Capacity of power costraied ad-hoc etworks Rohit Negi, Arjua Rajeswara egi@ece.cmu.edu, arjua@cmu.edu Departmet of Electrical ad Computer Egieerig Caregie Mello Uiversity Abstract hroughput capacity is a critical parameter for the desig ad evaluatio of ad-hoc wireless etworks. Cosider idetical radomly located odes, o a uit area, formig a adhoc wireless etwork. Assumig a fixed per ode trasmissio capability of bits per secod at a fixed rage, it has bee show that the uiform throughput capacity per ode r is Θ. log We cosider a alterate commuicatio model, with each ode costraied to a maximum trasmit power P 0 ad capable of utilizig W Hz of badwidth. Uder the limitig case W, such as i ultra wide bad etworks, the uiform throughput per ode is O log α upper boud ad Ω α / log α+/ achievable lower boud. hese bouds demostrate that throughput icreases with ode desity, i cotrast to previously published results. his is the result of the large badwidth, ad the assumed power ad rate adaptatio, which alleviate iterferece. hus, the sigificace of physical layer properties o the capacity of ad-hoc wireless etworks is demostrated. Keywords: Iformatio theory, ad-hoc etwork, etwork capacity, throughput, ultra-wide bad. I. INRODUCION Recetly, it has bee show [] that the uiform throughput capacity per ode of a ad hoc wireless etwork with odes decreases with as Θ. However, i this paper, log we show that uder a Ultra-Wide Bad UWB [] commuicatio model large badwidth, limited power, the uiform throughput capacity per ode icreases as Θ α /, where α is the distace loss expoet. he otatio used is elaborated later i the paper. hus, this paper shows that the assumptios about the physical layer of a ad hoc etwork ca dramatically affect the etwork capacity. Wireless commuicatio etworks cosist of odes that commuicate with each other over a wireless chael. Some wireless etworks have a wired ifrastructure of cotrollers, with odes coected to a cotroller over a wireless lik. Other etworks, such as ad-hoc etworks [3], are all wireless. Some saliet features of ad hoc etworks are speedy deploymet, low cost ad low maiteace. hese features led towards applicatios such as sesor etworks or military systems. he lack of cetralized cotrol is also advatageous for short-lived commercial etworks. Ad-hoc etwork desig is burdeed with the issues of schedulig at the lik layer ad relayig of data packets routig at the etwork layer. he wireless medium or chael is a resource shared amogst various odes. Schedulig or Medium Access Cotrol MAC is the process of providig access of the chael to the competig odes. he broadcast ature of the wireless medium ad the decetralized ature of ad-hoc etworks makes this schedulig problem very differet from that i ifrastructure etworks. Iterferece mitigatig techiques ad distributed protocols have bee cosidered i this regard [4],[5],[6]. Routig is the fuctioality of trasportig data from the source to the destiatio across a sequece of liks. I adhoc etworks, routig faces a umber of challeges, icludig the variability i topology due to the ureliable wireless lik [7] ad ode mobility. hese issues differ sigificatly from their couterparts i both cellular etworks ad wired etworks such as the Iteret, ad have bee extesively studied [8],[9],[0]. Recetly, there has bee sigificat iterest i computig the capacity of ad-hoc etworks [],[],[],[3]. Cosider a ad-hoc wireless etwork, where idetical odes o a uit area commuicate over a wireless chael, with possible cooperatio, to relay traffic. Assume that each lik operates at a fixed data rate, utilizig a fiite badwidth ad large power i.e., sigal-to-oise ratio. Uder these assumptios, it was show i [], that the per ode throughput capacity decreases as a fuctio of the umber of odes i a static etwork. Referece [] exploited ode mobility, to demostrate that a costat per ode throughput could be achieved i a mobile etwork. he issue of delay was addressed i [3], where the assumed mobility model was aalyzed to provide bouds o the delay experieced by the packets. he throughput icrease, by the utilizatio of directioal ateas, was studied i [4]. Simulatio-based results o the capacity of small adhoc etworks have bee show i [5]. Curretly deployed commercial wireless etworks are built usig either arrowbad e.g., cellular GSM or widebad e.g., 3G, 80. liks. However, the demad for higher data rates at short distaces has created a market for ultra-wide bad liks. UWB radios were desiged for covert military applicatios, as a spread spectrum techology that used a large amout of badwidth at extremely low powers, thus differetiatig them from arrowbad ad widebad radios. However, recet chages i U.S. federal regulatios have opeed up UWB for commercial applicatios. hus, there are curretly itese research ad developmet efforts uderway,
to desig ad stadardize commercial UWB radios [6], [7], [8]. For example, the UWB based IEEE 80.5.3.a stadard [9] is expected to support 00 to 500 Mbps depedig o the lik distace. Further, UWB radios will be iexpesive ad low power, makig them ideal for ad hoc wireless applicatios. Commercial deploymet of UWB etworks is expected to occur i the ear future [0]. Sice UWB radios possess properties extremely large badwidths ad low power that are drastically differet from existig commercial radios fiite badwidth ad large power, the questio arises whether the existig results o ad hoc etwork capacity are applicable to UWB etworks. Note that a system with ifiite badwidth does ot imply a arbitrarily large lik capacity, because of the fiite power costrait See Sectio IV. Specifically, i this paper, we assume a UWB commuicatio model where each lik operates over a relatively large badwidth W ad with a costrait P 0 o the maximum power of trasmissio. he ambiet Gaussia oise power spectral desity is ad the sigal power loss, with distace d, is/d α.hereα, is the distace-loss expoet []. Nodes are assumed to remai static. his commuicatio model is aalyzed to address the questio of etwork capacity. Upper ad lower bouds are derived, demostratig that the etwork capacity is a icreasig fuctio of ode desity, as Θ α /, where Θ stads for soft order, i cotrast to [], which shows a decreasig fuctio for capacity. As argued i Sectio III, this result is equally valid for low power, low data rate sesor etworks, although they may ot use a large badwidth. he rest of the paper is orgaized as follows. I Sectio II the required backgroud is reviewed. A short ituitive review of the relevat assumptios ad proofs from [] is preseted. he cocept of uiform covergece is discussed. Sectio III presets the UWB commuicatio model assumed, motivatig the eed for a aalysis that is differet from existig literature. I Sectio IV, importat characteristics of the alterate commuicatio model are preseted ad are used to show that the optimal MAC is Code Divisio Multiple Access CDMA. Sectio V derives a upper boud o the uiform throughput capacity. his is achieved by aalyzig a optimal routig scheme, uder a relaxed power costrait. I Sectio VI, a lower boud is derived by applyig some results of [] to the ew commuicatio model. It is thus show that the capacity r is Θ α /, i cotrast to [] where r is Θ. Badwidth scalig is addressed log i Sectio VII as a guidelie for practical implemetatios. Fially, coclusios are preseted i Sectio VIII. II. BACKGROUND he objective of this paper is to demostrate the effect of the UWB physical layer o ad hoc etwork capacity. his requires cotrastig the assumptios ad results of [] with our Here O,Ω ad Θ are the stadard order bouds. he soft order Θ is the same as a Θ boud with the powers of log eglected. results. herefore, for the sake of completeess, this sectio reviews the assumptios ad methods used i [], that are most relevat for such a compariso. Readers familiar with [] may skip this sectio. A. Network Model he assumed etwork model i [], relevat to this paper, is oe of Radom Networks, where the static odes o a uit area are i.i.d. idepedet ad idetically distributed ad distributed uiformly. o avoid edge effects, this uit area is cosidered to be o the surface of a sphere S. he odes commuicate over a wireless chael, with possible cooperatio, to relay traffic. Each of the odes has a idepedet radomly chose destiatio chose as the ode closest to a radom poit, i.e. uiformly ad idepedetly distributed. B. Performace Metric All odes require to sed traffic at a rate of r bits per secod to their correspodig destiatios. A uiform throughput r is feasible if there exists a schedulig ad relayig scheme by which every source-destiatio pair ca commuicate at a time-average of r bits per secod. he maximum feasible uiform throughput is the uiform throughput capacity, ad is the metric of choice. he motivatio for choosig this metric is a sese of fairess, sice all odes are assumed to be homogeeous i their capabilities ad requiremets. Sice the uderlyig etwork is radom, so is the capacity. he aim is to boud the radom capacity by fuctios of. hus, bouds are show that hold with w.h.p. the abbreviatio w.h.p. represets with high probability, i.e., with probability approachig as. Specifically, the uiform throughput capacity r is said to be of order Θf if there exists determiistic costats k >k 0 > 0 s.t. lim 0f is feasible = lim f is feasible <. k i,k i,c i,c i will be used to deote costats, with respect to. C. Commuicatio Model: I [], it was assumed that each ode ca trasmit at a rate of bits/s. he homogeeous odes share a commo rage equivaletly, power of trasmissio tr. his simplistic commuicatio model assumes that all operatig liks trasport data at a costat rate. he trasmissio by a legitimate trasmitter X i to its iteded receiver X j is successful, if their distaces are related as, X k X j + X i X j for every other X k trasmittig simultaeously. his iterferece criterio models a protocol, which specifies a guard zoe
, aroud the receiver where o other ode may trasmit, ad is termed as the protocol model. A secod model, more directly related to physical layer desig, works with the Sigal to Iterferece Noise Ratio at the receiver, with the assumptio of arbitrarily large. Note that the assumptio of arbitrarily large power is i sharp cotrast to our UWB model. he order results remai the same uder both the models. For simplicity this review assumes the protocol model. rasmissio Radius D. Network throughput With these assumptios it is prove i [] that the capacity of radom ad-hoc etworks r is Θ. 3 log hus the throughput per ode decreases with icreasig ode desity. he essetial reaso for this capacity decrease is the requiremet for all odes to share the wireless chael locally. his may be demostrated by a cotrast betwee the MAC ad routig requiremets. he mea source-destiatio distace is assumed to be L. he mea umber of hops take by packets L tr is. he total traffic geerated by all odes due to the L multi-hop relayig routig is r tr bits/s. his traffic is required to be served by all odes. However, the capacity of each ode is reduced by iterferece MAC, sice odes close to a receiver caot trasmit simultaeously. he iterferece radius is proportioal to the trasmissio radius. Sice the umber of iterferig odes is proportioal to the iterferece area uiform distributio, the capacity loss is quadratic i the trasmissio radius tr. hus, the available capacity reduces from to tr. he tradeoff betwee the routig requiremet ad the MAC restrictio yields that the capacity of the etwork r k tr L. 4 A lower limit o tr, due to the requiremet of etwork log coectivity, has bee derived as tr π.helower limit o tr esures that o ode i the etwork is isolated w.h.p.. his is required sice the performace metric is oe of uiform throughput capacity, which would be zero if there was eve a sigle starved ode. he applicatio of this limit to 4 results i the capacity upper boud as r k log. 5 E. Lower boud o provide a capacity lower boud feasibility, specific MAC ad routig schemes are chose. hese schemes achieve the same order of capacity as the upper boud. he lower ad upper boud prove the order boud 3 o r. o elaborate, it is required to specify schemes for both the MAC ad the routig o this radom etwork. Such a specificatio requires some structure o the radom etwork. Fig.. Routes Vorooi essellatio Motivated by cellular architectures, a tessellatio coverig by cells of the uit area is cosidered. Regularity i the tessellatio i the properties of every cell ecessitates a regular cell shape. However, sice the etwork is radom, some deviatio from the regular cell shape should be allowed to esure that the tessellatio may be made as fie as required. A Vorooi tessellatio V : of the surface S is a tessellatio that has the desired properties. he tessellatio such as i Figure has the followig properties : 00 log Every Vorooi cell V cotais a disk of area ad correspodig radius ρ. Every Vorooi cell is fully cotaied withi a circle of radius ρ. he existece of such a tessellatio was proved i []. he size of each cell, relative to the umber of odes, is importat. It is required that every cell cotai at-least oe ode, to esure that the routig scheme described below is feasible. It is this uiformity restrictio that results i the choice of cell size. he resultig cellular-like architecture imparts some otio of regularity o the radom etwork. MAC ad available capacity: With the Vorooi tessellatio, a MAC is defied that achieves a schedulig betwee the cells. he MAC esures that trasmissios from a cell do ot iterfere with trasmissios i simultaeously trasmittig cells. he radius of trasmissio is chose to be tr =8ρ to allow for direct trasmissios betwee adjacet cells cells sharig a poit ad withi a cell Figure. Cells cotaiig odes withi a distace of + tr are iterferig cells, sice a ode i oe cell may iterfere with the trasmissio i the iterferig cell. he distace betwee two odes i iterferig cells is upper bouded. Also, the area of each iterferig cell is lower bouded, by the tessellatio properties. he ratio of the maximum iterferece area ad the miimum
area of each cell is a costat, k 3. hus for every cell i the tessellatio the total umber of iterferig cells may be upper bouded by a costat k 3 which depeds oly o the parameter of the iterferece model. Cosequetly the graph defied by iterferece amogst cells, has a bouded degree of k 3. he chose MAC is a schedule of +k 3 slots, i which each cell is assiged oe slot to trasmit. his is possible sice a graph with degree ot greater tha k 3 may be colored by +k 3 colors []. hus, the cellular-like architecture is utilized to achieve a simple slotted MAC amogst the cells. herefore each cell has a available capacity of available capacity = bits/s. 6 +k 3 Routig: every source destiatio pair may be coected by a straight lie segmet segmet of a great circle o S, as i Figure. he packet routig scheme employed is as follows. Packets origiatig from a source are relayed from the cell cotaiig the source to the cell cotaiig the destiatio i a sequece of hops. I each hop, the packet is trasferred from oe cell to aother, i the order i which cells itersect the straight lie segmet coectig the source ad destiatio. A ode head/relay ode is chose radomly i each cell to relay all traffic. Withi a cell all sources sed traffic to the head ode ad destiatios receive traffic from the head ode. his choice of routig is idepedet of the MAC ad hece the two are aalyzed separately. o make relayig of traffic betwee cells feasible, it is required that every cell uiformity i the tessellatio V cotais at least oe ode w.h.p.. Asimple uio boud of the probabilities that every cell cotais at least oe ode is isufficiet, ad hece, a more itricate techique is required to provide this uiformity. he appedix reviews Vapik-Chervoekis VC theory, which provides the required uiform covergece i the aalysis of the uiform throughput capacity. Routig ad traffic to be carried: he VC theorem 33 obtaied from VC theory may be applied to the tessellatio of the etwork. A set of disks of fixed area have a VC dimesio of three. Each cell i the tessellatio cotais a disk of radius ρ ad is cotaied i a disk of radius ρ.he applicatio of the VC theorem to the set of disks cotaied i the cells yields Prob sup VɛV NV 00 log 50 log > 50 log, 7 where NV is the umber of odes i cell V.heresult applied to both the sets of disks implies that w.h.p., the etwork G is such that for every Vorooi cell i the tessellatio, the umber of odes per cell obeys 50 log NV 50 log. his result allows for the viability of the routig scheme, by guarateeig a ode i every cell that ca serve as the head ode, ad justifies the choice of cell size as 00 log. he traffic geerated due to this routig scheme is cosidered. he radom sequece of straight-lie segmets is i.i.d ad hece the weak law of large umbers may be applied to the routes which approximate these lie-segmets. he traffic to be carried by a cell is proportioal to the umber of straight lie segmets passig through the cell. he umber of routes itersectig every cell maybe bouded w.h.p.. hus, the traffic to be carried by every cell ca be upper bouded w.h.p. as sup raffic carried by cell V VɛV k 4 r log. 8 Bouds: he lower boud is derived by costraiig the traffic to be carried 8, obtaied from the routig requiremets, to be less tha the available capacity 6, obtaied from the MAC costrait. hus for radom etworks r = k 5 9 log bits/s is feasible w.h.p.. he upper boud, obtaied by the requiremet for coectivity, also preseted the same order 5 ad hece r is of order Θ. log III. POWER CONSRAINED NEWORKS Now we address the capacity of power costraied radom ad-hoc etworks. I cotrast to existig literature, the followig UWB Commuicatio Model is assumed: Power: Each ode is costraied to a maximum trasmit power of P 0. Badwidth : he uderlyig commuicatio system has a arbitrarily large badwidth W. he key characteristic of such a model is the low spectral P efficiecy i.e., 0 W, which implies a relatively large badwidth []. he results of this paper are applicable to all systems that have a low spectral efficiecy. hus, i particular, our results hold for two practical applicatios, UWB systems, where the badwidth used is of the order of a few GHz, such as i the IEEE 80.5.3a stadard [9]. For such a system, W, which implies that P 0 W. As metioed i the Itroductio, such systems are actively beig cosidered for commercial deploymet. Sesor etworks, with badwidths of the order of a few MHz or less, but which use very low power devices to exted battery life. For such a low power system, P 0 P 0 W 0, which implies that. Such etworks are beig cosidered i both military as well as commercial applicatios [3]. As oted i Sectio I, a ambiet Gaussia oise power spectral desity of ad a sigal power loss of /d α, with distace d, is assumed. Here α, is the distace loss expoet. Shadowig effects are ot cosidered i this model. Capacity-achievig Gaussia chael codes are assumed for
each lik. hus, each lik is assumed to support a data rate correspodig to the Shao capacity of that lik [4]. It is assumed that each ode ca trasmit ad receive simultaeously although this restrictio does ot affect the results, as show later. Also, each ode ca cotrol its trasmit power, as well as adapt its data rate to the lik coditio [5], [6]. Every ode may trasmit or receive, ad wishes to commuicate with a radomly chose destiatio chose as the ode closest to a radomly chose poit. It has bee show i [] that r, the uiform throughput capacity per ode, is a decreasig fuctio of 3, uder a simplistic fixed per-lik data rate. I our commuicatio model, the costraits o power ad badwidth are differet from [], the lik capacity explicitly depeds o distace, ad each lik is allowed to adapt its power ad rate. herefore, the results of [] are ot applicable i our case. hus, i cotrast, we show i this paper that uder the ew commuicatio model, the capacity of the ad-hoc etwork icreases as a fuctio of the ode desity! o demostrate this result, the characteristics of the commuicatio model are studied. his icludes a presetatio of the optimal MAC, followed by a aalysis of the routig problem that provides the required upper ad lower bouds o the uiform throughput capacity. IV. OPIMALIY OF CDMA MAC he iterferece problem i the ad hoc etwork is first addressed. It is show that the iterferece perceived by a receiver is bouded w.h.p., ad hece, a certai scalig of badwidth W, as a fuctio of, reders the iterferece egligible. his, implies that uder the limitig badwidth assumptio, a CDMA MAC scheme is optimal. i.e., all trasmitters trasmit at the same time, usig the etire badwidth. Here optimal is used i compariso to time/frequecy schedulig schemes as oted subsequetly. A. Badwidth Scalig Let X i deote the ode ad its positio. Let P ij 0 be the trasmit power chose by ode X i to trasmit to its chose receiver X j, over lik X i X j. he ode power costrait P 0 implies that P i = j P ij P 0. he wireless medium causes a power loss g ij, give by g ij = X, where i X j α other physical costats like atea gai have bee absorbed ito. he distace X i X j is defied as the legth of the segmet alog the great circle, coectig X i ad X j,o the surface S. he sigal-to-iterferece oise ratio at the receiver X j is [4] P ij g ij SINR = W + kɛi P, 0 kg kj where I is the set of all iterferig odes the set of all simultaeous trasmitters. It is required to boud the he Shao capacity r for a lik with Gaussia oise ad iterferece sources is, r = W log+sinr,wheresinr is the sigal-to-iterferece oise ratio of that lik. iterferece, so that a certai badwidth scalig ca reder the iterferece egligible with respect to ambiet oise. he problem stems from the fact that potetially, a ode arbitrarily close to the receiver i.e., X k s.t. X k X j 0 could cause arbitrarily large iterferece. his is, however, a very low probability evet. Specifically, let the radom variable d mi G deote the miimum distace o the surface S of the sphere betwee pairs of odes i a specific realizatio G of the the etwork. he followig lemma shows that d mi G caot be very small. Lemma : Proof: Prob Prob i>j d mi G < log d mi G < log = Prob X i X j < i>j Prob X i X j < log log c log. a π log, where a arises because the uiformly distributed ode X j has to lie withi a disc of radius cetered o X i. hus log w.h.p., d mi G of etwork G exceeds. log Notig that P k P 0, I ad g ij log α from, the total iterferece ca be bouded w.h.p. by P 0 log α. hus, settig W =Θ log α reders the iterferece egligible with respect to ambiet oise. Sectio VII discusses a practical badwidth scalig. he above badwidth scalig esures that there is o requiremet to schedule trasmitters, sice they cause egligible iterferece to each other. his badwidth scalig which implies that W,as, allows for a CDMA MAC, where all odes may trasmit simultaeously. It is proved below that the CDMA MAC is ideed a optimal MAC scheme for such a ad hoc etwork. B. Optimality of CDMA MAC he optimality of the CDMA MAC is i the sese that it performs at least as well as ay other optimal schedulig scheme, which assigs time slots ad frequecy bads to various odes DMA/FDMA, as show below. Sice the badwidth is arbitrarily large, each lik s Shao capacity r ij is proportioal to the received power, as below. r ij = lim W log + P ijg ij W W =P ijg ij. hroughout this paper, log deotes log e ad capacity is expressed i uits of ats [4]. As shows, although the badwidth is ifiite, the lik capacity is bouded, due to the
fiite power costrait, a classical result i commuicatio theory. Now, assume that there exists a DMA/FDMA schedulig scheme, which coupled with some routig scheme, achieves the maximum possible uiform throughput. Cosider the followig geeric partitio of the allotted badwidth W ad the time frame ormalized to uity, correspodig to this optimal DMA/FDMA solutio; {W k,k =,,...,K}, s.t. k W k = W, ad {f t,t=,,...,}, s.t. t f t =.he assumed optimal DMA/FDMA schedulig scheme partitios the total power P ij k,t f tp k,t ij assiged to lik X i X j,asp k,t ij s.t. is the power assiged to the = P ij. hus, P k,t ij likithet th time slot of legth f t, ad over the k th frequecy bad of badwidth W k. he followig theorem shows that a CDMA MAC is ideed optimal. heorem : For each lik X i X j,therater ij achieved usig a CDMA MAC scheme is ot less tha that achieved usig the optimal DMA/FDMA schedulig scheme. Proof: Cosider a particular lik X i X j. Sice the rate achieved is upper bouded by the capacity i the absece of iterferece, the rate achieved o this lik by the DMA/FDMA scheme is bouded as, r DMA/FDMA ij a K k= t= K k= t= f t W k log + P k,t ij g ij W k P k,t ij f g ij t = P ijg ij, 3 where a arises sice x log + C x C. P ij P 0 is the total power assiged to lik X i X j. Sice Pijgij is the rate achieved by the CDMA MAC scheme, the theorem is proved. hus, the optimal CDMA MAC scheme is assumed i the subsequet sectios. Essetially, this results i a clea separatio of the MAC ad routig problems, i.e., it remais to cosider optimal routig of the source-destiatio pairs, with the liks scheduled usig the CDMA MAC. Sice the badwidth is large, the key costrait is o loger badwidth, but rather the power of the odes. hus, ulike [], which aalyzed the distributio of badwidth amog the differet liks, i our case, the distributio of power amog the competig liks ad routes eeds to be aalyzed. V. AN UPPER BOUND ON HROUGHPU CAPACIY With a CDMA MAC, the optimal routig cosists of fidig source-destiatio routes for all sources, that achieve the uiform throughput capacity. he difficulty here is that, the per-ode power costrait results i a couplig betwee the route selectios for differet sources. However, iterestigly, as opposed to the classical routig problem i wired etworks [7], the costrait is ot i terms of the capacities of idividual liks, but rather, i terms of the total power trasmitted by each ode. A upper boud o throughput capacity is derived i this sectio, by aalyzig such a power-costraied routig problem. he upper boud Sectio II i [] was derived by boudig log π. the miimum trasmissio radius as tr It was show that violatio of the boud o tr would result i a isolated ode all eighbors beig beyod tr, causig the uiform throughput capacity to be zero. However, i our case, due to lik adaptatio, there is o cocept of ode isolatio, or etwork discoectivity, sice the lik capacity simply decreases with distace, but is always ozero. herefore, a more sophisticated method, which ca aalyze the optimal power-costraied routig i detail, is required to upper boud the throughput capacity. A. raffic Routig he routig problem is to fid a set of routes for each source-destiatio pair, ad to fid the power to be allotted to each lik alog these routes, to maximize the uiform throughput capacity of the etwork. Uder the optimal CDMA MAC, each lik s Shao capacity is proportioal to the received power. hus, r ij = P ijg ij P ij = r ij X i X j α. 4 he couplig of the various routes due to the per-ode power costrait complicates the routig aalysis. herefore, towards obtaiig a upper boud, the power costrait is relaxed from a costrait o each ode, to a average or equivaletly, total power costrait. hus, assume that P i P 0, P i 0 i, 5 i= istead of P i P 0 i. Cosider the source ode X i ad the set of all possible routes from this source to its fial destiatio recollect that each source is assumed to have exactly oe destiatio. Note that all liks o a specific route must operate at a equal data rate. For, if this were ot the case, a redistributio of power amogst the liks while maitaiig the total power utilized would result i a ew rate which is at least as large as the previous rate. Such a redistributio is possible due to the relaxed power costrait 5. herefore, each route ca be associated with a sigle rate. hus, assume some optimal power distributio amogst the set of all routes for a give source-destiatio pair, for each pair, that results i the maximum uiform throughput. Ca this power distributio be characterized? Cosider two routes correspodig to a give source-destiatio pair, X i Xi K, as R i = [Xi 0X i X i...xk i ] ad R i = [Xi 0X i Xi...Xi K ] where Xi 0 = X0 i = X i is the source ad Xi K = Xi K is the destiatio. Let r i ad ri be the rates achieved o these routes i.e., o every lik of each
route respectively. he route Ri is defied as the route for which K k= X k i X k i α 6 is the miimum of all possible routes from the source to its destiatio. i.e., Ri is the miimum power route for the chose source-destiatio pair. From 4, the total power used o these routes is respectively k=k P R i = r i Xi k X k i α, k= k=k P Ri = ri k= X k i X k i α. 7 If the power P R i is shifted from the route R i to Ri,by scalig the power of each lik Xi k X k+ i of Ri by a, the relaxed power costrait 5 would factor + P Ri P Ri still be met, while the achieved rate o Ri would be ot less tha r i +ri. his follows from 7 ad from the defiitio 6 of Ri. hus, uder the relaxed power costrait 5, it is sufficiet for each source-destiatio pair i to choose the miimum power route Ri to route all its traffic, soasto maximize its rate. Further, differet source-destiatio pairs make their choice idepedet of other pairs. Note that such a simplificatio i routig is ot possible with the origial per-ode power costrait P i P 0 i. herefore, the exact uiform throughput capacity r u uder the relaxed power costrait ca be obtaied by settig ri =ru, i, ad solvig i P R i =P 0, where P Ri is give by 7. his routig scheme will be referred to as Miimum Power Routig. I geeral, this may ot coicide with shortest-path routig. he uiform throughput capacity with the per-ode power costrait r satisfies r r u. he objective of this sectio is to upper boud r, which is achieved below by upper boudig r u. As a aside, the per-ode power-costraied routig problem may be posed as a covex optimizatio problem. he problem is similar to the classical joit optimal routig problem for wired etworks [7], but differs i that the costraits are per-ode power costraits, rather tha per-lik capacity costraits. Decetralized algorithms to solve the classical joit optimal routig problem [7], ad the resultig practical routig protocols, may poit to similar solutios for our powercostraied routig problem. his will be the subject of future ivestigatio. B. Maximum umber of odes o a route As described above, the Miimum Power Routig scheme chooses the miimum power route Ri 6 for each sourcedestiatio pair i. o boud r u, the maximum umber of hops i Ri is required. Ituitively, if it were possible for to have a large umber of short hops, the potetially R i Fig.. Source Coverage Regio CR Route 4 ρ Coverage regio of a Route Destiatio the throughput capacity ca become very large, due to rate adaptio. Deote D i as the distace betwee the source X i ad its destiatio Xi K measured o S. By the triagle iequality, the sum of the hop-legths L i of path Ri is bouded by D i as, k=k L i = k= X k i X k i X i Xi K = D i 8 Cosider the Vorooi tessellatio of the etwork, as described i Sectio II. Note that ρ is the radius of the circle with 00 log area o the surface of a sphere S. Also ote that 300 log 4ρ. 9 π his is because a circle of radius ρ o S has a area less tha πρ ad more tha π ρ. he followig lemma will be used to boud the umber of odes o Ri. Lemma 3: he umber of Vorooi cells N max that itersect a miimum power route Ri is upper bouded by 3 + 6Li 0. π log Proof: A particular ode X i is cosidered, alog with its correspodig optimal route Ri. Defie a regio CR i S as follows. YɛCR i iff ZɛR i s.t. Z Y 4ρ, here Y ad Z are poits o S. CRi defies a coverage regio aroud the route such that all cells itersectig the route have to be fully cotaied withi this coverage regio. We ow boud the area of the coverage regio. Correspodig to a route, each lik cotributes a bad rectagular regio of width 4ρ ad legth Xi k 4 ρ X k i to the coverage regio. Also, the edge liks cotribute a additioal two semi-circular regios of radius 4ρ. his is demostrated i Figure. hus, the total area is bouded as AreaCR i 300 log 300 log + L i 0 π 00 log he miimum area of a Vorooi cell is Sectio II. Sice the route ca oly itersect cells that are completely cotaied i CRi, the umber of such cells is upper bouded
as N max AreaCRi Miimum area of a Vorooi cell 3 + 6L i 0 π log 50 log, Usig the result 7, with probability exceedig every cell i the tessellatio cotais at most 50 log odes. hus, the maximum umber of odes o Ri is bouded w.h.p. as N odes max 50 log N max c log + c L i log C. Upper boud o throughput capacity he power P Ri see 7 utilized o R i is related to the legth of the route ad the rate achieved o that route. his relatio is obtaied as k=k P Ri = r u a N odes k= max r u X k i X k i α L i Nmax odes α b r L α i c log + c L i log α c r fl i, where a is because of the covexity of y α for α, K Nmax odes ad from 8. b is from ad because r u upper bouds r. c is from defiig fl i = L α i. c log +c L i log α As a fial step i derivig the upper boud, the expected total power required by all the routes, over the esemble of graphs G, is bouded by the total available power P 0. herefore, by symmetry, the expected power of each route P Ri is bouded by P 0 as P 0 EP R i a r EfD i b r ProbD i ε EfD i D i ε c r ε EfD i D i ε d c 3 ED i r, log α 3 where a is from, the fact that fl i is a icreasig fuctio ad from 8. Iequality b comes from the coditioal expectatio. Nodes are distributed uiformly o S, ad hece the probability that the distace betwee a source destiatio pair exceeds ε, is lower bouded by ε, which results i c. Whe L i exceeds a costat ε, thec L i log term i the deomiator of fl i domiates, resultig i d. Recollect that D i is the physical distace o S betwee the source i ad destiatio. Sice ED i is a costat, 3 results i r c 4 P 0 log α 4 with probability exceedig log log 50. hus, w.h.p., for sufficietly large c 5, lim Prob r =c 5 log α is feasible = 0. 5 his proves the upper boud, r =O log α. D. Area Scalig he area of the etwork has bee ormalized to uity i the aalysis above. hus, the ode desity icreases liearly with. Cosider a alterate sceario where the area of the etwork A icreases with, asa 0. his could represet a situatio such as smart homes, where the ode desity A 0 could be a costat. his results i a scalig of all distaces by A 0. he upper boud uder this area scalig may be obtaied by followig the argumets of the previous sectios. he probabilistic argumets for routig optimality ad the umber of Vorooi cells itersectig a route remai the same. hese argumets are idepedet of the absolute distaces. he distace scalig, however, affects the relatioship betwee power ad capacity. Followig the argumets for the upper boud, we ote the poit of departure is that, uder the ew scalig, EL i =Θ, ad so 4 must be modified to log α R C 5 P 0, 6 where R is the uiform throughput capacity uder the ew area scalig, with a correspodig modificatio i 5. VI. LOWER BOUND ON HROUGHPU CAPACIY o provide a lower boud o the capacity, the techiques reviewed i Sectio II will be useful. he MAC scheme is agai chose as the CDMA MAC, sice that was show to be optimal i Sectio IV. We eed to demostrate a feasible routig scheme to provide the lower boud. he routig scheme chose is the same as i []. hus, as reviewed i Sectio II, a route is selected for each source-destiatio pair by followig the miimum distace path segmets of great circles, as closely as possible. For such a routig scheme, the umber of routes itersectig ay cell maybe bouded w.h.p., similar to []. hus, the traffic to be carried by a cell may be upper bouded w.h.p. as 8, reproduced below for coveiece. sup VɛV raffic carried by cell V k 4 r log. 7 raffic is relayed from cell to cell till it reaches the cell of the destiatio ode. However, each relay ode has a limit o its available capacity. his limit arises due to the power costrait of the ode, ulike [], where the capacity limit
arose from the badwidth costrait of the etwork. Fromthe Vorooi tessellatio, we kow that every cell is cotaied i a disk of radius ρ, ad so the legth of each hop, to reach the ext relay ode, is at most 8ρ. hus, from 4, the relay ode has a total capacity r i bouded as, r i c 6P 0 log α. 8 he trade off betwee the traffic to be carried 7, obtaied from routig requiremets, ad the available capacity 8, obtaied by power costrait provides the lower boud. hus, from 7 ad 8, a uiform throughput r is feasible w.h.p. if k 4 r log c α 6P 0 α. 9 log hat is, Prob r =c 7 α log α+ is feasible =. 30 α /. log α+/ his proves the lower boud, r =Ω As a side ote, it was assumed that each ode ca trasmit ad receive simultaeously. However, this restrictio ca be easily circumveted, by assumig that each lik trasmits over oly half the badwidth chose radomly. he, the trasmissio by a ode ca be thought of as causig a erasure i its ow received sigal. hus, as log as each lik is ecoded with a rate- erasure correctio code [4], the throughput will reduce by a factor of at most two, thus satisfyig the same order bouds. Area Scalig : As i the case of the upper boud, we ca derive the lower boud uder the area scalig A = A 0,for which ode desity is costat. he available capacity 8 is altered due to the depedece of gai G ij where G ij is used to represet the gai uder the ew scalig o the absolute distace measure. Accoutig for the scalig, the uiform throughput is bouded as R C 7 log α+ 3 hus, for the case of costat ode desity, both the upper boud ad lower boud are decreasig with. he ituitive reaso for this is the explicit capacity-distace relatioship 4, where capacity decreases with distace. VII. PRACICAL BANDWIDH SCALING he large badwidth W = Θ log α, assumed i Sectio IV to prove the optimality of CDMA MAC is restrictive. However, practical badwidth scalig schemes ca be developed for the uit area ad area scalig cases, that require smaller badwidth. It oly eeds to be show that the lower boud is achievable with a smaller badwidth. Uit Area case: Let the badwidth scalig be W = W. Allot each ode a uique disjoit frequecy bad of badwidth W, disjoit with the bads of other odes i.e., a FDMA MAC. hus, the capacity of lik X i X j is W log + P ijg ij α. A badwidth of W W = O is sufficiet to esure that the capacity is approximately liear i the received power. herefore, the capacity uder this FDMA MAC approximates the CDMA MAC lik capacity with a badwidth W = Ω α+. Area scalig case: Whe the area is scaled as A = A 0, a efficiet badwidth scalig may be obtaied, by choosig a hybrid FDMA/CDMA MAC. Recollect that the area of each Vorooi cell is 00 log i this case, due to area scalig. Form a graph G, with odes represetig vertices, such that two vertices are coected if the correspodig odes are withi a distace of c log of each other, for some large costat c. he, the umber of cells that have a ode coected to a give ode is a costat. Each cell has at most 50 log odes. So, w.h.p., the degree of G is upper bouded as c log. Now, cosider a FDMA/CDMA scheme where the total available badwidth W = c W 0 log is partitioed equally ito c log disjoit frequecy bads FDMA. Oe bad of width W 0 is allotted to each ode, such that o two odes that are coected i G are allotted the same bad. A simple greedy algorithm ca achieve such a graph colorig []. hus, the MAC chose is FDMA locally, while CDMA is used to hadle the iterferece from outside the local regio. It eeds to be show that W 0 ca be chose so that the iterferece from odes usig the same frequecy bad all of which lie outside the local regio is redered egligible. o this ed, cosider the iterferece caused to a give receiver ode, by odes usig the same frequecy bad. Cosider the aulus regios formed by circles of radii ρ i = c 0 i log, i, cetered o the receiver ode uder cosideratio. he umber of Vorooi cells i each ρ aulus ca be upper bouded by c i+ ρ i 3 log. Eve with the pessimistic assumptio that every cell outside the circle of radius ρ has oe ode iterferig with the ceter ode it caot be more tha oe due to the local FDMA, the iterferece from each aulus caused to the ceter cell is ρ upper boud by c i+ ρ i 4 ρ α i log. Sice the total umber of auli is c 5 log, the total iterferece at the ceter ode is therefore upper bouded as, Iterferece c 4 = c 5 log i= ρ i+ ρ i ρ α i log c 5 c 6 log i + α log i α i= c 5 c 6 log α log i= 3 i α c 5 c 6 log 3 log i i= for ay α a c 7. 3
where a arises from the boud y i= i +logy. hus, a sufficietly large costat per-ode badwidth W 0 is sufficiet to reder iterferece egligible with respect to oise W 0, if α. Sice there are Θlog frequecy bads required, the total required system badwidth is W = Θlog. Ituitively, with area scalig, the closest ad therefore domiat iterferers are moved away, resultig i a smaller iterferece. However, as demostrated i Sectio VI, this area scalig results i a decreasig capacity fuctio. A practical badwidth scalig for α< does ot seem obvious. VIII. CONCLUSION I this paper, the capacity of a power costraied ad-hoc etwork with a arbitrarily large badwidth was studied. Examples of such a etwork iclude UWB ad sesor etworks. It was show that for such a etwork, cosistig of radomly distributed idetical odes over a uit area, with probability approachig oe as, the uiform throughput capacity r is O log α upper boud ad Ω α / log α+/ lower boud. hus, the throughput capacity r for such a radom ad-hoc etwork is Θ α /. Iterestigly, this boud demostrates a icreasig per-ode throughput, i compariso to the decreasig per-ode throughput show i []. he key reaso for this cotrastig result is that our model assumes fiite power, large badwidth, ad the explicit use of lik adaptatio. hus, the properties of the physical layer dramatically alter the ad hoc etwork capacity. Practical badwidth scalig results were derived to show that the assumptio of arbitrarily large badwidth is ot excessively restrictive. Further, the optimal MAC ad routig, which ca achieve etwork capacity, were specified - amely a CDMA MAC ad a power-costraied routig. Future work icludes desigig a decetralized routig scheme to implemet the power costraied routig. Simulatios to demostrate the applicability of this power costraied routig to realistic scearios will also be performed. IX. APPENDIX he ideas of Vapik-Chervoekis VC theory are reviewed i this sectio. A fiite set of poits X such as odes of size ad a set of subsets F such as Vorooi cells is cosidered. X is said to be shattered by F if for every subset B of X there is a set FɛF such that X F = B. F geerates all subsets of X. he VC-dimesio of F is defied as the supremum of the sizes of all fiite sets that ca be shattered by F [8],[9]. A uderlyig probability distributio PD is assumed. A i.i.d sequece X = X...X is chose with distributio PD. he relative frequecies of evets are NF = i= IX iɛf. Sets of fiite VC-dimesio obey a uiform covergece i the weak law of large umbers, i.e., relative frequecies of the evets coverge to their probabilities uiformly. Formally, VC HEOREM: If F is a set of fiite VC-dimesio, ad X i is a sequece of i.i.d. radom variables with commo probability distributio PD, the for every ε, δ > 0, Prob NF PDF ε F ɛf > δ, 33 for sufficietly large. he VC-theorem 33 is a stroger statemet tha the weak law of large umbers due to the uiformity over all evets. he proof [9] is developed by first defiig the growth fuctio m, which is the maximum umber of subsets of a sized sample X geerated by the set of evets F. Itis the proved, for sets with fiite VC-dimesio vc, that the growth fuctio is upper bouded as m vc + s.t. vc. 34 he probability that the relative frequecy of a particular evet F exceeds the mea by ε is expoetially decreasig i the sample size. o boud the probability over the class of evets F a uio boud is applied. However the uio is ow take over m evets sice oly m evets are distict. hus Prob sup FɛF NF PDF ε m e ε. 35 he fiite VC-dimesio implies m grows slower that the expoetial i, ad hece the weak law holds uiformly. 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