Capacity Bounds For Ad-Hoc Networks Using Directional Antennas

Transcription

1 Capacity Bouns Fo A-Hoc etwoks Using Diectional Antennas Akis Spyopoulos an Cauligi S. Raghavena Electical Engineeing - Systems Univesity of Souen Califonia Los Angeles, Califonia {spyo, aghu}@halcyon.usc.eu Abst Diectional antennas can be useful in significantly inceasing e capacity of wieless a hoc netwoks. Wi iectional antennas, inepenent communications between noes can occu in paallel, even if e noes ae wiin ange of each oe. Howeve, mutual intefeence by simultaneous tansmissions limits e maximum numbe of such concuent communications. Fuemoe, it poses bouns on e amount of one can achieve by using iectional antennas instea of omni-iectional ones. These bouns epen on e specific antenna type an its paametes, as well as highe laye potocol equiements. n is pape we calculate intefeencebase capacity bouns fo a geneic antenna moel as well as a eal-wol antenna moel an analyze how ese bouns ae affecte by impotant antenna paametes like gain an beamwi Keywos-component; capacity; a-hoc; iectional antennas fomatting;. TRODUCTO Wieless a-hoc netwoks ae multi-hop netwoks whee all noes coopeatively maintain netwok connectivity. The ability to be set up fast an opeate wiout e nee of any wie infastuctue (e.g. base stations, outes, etc.) makes em a pomising caniate fo militay, isaste elief, an law enfocement applications. Fuemoe, e gowing inteest in senso netwok applications has ceate a nee fo potocols an algoims fo lage-scale self-oganizing a-hoc netwoks, consisting of hunes o ousans of noes. Until ecently, most papes on wieless a-hoc netwoks wee assuming at all noes ae equippe wi omniiectional antennas. Howeve, uing e past couple of yeas ee has been a apily gowing inteest in e use of iectional antennas in a-hoc netwoks. The eason fo at is e potential benefits one coul have fom e ability of iectional antennas to concentate e aiate powe towas a specific iection. Those benefits ae in tems of highe oughput [] [] [3] [] [], bette enegy-efficiency [5], lowe intefeence, an moe secue communications. An impotant chaeistic of a wieless a-hoc netwok, o as a matte of f any netwok in geneal, is its capacity. Thee have been quite a few papes exploing e capacity of a-hoc netwoks whee noes have omni-iectional antennas [7] [], an e most well-known esult is by Kuma an upta []. Alough many papes following at wok expane is analysis o use simulation to confim e esults, none yet (to This wok is suppote by e DARA owe Awae Computing an Communications ogam une cont no. F335-C--33 e best of ou knowlege) has conucte any extensive capacity analysis fo wieless a-hoc netwoks using iectional antennas. n is pape we pefom a capacity analysis fo a-hoc netwoks consisting of noes at ae equippe wi iectional antennas. We show at intefeence fom concuent tansmissions limits e maximum achievable capacity. We povie bouns fo an abst an eal-wol linea aay iectional antenna moels an show how ese bouns ae affecte by impotant antenna paametes like gain an beamwi. n e next section, we biefly escibe e moels fo e antennas we e going to assume in ou subsequent analysis. The capacity analysis fo ese antenna moels follows in section. n section V, we illustate how e calculate capacity bouns ae elate to antenna type an paametes an povie esults fo a ange of values fo ose paametes. Finally, in section V we conclue e pape an iscuss open issues an futue wok.. DRECTOAL ATEA MODELS Fist, we e going to consie a simple geneic iectional antenna moel, which is usually efee to as e flat-toppe antenna moel in e liteatue. n is moel, a specific constant gain is assume wiin a specific angle θ, epesenting e main antenna beam, an a lowe constant gain is assume fo all oe iections. Fo ou analysis, we e going to assume at is equal to one an takes values between zeo an one, in oe to have e same tansmission ange as an omni-iectional antenna. This coul be achieve by eucing e tansmitting powe by a fo of /. We e going to call e suppession atio of e antenna. This moel albeit quite simplistic, can povie valuable insight on how e iectional antenna chaeistics affect e capacity of an a-hoc netwok consisting of noes utilizing iectional antennas. The natue of a-hoc netwoks poses cetain limitations on e types of antennas at coul be use in at context as well as ei espective paametes (i.e. gain an main beam angle). The size of e teminal (e.g. DA, senso, laptop) is a majo esticting fo an so is e nee to be able to quickly eiect e antenna. Fo ese easons, we choose a simple linea The gain of an antenna in a given iection is e atio of e input powe neee by an omni-iectional antenna an at of a iectional one, in oe to each e same istance.

2 aay antenna consisting of ipoles as ou eal-wol antenna moel. We will only consie en-fie aays at have a single main beam [9]. The nomalize gain fo an en-fie linea aay antenna as a function of θ, e angle fom e iection whee e antenna is pointing, is given by. π sin ( cosθ ) π Ν sin ( cosθ ) CAACTY AALYSS A. oblem Fomulation Assume we have wieless a-hoc noes anomly place in an unboune flat aea, which fo e sake of ou analysis we consie as extening to infinity. Let ρ be e unifom noe ensity in is aea an let enote e pobability at any noe is engage in communication at any point in time. The quantity ρ enotes e ive noe ensity, which we e going to be efeing to as ρ heeafte. We assume a geneal, paameteize goun popagation moel. f t is e tansmitting powe, t an ae e tansmitte an eceive antenna gains espectively, an e istance between sene an eceive en e signal powe at e eceive s aio is given by t c () When specific esults fo e two-ay goun popagation moel ae given, c a is eplace by h, whee h is e antenna elevation assume to be equal to.5m fo all antennas, an e attenuation fo is equal to. We e going to assume at a iectional vesion of e. Meia Access Contol potocol (DMAC) is being implemente by all a-hoc noes, in oe to access e shae channel using ei iectional antennas. This iectional. potocol was inepenently popose in [7] an []. t is essentially an aaptation of e collision avoiance an vitual caie sensing mechanisms use in e oiginal.[] fo a-hoc netwoks utilizing iectional antennas to communicate. Fo is pape to be self-containe, we will biefly summaize e potocol featues at ae neee fo ou analysis: All noes have two moes of opeation, iectional an omni-iectional. When noes ae ile, ey e listening to e meia omniiectionally. All non-boacast packets (i.e. RTS, CTS, DATA an ACK) ae tansmitte iectionally. On eception of an RTS packet a noe switches to iectional moe an points its antenna back to e tansmitting noe, base on e iection-of-aival of e RTS packet o knowlege of e location of at sene. Diectional vitual caie sensing is implemente as follows. Each noe keeps a iectional AV table wi a simila use to e AV value in.. When it oveheas an RTS o CTS packet, not estine to itself, it maks e iection-of-aival in its AV table as busy fo e time uation containe insie e packet. () When a noe has a packet to tansmit, it checks e iection of e intene ecipient in its AV table to see whee ee is any ongoing tansmission in at iection. f ee is, it backs off an ties again late. B. A Fist Level Capacity Compaison One impotant effect of using. wi omni-iectional antennas is e following. When two noes, say A an B, communicate en all noes insie a cicle egion of aius R (aio ange) aoun each of noes A an B, ae enee unable to communicate emselves wi any oe noe. We shall call is egion e silence egion of noes A an B. Consequently, if evey noe has a packet to sen to some oe noe, en all noes wiin ange of each oe will have to take tuns sening ei taffic. Assume ee is some highe laye scheuling potocol at ivies ese noes into K inepenent pais at nee to communicate wi each oe. Then, e available channel capacity, say C, is going to be effectively ivie into K equal chunks of size at most C/K. f noe ensity inceases an accoingly e numbe of noes in e silence egion, e pe-noe allocate capacity can become infinitesimally small. On e oe han, by using vey naow beam iectional antennas, one coul ague at we coul isolate e K inepenent pais fom e pevious example an have em all communicate at e same time. This way, each pai coul utilize all e available channel capacity. Fuemoe, by inceasing e noe ensity of e aea, one coul make e elative fom e use of iectional antennas abitay lage compae to using omni-iectional antennas. This is, in geneal, not tue, because it neglects e effects of intefeence. Alough most powe is aiate (eceive) ough e main lobe ee is still some powe aiate (eceive) fom e sie lobes as well. This powe is peceive as intefeence to (fom) oe communicating noes. f e signal-to-intefeence atio (SR) is high enough, en e eceiving noe may be able to captue e intene signal. eveeless, intefeence is an aitive (an anom phenomenon). A single intefeing noe may not be stong enough to pevent successful communication, but e sum of all intefeing signals fom concuently communicating noes coul easily gable each oes useful ata. Consequently, ee is a boun on how many noes one can pack insie a specific aea, wiout compomising e ability of inepenent pais of noes to communicate wi each oe at e same time. Altenatively, ee is a boun on e capacity impovement one can achieve by using iectional antennas in a-hoc netwoks in place of omni-iectional ones. Those capacity bouns ae e focus of is pape an ae going to be exploe in e est of is section. C. ntefeence-base capacity analysis: eneic Antenna Moel All noes ae assume to be equippe wi a flat-toppe iectional antenna an use DMAC to communicate. When a noe, say A, sens a iectional RTS o CTS packet, all noes insie e main beam of at noe an wiin ange R, ae going to successfully eceive e packet, povie ey e not busy emselves. Those noes will eefoe efain fom any

3 tansmission towas A s iection fo e uation inicate in e RTS o CTS packet an cause no iect intefeence. Howeve, ose noes wiin ange R ae fee to engage in communication towas some oe iection. Theefoe, ey may iniectly intefee wi A ough ei sie lobes, as long as ey e outsie a ange R, which is e ange at which even sie lobe intefeence woul be high enough to coupt A s signal on its own. All e oe noes, outsie ange R, ae potential iect intefeence souces. Fo noes lying outsie A s main beam, ose fue away an R ae potential iect intefeence souces, while ose between anges R an R may intefee ough ei sie lobes. Let be e eceive s powe eshol, t be e tansmitting powe, an be e attenuation fo. Then R, R an R ae given by R n Fig. we epict e aeas whee noes at can potentially intefee wi noe A ae containe. R R" A Azimu lane R Diect ntefeence niect ntefeence o ntefeence Figue : Aea whee potential intefeence souces lie Those noes at can iectly intefee wi A, may point ei antenna to any iection wi equal pobability. As a esult of at, e antenna gain of any intefeing signal is a anom vaiable. We choose to use e aveage antenna gain fo e intefeence signal [(π θ) + θ]/π fo ose noes. On e oe han, ose noes at can only intefee iniectly have a constant intefeence gain. We now nee to calculate e total amount of intefeence as peceive by noe A. Consie fist e noes insie e infinitesimally small ac-shape aea elimite by A s main beam an between istances an + fom A. Each noe in is aea is going to contibute an intefeing signal (). Similaly, each noe outsie A s main beam an at istance in [, + ] will contibute an intefeing signal (). () an () ae given by ( t c ) /, > R () ( c ) /, R < < R ( > t c ) /, R () " ( c ) /, R < < R The numbe of ive noes in any aea of size L is given by ρ L. Hence, e total amount of intefeence peceive by A, will be given by tot ( c / ), R R ( ), R" R ( ) t θ () ρ R + (π - θ) R" () ρ (3) () (5) f e noe wi which A is communicating is locate at istance, en e signal-to-intefeence atio (SR) at A will be given by c / SR () tot We explaine ealie at using iectional antennas in combination wi DMAC potocol, allows iffeent pais of noes to communicate in paallel, even if ey ae in ange of each oe. The question is how many such tansmissions like at can go on in paallel, wiout estoying each oes useful ata. Most aio eceives toay ae able to captue a useful signal between oe intefeing signals, as long as e SR at e eceive is above a specific eshol level SR, which epens on e coing scheme use. We can see fom () at e SR is invesely popotional to e ive noe ensity ρ. One coul eefoe claim at by eceasing e maximum istance up to which two noes can (on e aveage) successfully communicate it is possible to stay above SR fo any ρ. This way we coul abitaily incease e total peunit-of-aea communication capacity. Howeve, is is not accuate. Deceasing e tansmission ange inefinitely is not a goo pice an coul lea to opposite esults fom what one woul expect by using e pevious agument. The eason fo at is e inteion of uppe-laye potocols wi e physical an MAC laye, as well as wi one anoe. A shot tansmission ange implies at a packet has to go ough moe hops to each a specific estination incuing moe pocessing an lage elays. A shot effective tansmission ange implies also at each noe will have to fowa moe taffic at oesn t belong to at noe. This woul esult in a ecease of e available oughput an pocessing esouces at each noe has available to itself, in oe to hanle packets oiginating o estine to at noe, as shown in []. Fo ese easons, we will assume at is effective tansmission ange is ictate by highe laye potocols an will eefoe be consiee as given. n at case, e maximum ive noe ensity at allows noes to successfully communicate at a tansmission ange of at least, espite intefeence, is given by ρ max K c SR K ()θ + R R" ()(π -θ) Consie now a cicle aea of aius aoun any tansmitting noe X. We saw ealie at fo e omniiectional case,. woul ceate a silence egion of at least π aoun X. n oe wos, only one tansmission coul successfully take place in any aea of size π. On e oe han, accoing to ou pevious analysis, if iectional antennas wee use instea, ee coul be up to max anoe ρ π ive noes in e cicle aea aoun X, wiout isupting X s communication. Assume at ee exists some optimal scheuling algoim at specifies (7)

4 max ρ π / inepenent pais of noes, wiin e aea of π, at can communicate wi each oe in paallel. Then, e of using iectional antennas in place of omni-iectional ones is given by: ρ π CMAX () Finally, when e two-ay goun popagation moel is assume, e values fo e maximum ive noe ensity an maximum ae given by ρ C max() () MAX h SR max (9) () D. ntefeence-base capacity analysis: Linea Aay Antenna Moel The capacity analysis fo e case of linea aay antennas is simila to at fo e geneic antenna case. When a noe tansmits an RTS o CTS packet, e maximum ange until which suouning noes can successfully eceive it epens on e angle θ an we enote it as R (θ). Similaly, e istance up to which noes can iniectly intefee is enote by R. R (θ) an R ae given by R c, R ( ) R Ν () The aveage antenna gain fo a iect an an iniect intefeing signal an, espectively ae given by π π-θ/ θ, () θ θ () π π -θ h π SR t ( θ, ) ( θ, ) The total intefeence level an e signal-to-intefeence atio when using -element linea aay antennas ae given by θ c tot () ρ θ () R θ (3) π R Ι c + ρ θ R Ν c / SR() () tot () Finally, using simila aguments as in e case of e geneic antenna moel we can calculate e maximum ive noe ensity ρ max () an fom at e maximum capacity gain C MAX () as follows: ( θ, ) [( π θ) + θ][θ + ( π θ) + ( )] f π () f f θ/ W() V. RESULTS (5) A. eneic Antenna Moel n is section we e going to examine how e suppession-atio an beam-wi θ affect e maximum achievable capacity, base on (7) an (). n Fig. we can see how C MAX changes wi eceasing suppession-atio fo seveal fixe antenna beam-wis as well as iffeent effective tansmissions ange. Figue : Capacity gain as a function of antenna suppession atio fo fixe effective tansmission ange m (left) an fixe antenna beam-wi of 9 o (ight). t is evient fom e two gaphs at e maximum inceases exponentially wi lowe suppession atio fo any beam-wi an any effective tansmission ange. On e oe han, e is not as sensitive to e antenna beam-wi. π R C MAX - Capacity ain ( m) Capacity ain (.3) supession atio π/3 π/ π/ beam-wi θ πw() () - SR θ + π R R Capacity ain (.) beam-wi θ m m m m Figue 3: Capacity gain as a function of antenna beam-wi θ fo high suppession atios (left) an low suppession ations (ight) Contay to one s intuition, naowe beams esult in maginally highe s fo high suppession atios (>.). The eason fo at is e following. eaby noes at lie wiin e main antenna beam can only intefee wi ei sie lobes wi e ongoing communication, as we saw ealie. Oe noes, howeve, o not lean about e upcoming communication an may iectly intefee. Theefoe, e naowe e beam is e moe e potential neaby intefeence souces. This poblem gets aggavate by suppession atios, - Capacity ain (θ π/) θ supession atio m m m m

5 which ae not low enough to cancel any potential intefeence, fom such neaby souces. n Fig. 3 we look close at how e antenna beam-wi affects e potential s fo low (<.) an high suppession atios (>.), espectively. We can see at e sensitivity of e to e antenna beam-wi inceases wi eceasing suppession atios. Altenatively, e espective antenna beam-wi becomes an impotant fo only when e suppession-atio is low enough to cancel neaby intefeing tansmissions B. Linea Aay Antenna Moel We will analyze how e maximum behaves as a function of e numbe of elements in e aay. We have use e Maematica tool [] to numeically calculate all integals involve in ()-(5). n Fig. we epict e maximum achievable as a function of e numbe of elements in e aay. Results ae given fo bo e twoay goun popagation moel an secon popagation moel, whee e attenuation fo is equal to 3 an c a is still equal to h. 5 3 Capacity ain (two-ay goun) # of elements () Figue : Capacity gain as a function of e numbe of ipoles in linea aay antenna fo e two-ay goun popagation moel (left) an a less hash { 3, c a h } popagation moel (ight). t is evient at maximum inceases wi highe numbe of elements in e aay in bo cases, as expecte, since bo suppession atio an antenna beam-wi of a linea aay antenna ecease when inceases. Howeve, e moe complex antenna patten consiee, as well as e counteintuitive imp of antenna beam-wi fo high suppession atios, esults in a less smoo cuve compae to e ieal flat-toppe antenna moel. Fuemoe, we see at e popagation moel assume has an impotant effect on e achievable capacity. Specifically, it is evient in Fig. at fo a less hash envionment, e maximum capacity gains ae highe in evey case. This behavio is justifie by e lage silence egion, which esults fom e highe aio ange in combination wi e less attenuate useful signal eceive. V. DSCUSSO t is impotant to note at all capacity esults pesente in is pape, ae technology-base an not infomation eoetic. Specific assumptions ae mae about technology being use (e.g. DMAC potocol,., etc.), in oe to ealistically moel cuent pice in a-hoc netwoks. The implication of at is at eive capacity bouns apply to e technology an m m m m Capacity ain ( 3) # of elements () potocols being assume. n a ecent wok [], Kuma ties to eive geneal, infomation eoetic, bouns fo e capacity of wieless netwoks. Howeve, as e auos note, open questions still aboun. Anoe point to be mae is at iectional antennas o not change e scaling oe of a-hoc netwok capacity wi e total numbe of noes n, as shown in [] o []. Only e constants in font of e oe o change. Howeve, ese eoetical asymptotic capacity bouns eive in [] ae inheent to e nee fo multi-hop outing in a-hoc netwoks. V. COCLUSOS AD FUTURE WORK n is pape, we ve analyze how intefeence fom simultaneous tansmissions affects e capacity of a-hoc netwoks utilizing iectional antennas to communicate. We have calculate uppe bouns fo e s of using iectional antennas in place of omni-iectional ones, fo bo an ieal flat-toppe geneic antenna moel as well as a ealwol linea aay antenna moel. Fuemoe, we examine how impotant antenna paametes like gain an beam-wi affect ose bouns in evey case. Finally, we biefly illustate e impotance of e popagation moel assume in tems of e esulting netwok capacity. n futue wok, we e planning to incopoate smat antenna moels as well as Ricean an Rayleigh popagation moels [3] in ou analysis. REFERECES [] A. asipui, S. Ye, J. You, an R. E. Hiomoto, A MAC potocol fo mobile a hoc netwoks using iectional antennas, EEE Wieless Communications an etwoking Confeence (WCC ),. [] M. Takai, J. Matin, R. Bagoia, an A. Ren, Diectional Vitual Caie Sensing fo Diectional Antennas in Mobile A Hoc etwoks, oc. ACM MobiHoc, June [3] R. Roychouhuy, X. Yang, R. Ramanaan, an. Vaiya, Meium Access Contol in A Hoc etwoks Using Diectional Antennas, in oc. of MOBCOM, Semptembe. [] L. Bao, an J.J. acia-luna-aceves, Tansmission Scheuling in A Hoc etwoks wi Diectional Antennas, in oc. of ACM/EEE MOBCOM, Semptembe. [5] A. Spyopoulos, an C. S. Raghavena, Enegy Efficient Communication in A Hoc etwoks Using Diectional Antennas, in oc. EEE FOCOM, June. []. upta, an. R. Kuma, The capacity of wieless netwoks," EEE Tansions on nfomation Theoy, vol., pp. 3-, Mach. [7] J. Li, C. Blake, D. S. J. Decouto, H.. Lee, an R. Mois, Capacity of Wieless A Hoc etwoks, in oc. MOBCOM, July. [] S. Toumpis, an A.J. olsmi, Capacity Regions fo Wieless A Hoc etwoks, in oc. EEE CC, May. [9] C. A. Balanis, Antenna Theoy: Analysis an Design, n e. ew Yok: Wiley, 997. [] EEE Local an Metopolitan Aea etwok Stanas Committee, Wieless LA meium access contol (MAC) an physical laye (HY) specifications, EEE stana.-999, 999. [] S. Banyopahyay, K. Hasuike, S. Hoisawa, an S. Tawaa, An Aaptive MAC an Diectional Routing otocol fo A Hoc Wieless etwok Using ESAR Antenna, in oc. ACM MobiHoc. [] [3] T. S. Rappapot, Wieless Communications: inciples an ice, entice Hall, 99 [] L. L. Xie, an. R. Kuma, A etwok nfomation Theoy fo Wieless Communication: Scaling Laws an Optimal Opeation, submitte to EEE Tansions on nfomation Theoy, Apil.