IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY

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1 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY Capacity ad Delay of Hybrid ireless Broadbad Access Networks Pa Li, Chi Zhag, Studet Member, IEEE, ad Yuguag Fag, Fellow, IEEE Abstract A optical etwork is too costly to act as a broadbad access etwork. O the other had, a pure wireless ad hoc etwork with odes ad total badwidth of bits per secod caot provide satisfactory broadbad services sice the perode throughput dimiishes as the umber of users goes large. I this paper, we propose a hybrid wireless etwork, which is a itegrated wireless ad optical etwork, as the broadbad access etwork. Specifically, we assume a hybrid wireless etwork cosistig of radomly distributed ormal odes, ad m regularly placed base statios coected via a optical etwork. A source ode trasmits to its destiatio oly with the help of ormal odes, i.e., i the ad hoc mode, if the destiatio ca be reached withi L (L ) hops from the source. Otherwise, the trasmissio will be carried out i the ifrastructure mode, i.e., with the help of base statios. Two trasmissio modes share the same badwidth of bits/sec. e first study the throughput capacity of such a hybrid wireless etwork, ad observe that the throughput capacity greatly depeds o the maximum hop cout L ad the umber of base statios m. e show that the throughput capacity of a hybrid wireless etwork ca scale liearly with oly if m =Ω(), ad whe we assig all the badwidth to the ifrastructure mode traffics. e the ivestigate the delay i hybrid wireless etworks. e fid that the average packet delay ca be maitaied as low as Θ() eve whe the per-ode throughput capacity is Θ( Idex Terms Hybrid wireless etworks; capacity; delay; broadbad access etworks. I. INTRODUCTION THE INTERNET S pheomeal growth has triggered great icrease o demads for broadbad services. Thus, how to desig a broadbad access etwork to provide broadbad services is essetial to the further success of the Iteret. Optical etworks ca provide high badwidth ad low etwork delay [3] [25] [26]. However, they are too costly to act as broadbad access etworks. Sice wireless etworks ca be deployed easily ad quickly with low cost, we the tur to them for help. I their semial paper [2], Gupta ad Kumar show that the per-ode throughput capacity i radom wireless ad hoc etworks is Θ( log ) bits/sec, which meas radom ad hoc etworks caot scale. Later o, Buragohai et al. [4] study Mauscript received 5 Jauary 2008; revised 4 July This work was partially supported by the Natioal Sciece Foudatio uder grats CNS ad DBI The work of Fag was also partially supported by the Project uder B08038 with Xidia Uiversity, Chia. P. Li, C. Zhag ad Y. Fag is with Departmet of Electrical ad Computer Egieerig, Uiversity of Florida, Gaiesville, FL 326 ( {lipaleo@,zhagchi@,fag@ece.}ufl.edu Y. Fag is also a Chagjiag Scholar Chair Professor with Natioal Key Laboratory of Itegrated Services Networks, Xidia Uiversity, Xi a, Chia. Digital Object Idetifier 0.09/JSAC I this paper, we use the Kuth s otatios [3]: f() =O(g()) meas f() is asymptotically upper bouded by g(); f() = Ω(g()) meas f() is asymptotically lower bouded by g(); f() = Θ(g()) meas f() is asymptotically tight bouded by g(); f() =o(g()) meas f() is asymptotically egligible with respect to g(); f() =ω(g()) meas f() is asymptotically domiat with respect to g( the throughput capacity i grid etworks where there are odes ad the average source-destiatio distace is d. They show that the Ω(/d) throughput ca be achieved. Thus, grid etworks caot scale either sice d = ω() formostofthe cases. The work i [2] deals with dese etworks, i.e., the area is fixed ad the ode desity icreases liearly as the umber of odes, ad the authors assume the whole etwork is coected. Dousse et al. [6] study the throughput capacity i exteded etworks where the desity of odes is fixed ad the area icreases liearly with the umber of odes. They show that by allowig a arbitrary small fractio of the odes to be discoected i 2-dimesioal exteded etworks, a ovaishig rate ca be achieved for each ode. Ozgur et al. [22] also ivestigate the throughput capacity of a coected ad hoc etwork. Their results show that by itelliget ode cooperatio ad distributed MIMO commuicatio, the dese etworks ca scale liearly with the umber of odes, ad the exteded etworks scale as 2 α/2 for 2 α<3 ad for α 3, whereα is the path loss expoet i power propagatio model. Moreover, Duarte-Melo et al. [7] study the case of semi-exteded etworks, where both ode desity ad the etwork area icrease as the umber of odes icreases. Specifically, they assume the etwork area is a disk of radius γ, 0 <γ< 2. ith a (+d) propagatio model, they show α that the per-ode throughput capacity is Ω( ),i.e.,semiexteded etworks caot scale. γ Sice we prefer the etwork to be coected, ad the odes i the etwork to be loosely coupled as well, the results for pure ad hoc etworks are pessimistic, i.e., they caot scale as the umber of odes. This meas pure wireless ad hoc etworks caot provide satisfactory broadbad service whe the umber of etwork users goes large. I this study, we propose to use hybrid wireless etworks as the broadbad access etworks, which are also called multihop cellular etworks [8]. Hybrid wireless etworks ca be oe-dimesioal, two-dimesioal with strip area [7], or two-dimesioal with square area [6]. Traffic patter i the etwork ca be asymmetric [27] or symmetric [4] [23] [28]. I this study, we oly focus o two-dimesioal square hybrid wireless etworks with symmetric traffics. e first ivestigate the throughput capacity of hybrid wireless etworks. Kozat ad Tassiulas [4] study the throughput capacity of hybrid wireless etworks where both ad hoc odes ad access poits are radomly distributed. They show that the per-ode throughput capacity ca be Θ(/ log ) bits per secod if the umber of access poits scales liearly with the umber of odes, which meas the etwork caot scale. Similar results are also reported i []. Zemliaov ad Veciaa [28] ivestigate the throughput capacity of hybrid wireless /08/$25.00 c 2008 IEEE Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

2 8 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 etworks where ad hoc odes are radomly distributed ad base statios are arbitrarily placed. They show that the perode throughput capacity depeds o the umber of base statios, but the etwork still caot scale. Assumig odes are radomly distributed ad m base statios are regularly placed, Liu et al. [6] study the throughput capacity of hybrid wireless etworks. They cosider two differet routig strategies. Uder k-earest-cell routig strategy, if m grows asymptotically slower tha, the maximum per-ode throughput capacity is Θ( log /m ), adthe 2 beefit of addig base statios is isigificat. However, if m grows asymptotically faster tha, the maximum per-ode throughput capacity is Θ( m ), which icreases liearly with the umber of base statios. Uder probabilistic routig strategy, similar results are obtaied. The threshold of the umber of base statios above which the per-ode throughput capacity icreases liearly with m is log. Thus, the etwork ca scale if m =Ω( Actually, the authors i [6] assume k =0, i.e., a ode trasmits to its destiatio i the ad hoc mode oly if it is i the same cell. However, the 0-earest-cell routig strategy caot efficietly make use of the wireless chael. For example, if a source ode ad its destiatio are withi oehop distace of each other, but they are ot i the same cell, the they caot directly commuicate i the ad hoc mode accordig to the routig strategy. Istead, the trasmissios betwee these two odes ca oly be carried out through base statios. Besides, the case k = 0 is too specific. To provide a solutio to a more geeral case, ad to better utilize the badwidth resource, Pei et al. [23] propose to use the L-maximum-hop routig strategy. Ufortuately, the capacity bouds derived i [23] are ot tight. I this paper, we revisit the throughput capacity problem i hybrid wireless etworks by usig the L-maximum-hop resource allocatio strategy i [23] 2. Specifically, a source ode trasmits to its destiatio i the ad hoc mode if the destiatio ca be reached from the source withi L (L ) hops. Otherwise, the trasmissio will be carried out i the ifrastructure mode. Assumig a total badwidth of bits/sec is split ito three parts, i.e., for ad hoc mode, 2 for uplik i the ifrastructure mode, ad 3 for dowlik i the ifrastructure mode, we show that: ) whe L =Ω( 3 ), the throughput capacity of the log 2 3 etwork is Θ( L log )+Θ(m 2 Ifm =Ω( L log ), we ca have higher throughput whe all the traffics are carried i the ifrastructure mode. The per-ode throughput capacity icreases liearly with the umber of base statios m, ad the etwork ca scale oly if m =Ω( Ifm = O( L log ), we ca achieve higher throughput whe all the traffics are carried i the ad hoc mode, ad the etwork caot scale. 2) whe L = o( 3 ), the throughput capacity of log 2 3 the etwork is Θ(L 2 log )+Θ(m 2 If m = Ω(L 2 log ), we ca have higher throughput whe all 2 e call it a resource allocatio strategy istead of a routig strategy because we oly decide which kid of resource a trasmissio uses, ad do ot specify how to choose a route from a source to a destiatio. the traffics are carried i the ifrastructure mode. The per-ode throughput capacity icreases liearly with the umber of base statios m, ad the etwork ca scale oly if m =Ω( Ifm = O(L 2 log ), we ca achieve higher throughput whe all the traffics are carried i the ad hoc mode, ad the etwork caot scale. It ca be easily show that the results obtaied i [6] uder the 0-earest-cell routig strategy is just a special case i 2) preseted above. Furthermore, we also compare the throughput capacity of our hybrid wireless etworks with that of pure ad hoc etworks. The capacity gai is clearly show. I additio to the throughput capacity, the packet delay is also a importat issue i the etwork. [2], [8], [0], [5], [9], ad [20] propose to utilize odes mobility to deliver packets. Each packet is oly relayed for very few times before arrivig at the destiatio. For example, i [0], each packet is at most relayed oce, i.e., relayed by at most oe relayig ode. They fid that there is a trade-off betwee the capacity ad the delay. Specifically, i pure ad hoc etworks, the capacity ca oly be icreased at the cost of greatly icreased delay. Moreover, Gamal et al. [9] show that usig mobility to icrease throughput, eve slightly, would lead to a abrupt ad large icrease i delay. I this paper, we also study the delay i hybrid wireless etworks. e fid that by addig base statios i pure ad hoc etworks, the capacity ca be improved without icreasig the delay. Particularly, i hybrid wireless etworks, the average packet delay ca be maitaied as low as Θ() eve whe the per-ode throughput capacity is Θ( The rest of this paper is orgaized as follows. I Sectio II we itroduce some defiitios. Sectio III gives the hybrid wireless etwork model, icludig the etwork architecture, the iterferece model, ad the resource allocatio strategy. I Sectio IV ad Sectio V, we derive the throughput capacity ad delay of hybrid wireless etworks, respectively. e fially coclude this paper i Sectio VI. II. DEFINITIONS Throughput: As defied i the usual way, the time average of the umber of bits per secod that ca be trasmitted by each ode to its destiatio is called the per-ode throughput. The sum of per-ode throughput over all the odes i a etwork is called the throughput of the etwork. Feasible Throughput: e say that the throughput of a etwork, deoted by λ(), is feasible if there exists a spatial ad temporal schedulig scheme that yields a aggregate etwork throughput of λ() bits/sec. Throughput Capacity of A Network: e say that the throughput capacity of a etwork ( [4]) is of order O(f()) bits per secod if there is a determiistic costat c < + such that lim if Prob(λ() =c f() is feasible) <, + ad is of order Θ(f()) bits per secod if there are determiistic costats 0 <c 2 <c 3 < + such that lim if Prob(λ() =c 2f() is feasible) =, + lim if Prob(λ() =c 3f() is feasible) <. + Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

3 LI et al.: CAPACITY AND DELAY OF HYBRID IRELESS BROADBAND ACCESS NETORKS 9 Average Packet Delay of A Network: The delay of a packet i a etwork is the time it takes the packet to reach the destiatio after it leaves the source. As i [5], [20], we do ot cosider the queuig delay at the source ode sice we are more iterested i the etwork delay. The average packet delay of a etwork is obtaied by averagig over all trasmitted packets i the etwork. Besides, we also assume the packet size scales as the per-ode throughput 3. III. HYBRID IRELESS NETORK MODEL A. Network Architecture e cosider a two-tier hybrid wireless etwork o the surface of a torus of uit area. The low tier is composed of ormal odes, ad the higher tier cosists of m base statios, respectively. The assumptio of a torus eables us to avoid techicalities arisig out of edge effects, but the results derived i the paper are applicable for odes located o a uit square as well. e assume odes are uiformly ad idepedetly distributed. They have the same trasmissio power, ad hece the same trasmissio rage deoted by r( e follow the process i [0] to choose radom seder-receiver pairs so that each ode is a source ode for oe flow ad a destiatio ode for at most O() flows. The m base statios are regularly placed i the etwork, dividig the area ito a hexagoal tessellatio, which is exactly the classical 7-cell reuse model as described i [24]. Each hexago is called a cell ad there is oe base statio i the ceter of each cell. Base statios do ot serve as data sources or data destiatios. Istead, they oly help relay the packets for the ormal odes. Furthermore, we also assume base statios are iter-coected by a optical etwork, i which the lik badwidth is large eough. Thus, the wired etwork has o badwidth costraits. B. Iterferece Model e employ the Protocol Model i [2] as the iterferece model. Suppose ode X i trasmits to aother ode X j. X i ad X j also deote the positios of these two odes. The, the trasmissio is successful if the followig two coditios are satisfied: ) The distace betwee X i ad X j is o more tha r(), the trasmissio rage of the odes, i.e., X i X j r( 2) The positios of other trasmitters X k simultaeously trasmittig over the same chael should satisfy: X k X j ( + Δ)r( The quatity Δ > 0 models situatios where a guard zoe is specified by the protocol to prevet a eighborig ode from trasmittig o the same chael at the same time. It also allows for imprecisio i the achieved rage of trasmissios. 3 As poited out i [9], uder this assumptio, queuig delay at source ode ca actually be a costat, which gives us aother reaso to focus o etwork delay. C. Resource Allocatio Strategy I hybrid wireless etworks, packets ca be trasmitted i two modes: ad hoc mode ad ifrastructure mode. I the ad hoc mode, packets are forwarded from the source to the destiatio with oly the help of ormal odes, i.e., without the help of base statios. hile i the ifrastructure mode, packets are first trasmitted from the source to the wired etwork, ad the to the destiatio. I this paper, we cosider a L-maximum-hop (L ) resource allocatio strategy. I particular, if a destiatio ode ca be reached withi L hops from a source ode, the the packets betwee this source ad destiatio pair are trasmitted i the ad hoc mode. Otherwise, packets are trasmitted i the ifrastructure mode. Moreover, we assume a total badwidth of bits/sec 4, which is split ito three frequecy bads, i.e., for ad hoc mode, 2 for uplik for ifrastructure mode, ad 3 for dowlik for ifrastructure mode, respectively. Sice the uplik has the same amout of traffic as the dowlik, we have 2 = 3. Thus, = IV. CAPACITY OF HYBRID IRELESS NETORKS UNDER L-MAXIMUM-HOP RESOURCE ALLOCATION STRATEGY I this sectio, we derive the capacity of hybrid wireless etworks uder L-maximum-hop resource allocatio strategy. e assume all odes are equipped with omidirectioal ateas. Recall that the trasmissios i the ad hoc mode, the uplik ad dowlik trasmissios i the ifrastructure mode use differet frequecy bads, i.e.,, 2,ad 3, respectively. e assume there is o iterferece betwee these three types of traffics, ad they ca be carried out simultaeously. Thus, the throughput capacity of the etwork with odes ad m base statios, deoted by λ(, m), ca be represeted as λ(, m) =λ a (, m)+λ i (, m) where λ a (, m) ad λ i (, m) deote the throughput capacity cotributed by the ad hoc mode trasmissios ad the ifrastructure mode trasmissios, respectively. Notice that λ(, m), λ a (, m), adλ i (, m) all deote the aggregated throughput capacity. e use λ 0 (, m), λ 0 a(, m), adλ 0 i (, m) to deote the correspodig per-ode throughput capacity, respectively. A. Ad Hoc Mode Throughput Capacity Cosider the low tier etwork compoet, i.e., uiformly ad idepedetly distributed odes o a plaar torus. e first itroduce some of the defiitios ad results i [2], listed as follows. c i s are used to deote determiistic costats idepedet of. Vorooi Tessellatio [2]: Give a set of poits i a plae, Vorooi tessellatio divides the domai ito a set of polygoal regios, the boudaries of which are the perpedicular bisectors of the lies joiig the poits. 4 Notice that badwith has several meaigs. I sigal processig, it is a measure of the width of a rage of frequecies, measured i hertz. Here, however, we refer it to be a rate of data trasfer, measured i bits per secod. Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

4 20 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 Lemma 4. i [2]: For every ε>0, there is a Vorooi tessellatio with the property that every Vorooi cell cotais a disk of radius ε ad is cotaied i a disk of radius 2ε. The for the odes, we ca costruct a Vorooi tessellatio V for which (V) Every Vorooi cell cotais a disk of area 00 log /. (V2) Every Vorooi cell is cotaied i a disk of radius 2ρ(), where ρ() := the radius of a disk of area 00 log. Adjacet Vorooi Cells: e say two Vorooi cells are adjacet if they share a commo poit (every Vorooi cell is aclosedset e choose the rage r() of each trasmissio so that r() =8ρ(), which allows direct commuicatio withi a Vorooi cell ad betwee adjacet Vorooi cells. Iterferig Neighbors: we say two cells are iterferig eighbors if there is a poit i oe cell which is withi a distace (2 + Δ)r() of some poit i the other cell. Lemma 4.3 i [2]: he all odes i the etwork use omidirectioal ateas, every cell i V has o more tha c iterferig eighbors, where c depeds oly o Δ ad grows o faster tha liearly i ( + Δ) 2. Proof: Let V be a Vorooi cell. If V is a iterferig eighborig Vorooi cell, there must be two poits, oe i V ad the other i V, which are o more tha (2+Δ)r() uits apart. From (V2), the diameter of a cell is bouded by 4ρ( Hece V, ad similarly every other iterferig eighbor i the Protocol Model, must be cotaied withi a commo large disk D of radius 6ρ() +(2+Δ)r( Such a disk caot cotai more tha c 2 = [6ρ()+(2+Δ)r()]2 ρ 2 () =(22+8Δ) 2 O((+Δ) 2 ) disks of radius ρ( By (V), there ca therefore be o more tha this umber of cells withi D. Thus, c = c 2 is the a upper boud o the umber of iterferig eighbors of the cell. Lemma 4.4 i [2]: I the Protocol Model, there is a schedule for trasmittig packets such that i every ( + c ) slots, each cell i the tessellatio V gets oe slot for packet trasmissio, ad all trasmissios are successfully received withi a distace r() from their trasmitters. e first derive a lower boud o the per-ode throughput capacity by choosig the routes of packets to approximate the straightlie coectig the source ad the destiatio. Deote the straight lie coectig a source ode X i ad a destiatio ode Y i as L i. Uder the L-maximum-hop resource allocatio strategy, we ow boud the probability that L i itersects a give Vorooi cell V. Lemma : For segmet L i ad Vorooi cell V, uder the L-maximum-hop routig strategy, Prob(L i itersects V ad L i is usig ) c 3 L 3 ( log )2. Proof: As metioed before, Vorooi cell V is cotaied 400 log π i a disk of radius 2ρ(), i.e.,. Suppose X i lies at a distace x from the ceter of this disk as show i Fig., the the agel α subteded at X i by the disk is o X i Lr( ) 2 ρ( ) Fig.. Illustratio for calculatig the probability that L i itersects Vorooi cell V. more tha c4 log x. The area of the sector formed is o more tha c5l2 r 2 ()α 2π.IfY i does ot lie i the sector, the the lie L i caot itersect the disk cotaiig the cell V. Thus, the probability that L i itersects the disk is o more tha c6l2 x ( log ) 3 2. Sice X i is uiformly distributed o the plae of uit disk, the probability desity that it is at a distace x from the ceter of the disk is bouded above by 2c 7 πx. Besides, i order for L i to itersect V, we eed 2ρ() x Lr( As a result, we ca obtai Prob(L i itersects V ad L i is usig ) 8L 00 log π c 6 L log x (log ) 3 2 2c7 πxdx π c 3 L 3 ( log )2. Sice there are lies {L i } i=, coectig X i ad Y i,the mea umber of lies passig through a Vorooi cell that use frequecy bad is bouded as follows: E(Number of lies i {L i } i= itersects V ad L i is usig ) c 3 L 3 log2. Notice that routes follow lies. By exploitig uiform covergece i the law of large umbers alog the lie i [2], we have the followig two results. Lemma 2: There is a δ () 0 such that Prob ( sup (Number of lies L i itersectig V ad L i V V is usig ) c 3 L 3 log2 ) δ ( Note that the traffic hadled by a cell is proportioal to the umber of lies passig through it. Sice each lie o frequecy bad carries traffic of rate λ 0 a (, m) bits per secod, we have the followig boud. Lemma 3: There is a δ () 0 such that Prob ( sup (Traffic eedig to be carried by cell V ) V V c 3 λ 0 a (, m)l3 log2 ) δ ( This implies that the rate at which each cell eeds to trasmit is less tha c 3 λ 0 a (, m)l3 log2 with high probability. This rate ca be accommodated by all cells if it is less tha the rate available, i.e., if c 3 λ 0 a(, m)l 3 log2 c 2. Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

5 LI et al.: CAPACITY AND DELAY OF HYBRID IRELESS BROADBAND ACCESS NETORKS 2 Thus, we arrive at a lower boud o the per-ode throughput capacity cotributed by ad hoc mode trasmissios, as show i the followig lemma. Lemma 4: For ad hoc mode trasmissios, uder the L- maximum-hop resource allocatio strategy, ) whe L =Ω( 3 ), there is a determiistic costat log 2 3 c>0 ot depedig o, Δ, or, such that λ 0 a (, m) = c ( + Δ) 2 L 3 log 2 bits per secod is feasible with high probability, i.e., λ 0 a (, m) =Ω( L 3 log 2 2) ad whe L = o( 3 ), there is a determiistic log 2 3 costat c>0 ot depedig o, Δ, or,such that λ 0 a(, m) =. bits per secod is feasible with high probability. Next, we fid a upper boud o the per-ode throughput capacity. Lemma 5.4 i [2]: The umber of simultaeous trasmissios o ay particular chael is o more tha 4 N max = c 8 πδ 2 r 2 () i the Protocol Model. Uder the L-maximum-hop resource allocatio strategy, the mea umber of hops take by a packet trasmitted i the ad hoc mode, deoted by h, is calculated as follows: h πr 2 () πl 2 r 2 () +2 3πr 2 () πl 2 r 2 () L [L2 (L ) 2 ]πr 2 () πl 2 r 2 () = 4L3 +3L 2 L 6L 2 Sice each source geerates λ 0 a(, m) bits per secod, there are sources, each of which trasmits to its destiatio i ad hoc mode with a probability of πl 2 r 2 (), the the total umber of bits per secod served by the etire etwork eeds to be at least πl 2 r 2 () hλ 0 a (, m To esure that all the required traffic is carried, we therefore eed Thus, πl 2 r 2 () hλ 0 a(, m) N max. λ 0 a(, m) log c 9 Δ 2 L 3 r 4 (). Sice r() > π with high probability [], the we obtai is ecessary to guaratee coectivity λ 0 c a(, m) Δ 2 L 3 log 2. Besides, we also have λ 0 a (, m). Thus, we arrive at the followig lemma. Lemma 5: For ad hoc mode trasmissios, uder the L- maximum-hop resource allocatio strategy, ) whe L = Ω( 3 ), a upper boud o per-ode log 2 3 throughput capacity is λ 0 a (, m) = c Δ 2 L 3 log 2 bits per secod, where c < +, ot depedig o, Δ, or, 2) ad whe L = o( 3 ), a upper boud o per-ode log 2 3 throughput capacity is λ 0 a(, m) =. Notice that the probability that oe ode will trasmit to its destiatio ode i ad hoc mode is πl 2 r 2 ( LetN i ( j ) be a radom variable defied as follows: N i =, source ode i trasmits to its destiatio ode i ad hoc mode; 0, otherwise. Let N T be a radom variable defied as the total umber of source odes trasmittig i ad hoc mode, i.e., N T = i= N i. Thus, the expected umber source odes i ad hoc mode is: Sice f(n i Θ( E(N T )=E( N i )= i= E(N i i= = ) = πl 2 r 2 (), ad r() eeds to be log ) to make the etwork coected [], we have5 E(N i )= πl 2 r 2 () +0 ( πl 2 r 2 ()) = πl 2 log. Thus, E(N T )= πl 2 log = πl2 log. Recall the Cheroff bouds [5]: For ay δ>0, P [N T > ( + δ)πl 2 log ] < ( For ay 0 <δ<, e δ )πl ( + δ) +δ P [N T < ( δ)πl 2 log ] <e 2 δ2 πl 2 log. From the above, we ca obtai for ay 0 <δ<, P [ N T πl 2 log >δπl 2 log ] <e θπl2 log. 2 log. where θ > 0. So,as, the total umber of source odes trasmittig i ad hoc mode is equal to πl 2 log with probability. Thus, the total ad hoc mode traffic is πl 2 r 2 ()λ 0 a (, m), i.e., c L 2 log. Combiig Lemma 4 ad Lemma 5 leads to the followig theorem. q 5 Note r() =Θ(q log ) meas r() log =cr where 0 <c r < +. e igore c r i the followig derivatios for simplicity, which will ot chage our fial results. Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

6 22 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 Theorem : Uder the L-maximum-hop routig strategy, the throughput capacity of the etwork cotributed by ad hoc mode trasmissios is Θ( L log ), L =Ω( 3 ); log λ a (, m) = 2 3 Θ((L 2 log ) ), L = o( 3 log 2 3 B. Ifrastructure Mode Throughput Capacity e the derive the throughput capacity cotributed by trasmissios i the ifrastructure mode. Notice that each packet trasmitted from a source to its destiatio i ifrastructure mode will use oe uplik ad oe dowlik, ad hece it should be couted oly oce for the throughput capacity. Sice the badwidth for uplik is 2 bits/sec, the throughput capacity per cell, deoted by λ c i (, m), is upper bouded by 2. As we metioed before, the base statios divide the area ito a hexago tessellatio, i.e., a 7-cell frequecy reuse patter. Thus, the throughput capacity per cell is lower bouded by 7 2. Fially, we obtai the followig theorem. Theorem 2: Uder the L-maximum-hop routig strategy, the throughput capacity of the etwork cotributed by ifrastructure mode trasmissios is λ i (, m) =Θ(m 2 Proof: e have show that λ c i (, m) =Θ( 2 There are m cells, which leads to λ i (, m) =Θ(m 2 C. Throughput Capacity of the Network From Theorem ad Theorem 2, we ca obtai the followig theorem. Theorem 3: Uder the L-maximum-hop resource allocatio strategy, the throughput capacity of the etwork is Θ( L log )+Θ(m 2), L =Ω( 3 ); log λ a (, m) = 2 3 Θ(L 2 log )+Θ(m 2 ), L = o( 3 log 2 3 Case : L =Ω( 3 log 2 3 Accordig to Theorem 3, we have λ(, m) = Θ( L log )+Θ(m 2 If m =Ω( L log ), the we ca have higher throughput whe =0, i.e., 2 = /2, ad λ max (, m) = Θ(m ), ad hece, { Θ( ), if m =Ω(); λ 0 max(, m) = Θ( m ), if m = o( If m = o( L log ), the we ca have higher throughput whe 2 =0, i.e., =,ad λ max (, m) = Θ( L log ), ad hece, λ 0 max(, m) = Θ( L log Sice L = Ω( 3 ),thel log, ad hece log 2 3 λ 0 max(, m) 0 as, which meas the per-ode throughput capacity dimiishes as goes large ad the etwork caot scale. Case 2: L = o( 3 log 2 3 Accordig to Theorem 3, we have λ(, m) = Θ(L 2 log )+Θ(m 2 If m =Ω(L 2 log ), the we ca have higher throughput whe =0, i.e., 2 = /2, ad λ max (, m) = Θ(m ), ad hece, { Θ( ), if m =Ω(); λ 0 max (, m) = ), if m = o( Θ( m If m = o(l 2 log ), the we ca have higher throughput whe 2 =0, i.e., =,ad ad hece, λ max (, m) = Θ(L 2 log ), λ 0 max (, m) = log Θ(L2 Sice L = o( 3 ),the L 2 log log as,which meas the per-ode throughput capacity dimiishes as goes large ad the etwork caot scale. From the above, we arrive at the followig results. Corollary : Uder the L-maximum-hop resource allocatio strategy, ) whe L =Ω( 3 ), (i) if m =Ω( log 2 3 L log ), we ca have higher throughput whe =0, ad the etwork ca scale oly if m =Ω(); (ii) if m = o( L log ), we ca have higher throughput whe 2 =0,adtheetwork caot scale. 2) whe L = o( 3 ), (i) if m log 2 3 =Ω(L2 log ), we ca have higher throughput whe =0,adtheetwork ca scale oly if m =Ω(); (ii) if m = o(l 2 log ), we ca have higher throughput whe 2 =0,adthe etwork caot scale. D. Comparisos with Pure Ad Hoc Networks Gupta ad Kumar have show i [2] that for pure ad hoc etworks, whe each ode radomly chooses aother ode as its destiatio with o limit to the maximum umber of hops, the per-ode throughput capacity is Θ( log ediscuss i the followig whether the per-ode throughput capacity ca be ehaced by placig some base statios i the etwork ad the impacts of L ad m o the throughput capacity of hybrid wireless etworks. Case : L =Ω( 3 log 2 3 ) If m =Ω( L log ), If m =Ω(), theλ 0 max (, m) =Θ( If m = o(), the λ 0 max (, m) = Θ(m ) = Sice the trasmissio rage of the Ω( L log Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

7 LI et al.: CAPACITY AND DELAY OF HYBRID IRELESS BROADBAND ACCESS NETORKS 23 log π odes satisfies r() > as metioed before, the L = O( log Thus, we obtai λ 0 max(, m) = Ω( log Moreover, whe L = Θ( 3 ), we ca obtai log 2 3 λ0 max(, m) = 3 Ω( ),adwhel = ω( ), we ca 3 log 3 log 2 3 obtai λ 0 max (, m) =Ω(o( ) 3 log 3 2) If m = o( L log ),wehaveshowthatλ0 max(, m) = Θ( L log Sice L = O( log ), we have λ 0 max(, m) = Ω( log Furthermore, whe L = Θ( 3 ), we have log 2 3 λ0 max(, m) = Θ( ), 3 log 3 ad whe L = ω( 3 ), we have log 2 3 λ0 max (, m) = o( 3 log 3 I this case, we limit L to Ω( 3 Ifm = o( log 2 3 L log ), the icrease of base statios does ot icrease the per-ode throughput capacity. But, the throughput capacity of hybrid wireless etworks is greater tha that of pure ad hoc etworks. If we add some more base statios i the etwork so that m is lower bouded by L log but upper bouded by, the per-ode throughput capacity icreases liearly with the umber of base statios, which is Θ( m ), ad also greater tha that of pure ad hoc etworks. Moreover, if we keep addig base statios i the etwork such that m = Ω(), the per-ode throughput capacity will reach its maximum, i.e., Θ( Besides, we also observe that except for the case that m =Ω(), the throughput capacity of hybrid wireless etworks always gets smaller as the maximum umber of hops L icreases. Case 2: L = o( 3 log 2 3 e otice that i this case, L 2 log = o( 2 3 log 3 ) If m =Ω(L 2 log ), If m =Ω(), wehaveshowthatλ 0 max(, m) = Θ( If m = o(), λ 0 max (, m) = Θ(m Thus, if m =Ω( 2 3 log 3 ),theλ0 max(, m) =Ω( ); 3 log 3 ad if m = o( 2 3 ), the log 3 λ0 max (, m) = 4 o( Furthermore, whe L = Ω( ), 3 log 3 log 3 4 we obtai that λ 0 max(, m) = Ω( log ), ad whe L = o( 4 ), we obtai that log 3 4 λ0 max(, m) = Ω(o( log ) 2) If m = o(l 2 log ) =o( 2 3 log 3 ),theλ0 max (, m) = Θ( L2 log ) = o( Moreover, whe L = 3 log 3 Ω( 4 ), we obtai that log 3 4 λ0 max (, m) =Ω( log ), ad whe L = o( 4 ), we obtai that log 3 4 λ0 max(, m) = o( log I this case, we further limit L to o( 3 If m = log 2 3 o(l 2 log ), the icrease of base statios does ot icrease the per-ode throughput capacity, which is o( If 3 log 3 we add some more base statios i the etwork so that m =Ω(L 2 log ), the per-ode throughput capacity icreases liearly with the umber of base statios. If m =Ω( 2 3 ), log 3 the the per-ode throughput capacity will be lower bouded by.ifm =Ω(), the per-ode throughput capacity 3 log 3 will reach its maximum, i.e., Θ( Besides, we observe that except for the case that m =Ω(), the throughput capacity of hybrid wireless etworks gets larger as the maximum umber of hops L icreases, which is quite differet from that i Case. Notice that whe L is as small as o( 4 ), the capacity log 3 4 of hybrid wireless etworks ca be eve smaller tha that of pure ad hoc etworks. This is because i this case too may odes share the resource i the ifrastructure mode, which sigificatly limits the etwork capacity. From the above, we also fid that addig base statios ito the pure ad hoc etworks ca have sigificat impacts o the etwork capacity oly if m =Ω( L log ) whe L =Ω( 3 ), log 2 3 or m =Ω(L 2 log ) whe L = o( 3 Iotherwords, log 2 3 as the maximum hop cout L icreases, the threshold of m, above which the capacity of hybrid wireless etworks icreases liearly with m, first icreases whe L is small, ad the decreases whe L is large. E. More Discussios I [6], the authors use a routig strategy such that a source ode trasmits to its destiatio ode i the ad hoc mode oly if the destiatio is i the same cell as the source. They show that uder this routig strategy, the maximum throughput capacity icreases liearly with the umber of base statios m if m =Ω( e show i the followig that this ca be cosidered as a special case i our aalysis. he m = Ω( ), we ca obtai that L = O( m /r()) = O( 4 log ), ad hece m = Ω(L 2 log Actually, this is icluded i the case that L = o( 3 ) log 2 3 as we have show before. Accordig to our results, whe m = Ω( ) = Ω(L 2 log ), the maximum throughput capacity icreases liearly with m, which is the same as that i [6]. V. DELAY IN HYBRID IRELESS NETORKS UNDER L-MAXIMUM-HOP RESOURCE ALLOCATION STRATEGY I the literature, there are some works ( [2], [8], [9], [5], [9], [20]) about the trade-off betwee capacity ad delay i mobile ad hoc etworks. They show that by usig mobility to icrease the capacity of the etwork, the delay will also be icreased. Recall that i static radom ad hoc etworks, the per-ode throughput capacity is Θ( log ), ad the average packet delay is Θ( log Gammaletal.[9]showthatwhe the capacity of mobile radom ad hoc etworks icreases to Θ(), the average packet delay icreases to Θ( log Ithis Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

8 24 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 TABLE I THROUGHPUT CAPACITY AND PACKET DELAY IN HYBRID IRELESS NETORKS. L m Per-ode Throughput Capacity Average Packet Delay Ω() Θ( ) Ω( 3 log 2 3 ) (Ω( L log ),o()) Θ( m ) Θ() o( L log ) Θ( L log ) Ω() Θ( ) Θ(L) o( 3 log 2 ) (Ω(L 2 log ),o()) Θ( m ) Θ() 3 o(l 2 log ) Θ( L2 log ) Θ(L) sectio, we study the delay i hybrid wireless etworks to see whether there also exists such a trade-off. e first preset a fudametal result, which will be used frequetly, as follows. Lemma 6: Uder L-maximum-hop resource allocatio strategy, if we choose the route of packets to approximate the straightlie coectig the source ad the destiatio, for packets trasmitted i the ad hoc mode, the average delay is Θ(L), ad for packets trasmitted i the ifrastructure mode, the average delay is Θ( Recall that the umber of trasmitters i the ad hoc mode is πl 2 log with probability. The, the umber of trasmitters i the ifrastructure mode is πl 2 log with probability. Thus, the average delay of all the packets i the hybrid etwork, deoted by D 0 (), is D 0 () = Θ( πl2 log L +( πl 2 log ) ) = Θ( L3 log + Case : L =Ω( 3 log 2 3 As we metioed before, the average delay of all the packets i the hybrid etwork is D 0 () =Θ( L3 log +)=Ω( ) If m = Ω( L log ), we ca have higher throughput whe all the badwidth is assiged to the traffics i the ifrastructure mode, i.e., there are o traffics i the ad hoc mode. Thus, the average delay of all the packets is D 0 max() =Θ( 2) m = o( L log ), we ca have higher throughput whe all the badwidth is assiged to the ad mode traffics. I this case, the average delay of all the packets is D 0 max () =Θ(L) =Ω( 3 log 2 3 Besides, sice L = O( log ), we also have Dmax 0 () =O( log Thus, we observe that i this case, as we put more ad more base statios i the etwork so that m icreases from o( L log ) to Ω( L log ), the delay decreases while the capacity icreases as we show i Sectio IV-D. Case 2: L = o( 3 log 2 3 The same as before, the average delay of all the packets i the hybrid etwork is D 0 () =Θ( L3 log +)=Θ( ) If m = Ω(L 2 log ), we ca have higher throughput whe all the badwidth is assiged to the traffics i the ifrastructure mode, ad the average delay of all the packets is Dmax() 0 =Θ( 2) m = o(l 2 log ), we ca have higher throughput whe all the badwidth is assiged to the ad mode traffics, ad the average delay of all the packets is Dmax 0 () =Θ(L) =o( 3 log 2 3 Besides, sice L =Ω(),wehave Dmax 0 () =Θ(L) =Ω( Similar to that whe L =Θ( 3 log 2 3 ),asm icreases from o(l 2 log ) to Ω(L 2 log ), the delay decreases while the capacity icreases. I coclusio, we fid that the smaller the maximum hop umber L is, the smaller the average packet delay is. Specifically, whe L = Ω( 3 ), the average delay is log 2 3 lower bouded by 3 if m = o( log 2 3 L log ), ad is Θ() if m = Ω( L log Besides, whe L = o( 3 ), the log 2 3 delay is o( 3 ) if m = log 2 3 o(l2 log ), ad is Θ() if m =Ω(L 2 log Combiig the results of the throughput capacity ad the correspodig delay, we arrive at Table I, from which we observe that i hybrid wireless etworks, by addig base statios to help carry out trasmissios, the per-ode throughput capacity ca achieve Θ( ) while the average packet delay is kept as low as Θ( VI. CONCLUSION I this paper, we study the throughput capacity ad the average packet delay i hybrid wireless etworks. e fid that for most of the cases, hybrid wireless etworks have greater throughput capacity ad smaller average packet delay tha pure ad hoc etworks. Moreover, we observe that whe m = Ω(), the per-ode throughput capacity ca be Θ( ) while the average packet delay is maitaied as low as Θ( Oly Authorized licesed use limited to: Uiversity of Florida. Dowloaded o Jue 20, 2009 at 03:00 from IEEE Xplore. Restrictios apply.

9 LI et al.: CAPACITY AND DELAY OF HYBRID IRELESS BROADBAND ACCESS NETORKS 25 whe L = o( 4 ) ad m = log 3 4 o(l2 log ), hybrid wireless etworks have smaller throughput capacity tha pure ad hoc etworks. This is because i this case, there are a small umber of base statios while too may odes share the badwidth i the ifrastructure mode. e also otice that we eed to assig all the badwidth to either ad hoc mode trasmissios or ifrastructure mode trasmissios i order to have higher throughput. I either case, oe of the two mode trasmissios will get o badwidth at all. I order to avoid this situatio, we ca assig some miimum amout of badwidth to each mode, as suggested i [6]. Sice hybrid wireless etworks ca provide high throughput capacity ad low packet delay, we ca fially coclude that wireless hybrid etworks is a good solutio to broadbad access etworks. REFERENCES [] A. Agarwal ad P. Kumar. Capacity bouds for ad hoc ad hybrid wireless etworks. 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I Proceedig of the IEEE/ACM Iteratioal symposium o Modelig, Aalysis, ad Simulatio of Computer ad Telecommuicatio Systems (MAS- COTS 04), Voledam, Netherlads, October [2] A. Okabe, B. Boots, K. Sugihara, ad S. Chiu. Spatial tessellatios : cocepts ad applicatios of Vorooi diagrams. iley Series i Probability ad Statistics, [22] A. Ozgur, O. Leveque, ad D. Tse. How does the iformatio capacity of ad hoc etworks scale? I Proc. Forty-fourth Aual Allerto Coferece o Commuicatio, Cotrol ad Computig, Moticello, IL, USA, September [23] Y. Pei, J. Modestio, ad X. ag. O the throughput capacity of hybrid wireless etworks usig a l-maximum-hop routig strategy. I Proc. IEEE Vehicular Techology Coferece, [24] T. Rappaport. ireless Commuicatios: Priciples ad Practice (Secod Editio Pretice-Hall PTR, [25] S. Sarkar, S. Dixit, ad B. Mukherjee. A evolutio sceario of a broadbad access etwork usig r-soa-based wdm-po techologies. J. Lightwave Techol., 25(): , [26] S. Sarkar, S. Dixit, ad B. Mukherjee. Hybrid wireless-optical broadbad-access etwork (woba): A review of relevat challeges. J. Lightwave Techol., 25(): , [27] S. Toumpis. Capacity bouds for three classes of wireless etworks. I Proc. ACM MobiHoc, Roppogi Hills, Tokyo, Japa, May [28] A. Zemliaov ad G. Veciaa. Capacity of ad hoc wireless etworks with ifrastructure support. IEEE J. Sel. Areas Commu., 23(3), March Pa Li (IEEE, S 06 / ACM 07) received his B.E. i Electrical Egieerig from Huazhog Uiversity of Sciece ad Techology, uha, Chia, i He is workig towards his Ph.D. degree i the Departmet of Electrical ad Computer Egieerig at Uiversity of Florida. His research iterests iclude capacity ad coectivity aalysis, medium access cotrol, routig algorithms, ad cross-layer desig i wireless etworks. He is a studet member of the IEEE ad the ACM. Chi Zhag (IEEE, S 06) received the B.E. ad M.E. degrees i Electrical Egieerig from Huazhog Uiversity of Sciece ad Techology, uha, Chia, i July 999 ad Jauary 2002, respectively. Sice September 2004, he has bee workig towards the Ph.D. degree i the Departmet of Electrical ad Computer Egieerig at Uiversity of Florida, Gaiesville, Florida, USA. His research iterests are etwork ad distributed system security, wireless etworkig, ad mobile computig. Yuguag Michael Fag (IEEE, S 92-M 94- S 96-M 97-SM 99-F 08) received a Ph.D. degree i Systems Egieerig from Case ester Reserve Uiversity i Jauary 994 ad a Ph.D degree i Electrical Egieerig from Bosto Uiversity i May 997. He was a assistat professor i the Departmet of Electrical ad Computer Egieerig at New Jersey Istitute of Techology from July 998 to May He the joied the Departmet of Electrical ad Computer Egieerig at Uiversity of Florida i May 2000 as a assistat professor, got a early promotio to a associate professor with teure i August 2003 ad to a full professor i August He holds a Uiversity of Florida Research Foudatio (UFRF) Professorship from 2006 to Authorized licesed use limited to: Uiversity of Florida. 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