adjaet hael blok is radomly assiged. We show that the per-low apaity or this hael assigmet model is Θ(W log ). We the show how the adjaet (,2) assigmet maps to the ase o utued radios. We also osider a radom (, ) assigmet where eah ode may swith betwee a pre-assiged radom subset o haels. For this model, we prove that per-low apaity is O(W prd log ) (where p rd = ( )( )...( + ))ad Ω(W log ). Idex Terms Multi-hael, swithig ostraits, oetivity, apaity, adjaet (, ) assigmet, radom (, ) assigmet, detour-routig. I. INTRODUCTION Earlier work o protools or multi-hael wireless etworks [] has assumed that eah ode is apable o swithig o all haels. This assumptio may be halleged by emergig paradigms i wireless etworkig, suh as evisioed large-sale deploymet o extremely iexpesive wireless devies embedded i the eviromet, ad dyami spetrum aess via ogitive radio. We briely summarize some suh searios: The eed or low-ost, low-power radio traseivers to be used i iexpesive sesor odes a give rise to may situatios ivolvig ostraied swithig. Hardware omplexity (ad hee ost), ad/or power osumptio may be sigiiatly redued i eah ode operates oly i a small spetral rage, ad swithes betwee This researh is supported i part by US Army Researh Oie grat W9NF-05--0246, NSF grat CNS 06-27074, ad a Vodaoe Graduate Fellowship. a small subset o adjaet haels (e.g., i the traseiver uses a osillator with limited tuability). However, i more spetrum is available tha a sigle devie a utilize, it may be possible at time o mauature to lok dieret devies o to dieret requey rages. Also, potetially a traseiver may have a RF hael seletor omprisig a bak o swithable ilters [2], rom whih it may selet oe to use or trasmissio/reeptio. I ogitive radio etworks, give a multi-hop etwork o seodary users attemptig to utilize uused spetrum, some haels may be loally uusable due to the presee o a ative primary user i the viiity. Thus, there is eed to address the issue o multi-hael etwork perormae i the presee o ostraits o haelswithig, both i terms o determiig how asymptoti trasport apaity is aeted by the ostraits, ad desigig protools or eiiet hael-oordiatio, ad data-traser. It has bee proposed i [3] that extremely iexpesive wireless devies a be mauatured i it is possible to hadle utued radios whose operatig requey may lie radomly withi some bad. Also osidered i [3] is the possibility that eah devie may have a small umber o suh utued radios, ad a radom etwork odig based approah is proposed to relay iormatio betwee a sigle soure-destiatio pair. Some work o ogitive radio has addressed the issue o oordiatio i the ae o restrited ad variable hael availability at idividual odes due to ative primary users [4], [5]. However, o ormal theoretial models have bee developed or the various types o swithig ostraits eoutered i these previous works, ad i other atiipated searios, ad the impat o the ostraits o etwork perormae i a geeral multi-hop settig has ot bee quatiied. I this paper we preset a iitial oudatio or this domai by itroduig some models or ostraied hael assigmet, ad explorig issues o oetivity ad trasport apaity or some o these models. We osider a adjaet (, ) hael assigmet model, ad show that the per-low apaity or this ase is 0743-66X/07/$25.00 2007 IEEE 785
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. Θ(W log ). We the use the results or this model to obtai asymptoti apaity results or utued radios with radom soure-destiatio pairs. We also osider a radom (, ) assigmet model. For this model, we prove that per-low apaity is O(W prd log ) (p rd is deied i Setio XI) ad Ω(W log ). We also briely disuss a spatially orrelated hael assigmet model. Due to pauity o spae, we are oly able to provide highlevel proo skethes i this paper, ad most lemmas/theorems are stated without proo. Please see [7] or detailed proos. II. SOME MODELS FOR CONSTRAINED CHANNEL ASSIGNMENT I this setio we elaborate o some o the models or ostraied hael assigmet that we propose. These models assume that odes possess oly oe iterae eah, there are haels available, ad all haels are orthogoal. However, they may potetially be exteded to the ase where multiple iteraes are available at eah ode 2. A. Adjaet (, ) Assigmet We itrodue a assigmet model wherei a ode a swith betwee a set o otiguous haels (2 ). Thus, i the requey bad is divided ito haels umbered, 2,..., i order o ireasig requey, the, at mauature/pre-deploymet time, eah ode is assiged a blok loatio i uiormly at radom rom {,..., + } ad thereater it a swith betwee the set {i,...,i + }. This model is relevat whe eah idividual ode has a traeiver with limited tuability, ad thus may oly swith betwee a small set o otiguous haels. It is also possible to establish a mappig betwee speii istaes o this model, ad the ase o utued radios (see Setio X). B. Radom (, ) Assigmet I this assigmet model, a ode is assiged a subset o haels (2 ) uiormly at radom rom the set o all possible hael subsets o size. This model a apture situatios where tiy low-ost sesor odes may be equipped with a traseiver havig a bak o ilters (e.g., suh a desig has bee proposed i [2]). Oe a evisage searios where eah ilter operates o some radom hael determied at time o mauature. C. Spatially Correlated Chael Assigmet I this model, a set o N pseudo-odes is plaed radomly i the etwork, i additio to the regular etwork odes. Eah pseudo-ode is assiged a radomly hose hael. All etwork odes withi a distae R o a pseudoode with assiged hael i are bloked rom usig hael i. This We have reetly obtaied ew results showig that apaity with radom (, ) assigmet is Θ(W prd log ),or = O(log). Please see [6]. 2 I these models, we assume that 2, as = is the sigle hael ase i whih = = is the oly possibility. I Setio VI, we explai why we do ot allow = or 2. model aptures hael uavailability due to a ative primary user i the viiity i ogitive radio etworks, as well as situatios where a exteral soure o oise leads to poor hael quality i a ertai regio. III. NETWORK MODEL I the assumed etwork model, odes are loated uiormly at radom i a uit area toroidal regio. Nodes use a ommo trasmissio rage r(). Itereree is modeled usig the Protool Model [8]. There are available haels o badwidth W eah. We ous o the ase where the total umber o available haels = O(log ). This is justiiable beause i large sale deploymets, the umber o odes will typially be muh larger tha the umber o available haels. Besides, whe = ω(log ), there is a large apaity degradatio eve with uostraied hael swithig (as show i []), thus makig haelizatio a ireasig liability, ad ostraied swithig may lead to additioal degradatio, ad potetially uaeptable perormae. As i [8], eah ode is soure o exatly oe low. It hooses a poit uiormly at radom (we shall reer to these poits as pseudo-destiatios throughout this paper), ad selets the ode (other tha itsel) lyig losest to that poit as its destiatio. IV. NOTATION AND TERMINOLOGY We use stadard asymptoti otatio [9]. Whe () = O(g()), ay utio h() =O( ()) is also O(g()). We ote reer to suh a situatio as h()=o( ()) = O(g()). We ote reer to results as holdig with high probability (w.h.p.), by whih we mea with probability as. As i [8], we say that the per low etwork throughput is λ() i eah low i the etwork a be guarateed a throughput o at least λ() with probability as. Wheever we use log without expliitly speiyig the base, we imply the atural logarithm. V. RELATED WORK It was show by Gupta ad Kumar [8] that or a siglehael sigle-iterae seario, i a arbitrary etwork, the per low apaity sales as Θ( W ) bit-m/s per low, while i a radom etwork, it sales as Θ( log ) bits/s. It was also show i [8] that i the available badwidth W is split ito haels, with eah ode havig a dediated iterae per hael, the results remai the same. The throughput-delay trade-o was studied i [0], ad it was show that the optimal trade-o is give by D() = Θ(T ()) where D() is delay, ad T () is throughput. The apaity o ultra-widebad (UWB) etworks was studied i [], ad [2]. I the multi-hael otext, a iterestig seario arises whe the umber o iteraes m at eah ode may be smaller tha the umber o available haels. This issue was aalyzed i [] ad it was show that the apaity results are a utio o the hael-to-iterae ratio m. It was also show that i the radom etwork ase, there are three distit apaity regios: whe m = O(log), the per-low apaity is W 786
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. (( W log, whe m = Ω(log) ad also O loglog log low apaity is Θ(W m ), ad whe m = Ω ( ) ) 2, the per ) ) 2, ( loglog log the per-low apaity is Θ( Wmloglog log ). Aother relevat body o work is that o bod perolatio i wireless etworks, e.g. [3]. The ostraied assigmets osidered by us also lead to odes withi rage beig able to ommuiate oly with a ertai probability. However, ulike perolatio, i our ase the probabilities are ot idepedet or all odes pairs. A multi-hael multi-hop etwork arhiteture has bee osidered i [4] i whih eah ode has a sigle traseiver, ad odes have a quieset hael to whih they tue whe ot trasmittig. A ode wishig to ommuiate with a destiatio tues to its quieset hael, ad trasmits the paket to a eighbor whose quieset hael is the same as that o the destiatio. Thereater, the paket proeeds towards the destiatio o the quieset hael. This has some similarity to our model ad ostrutios i that a low seeks to trasitio to a target destiatio hael (see Setios IX ad XIII or our ostrutios). However, i their ase, the trasitio a happe trivially at the very irst hop, sie the soure ode is always apable o tuig to the destiatio s quieset hael. I our models odes a oly swith o some haels, ad this eeds to be take ito aout. VI. UPPER BOUNDS ON CAPACITY Some geeral ostraits o the apaity o the etwork (or ay hael assigmet model) are as ollows: a) Soure-Destiatio Costrait or = : I =, but >, the a soure ad its destiatio should have the same hael or ommuiatio betwee them to be possible. This may ot always happe i the haels are assiged radomly. To illustrate, osider the lass o assigmet models where the assigmet to idividual odes is i.i.d. Suppose, Pr[i ad dst(i) share a hael ] p. I the trai model is suh that ay sigle ode a be the destiatio o oly upto D() lows, the we argue thus: We a obtai at least 2D() pairs with distit odes (thus leadig to idepedet probabilities). The probability that at least oe o the soure-destiatio pairs have dieret haels a be lower bouded by the probability that at least oe o these distit pairs do ot share a ommo hael, ad this is at least p ( ) 2D(). Whe log p = ω( 2D() ), it grows to, as. Thus, the etwork apaity would be 0. For the adjaet (, ) ad radom (, ) assigmets studied i this paper, this oditio holds whe >, ad so = whe > yields zero apaity. Whe >, as i the rest o this paper, this ostrait does ot apply. b) Coetivity Costrait: Suppose the eesary oditio or oetivity is that r()=ω(g()). Thus, the spatial re-use i the etwork is limited to O( ) ourret (g()) 2 trasmissios o ay sigle hael. Besides, eah souredestiatio is separated by average Θ() distae (see [8] or details) ad hee average Θ( r() ) hops. Thus per low throughput is limited to O( W r() ). ) Itereree ad Destiatio Bottleek Costrait: I [], it was established that the per low apaity is ostraied to O(W ), whe sigle-iterae odes a swith to ay hael. It was also show that i some ode a be the destiatio o upto D() lows, the per-low throughput is ostraied to be O( W D() ). These upper bouds also apply to the adjaet (, )-assigmet ase, sie whatever is ahievable with adjaet (, ) assigmet, is also ahievable whe odes a swith to ay hael. Note that sie we are oly iterested i the regio = O(log ), the oetivity ostrait is asymptotially domiat. VII. ADJACENT (, ) CHANNEL ASSIGNMENT Reall that i this model, the requey bad is divided ito haels umbered, 2,..., i order o ireasig requey, but a idividual ode a oly use haels (where 2 ). At deploymet time, eah ode is assiged a blok loatio i uiormly at radom rom,..., + ad thereater it a swith betwee the set i,...,i +. Thus, the probability that a ode is apable o swithig to hael i is give by p ad s j mi{i, i+,, +} (i) = +, sie hael i ours i mi{i, i +,, + } bloks, ad eah blok is radomly hose with probability +. Let us all haels with p ad s j (i) 2 the preerred haels. The, oe a see that, or ay set o otiguous haels, at least j 2 o the haels have pad s (i) 2. Hee, eah ode a swith o x 2 2 preerred haels. Also ote that o-preerred haels oly our at the riges o the requey bad. The probability that a ode with blok loatio i shares a hael with aother radomly hose ode is give by (+mi{i, }+mi{ + i, }) + p ad j (i)=. Sie blok loatios are hose uiormly at radom rom,..., +, the probability that two radomly hose odes share at least oe hael is give by: p ad j = It a be see that Thus, + + mi{, +} mi{, +} + p ad j i= + p ad j (i) mi{2, +} +. A. Neessary Coditio or Coetivity p ad j (i) () mi{2, +} +. A adaptatio o the proo tehiques used to obtai the eessary oditio or oetivity i [5], eables oe to hadle oetivity with adjaet (, ) assigmet. Theorem : With a adjaet (, ) hael assigmet (whe = O(log)), i p = mi{ 2 +,}, ad πr2 () = (log+b()) p, where b = lim b() < + the: lim ipr[ disoetio ] e b ( e b ) > 0 787
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. where by disoetio we imply the evet that there is a partitio o the etwork. The proo is omitted due to spae ostraits. Please see [7]. B. Suiiet Coditio or Coetivity It a be show that settig r() =a log,orsome suitable ostat a, suies or oetivity. This will be evidet rom our lower boud ostrutio or apaity, ad the proo is hee ot preseted separately. VIII. ADJACENT (, ) ASSIGNMENT: CAPACITY UPPER BOUND We proved that the eessary oditio or oetivity implies r()=ω( log ). The by the oetivity ostrait metioed i Setio VI, the per low throughput is limited to O(W log ) (reall that, as i [5], the disoetio evets osidered ivolved idividual odes gettig isolated, ad thus some soure ode would be uable to ommuiate with its destiatio). IX. ADJACENT (, ) ASSIGNMENT: CAPACITY LOWER BOUND We preset a ostrutive proo that ahieves Ω(W log ). This ostrutio has similarity to the ostrutios i [8], [0], ad [], but must ow hadle the ostrait that a ode may ot swith o all haels. The surae o the uit torus is divided ito square ells o area a() eah. The trasmissio rage r() is set to 8a(), thereby esurig that ay ode i a give ell is withi rage o ay other ode i ay adjoiig ell. Sie we utilize the Protool Model [8], a ode C a potetially iterere with a ogoig trasmissio rom ode A to ode B, oly i BC ( + )r(). Thus, a trasmissio by A i a give ell a oly be aeted by trasmissios i ells with some poit withi a distae (2 + )r() rom it, ad all suh ells must lie withi a irle o radius O(( + )r()). Sie is idepedet o, the umber o ells that iterere with a give ell is oly some ostat (say β). We hoose a()= 00log (i.e. r()= 800log ). Lemma : Suppose we are give a uit toroidal regio with poits loated uiormly at radom, ad the regio is sub-divided ito axis-parallel square ells o area a() eah. I a()= 00α()log, the eah ell, α() 00log has at least 00α()log 50log 50α()log poits ad at most 00α()log + 50log 50α()log poits, with probability at least 50log. Thus, by Lemma, the umber o odes i ay ell lies betwee 50log 50log. ad 50log with probability at least Lemma 2: I there are at least 50log odes i every ell D, the there are at least 2log odes i eah ell o eah o the preerred haels, with probability at least q, where q = O( ). 2 Lemma 3: I there are at least 50log odes i every ell D, the, or all adjaet preerred haels i ad i +, there are at least 2log odes i the ell havig both haels i ad i +, with probability at least q 2, where q 2 = O( ). 2 Lemma 4: I there are at least 50log odes i every ell, ad i i ad i + x are both preerred haels, where x 2, the there are at least 2log odes i the ell havig both haels i ad i + x, with probability at least q 3, where q 3 = O( ). 2 A. Routig Let us deote the soure o a low as S, the pseudodestiatio as D, ad the atual destiatio as D. I there were o ostraits o swithig, we ould have used a routig strategy similar to that i [8], i whih a low traverses the ells iterseted by the straight lie SD, ad thereater eeds to take at most oe extra-hop to reah the atual destiatio D, whih must eessarily lie either i the same ell as D or i oe o the 8 adjaet ells. I that were the ase, it a be laimed that: Lemma 5: The umber o SD D routes that traverse ay ell is O( a()). We shall hereater reer to this routig as straight-lie routig, as it basially omprises a straight-lie exept or the last hop. Lemma 6: No ode is the destiatio o more tha O(log ) = O(a()) lows. For adjaet (, ) assigmet, we aot stipulate that all lows be routed alog the (almost) straight-lie path SD D. This is beause the low is required to traverse a miimum umber o hops to be able to guaratee that it a swith rom soure hael to destiatio hael w.h.p. We elaborate urther o this issue. Chael Seletio ad Trasitio Strategy: Iitially, ater eah soure has hose a radom destiatio, the lows are proessed i tur ad eah is assiged a iitial soure hael, as well as a target destiatio hael. Suppose the soure S o a low is assiged hael set (i,...,i + ), while the destiatio D has ( j,..., j + ). The low hooses oe o the x preerred haels available at the soure uiormly at radom. Let us deote it by l. It also hooses oe o the y 2 preerred haels available at the destiatio (let us all it r) as the hael o whih the low reahes the destiatio. The destiatio hael hoie may be made i ay maer, e.g. we may make a i.i.d. hoie amogst all haels available at the destiatio. We assume, without loss o geerality, that l r. Suppose r l = k 2 + m(0 m < 2 ). Thus k = r l m 2 2 = 2( ) 4.Note that give two preerred haels l ad r all haels l i r must also eessarily be preerred. The, rom Lemma 4, it is always possible to trasitio rom l to r i at most k + steps: l l + 2,l + 2 l +2 2,...,l +k 2 l + k 2 + m = r. Thus, the route passes through a sequee o odes x,x 2,...x k suh that x ad x 2 share hael l, x 2 ad 788
S This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. Fig.. D P Illustratio o detour routig x 3 share hael l + 2 ad so o. Whe l r, the trasitios are o the orm l l 2,...,r. Thus, we stipulate that the straight-lie path be ollowed i either the hose soure ad destiatio haels are the same, or i the straight-lie segmet SD omprises h 4 itermediate hops. I S ad D (hee also D) lie lose to eah other, the hop-legth o the straight lie ell-to-ell path a be muh smaller. I this ase, a detour path is hose. Cosider a irle o radius 4 r() etered at S. Choose a poit o this irle, say P. I the osidered = O(log) regime, P a be ay poit o the irle. The the route is obtaied by traversig ells alog SP ad the PD D. This esures that the route has at least the miimum required hop-legth (provided by segmet SP). This situatio is illustrated i Fig.. A o-detour-routed low is iitially i a progress-osoure-hael mode, ad keeps to the soure hael till there are oly 4 itermediate hops let to the destiatio. At this poit, it eters trasitio mode, ad starts makig hael trasitios alog the remaiig hops, till it has trasitioed ito its hose destiatio hael. Thereater, it remais o that hael. Whe a low eters a ell i progress-osoure-hael mode, amogst all odes i that ell apable o swithig o that hael, it is assiged to the ode whih has the least umber o lows assiged to it o that hael so ar. A detour-routed low is always i trasitio mode. Lemma 7: Give that the high probability evet i Lemma 4 holds, suppose a low is o preerred soure hael i ad eeds to ially be o preerred destiatio hael j. The ater havig traversed h 4 + ells (reall that 2 ), it is guarateed to have made the trasitio. Lemma 8: The legth o ay route ireases by oly O( )= O(log) hops due to detour routig. The average route legth ireases by O(log ) hops. Lemma 9: I the umber o distit lows traversig ay ell is x with pure straight-lie routig, it is x + O( 2 r 2 ()) = 2 x + O(log 4 ) eve with detour routig. Lemma 0: The umber o distit lows traversig ay ell is O( a() eve with detour routig. Lemma : The umber o lows traversig ay ell i trasitio mode is O(log 4 B. Balaig Load withi a Cell Per-Chael Load: Reall that eah ell has O(a()) odes w.h.p., ad O( a()) lows traversig it w.h.p. Lemma 2: The umber o lows that eter ay ell o ay sigle hael is O( a() Lemma 3: The umber o lows that leave ay give ell o ay sigle hael is O( a() Per-Node Load: Lemma 4: The umber o lows that are assiged to ay oe ode i ay ell is O( a() C. Trasmissio Shedule As oted earlier, eah ell a ae itereree rom at most a ostat umber β o earby ells. Thus, i we osider the resultat ell-itereree graph, it has a hromati umber at most + β. We a hee ostrut a global shedule havig + β uit time slots i eah roud. I ay slot, i a ell is ative, the all itererig ells are iative. The ext issue is that o itra-ell shedulig. We eed to shedule trasmissios durig the ell s slot, so as to esure that at ay time istat, there is at most oe trasmissio o ay give hael i the ell. Besides, we also eed to esure that o ode is expeted to trasmit or reeive more tha oe paket at ay time istat. We use the ollowig proedure to obtai a itra-ell shedule: We ostrut a olit graph based o the odes i the ative ell, ad its adjaet ells (ote that the hop-seder o eah low shall lie i the ative ell, ad the hop-reeiver shall lie i oe o the adjaet ells), as ollows: we reate a separate vertex or eah low that requires a hop-trasmissio i the ell (ote that we outed possible repeat traversals by detour-routed lows separately i Lemma, ad ow a twietraversal a be treated like two distit lows or shedulig purposes). Sie the low has a assiged hael o whih it operates i that partiular hop, eah vertex i the graph has a impliit assoiated hael. Besides, eah low (ad hee its vertex) has a assoiated pair o odes orrespodig to the hop-edpoits. Two verties are oeted by a edge i () they have the same assoiated hael, or (2) at least oe o their assoiated odes is the same. The shedulig problem thus redues to obtaiig a vertex-olorig o this graph. I we have a vertex olorig, the it esures that () a ode is ever simultaeously sedig/reeivig or more tha oe low (2) o two lows o the same hael are ative simultaeously. The umber o eighbors o a graph vertex is upper bouded by the umber o lows eterig/leavig the ative ell o that hael, ad the umber o lows assiged to the low s two hop edpoits (both hop-seder ad hop-reeiver). Thus, it a be see rom Lemmas 2, 3 ad 4 that the degree o the olit graph is O( a() ). Sie ay graph with maximum degree d is vertex-olorable i at most d + olors, the olit graph a be olored i O( a() ) olors. 789
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. log Thus the ell-slot is divided ito O( a() )=O( ) equal legth subslots, ad all lows i the ell get a slot or trasmissio. This yields that eah low will get Ω(W log ) throughput. We thus obtai the ollowig theorem: Theorem 2: With a adjaet (, )-hael assigmet, the etwork apaity is Θ(W ) per low. log X. THE CASE OF UNTUNED RADIOS The utued hael model is as ollows: eah ode possesses a traseiver with arrier requey uiormly distributed i the rage (F,F 2 ), ad admits a spetral badwidth B. Let = F 2 F B. The is the maximum umber o disjoit haels that ould be possible. However the haels are utued ad hee partially ovelappig, rather tha disjoit. As per the assumptio i [3], two odes a ommuiate diretly i the arrier requey o oe is admitted by the other, i.e., i there is at least 50% overlap betwee two haels, ommuiatio is possible. We osider the issue o apaity o a radomly deployed etwork o odes, where eah ode has a utued radio, ad eah ode is the soure o oe low, with a radomly hose destiatio. Eve though eah ode oly possesses a sigle radio ad stays o a sigle sub-bad, due to the partial overlap betwee sub-bads, it is still possible to esure that ay pair o odes will be oeted via some path. Cotrast this to the ase o orthogoal haels, where we argued i Setio VI that whe =, ad >, some pairs o odes are disoeted rom eah other beause they do ot share a hael. It is possible to map the partial overlap eature o the utued hael ase to adjaet (2+2,3) ad (4+,2) assigmet. Note that = 2 allows or all odes to be oeted, eve with orthogoal haels. We map the utued radio seario to a seario havig (2 + 2, 3) adjaet hael assigmet. We perorm a virtual haelizatio o the bad (F,F 2 ) ito 2 orthogoal sub-bads. We add a additioal (virtual) sub-bad o the same width at eah ed o the bad, to get 2+2 orthogoal haels, umbered,...,2+2. Thus ad 2 + 2 are the artiiially added haels. I a radio s arrier requey lies withi virtual hael i, it is assoiated with virtual hael blok (i,i,i + ), ad i is alled its primary virtual hael. Thus the primary hael a oly be oe o, 2,..., 2 (sie the arrier requey a oly all i 2,.., 2 + ). I a ode s primary hael is i, it is apable o ommuiatig with all odes with primary virtual hael i 2 j i + 2 i the virtual haelizatio. I the atual situatio, the ode with the utued radio would be able to ommuiate with some subset o those odes. Thus, i a pair o odes aot ommuiate diretly i the virtual haelizatio, they aot do so i the atual situatio either, ad disoetio evets i the ormer are preserved i the latter. The probability that a ode has virtual hael blok ( j, j +, j + 2) is 2, i.e., the same as or adjaet (2 + 2, 3) assigmet, ad the eessary oditio or the (virtual) (2 + 2,3) assigmet otiues to hold or the orrespodig utued radio ase. This yields a upper boud o apaity o O(W log ). It a be show that a shedule ostruted or a adjaet (4 +,2) assigmet a be used almost as-is with utued radios (exept that the umber o subslots i the ell-slot must irease by a ator o to avoid itereree due to overlap). We perorm a virtual haelizatio o the bad (F,F 2 ) ito 4 + orthogoal sub-bads. I a radio s arrier requey lies withi virtual hael i, it is assoiated with virtual hael blok (i, i + ), ad i is alled its primary virtual hael. Note that i a ode s primary hael is i, itisalways apable o ommuiatig with all odes with primary virtual hael i j i +, but we will preted that it a oly ommuiate with those havig i or i +. Thus, i a pair o odes share a hael i the virtual haelizatio, the they are always apable o diret ommuiatio i the atual utued radio situatio. The probability that a radio has virtual hael blok (i,i + ) is 4, same as or adjaet (4 +,2) assigmet. I the adjaet (4 +,2) assigmet, all hael are orthogoal ad a operate ourretly. With utued radios, we assume two odes a iterere i there is some spetral overlap. Thus, a trasmissio by a ode o arrier requey F a iterere with trasmissios by odes with arrier requey i the rage (F B, F + B). Hee, the trasmissio shedule or utued radios is made to ollow the additioal ostrait that i a ode with primary virtual hael i is ative the o ode with primary hael i 5 j i + 5 should be ative simultaeously. This would derease apaity by a ator o, but would ot aet the order o the asymptoti results. Also, i the atual etwork ivolvig utued radios, a traseiver a use upto B = F 2 F spetral badwidth, while i the adjaet (4 +,2) ase, it would be F 2 F 4+, leadig to the possibility o havig a higher data-rate i the ormer, give the same trasmissio power, modulatio, et. However this a oly aet apaity by a small ostat ator, whih does ot aet the order o the results. I the adjaet (4 +,2) ase, our ostrutio perorms trasitios to esure that a soure o haels (i,i + ) ad a destiatio o haels (i + j,i + j + ) a ommuiate. I the utued radio ase, trasitioig is doe through odes that provide the required virtual hael pair, ad the same trasitio strategy as or (4 +, 2) assigmet otiues to work. Hee the apaity is Ω(W log ) per low. We re-emphasize that eve though =, the utued ature o the radios allows or a progressive shit i the requey over whih the paket gets trasmitted, thereby allowig a step-by-step trasitio rom the soure s arrier requey to a requey admitted by the destiatio. The adjaet (, ) model aptures this progressive requey-shit harateristi, ad is thus able to model the utued radio situatio. From the upper ad lower bouds proved i this setio, it ollows that the apaity o the utued radio etwork, whe 790
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. = O(log), isθ(w log ) per low. XI. RANDOM (, ) ASSIGNMENT I this assigmet model, a ode is assiged a subset o haels uiormly at radom rom the set o all possible hael subsets o size. Thus the probability that a ode is apable o swithig o a give hael i is p rd s (i) = = p rd s, i, ad the probability that two odes share at least oe hael is give by p rd = ( )( )...( + ). A. Neessary Coditio or Coetivity Theorem 3: With a radom (, ) hael assigmet (whe = O(log)), i πr 2 ()= (log+b()) p, where p = p rd = ( )( )...( b = lim b() < + the: + ), ad = O(log), ad lim ipr[ disoetio ] e b ( e b ) > 0 where by disoetio we imply the evet that there is a partitio o the etwork. B. Suiiet Coditio or Coetivity Theorem 4: With radom (, ) assigmet (whe = O(log)), i πr 2 ()= 800πlog p rd, the: Pr[ etwork is oeted ] Proo: We preset a ostrutio based o a otio o per-ode bakboes. Cosider a subdivisio o the toroidal uit area ito square ells o area a() = 00log p rd. The by settig α()= p rd i Lemma there are at least 50log p rd odes i eah ell with high probability. Set r()= 8a(). The a ode i ay give ell has all odes i adjaet ells withi its rage. Withi eah ell, hoose 2log p rd odes uiormly at radom, ad set them apart as trasitio ailitators (the meaig o this term shall beome lear later). This leaves 48log at least p rd odes i eah ell that a at as bakboe adidates. Cosider ay ode i ay give ell. The probability that it a ommuiate to ay other radom ode i its rage is p rd. The the probability that i a adjaet ell, there is o bakboe adidate ode with whih it a ommuiate is less tha ( p rd ) 48log p rd =. The probability e 48log that a give ode aot ommuiate 48 with ay ode i 8 some adjaet ell is thus at most (as there are upto 8 adjaet ells per ode). By applyig 48 the uio boud over all odes, the probability that at least oe ode is uable to ommuiate with ay bakboe adidate ode i at least 8 oe o its adjaet ells is at most. We assoiate with eah ode x a 47 set o odes B(x) alled the primary bakboe or x. B(x) is ostituted as ollows. Throughout the proedure, ells that are already overed by the uder-ostrutio bakboe are reerred to as illed ells. x is by deault a member o B(x), ad its ell is the irst illed ell. From eah adjaet ell, amogst all bakboe adidate odes sharig at least oe ommo hael with x, oe is hose uiormly at radom is added to B(x). Thereater, rom eah ell borderig a illed ell, o all odes sharig at least oe ommo hael with some ode already i B(x), oe is hose uiormly at radom, ad is added to B(x); the ell gets added to the set o illed ells. This proess otiues iteratively, till there is oe ode rom every ell i B(x). From our earlier observatios, or all odes x, B(x) evetually overs all ells with probability at least 8. Now osider ay pair o odes x ad 47 y. IB(x) B(y) φ, i.e., the two bakboes have a ommo ode, the x ad y are obviously oeted, as oe a proeed rom x o B(x) towards oe o the itersetio odes, ad thee to y o B(y), ad vie-versa. Suppose, the two bakboes are disjoit. The x ad y are still oeted i there is some ell suh that the member o B(x) i that ell (let us all it q x ) a ommuiate with the member o B(y) i that ell (let us all it q y ), either diretly, or through a third ode. q x ad q y a ommuiate diretly with probability i they share a ommo hael. Thus the ase o iterest is oe i whih o ell has q x ad q y sharig a hael. I they do ot share a ommo hael, we osider the evet that there exists a third ode z amogst the trasitio ailitators i the ell through whom they a ommuiate. Note that, or two give bakboes B(x) ad B(y), the probability that i a etwork ell, give q x ad q y that do ot share a hael, they a both ommuiate with a third ode z that did ot partiipate i bakboe ormatio ad is kow to lie i the same ell, is idepedet aross ells. Thereore, the overall probability a be lower-bouded by obtaiig or oe ell the probability o q x ad q y ommuiatig via a third ode z, give they have o ommo hael, osiderig that eah ell has at least 2log p rd possibilities or z, ad treatig it as idepedet aross ells. We elaborate this urther. Let q x have the set o haels C(q x )={ x,..., x }, ad q y have the set o haels C(q y )={ y,..., y }, suh that C(q x ) C(q y )=φ. Cosider a third ode z amogst the trasitio ailitators i the same ell as q x ad q y. We desire z to have at least oe hael ommo with both C(q x ) ad C(q y ). The let us merely osider the possibility that z eumerates its haels i some order, ad the ispets the irst two haels, hekig the irst oe or membership i C(q x ), ad hekig the seod oe or membership i ( )( ) C(q y ). This probability is > 2. Thus q 2 x ad q y a ommuiate through z with probability p z > 2 = Ω( ). 2 log 2 There are 2log p rd possibilities or z withi that ell, ad all the possible z odes have i.i.d hael assigmets. Thus, the probability that q x ad q y aot ommuiate through ay z i the ell is at most ( p z ) 2log p rd, ad the probability they a ideed do so is p xy > ( p z ) 2log p rd. Thus, the probability that this happes i oe o the p rd 00log ells is at most ( p xy) prd p rd 00log e Ω( log 2 ) 00log < ( p z ) 2log a() = p rd p rd 00log < ( ) 2log 2 p rd (reall that = O(log)). ( Applyig uio boud over all 2) < 2 2 ode pairs, the probability that some pair o odes are ot oeted is at 79
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. most 2 e Ω( log 2 ) 2 < 2 e Ω( log 2 )+2log 0. Thus the probability o a oeted etwork overges to. XII. RANDOM (, ) ASSIGNMENT: CAPACITY UPPER BOUND Sie the eessary oditio or oetivity requires that r()=ω( log p rd ), the per low apaity is O(W prd log ) rom the disussio o the oetivity upper boud i Setio VI. XIII. RANDOM (, ) ASSIGNMENT: CAPACITY LOWER BOUND We preset a ostrutive proo that ahieves Ω(W log ). This ostrutio is quite similar to that or adjaet (, ) assigmet. The surae o the uit torus is divided ito square ells o area a() eah. The trasmissio rage is set to 8a(), thereby esurig that ay ode i a give ell is withi rage o ay other ode i ay adjoiig ell. As disussed or the adjaet assigmet ase, the umber o ells that iterere with a give ell is oly some ostat (say β). We hoose a()= 00log (resultatly 800log r()= ). Thus, Lemma applies or this ase too. Lemma 5: I there are 50log odes i every ell, the there are at least 25log odes i eah ell o eah o the haels, with probability at least q, where q = O( ). 4 A. Routig Observe that Lemmas 5 ad 6 stated i Setio IX or SD D routig are appliable here too. I ase o radom (, ) assigmet, as with adjaet assigmet, we aot stipulate that all lows be routed alog the straight-lie path SD D. A low may be required to traverse a miimum umber o hops to be able to esure that it will id a opportuity to make the swith rom soure hael to destiatio hael. Chael Seletio ad Trasitio Strategy: Iitially, ater eah soure has hose a radom destiatio, the lows are proessed i tur ad eah is assiged a iitial soure hael, as well as a target destiatio hael. The soure hael or a low origiatig at ode S is hose aordig to the uiorm distributio rom the haels available at S. The destiatio hael may be hose rom amogst the haels available at destiatio D i ay maer, e.g., it may be the oe with the smallest umber o iomig lows assiged to it so ar. We stipulate that a o-detour-routed low is iitially i a progress-o-soure-hael mode, ad keeps to the soure hael till there are oly 25 4 itermediate hops let to the destiatio. At this poit, it eters a ready-or-trasitio mode, ad atively seeks opportuities to make a hael trasitio alog the remaiig hops. It makes use o the irst opportuity that presets itsel, i.e., i a ode i a o-route ell provides the soure-destiatio hael pair, the low is assiged to that ode or relayig (the ode reeived it o the soure hael, ad orwards it o the destiatio hael). Oe it has made the trasitio, it remais o the destiatio hael. Durig the progress-o-soure-hael phase, the ext hop ode is hose to be the ode i the ext ell whih has the smallest umber o lows assiged so ar o that hael, amogst all odes that a swith o the soure hael. I the ready-or-trasitio phase, it may be assiged to ay eligible ode that provides either the trasitio opportuity, or the soure hael (or lows yet to id a trasitio), or the destiatio hael (or lows that have already trasitioed ito their destiatio hael). A detour-routed low is always i ready-or-trasitio mode. Lemma 6: Suppose a low is o soure hael i ad eeds to ially be o destiatio hael j. The ater havig traversed h 2( ) ( ) distit ells (reall that 2, a hee h = O(log)), it will have oud a opportuity to make the trasitio w.h.p. 2( ) 25( ) 4 Note that 25. Thus, the (almost) straight-lie SD D path is ollowed i either soure ad destiatio haels are the same, or i the straight-lie segmet SD provides h 25 4 itermediate hops. I S ad D (hee also D) lie lose to eah other, the hop-legth o the straight lie ell-toell path a be muh smaller. I this ase, a detour path is hose. Cosider a irle o radius 25 4 r() etered at S. Choose ay poit o this irle, say P, so log as P does ot lie i the same ell as D (this guaratees at least oe itermediate 4 hop eve i 25 ). The the route is obtaied by traversig ells alog SP ad the PD. This esures that the route has at least the miimum required hop-legth (sie the segmet SP always provides at heast 25 4 distit hops(ells). This situatio is illustrated i Fig.. Lemma 7: The umber o distit lows traversig ay ell is O( a()) eve with detour routig. Lemma 8: The umber o lows traversig ay ell i ready-or-trasitio mode is O(log 4 B. Balaig Load withi a Cell Per-Chael Load: Reall that eah ell has O(a()) odes w.h.p., ad O( a()) lows traversig it w.h.p. Lemma 9: The umber o lows that eter ay ell o ay sigle hael is O( a() Lemma 20: The umber o lows that leave ay give ell o ay sigle hael is O( a() Per-Node Load: Lemma 2: The umber o lows that are assiged to ay oe ode i ay ell is O( a() C. Trasmissio Shedule The trasmissio shedule is obtaied i a maer similar to Setio IX-C. First, we obtai a global iter-ell shedule, ad the ostrut a olit graph or itra-ell shedulig. Thus, it a be see rom Lemmas 9, 20 ad 2 that the degree o the olit graph is O( a() ). Thus the graph a be olored i O( a() ) olors. Thus the ell-slot is divided 792
This ull text paper was peer reviewed at the diretio o IEEE Commuiatios Soiety subjet matter experts or publiatio i the IEEE INFOCOM 2007 proeedigs. log ito O( a() )=O( ) equal legth subslots, ad all traversig lows get aslot or trasmissio. This yields that eah low will get Ω( logw) throughput. We thus obtai the ollowig theorem: Theorem 5: With a radom (, ) hael assigmet, the desribed ostrutio ahieves throughput o Ω(W log ) per low. XIV. DISCUSSION The lower boud ostrutios or the two assigmet models yield iterestig isights. As is ituitive, whe all odes aot swith o all haels, the trasmissio rage eeds to be larger to preserve etwork oetivity, leadig to a apaity degradatio. Also, it may o loger be possible to use the shortest route towards the destiatio, ad a low may eed to take a iruitous path (detour routig) i order to esure that the destiatio is reahed. However, whe the umber o haels is muh smaller tha the umber o odes, the irease i the legth o the routes is ot asymptotially sigiiat. Takig all ators ito aout, whe = O(log), givea suiietly dese etwork, it is beeiial to attempt to use all haels by assigig dieret hael subsets to dieret odes, rather tha ollow the aive approah o usig the same haels at all odes. I the latter ase, the per-low apaity would be redued to Θ(W ). Thus the use-all-haels log approah outperorms the -ommo-haels approah by a ator o. As a example, eve whe = 2, utilizig all haels yields a apaity o the order o haels. As metioed earlier, we have reetly obtaied ew results [6] showig that radom (, ) apaity is Θ(W prd log ), whih overges muh aster to the uostraied apaity. It is also to be oted that whe =, our models redue to the uostraied swithig model i [] with a sigle iterae per ode. For this ase, our per-low apaity results yield Θ( W log ), as also obtaied i [] or m = O(log). However, we are able to ahieve the optimal apaity by usig a muh simpler radom low-hael mappig. We also ote that the tehiques usig radom low-hael assigmet ad detour routig, whih were devised or the models i this paper, a be applied to other situatios, e.g., the determiisti ixed assigmet osidered i [6]. Aother iterestig isight is yielded by the results or radom (, ) assigmet. Note that a trasmissio rage o log Θ( p rd ) is both eessary ad suiiet or oetivity. However, at this trasmissio rage, it is possible that some ells may have some haels missig. Thus, the subgraph idued by a ertai hael (obtaied by retaiig oly odes apable o swithig o that hael, ad assumig this is the oly hael they a use) may ot eessarily be oeted, but the overall etwork graph is always oeted at this trasmissio rage. This may perhaps at times make it eessary (due to oetivity oers) to shedule dieret liks o a low o dieret haels, eve i the soure ad destiatio log share a hael. Note that i we set r()=θ( ), the a soure-destiatio pair that share a hael always have a route with all liks usig that hael (though it is ot apaity-optimal to use it with radom (, ) assigmet), sie eah hael is available o some odes i eah ell. XV. CONCLUSION I this paper we have preseted a ase or the study o multi-hael etworks with hael swithig ostraits. We itrodued some models or hael swithig ostraits, ad preseted oetivity ad apaity results or two suh models, viz. adjaet (, ) assigmet, ad radom (, )- assigmet, whe = O(log ). While origially derived or haelizatio i the requey domai, our results a also be iterpreted i the time domai, ad provide isights about eergy-apaity trade-os i etworks with low-duty-yle odes. Furthermore, we believe that there is sigiiat potetial or extesio o the urret models, as well as study o a wider rage o swithig ostraits. REFERENCES [] P. Kyasaur ad N. H. Vaidya, Capaity o multi-hael wireless etworks: impat o umber o haels ad iteraes, i Pro. o MobiCom 05. 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