On the Capacity of Hybrid Wireless Networks

Transcription

1 O the Capacity of Hybrid ireless Networks Beyua Liu,ZheLiu +,DoTowsley Departmet of Computer Sciece Uiversity of Massachusetts Amherst, MA IBM T.J. atso Research Ceter P.O. Box 704 Yorktow Heights, NY 0598 Abstract e study the throughput capacity of hybrid wireless etworks. A hybrid etwork is formed by placig a sparse etwork of base statios i a ad hoc etwork. These base statios are assumed to be coected by a high-badwidth wired etwork ad act as relays for wireless odes. They are ot data sources or data receivers. Hybrid etworks preset a tradeoff betwee traditioal cellular etworks ad pure ad hoc etworks i that data may be forwarded i a multi-hop fashio or through the ifrastructure. It has bee show that the capacity of a radom ad hoc etwork does ot scale well with the umber of odes i the system []. I this work, we cosider two differet routig strategies ad study the scalig behavior of the throughput capacity of a hybrid etwork. Aalytical expressios of the throughput capacity are obtaied. For a hybrid etwork of odes ad m base statios, the results show that if m grows asymptotically slower tha, the beefit of addig base statios o capacity is isigificat. However, if m grows faster tha, the throughput capacity icreases liearly with the umber of base statios, providig a effective improvemet over a pure ad hoc etwork. Therefore, i order to achieve o-egligible capacity gai, the ivestmet i the wired ifrastructure should be high eough. Keywords: Throughput capacity, hybrid wireless etworks, ad hoc etworks. I. INTRODUCTION Throughput capacity is a key characteristic of wireless etworks. It represets the log-term achievable data trasmissio rate that a etwork ca support. The throughput capacity of a wireless etwork depeds o may aspects of the etwork: etwork architecture, power ad badwidth costraits, routig strategy, radio iterferece, etc. A good uderstadig of the capacities of differet etwork architectures allows a desiger to choose a architecture appropriate for his or her specific purpose. Several etwork models are available for wireless data etworks. I a wireless cellular etwork or a wireless LAN, odes commuicate with each other through base statios or access poits. A ode first coects to the earest base statio or access poit i order to commuicate with other odes. A This research has bee supported i part by NSF uder awards ANI , EIA 00809, ad NSF ITR base statio (access poit) serves as a commuicatio gateway for all the odes i its cell (basic service area). I situatios where there is o fixed ifrastructure, for example, battle fields, catastrophe cotrol, wireless ad hoc etworks become valuable alteratives to wireless cellular etworks or wireless LANs for odes to commuicate with each other. A ad hoc etwork is a commuicatio etwork formed by a collectio of odes without the aid of ay fixed ifrastructure. I a ad hoc etwork, due to the lack of ifrastructure ad the limited trasmissio rage of each ode, data eeds to be routed to the destiatio by the odes i a multi-hop fashio. I a recet study [2], the authors proposed a hybrid etwork model to improve etwork coectivity. I the model, a sparse etwork of base statios coected by a wired etwork is placed withi a ad hoc etwork. The resultig etwork cosists of ormal odes ad some well-coected base statios. It is called a hybrid etwork sice it presets a tradeoff betwee traditioal cellular etworks ad pure ad hoc etworks. I ad hoc etworks, there is o ifrastructure, data ca oly be forwarded by the odes i a multi-hop fashio. I cellular etworks, data are always forwarded through the ifrastructure. hile i a hybrid etwork, data may be forwarded i a multihop fashio or through the ifrastructure. Commuicatios usig multi-hop forwardig ad commuicatios usig the ifrastructure coexist i the etwork. It is of great iterest to uderstad what performace gais ca be achieved by the hybrid etworks. hile the capacity performace of cellular etworks has bee well studied [3], researchers have started to ivestigate the capacity of wireless ad hoc etworks oly recetly. I [], Gupta ad Kumar studied the throughput capacity of a radom wireless etwork, where fixed odes are radomly placed i the etwork ad each ode seds data to a radomly chose destiatio. The throughput capacity per ode is show to be Θ( log ), as approaches ifiity, where is the umber of odes i the etwork (the same below) ad is the commo trasmissio rate of each ode over the wireless chael. Thus the aggregate throughput capacity of all the

2 odes i the etwork is Θ( log). I [4], Gupta ad Kumar studied the capacity of a radom three-dimesioal wireless ad hoc etwork, ad showed that the aggregate throughput scales ( 2 ) as Θ( 3 log). I [5], Grossglauser ad Tse proposed a scheme that takes advatage of the mobility of the odes. By allowig oly oe-hop relayig, the scheme achieves a aggregate throughput capacity of O() at the cost of ubouded delay ad buffer requiremet. Gastpar ad Vetterli studied the capacity uder a differet traffic patter i [6]. There is oly oe active source ad destiatio pair, while all other odes serve as relay, assistig the trasmissio betwee the source ad destiatio odes. The capacity is show to scale as O(log ). I [7], Li et al. examied the effect of IEEE 802. o etwork capacity ad preseted specific criteria of the traffic patter that makes the capacity scale with the etwork size. The above studies all focus o the capacities of pure ad hoc etwork models. It is ot clear how much capacity gai a etwork ca achieve by addig a certai umber of base statios to a ad hoc etwork ad formig a hybrid etwork. Ituitively, o oe had, the ifrastructure helps to reduce the wireless trasmissios, resultig i less iterferece ad a higher capacity. O the other had, too much use of the ifrastructure may cause hot spots aroud base statios ad iefficiet use of spatial cocurrecy, leadig to a sub-optimal capacity. It is the purpose of this work to study the capacity of hybrid etworks. I particular, we are iterested i the followig questios: How does the throughput capacity scale with the umber of odes ad the umber of base statios? How does the capacity of a hybrid etwork model compare to that of a pure ad hoc etwork? I this paper, we cosider two differet routig strategies ad obtai the aalytical expressios of the throughput capacity of hybrid etworks. Moreover, we derived the maximum throughput capacities ad the coditios to achieve them. e assume a hybrid etwork of m base statios ad odes, each capable of trasmittig at bits/sec over the wireless chael. I the first routig strategy, a ode seds data through the ifrastructure if the destiatio is outside of the cell where the source is located. Otherwise, the data are forwarded i a multi-hop fashio as i a ad hoc etwork. Uder this strategy, if m grows asymptotically slower ) tha, the maximum throughput capacity is Θ( log. I this case, the beefit of addig base statios is isigificat. However, i the case where base statios ca be added at a speed asymptotically faster tha, the maximum capacity is Θ(m ), which icreases liearly with the umber of base statios. Similar results are obtaied for a probabilistic routig strategy. I the strategy, a ode chooses whether to use ifrastructure to sed data accordig to some probability. Uder this routig strategy, if m grows slower tha log, the maximum throughput capacity has the same asymptotic behavior as a pure ad hoc etwork. There is o beefit to use the ifrastructure i this case. If m grows faster tha log, the maximum throughput capacity scales as Θ(m ), which icreases liearly with the umber of base statios. For both routig strategies, if the umber of base statios scales slower tha some threshold, the throughput capacity is domiated by the cotributio of ad hoc mode trasmissios. The beefit of addig base statios is miimal. If the umber of base statios scales faster tha the threshold, the capacity cotributed by the ifrastructure domiates the overall etwork throughput capacity. I this case, the maximum throughput capacity scales liearly with the umber of base statios, providig a effective improvemet over pure ad hoc etworks. Therefore, i order to achieve o-egligible capacity gai, the ivestmet i the wired ifrastructure should be high eough: the umber of base statios should be at least for the first routig strategy ad log for the probabilistic routig strategy. The rest of the paper is orgaized as follows: I Sectio II, we describe the hybrid wireless etwork model. I Sectio III ad IV, we preset the aalytical results of throughput capacity of a hybrid etwork, uder the two differet routig strategies. I Sectio V, we draw the coclusios. II. HYBRID NETORK MODEL I this sectio, we preset the hybrid wireless etwork model ad defie the throughput capacity of hybrid etworks. A. Network Compoets A hybrid etwork cosists of two compoets. The first compoet is a ad hoc etwork cotaiig oly ormal odes, the same as the model defied i []. The secod compoet is a sparse etwork of base statios. The base statios are assumed to be coected together by a wired etwork ad are placed withi the ad hoc etwork i a regular patter. e scale space ad assume that a populatio of odes are radomly, i.e., idepedetly ad uiformly, located withi a disk of area square meter i the plae. e further assume that the odes are homogeeous, employig the same trasmissio rage or power. Every ode is a data source. The destiatio for each ode is idepedetly chose as the ode earest to a radomly located poit withi the uit area disk. I additio to the odes i the etwork, a sparse etwork of m base statios is regularly placed i the uit area disk. The base statios divide the area ito a hexago tessellatio, as show i Fig. As i a cellular etwork, each hexago is called a cell ad there is a base statio i the ceter of each cell. Ulike ormal odes, the base statios are either data sources or data receivers. They are added as relay odes to improve etwork performace ad they oly egage i routig ad forwardig data for ormal odes. The base statios are assumed to be coected together by a wired etwork. Furthermore, we assume the lik badwidth i the wired etwork are all large eough so that there are o badwidth costraits i the wired etwork. e also assume there are o power costraits for the base statios.

3 Fig.. Base Statio Hybrid ireless Network Model B. Iterferece Model All the odes ad the base statios share a commo wireless chael. e assume a time-divisio multiplexig (TDMA) scheme for the data trasmissio over the wireless chael. Time is divided ito slots of fixed duratios. I each time slot, a ode is scheduled to sed data. A ode caot trasmit ad receive data simultaeously ad a ode ca oly receive data from oe other ode at the same time. The wireless trasmissios i the etwork are assumed to be homogeeous. Nodes icludig the base statios employ the same trasmissio rage, deoted by r. For the iterferece model, we adopt the Protocol Model itroduced i []. A trasmissio from ode X i is successfully received by ode X j if the followig two coditios are satisfied: ) Node X j is withi the trasmissio rage of ode X i, i.e., X i X j r where X i X j represets the distace betwee ode X i ad ode X j i the plae. 2) For every other ode X k that is simultaeously trasmittig over the same chael, X k X j ( + ) X i X j This coditio guaratees a guard zoe aroud the receivig ode to prevet a eighborig ode from trasmittig o the same chael at the same time. The radius of the guard zoe is ( + ) times the distace betwee the seder ad receiver. The parameter defies the size of the guard zoe ad we require that > 0. C. Routig Strategy I a hybrid etwork, there are two trasmissio modes: ad hoc mode ad ifrastructure mode. I the ad hoc mode, data are forwarded from the source to the destiatio i a multi-hop fashio without usig ay ifrastructure. I the ifrastructure mode, data are forwarded through the ifrastructure. It ca be show that i terms of throughput capacity, it is optimal to eter ad exit the ifrastructure oly oce. Also, it is optimal for a ode to commuicate with the earest base statio i order to reach the ifrastructure. Deote the base statio earest to ode X i as B(X i ). I this work, by ifrastructure mode we mea that data are first trasmitted from the source (X s )tob(x s ) over the wireless chael; the base statio the trasmits the data through the wired ifrastructure to B(X d ), which fially trasmits the data to the destiatio X d. I this work, we cosider two routig strategies. I the first routig strategy, if the destiatio is located i the same cell as the source ode, data are forwarded i the ad hoc mode. Otherwise, data are forwarded i the ifrastructure mode. Sice the destiatio for a source ode is radomly chose i the uit area disk, the probability that a ode commits to itra-cell commuicatios is /m; the probability that a ode commits to iter-cell commuicatios is /m. e ca geeralize the routig strategy to represet a family of routig strategies by relaxig the coditio that the ad hoc mode is chose to sed data. Istead of requirig the destiatio be located i the same cell as the source, a ode uses ad hoc mode to sed data as log as the destiatio is located withi k earest eighborig cells from the source ode, where k 0 defies the rage withi which ad hoc mode trasmissios should be used. e call this family of routig strategies the k-earest-cell routig strategies. The secod routig strategy is a probabilistic routig strategy. A trasmissio mode is idepedetly chose for each source destiatio pair. ith probability p, the ad hoc mode is employed, ad with probability p, the ifrastructure mode is used. By varyig the probability p, a family of probabilistic routig strategies ca be obtaied. e assume each ode ca trasmit at bits/sec over the commo wireless chael. e divide the wireless chael so that ad hoc mode trasmissios ad ifrastructure mode trasmissios go through differet sub-chaels. e further divide the sub-chael for ifrastructure mode trasmissios ito uplik ad dowlik parts, accordig to the directio of the trasmissios relative to the base statio. Sice itracell traffic, uplik traffic ad dowlik traffic use differet sub-chaels, there is o iterferece betwee the three types of traffics. The badwidth assiged to itra-cell, uplik, ad dowlik sub-chaels are, 2,ad 3, respectively. The trasmissio rates should sum to, i.e., 3 i= i =.Sice there are same amout of uplik traffic ad dowlik traffic, we let 2 = 3. D. Defiitio of Throughput Capacity To make the formulas more cocise, we preset the aggregate throughput capacity of the whole etwork istead of the throughput capacity of each ode. Note that the throughput capacity per ode is simply / of the aggregate throughput capacity of the whole etwork. I this paper, we adopt the asymptotic otatios defied i [8]. e ow defie the feasible aggregate throughput ad the aggregate throughput capacity of the hybrid etwork model.

4 Defiitio : Feasible Aggregate Throughput. For a hybrid etwork of odes ad m base statios, a aggregate throughput of T(,m) bits/sec is feasible if by trasmittig data i the ad hoc or ifrastructure mode, there is a spatial ad temporal schedulig scheme that yields a aggregate etwork throughput of T(,m) bits/sec o average. Here the aggregate throughput is the sum of the idividual throughputs from each ode to its chose destiatio. Defiitio 2: Aggregate Throughput Capacity of Hybrid Networks. The aggregate throughput capacity of a hybrid wireless etwork is of order Θ( f (,m)) bits/sec if there are determiistic costats c > 0, ad c < + such that lim Prob(T(,m) = cf(,m) is feasible) = lim ifprob(t(,m) = c f (,m) is feasible) < III. CAPACITY OF IRELESS HYBRID NETORKS UNDER K-NEAREST-CELL ROUTING STRATEGIES I this sectio, we derive the throughput capacity of the hybrid etwork uder the k-earest-cell routig strategies. I particular, we aalyze the case where k = 0, i.e., a ode seds data i ad hoc mode if the destiatio is located i the same cell as the source. e cojecture that the results hold for the family of routig strategies whe k is a costat. After this, we compare the capacity of hybrid etworks to the capacity of pure ad hoc etworks. A. Throughput Capacity Sice the itra-cell, uplik ad dowlik traffics are trasmitted i three differet sub-chaels, there is o iterferece betwee the three types of traffic. However, withi a subchael, iterferece exists betwee the same type of traffic i differet cells. Fortuately, the effect of this iterferece is miimal. There is a spatial trasmissio schedule that yields efficiet frequecy reuse. More specifically, the cells ca be spatially divided ito a costat umber of differet groups. Trasmissios i the cells of the same group do ot iterfere with each other. If the groups are scheduled to trasmit i a roud robi fashio, each cell will be able to trasmit oce every fixed amout of time without iterferig with each other. The degradatio of etwork capacity due to the iterferece betwee the same types of traffic is thus bouded by a costat factor. e ow provide the formal proof. e adopt the otio of iterferig eighbors itroduced i [], ad compute the umber of cells that ca be affected by a trasmissio i oe cell. Two cells are defied to be iterferig eighbors if there is a poit i oe cell which is withi a distace (2 + )r of some poit i the other cell, where r is the trasmissio rage of the odes. By the defiitio of the Protocol Iterferece Model, if two cells are ot iterferig eighbors, trasmissios i oe cell do ot iterfere with trasmissios i the other cell. Lemma : Each cell has o more tha c iterferig eighbors, where c is a costat that oly depeds o. Proof. e deote the legth of each side of a cell (hexago) as l ad assume that l = c r. Therefore, each cell is cotaied by a disk of radius c r ad cotais a disk of radius 3 2 c r. If a cell H is a iterferig eighbor of a cell H, oe poit i H must be withi a distace of (2 + )r of a poit i H. Therefore, for the Protocol Iterferece Model, all the iterferig cells of H must be cotaied by a disk D of radius 3c r +(2 + )r. e kow that each cell cotais a disk of radius 3 2 c r. The area of each cell is the larger tha the area of the cotaied disk. The umber of cells cotaied i disk D is thus bouded by: ( ) 2 c = π((3c )r) 2 3 /π 2 c r = 4 ( ) 3c c Lemma 2: I the Protocol Model, there is a spatial schedulig policy such that each cell gets oe slot to trasmit data i every ( + c) slots. Proof. e costruct the followig graph. Each cell is represeted by a vertex ad edges are added betwee iterferig eighbors. It is a well kow fact i graph theory that a graph of degree o more tha c ca have its vertices colored by usig o more tha ( + c) colors, while o two adjacet vertices have the same color. Therefore, the cells i the etwork ca be colored with o more tha ( + c) colors, while o two iterferig eighbors have the same color. e allow cells of the same color to trasmit i the same time slot. Trasmissios from o-iterferig cells do ot iterfere with each other. Therefore, there exist schedulig schemes where each cell receives a slot for trasmissio every ( + c) slots. The degradatio of etwork capacity due to the iterferece betwee the same type of traffic is bouded by a costat factor. Now we derive the aggregate throughput capacity of the hybrid etwork model. Depedig o the asymptotic behavior of m as a fuctio of, the hybrid etwork exhibits differet capacities. Theorem : For a hybrid etwork of odes ad m base statios, if m = o( ), uder the routig protocol ad chael allocatio scheme, the aggregate throughput capacity of the hybrid etwork is: ( ) T (,m)=θ log + m 2 () Proof. To derive the aggregate etwork throughput capacity, we first compute the per cell capacity cotributed by the ad hoc mode trasmissios (T a ) ad the per cell capacity cotributed by the ifrastructure mode trasmissios (T i ), respectively. Cosider a arbitrary cell k, lety i be a radom variable that represets whether ode i ( i ) ad its destiatio are

5 both located i cell k. The radom variables are defied as follows: { both ode i ad its destiatio are i cell k Y i = (2) 0 otherwise I a hybrid etwork, there are m cells. Nodes ad the correspodig destiatios are radomly ad idepedetly ad placed i the uit area disk. The probability that a ode i is located i cell k is /m; the probability that the destiatio of ode i is located i cell k is also /m. Therefore, E[Y i ]=/. e the defie a radom variable N k = i= Y i, represetig the umber of source ad destiatio pairs commuicatig usig the ad hoc mode withi cell k. Sice{Y i } is a i.i.d. sequece of radom variables with E[Y i ]=/. By Strog Law of Large Numbers, with probability, N k = i= Y i as (3) m2 Give m = o( ), we have lim /, ad thus lim N k. Accordig to [], for a radom ad hoc etwork of N k odes ad a commo trasmissio rate of, the per ode capacity is Θ( ), as N logn k goes to k ifiity. Therefore, the capacity of cell k cotributed by ad hoc trasmissios is T a (N k )=Θ( logn k ). Deote c 2 = lim if By (3), we have lim lim if T a (N k ) T a (N k ) logn k c 3 = lim sup T a(n k ) logn k / log(/ ) lim sup T a (N k ) / log(/ ) = lim N k logn k / log(/ ) if =. Therefore, T a (N k ) logn k = lim sup T a(n k ) logn k logn k / log(/ ) logn k / log(/ ) = c 2 = c 3 /m The term lim T a (N k )/ 2 log(/ ) is upper ad lower bouded by two costats. Therefore, the per cell throughput capacity cotributed by the ad hoc mode commuicatios is ( ) m T a = Θ 2 log (4) Now we calculate the capacity cotributed by the ifrastructure mode commuicatios. Sice the same packet has to go through a uplik ad a dowlik whe trasmitted i the ifrastructure mode, it should be oly couted oce i the throughput capacity. e cosider uplik throughput, for example. Sice all the ifrastructure mode traffic has to go through the base statio ad the base statio ca oly receive data at the rate of 2 bits/sec at ay time. T i is upper bouded by 2. For the lower boud, if each ode i the ifrastructure mode employs a trasmissio rage of l (the side legth of each cell), there is a schedule for each ode to trasmit to the base statio i a roud robi fashio, yieldig a throughput of 2. Therefore, T i = Θ( 2 ). (5) Sice there is o iterferece betwee the the ad hoc mode ad the ifrastructure mode, the aggregate throughput capacity of a cell is just T a + T i.ilemma2weprovedthatthereis a schedulig policy such that each cell gets a slot to trasmit i every costat umber ( + c) of time slots. Therefore, the aggregate capacity of the etwork is T (,m) = Θ(mT a + mt i ) ( ) = Θ log + m 2 Note that the above capacity results correspod to the specific chael allocatio scheme. To obtai the maximum capacity, we must maximize the capacity over differet chael allocatio schemes. More specifically, the maximum capacity should be obtaied over all possible combiatios of ad 2. Corollary : The aggregate throughput capacity is maximized whe 2 / 0. Ad the correspodig capacity is: ( ) T (,m)=θ log (6) Proof. Sice = ad 2 = 3,wehave = 2 2. Replace i () with 2 2. ( T(,m) = Θ c 4 log m + c 4 2c 4 )m 2 log 2 ( = Θ log m + c 6 )m 2 log 2 Sice m = o( ), / goes to ifiity as icreases. /m Hece, c 6 2 < 0 whe is large eough. The log(/ ) throughput capacity is maximized if 2 / 0. Ad the correspodig capacity is ( ) T (,m)=θ log

6 Maximum capacity ( ) T (,km) = Θ log k 2 m ( 2 ( )) Θ log + logk log m ( ) 2 = T (,m) + logk log (7) Number of base statios (m) Fig. 2. Maximum aggregate throughput capacity for m = o( ) The maximum capacity is achieved whe 2 / 0or equivaletly, /. is the badwidth assiged to carry the itra-cell traffic via ad hoc mode i each cell. Therefore, the coditio implies that i order to maximize the throughput capacity of the etwork, we should assig most of the wireless chael badwidth to carry itra-cell traffic. Ad oly a miimal amout of chael badwidth should be assiged to carry iter-cell traffic. A ituitive explaatio for the badwidth assigmet to achieve maximum capacity is as follows. The capacity cotributed by ad hoc mode trasmissios icreases at a rate of m as 2 icreases, while the capacity cotributed by ifrastructure mode trasmissios icreases at a rate of roughly as icreases. he m = o( ), it is more beeficial to assig badwidth to ad hoc mode trasmissios. Fig. 2 presets a umerical example to illustrate the behavior of the maximum aggregate throughput capacity as the umber of base statios icreases. I this example, the umber of odes i the etwork is fixed at =,000,000 ad we icrease the umber of base statio (m) from to 00. Note that the y-axis i the plot is the scalig term /log i (6). The real throughput capacity should be scaled by a costat factor. However, this does ot affect the tred of the capacity. e ca observe that as the umber of base statios icreases, the maximum throughput capacity icreases. However, the icrease of the maximum capacity is domiated by a logarithmic term of m, the umber of base statios. More specifically, if we icrease the umber of base statios from m to km, the maximum capacity of the resultig hybrid etwork is I the above derivatio, we used the fact that log >> sice / >>. Therefore, the additio of km base statios oly provides a less tha logk-fold icrease i the maximum capacity. Note that the above capacity result is oly valid whe the umber of odes commuicatig usig ad hoc mode is large. I the case where m = Ω(), / is upper bouded by a costat. The capacity result i [, Mai Result 3] caot be applied to obtai T a. I this case, the capacity ca be obtaied i a differet way. Theore: For a hybrid etwork of odes ad m base statios, if m = Ω( ), uder the routig protocol ad chael allocatio scheme, the aggregate throughput capacity of the hybrid etwork is: T (.m)=o ( ) + Θ(m2 ) (8) Proof. The value of / does ot chage the the capacity cotributed by the ifrastructure mode. e have T i = Θ( 2 ). However, if m = Ω( ), / is upper bouded by a costat, we caot use [, Mai Result3] to derive T a. Accordig to [, Theore.], if odes are optimally placed i a disk of area A, each trasmissio s rage is optimally chose, ad the average distace traversed by a packet is L, the etwork trasport capacity (bit-distace product per uit time) is bouded as follows: 8 λl π A bit-meters/sec (9) Cosider a cell i a hybrid etwork of odes ad m base statios, the area of the cell is A = /m ad the average distace traversed ( by a packet usig ad hoc mode withi the /m ) ( A ) cell is L = Θ = Θ. Recall that N k = i= Y i is a radom variable represetig the umber of source ad destiatio pairs withi cell k. By defiitio, the aggregate throughput capacity of a radom ad hoc etwork is ecessarily smaller tha that of the optimal etwork. Therefore, the capacity of cell k cotributed by ad hoc mode trasmissios is 8 T a (N k ) λn k π ) = O(

7 From (3), we kow that with probability, N k / / as. Hece, lim / =, ad lim T a (N k )= ) O(. Therefore, T (,m) = Θ(mT a + mt i ) = O ( ) + Θ(m2 ) Sice m = Ω( ), the aggregate throughput capacity is maximized whe 2 / /2, resultig i a maximum capacity of Θ(m ). Therefore, for the case m = Ω( ), we have the followig corollary. Corollary 2: For a hybrid etwork of odes ad m base statios, if m = Ω( ), the aggregate throughput capacity is maximized whe 2 / /2, ad the maximum capacity is: T (,m)=θ(m ) (0) I this case, it is more effective to assig badwidth to carry iter-cell traffic ad the maximum capacity is obtaied whe 2 / /2, or equivaletly, / 0. I other words, i order to maximize the throughput capacity, almost all of the chael badwidth should be assiged to uplik ad dowlik sub-chaels to carry iter-cell trasmissios, while the itra-cell ad hoc mode trasmissios are suppressed. Note that the optimal chael assigmet to achieve maximum throughput capacity is the opposite to the case where m = o( ). Compared to the logarithmic growth whe m = o( ), the maximum throughput capacity icreases liearly with the umber of base statios i this case. B. Compariso to pure ad hoc etworks Now we compare the maximum capacity of a hybrid etwork to the capacity of a pure ad hoc etwork. Sice a hybrid etwork cotais base statios which are coected by a high lik badwidth wired etwork, it is ot fair to compare the capacity of a hybrid etwork to the capacity of a ad hoc etwork directly. The purpose of this sectio is to ivestigate the beefit of costructig a hybrid etwork if some umber of base statios ca be added to a pure ad hoc etwork. e defie the capacity gai factor to quatify the beefit of itroducig ifrastructure i ad hoc etworks. Defiitio 3: Capacity gai factor. The capacity gai factor g(,m) of a hybrid etwork of odes ad m base statios is the ratio of the maximum throughput capacity of the hybrid etwork to the throughput capacity of a ad hoc etwork of odes, i.e., g(,m)=t(,m)/t a (), wheret a () represets the aggregate throughput capacity of a ad hoc etwork of odes. Gupta ad Kumar have show i [], for a ad hoc wireless etwork of odes, if the odes are radomly placed ad the destiatio of each ode is radomly chose, the aggregate throughput capacity as is ( ) T a ()=Θ log () where is the commo trasmissio rate of the odes over the wireless chael. Note that splittig the chael ito several sub-chaels does ot chage the capacity results. Based o the capacity result, Gupta ad Kumar poited out that if m additioal homogeeous odes are deployed as pure relays i radom positios, the aggregate throughput capacity becomes Θ( +m log(+m) ). Therefore, the additio of k pure relay odes merely provides a less tha k + -fold capacity gai. Note that there is o wired liks betwee the relay odes i Gupta ad Kumar s model. hile i our model, we assume base statios are coected by high-badwidth wired liks. I the previous sectio, we obtaied the maximum aggregate capacity of the hybrid etwork. Depedig o the scalig behaviors of m relative to, the capacity results have differet depedecies o ad m. I the followig, we preset the capacity comparisos for the two differet regimes of m.. m = o( ). he the umber of odes () goes to ifiity, the maximum capacity of a hybrid etwork of odes ad m base statios is obtaied whe 2 / 0 ad the capacity scales as Θ( log ), while the capacity of a pure ad hoc etwork of odes scales as Θ( log ). The capacity gai factor is ( ) log g(, m) = Θ log(/ ) = Θ (2) 2logm log e ow derive the capacity gai whe the umber of base statios scales as a polyomial of the umber of odes. Assume m = α where 0 < α < /2, from (2), we have ( ) g(, m)=θ 2α (3) Note that the gai factor is idepedet of ad m, it remais a costat as m goes to ifiity i the form of α. The gai factor is a icreasig fuctio of α, asshowi Fig. 3. If the umber of base statios grows asymptotically faster, the capacity gai is larger. Therefore, if m is bouded by two polyomials of, α ad α 2 where α < α 2,the capacity gai is bouded by two costats: c 7 c g(,m) 7 (4) 2α 2α2 where c 7 represets the costat factor i (2).

8 Capacity gai factor Fig. 3. Expoet «µ Capacity gai factor as a fuctio of α ( ) log g(, m)=θ m (5) Agai, we are iterested i the scalig behavior of capacity gai whe m scales as a polyomial of. Assume m = α where /2 α, a simple derivatio yields g(,m)= α m 2α logm (6) If α = /2, i.e., m = Θ( ), wehaveg(,m)=θ ( logm ). The capacity gai grows early logarithmically with the umber of base statios. If α > /2, 2α > 0, the capacity gai grows polyomially with the umber of base statios. I a extreme case where α =, we have m = Θ(), ad g = Θ ( mlogm ). The per ode throughput capacity is Capacity gai factor Number of base statios (m) Fig. 4. Capacity gai factor Above we show that the capacity gai is a costat if m scales as a polyomial of. Now we show the beefit of placig more base statios o etwork capacity, if the umber of odes i the etwork is fixed. e use the same example as i the previous subsectio, there are,000,000 odes ad the umber of base statios varies from to 00. Accordig to (7), icreasig the umber of base statios from m to km oly provides a less tha logk fold icrease i the maximum capacity. The throughput capacity of the pure ad hoc etwork does ot chage sice is fixed. As a result, the capacity gai factor oly icreases logarithmically with the umber of base statios, as show i Fig m = Ω( ). I this sceario, the maximum aggregate capacity of the hybrid etworks is achieved whe / 0, ad the maximum capacity scales liearly with the umber of base statios. The capacity gai factor is λ(,m)=θ() This suggests that if the umber of base statios grows asymptotically at the same speed as the umber of odes, each ode gets a costat throughput capacity. The reaso is that i this case, each base statio serves a costat umber of odes. Therefore, each ode ca coect to the wired etwork usig a costat share of the badwidth, resultig i a Θ() per ode capacity or a Θ() aggregate capacity. IV. CAPACITY OF IRELESS HYBRID NETORKS UNDER PROBABILISTIC ROUTING STRATEGIES I this sectio we briefly preset the throughput capacity of a hybrid etwork uder the probabilistic routig strategies. e skip some of the derivatios sice the techiques are very similar to those used i Sectio III. e first compute the throughput capacity cotributed by the ad hoc mode trasmissios. Let Z i be a radom variable that represets whether ode i chooses to sed data usig ad hoc mode. { ode i chooses ad hoc mode Z i = 0 otherwise (7) Accordig to the routig strategy, {Z i } is a i.i.d. sequece of radom variables with E[Z i ]=p. The radom variable N a = i= Z i represets the umber of odes that choose to sed data usig ad hoc mode i the etwork. By Strog Law of Large Numbers, with probability, N a = i= Z i p as Usig the same techique as we used to derive (4) i Sectio III, the throughput capacity cotributed by ad hoc mode trasmissios ca be obtaied as follows. ( ) p T a = Θ log(p) (8)

9 I each cell, i order to utilize the capacity provided by the ifrastructure, at least oe ode must choose the ifrastructure mode to sed data. Otherwise, the base statio does ot receive or forward data. The uplik ad dowlik sub-chaels are ot used ad the badwidth is wasted. Actually, if p <, the probability that at least oe ode chooses to sed data i ifrastructure mode approaches as goes to ifiity. e defie the radom variable Z i to be the opposite of Z i, i.e., Z i = if ode i chooses the ifrastructure mode ad Z i = 0 otherwise. For cell k, deote the umber of odes i the cell as N k,wehavelim N k / /m ad thus lim N k. The capacity of cell k cotributed by the ifrastructure mode is [ ( )] T i (N k ) = Θ( 2 )E Z i i= ( ) = Θ( 2 )P Z i i= = Θ( 2 ) ( p N ) k Sice lim N k, ifp <, lim T i(n k )=Θ( 2 ) There are a total umber of m cells i the etwork. For the whole etwork, the throughput capacity cotributed by ifrastructure mode trasmissios is Θ(m 2 ). Combie this with (8), we have the followig theorem. Theorem 3: For a hybrid etwork of odes ad m base statios, uder the probabilistic routig strategy, the aggregate throughput capacity of the hybrid etwork is: ( ) p T(,m)=Θ log(p) + m 2 (9) As ca be see from (9), for ay chael allocatio scheme, the throughput capacity is maximized whe p, which implies that almost all the odes should choose ad hoc mode i order to maximize the capacity. This is because the badwidth of a base statio is fully utilized as log as there is a ode commuicatig usig ifrastructure mode. The badwidth of the base statio is shared amog the odes that use it to forward data. More odes commuicatig through the base statio does ot icrease the capacity cotributed by the ifrastructure. Sice p is strictly less tha, as goes to ifiity, the probability that at least oe ode chooses the ifrastructure mode approaches. Therefore, the badwidth of the base statios is guarateed to be fully utilized whe goes to ifiity, p implies almost all the odes should commuicate usig ad hoc mode i order to fully take advatage of the spacial cocurrecy. To derive the maximum throughput capacity, we should accout for all chael allocatio schemes. ith the same techique as we used to derive (6), we obtai the followig corollaries. ) Corollary 3: If m = o( log, the aggregated capacity is maximized whe 2 / 0 ad p. The correspodig capacity is Θ( log ). Note that the maximum capacity i this case has the same asymptotic behavior as the capacity of a pure ad hoc etwork. he m grows asymptotically slower tha log,thereis o sigificat beefit to use the ifrastructure i terms of the throughput capacity. ) Corollary 4: If m = ω( log, the aggregate capacity is maximized whe / 0 ad p. The correspodig capacity is Θ(m ). he the umber of base statio grows faster tha log, the maximum throughput capacity icreases liearly with the umber of base statios. Compared to a pure ad hoc) etwork, the capacity gai factor is g(, m) =Θ ( m log Ad if m = α where /2 α, g(,m)= α m 2α logm, which icreases polyomially with the umber of base statios. Note that this is the same as the case whe m = Ω( ) uder the k-earest-eighbor routig strategy. V. CONCLUSIONS I this paper, we studied the throughput capacity of hybrid wireless etworks. A hybrid etwork cosists of a ad hoc etwork ad a sparse etwork of base statios. The base statios are coected by a wired etwork ad placed i the ad hoc etwork i a regular patter. Data may be forwarded i a multi-hop fashio as i ad hoc etworks or forwarded through the ifrastructure as i cellular etworks. The goal of this paper is to ivestigate the beefit of the ifrastructure to the throughput capacity ad derive the asymptotic capacity of hybrid etworks. e cosider a hybrid etwork of m base statios ad odes, each capable of trasmittig at bits/sec over the commo wireless chael. Uder the k-earest-cell routig strategies, if m grows slower tha, ) the maximum aggregate throughput capacity is Θ( log. I this case, the beefit of addig base statios is isigificat. However, if base statios ca be added at a speed asymptotically faster tha, the maximum throughput capacity scales as Θ(m ), which icreases liearly with the umber of base statios. I a probabilistic routig strategy, a trasmissio mode is idepedetly chose for each source destiatio pair with certai probability. Uder this strategy, if m grows slower tha ( log, the maximum throughput capacity is Θ log ), which is of the same asymptotic behavior as the capacity of a pure ad hoc etwork. There is o beefit to use the ifrastructure i this case. If m grows faster tha log,

10 the maximum throughput capacity scales as Θ(m ), which icreases liearly with the umber of base statios. For both routig strategies, there is a threshold for the scalig of the umber of base statios (m) with respect to the umber of odes (), where the maximum capacity chages the asymptotic behavior. he the umber of base statios scales slower tha the threshold, the capacity is domiated by the cotributio of ad hoc mode trasmissios. I this case, the effect of addig base statios o capacity is miimal. he the umber of base statios scales faster tha the threshold, the capacity is domiated by the cotributio of the ifrastructure. I this case, the maximum throughput capacity scales liearly with the umber of base statios, providig a effective improvemet over pure ad hoc etworks. Therefore, i order to achieve o-egligible capacity gai, the ivestmet i the wired ifrastructure should be high eough: the umber of base statios should be at least log for the k-earest-cell routig strategies ad for the probabilistic routig strategies. The maximum throughput capacities are achieved whe / 0 or /. Recall that is the chael badwidth assiged to carry ad hoc mode trasmissios. The coditios suggest that i order to maximize the throughput capacity, oe of the two trasmissio modes will get almost all of the badwidth while the other will get zero. I either case, some of the odes will ot get ay badwidth to sed data. Oe way to avoid this situatio is to assig some miimum amout of badwidth to each sub-chael. I this case, the maximum capacities would be achieved whe takes its miimum or maximum possible values. If the previous requiremet is / 0, the ew coditio would be that takes the miimum value assiged to the ad hoc mode sub-chael. Note that this does ot chage the domiatig scalig behavior of the maximum capacity. Our results about the scalig of the maximum capacity ad the compariso to pure ad hoc etworks still hold. REFERENCES [] P. Gupta ad P. R. Kumar, The capacity of wireless etworks, IEEE Trasactios o Iformatio Theory, vol. 46, o. 2, Mar [2] P. T. Olivier Dousse ad M. Hasler, Coectivity i ad-hoc ad hybrid etworks, i Proc. IEEE Ifocom, [3] T. S. Rappaport, ireless Commuicatios: Priciple ad Practice, Secod Editio. Pretice Hall, [4] P. Gupta ad P. Kumar, Iterets i the sky: The capacity of three dimesioal wireless etworks, Commuicatios i Iformatio ad Systems, vol., o., pp , 200. [5] M. Grossglauser ad D. N. C. Tse, Mobility icreases the capacity of ad-hoc wireless etworks, i Proc. IEEE Ifocom, 200. [6] M. Gastpar ad M. Vetterli, O the capacity of wireless etworks: the relay case, i Proc. IEEE Ifocom, [7] J. Li, C. Blake, D. S. J. D. Couto, H. I. Lee, ad R. Morris, Capacity of ad hoc wireless etworks, i Mobile Computig ad Networkig, 200, pp [8] T. H. Corme, C. E. Leiserso, ad R. L. Rivest, Itroductio to Algorithms. MIT Press, 990.