Throughput and Delay Analysis of Hybrid Wireless Networks with Multi-Hop Uplinks

Transcription

1 This paper was preseted as part of the ai techical progra at IEEE INFOCOM 0 Throughput ad Delay Aalysis of Hybrid Wireless Networks with Multi-Hop Upliks Devu Maikata Shila, Yu Cheg ad Tricha Ajali Dept. of Electrical ad Coputer Egieerig, Illiois Istitute of Techology, Chicago, USA Abstract How uch iforatio ca oe sed through a rado ad hoc etwork of odes, if overlaid with a cellular architecture of base statios? This etwork odel is cooly referred to as hybrid wireless etworks ad our paper aalyzes the above questio by characterizig its throughput capacity. Although several research efforts related to throughput capacity exist i the area of hybrid wireless etworks, ost of these solutios uder-explore the capacity aalysis. Their results particularly idicate that oe ca realize oly a less tha log or o gai o capacity, as copared to pure ad hoc etworks, whe scales slower tha soe threshold. This usatisfyig capacity gai is due to the fact that the base statios were ot properly exploited while forulatig the capacity aalysis. Moreover, these research efforts also assue a oe-hop wireless uplik betwee a ode ad its associated base statio. Nevertheless, with those powercostraied wireless odes, this assuptio clearly idicates a urealistic sceario. I this paper, we establish the bouds o capacity ad delay by resolvig the issues i existig efforts ad at the heart of our aalysis lies a siple routig policy kow as sae cell routig policy. Our fidigs particularly stipulate that whether = O( ) or Ω( ), each ode ca realize a log log throughput that scales, subliearly or liearly, with. Thisis i fact a sigificat result as opposed to previous efforts which clais that if grows slower tha soe threshold, the beefit of augetig those base statios to the origial ad hoc etwork is isigificat. Our aalysis also shows that for a axiu per ode throughput Λ(, ), the average ed-to-ed delay i a hybrid etwork ca be bouded by Θ(Λ(, ) ), which has a iverse relatioship to. I. INTRODUCTION There has bee sigificat iterest i the past o uderstadig how the capacity of ulti-hop wireless etworks scales with the uber of odes i the etwork []-[5]. I their pioeerig work [], Gupta ad Kuar studied the capacity of pure ad hoc wireless etworks i the liit as the uber of odes grows to a large level. Their results aily prove that whe odes are radoly (or arbitrarily) deployed i a plaar disk of uit area, the aout of iforatio that ca be trasitted by each source-destiatio (S-D) pair becoes vaishigly sall, as. This perforace liitatio o throughput with icreasig is due to the icreasig uber of hops betwee each S-D pair, which i tur iplies that those odes servig as relays essetially sped ost of the tie relayig traffic fro other odes. Thus, it follows that by decreasig the uber of hops betwee each S-D pair ad correspodigly the traffic relayed by the odes, oe ca greatly iprove the perforace of ulti-hop wireless etworks. This work was supported i part by NSF grat CNS A sigificat uber of works, [4]-[5],[3] ad the refereces therei, alog the taget of decreasig hops have bee proposed i the area of ulti-hop wireless etworks. More specifically, Grossglauser ad Tse [4] proved that a costat throughput scalig of Θ() per S-D pair ca be achieved, if each packet is costraied to take O() hops to destiatio by utilizig the obility characteristics of a ode. Gaal et.al. [5] however showed that this ehaced throughput scalig obtaied uder ode obility actually coes at the cost of delay ad as a result, eve a slight depedece o obility will lead to a abrupt ad icreased delay. Thus it appears that oe should target their efforts o buildig wireless etwork odels that ca furish each ode with a higher throughput, while possibly keepig the delay sall. It has bee recetly recogized that addig base statios to pure ad hoc wireless etworks, cooly referred to as hybrid wireless etworks, ca ideed reder larger beefits i ters of both capacity ad delay. Oe ca evisage these base statios as a eas to carry all the log distace trasissios fro a source ode, through the wired etwork, to its iteded destiatio. Ad sice each wireless ode is coitted to leverage ulti-hop trasissios oly for short distaces, we have saller uber of wireless hops per S-D pair ad correspodigly, a larger per ode throughput. As delay also icreases with the hops, it is obvious that by liitig the uber of hops oe ca realize a saller delay for hybrid wireless etworks, without sacrificig the per ode throughput. Several capacity related works exist i the area of hybrid wireless etworks. I [6], Liu et al. first studied the throughput capacity of hybrid wireless etworks uder two differet routig strategies. Specifically i k-earest cell routig strategy, it is show that if grows asyptotically slower tha,the axiu per ode capacity scales as Θ(W log / ).It is ot hard to see that the beefits obtaied o capacity due to the additio of base statios i this regio are isigificat. O the other had, if grows asyptotically faster tha, the axiu per ode throughput capacity scales as Θ( W ) which i tur offers a better throughput gai depedet o. Iportatly, ote that i the regio = O( ), oe ca attai oly less tha log k-fold beefit o capacity as the uber of base statios are icreased fro to k. We further otice siilar capacity liited figures i [8]-[] i.e., especially whe grows asyptotically slower tha soe threshold. The otivatio for this work is to fudaetally uderstad //$ IEEE 46

2 whether the capacity of the hybrid etwork has bee fully explored, especially stiulated by the usatisfyig gai whe scales slower tha soe threshold. By studyig these existig efforts i depth, we idetify the followig issues i the existig capacity aalysis: ISSUE I: I [6],[8]-[], their capacity aalysis fails to accout for the fact that addig base statios to the origial ad hoc wireless etworks plays a critical role i decreasig the uber of hops betwee each S-D pair ad thereby, the aout of traffic flowig through each relayig ode. Oe ay observe that failure to iclude this aspect i capacity aalysis ca lead to iaccurate results; ISSUE II: I [6]-[], each ode i the etwork is assued to egotiate with its base statio usig a oe-hop wireless uplik. This iplies that those power costraied wireless odes has to trasit at higher power levels to reach their associated base statios. However, such assuptios are ot feasible i practice, especially whe wireless odes are cofigured to trasit at < 00 Watts as opposed to the base statios that ca trasit at 0 60Watts []; ISSUE III: I [6], toforulate the fial capacity expressio, the paraeter i W log (capacity of pure ad hoc etwork) is siply substituted with, which is the uber of odes couicatig i ad hoc ode. However, those substitutios idirectly iply that odes are restricted to couicate oly with a liited uber of earby eighbors ad thus particularly durig the breakdow of ay ifrastructure odes, several discoected copoets will be created i the etwork. I [7], authors revisit the throughput capacity proble i hybrid wireless etworks based o a L-Maxiu hop routig strategy. Though their aalysis provide better capacity figures i copariso to [6], authors agai assue a oehop wireless uplik betwee a ode ad its associated base statio [ISSUE II]. Motivated by these shortcoigs i existig studies, we particularly address the followig two questios i this paper: Ca we desig a better schee that ca resolve the issues i previous efforts, while possibly providig a better asyptotic capacity ad delay scalig? How uch iforatio ca oe sed over a rado ad hoc etwork of odes if overlaid with a cellular architecture of base statios? This paper attepts to aswer these two questios by cosiderig a siple ad practical routig policy referred to as sae cell routig policy. I this policy, a source ode route its packets to the destiatio i ad hoc ode oly if both the source ad its destiatio are located i the sae cell. Otherwise, the packets are iitially trasitted i ad hoc ode to the base statio which evetually forwards all the packets to the destiatio as i a cellular etwork. Let W bits/sec be the total badwidth ad the for a hybrid etwork with odes ad base statios, we idetify the followig two regies: = O( log ). I this regie, a per ode throughput capacity of Θ(W log ) is achieved. It also follows that if the uber of base statios are icreased fro to k, we ca actually obtai a gai of k o capacity as opposed to [6], which oly provides a less tha log kfold icrease o capacity. The average delay is bouded by Θ( log ). = Ω( log ). I this regie, a per ode throughput capacity of Θ(W ) is achieved. The average delay is bouded by Θ(). It is obvious that idepedet of the regies defied by, the throughput capacity available to each ode scales with the uber of base statios. We are particularly iterested i the regio = O( log ), where we observe that the capacity icreases subliearly with the uber of base statios. This is a iterestig ad cotrastig result to [6],[9]- [], which states that if grows asyptotically slower tha soe threshold, there is o beefit i augetig those base statios to the origial ad hoc etwork. More iportatly, as copared to previous research efforts []-[4],[6]-[7], we clearly deostrate that our desig ca guaratee higher beefits o both capacity ad delay irrespective of the regies defied by. We suarize the distiguishig aspects of our work as follows: (a) NUMBER OF HOPS: I cotrast to previous efforts, our aalysis accurately odels the effect of uber of hops betwee each S-D pair o the capacity ad delay of a hybrid wireless etwork; (b) MULTI-HOP UPLINKS: We cosider a ore practical approach i which the power-costraied wireless odes are allowed to egotiate with their associated base statios usig ultiple hops; ad (c) FAILURE TOLERANT: We assue a stad-aloe ad hoc etwork ad as a result, eve i the failure of ay base statios, odes ca still route the packets to its chose destiatios. Roadap. The rest of the paper is orgaized as follows: I sectio II, we iitially preset several related research efforts ad the provide a detailed descriptio of our hybrid wireless etwork odel. Sectio III presets the upper ad lower bouds o the throughput capacity ad delay of a hybrid etwork uder sae cell routig policy. I sectio IV, we ivestigate the efficiecy of our odel by coparig with several existig literatures. Sectio V fially cocludes this paper. II. BACKGROUND AND NETWORK MODEL A. Related Work Due to space costraits, we briefly discuss soe related works i the area of hybrid wireless etworks. I [8], authors study the throughput capacity where both the ad hoc ad ifrastructure odes are radoly distributed. They aily show that each ode ca achieve a axiu throughput of Θ(W/ log ) if scales liearly with. Followig this work, Zeliaov et. al [9] ivestigates the capacity uder a differet odel where the ad hoc odes are radoly distributed ad the base statios are arbitrarily placed. Their results pricipally show that whe = O( / log ), the per ode throughput aps to the capacity of pure ad hoc etworks ad thus, there is o beefit i addig those base statios to the ad hoc etwork i this regio. O the other had, whe =Ω( / log ), they observe siilar results reported i [6]. I [0], authors 463

3 ivestigate the ipact of etwork diesioality ad geoetry o the capacity of these etworks. Though they observe iterestig capacity figures for -diesioal strip, their results for -diesioal etwork shows that there is o beefit o capacity fro deployig base statios as log as = O( ). Siilar results are also reported i []. Iportatly, i [8]- [], authors overlook the effect of uber of hops betwee each S-D pair o the throughput capacity (ISSUE I). These works also assue a oe-hop wireless uplik betwee odes ad their base statios (ISSUE II). Oe ay otice that these aspects ca i fact lead to iaccurate as well as ipractical solutios. Besides, while coparig with these solutios oe ca also observe the larger gai obtaied i eployig our desig. I [], authors study the capacity whe access poits are regularly placed o the pure ad hoc etwork. Ufortuately, the results established i that paper are ot validated with eough proofs. B. Hybrid Wireless Network Model We cosider a hybrid wireless etwork of wireless odes, overlaid with a cellular architecture of base statios o a plaar torus of uit area. Oe ay also ote that though we etio base statios i this paper, the results are applicable to all ifrastructure odes that are coected by a wired etwork such as Access Poits (AP), Fetocells etc. Fig. depicts the settig of such a hybrid wireless etwork odel i a plae. I particular, a hybrid wireless etwork cosists of two layers, a ad hoc layer ad a cellular layer. I the ad hoc layer, we assue that wireless odes are uiforly ad idepedetly (radoly) distributed o the surface of a uit area torus, siilar to the oe proposed by Gupta ad Kuar [], ad each ode leverages sae trasissio power to couicate with its eighborig odes or base statios. As i [4], we also cosider that each ode is a source of exactly oe flow ad a destiatio ode for at ost O() flows. Lastly, we assue a stad-aloe wireless etwork at the ad hoc layer. As a result, eve i the absece of ay base statios, odes ca still egage i couicatio with its chose destiatios. This assuptio solves ISSUE III i [6]. I the cellular layer, we regularly deploy base statios, at the top of ad hoc layer, i such a aer that it tessellates the plae ito equal-sized squares of area. For the sake of clarity of proofs, we assue a square tessellatio istead of hexagoal tessellatio. Oe ay ote that if a hexagoal tessellatio is cosidered, the area of each cell would still be k, where k is a costat. This i tur iplies that all the scalig results derived i this paper will hold true for hexagoal tessellatio as well. Besides, this odel is also siilar to the work i [5], where the authors cosider a square tessellatio to aalyze the perforace of eployig obility i pure ad hoc etworks. Next, as i a cellular cocept, each square is called a cell ad we place oe base statio at the ceter of each cell. Ulike wireless odes, base statios either serve as data sources or as data receivers. Istead, they serve The assuptio of a torus eables us to avoid techicalities arisig out of edge effects, but the scalig results derived here holds for uit square or disk as well. as relays to forward the traffic for wireless odes i the ad hoc layer. Moreover, the base statios are also assued to be coected to each other with a very high badwidth etwork so that there are o bottleecks associated with the base statios. I cotrast to wireless odes, we also assue that there are o power costraits for the base statios. Fially, to esure that the utual iterferece betwee base statios reais below a threshold, we assue that adjacet cells eploy a frequecy reuse policy siilar to a cellular etwork []; See Sectio III-C for details. C. Protocol Iterferece Model To study the ipact of wireless iterferece i hybrid wireless etworks, we adopt the protocol odel i []. Suppose that ode v i trasits to aother ode v j. I the so-called protocol odel, a trasissio fro ode v i is successfully received by ode v j satisfied. if the followig two coditios are The distace betwee ode v i ad v j, v i v j,iso ore tha the trasissio rage of the odes r(), i.e.,. v i v j r(). For every other ode v k that is siultaeously trasittig over the sae chael, ode v j should lies outside the iterferece regio of v k, i.e., v k v j ( + Δ)r(). The last coditio esures a exclusio regio [] aroud the receivig ode to prevet a eighborig ode fro trasittig o the sae chael at the sae tie. The paraeter Δ defies the size of the exclusio regio ad hece, Δ > 0. D. Routig Policy for Hybrid Wireless Networks I a hybrid wireless etwork, each ode i the ad hoc layer ca couicate with its chose destiatio i the correspodig layer usig two odes, purely ad hoc ode ad hybrid ode. I a purely ad hoc ode, the source ode trasits the data to its iteded destiatio usig ultiple hops, that is without relyig o the base statios i the cellular layer. O the other had, i a hybrid ode, the source ode iitially trasits the data to its earest base statio i a ultihop fashio (i.e., purely ad hoc ode) ad evetually, the base statio forwards the data through the wired etwork to the destiatio ode as i a cellular etwork. Oe ay agai ote that the approach of utilizig ultiple hops by source ode to coect to its earest base statio i a hybrid ode is differet fro the existig solutios [6]-[], where the authors assue a oe-hop wireless uplik betwee a source ode ad its base statio. Takig ito accout ulti-hop upliks resolves ISSUE II see i prior schees ad as a result, we call our desig as practical. This paper iitially cosiders a siple routig policy called as sae cell routig policy for hybrid wireless etworks. I this policy, a source ode trasits the data to its destiatio i purely ad hoc ode oly if both the source ad destiatio are located i the sae cell. Otherwise, data is traversed to the destiatio i hybrid ode. However, oe ay ote that eve though the source ad destiatio lie withi oe-hop 464

4 distace of each other but are located i two differet cells, the data will be forwarded i hybrid ode accordig to the sae cell routig policy. This i tur ca cause iefficiet utilizatio of the wireless badwidth, as poited out i [7]. As a result, besides sae cell routig policy, we further aalyze aother routig policy called as D legth routig policy; See Appedix A. I this policy, a source ode routes the data to its destiatio i purely ad hoc ode oly if the destiatio ca be reached withi D (i.e., D) distace fro the source. Otherwise, the data will be carried to the destiatio i hybrid ode. Our goal is to fid optial D ad iterestigly, our results deostrate that the axiu capacity bouds ca be realized at D = O(/ ), siilar to what is achieved uder sae cell routig policy. This clearly shows the effectiveess of our desig ad aalysis. Each ode is assued to trasit at a axiu data rate of W bits per secod over a coo wireless chael of badwidth W. We further partitio this wireless chael ito three subchaels each of badwidth, W a for purely ad hoc trasissios, W u for uplik trasissios to the base statios ad W d for dowlik trasissios fro the base statios, respectively. I fact, such a partitio will allow us to carry above three trasissios siultaeously i a etwork without causig iterferece to each other. Nevertheless, the trasissios occurrig i the sae subchael will still cause iterferece to each other. I additio, sice the aout of traffic i the uplik ad dowlik chaels are the sae, we ca write W u = W d. As a result, the su of the trasissio rates due to purely ad hoc trasissios as well as base statio related trasissios ca be expressed as W = W a +W u. Oe ay also ote that the badwidth W a assiged for purely ad hoc trasissios iclude ot oly the traffic fro source to destiatio i the purely ad hoc ode but also the traffic fro source to the base statio i the hybrid ode. I the sequel, by purely ad hoc trasissios, uless stated, we refer to the trasissios occurrig i ulti-hop fashio i purely ad hoc ode as well as i hybrid ode. E. Defiitios Throughput. A per-ode throughput of Λ(, ) bits per secod, for a hybrid wireless etwork of odes ad base statios is said to be achievable, if every ode ca trasit data to its chose destiatio at a rate of Λ(, ) bits per secod. I this paper, the throughput capacity of the hybrid wireless etwork with odes ad base statios are expressed by Λ(, ) = Λ a (, ) +Λ b (, ), where Λ a (, ) ad Λ b (, ) deote the throughput capacity cotributed by the purely ad hoc ode trasissios ad the base statio relative trasissios (i.e., uplik ad dowlik) respectively. Further, sice there are total of source-destiatio pairs, we defie the etwork capacity to be Λ(, ). Average Delay of Hybrid Networks. The delay of a packet is the tie it takes for the packet to reach the destiatio fro the source. Thus, the per packet delay is the su of the ties a packet speds at each relay ode. As i previous capacity ad delay studies [4]-[5], we scale the packet size by the per ode capacity so the trasissio delay at each ode is costat. Hece, the per packet delay correspods to the uber of hops eeded to reach its destiatio. The average packet delay of a hybrid etwork D(, ) is the obtaied by averagig over all trasitted packets i the etwork due to the wireless trasissios. We do ot accout for the delay caused by the trasissios through the wired etwork, as a high badwidth wired backboe etwork is assued. III. CAPACITY AND DELAY OF HYBRID WIRELESS NETWORKS UNDER SAME CELL ROUTING POLICY This sectio establishes the upper ad lower bouds o the throughput capacity ad delay of hybrid wireless etworks uder sae cell routig policy. The related theores are stated as follows. Theore. For a hybrid etwork with odes ad base statios, the throughput capacity Λ(, ) furished to each ode uder the sae cell routig policy is : Λ(, ) =Θ ( log W a ) +Θ( W u where Λ a (, ) = log W a ad Λ b (, ) = W u. Theore. For a hybrid etwork with odes ad base statios, the average delay D(, ) of each packet uder the sae cell routig policy is: { Θ( D(, ) = Θ() log ) = O( log ) ) () =Ω( log ) () As the purely ad hoc ode ad base statio relative trasissios are carried i two differet subchaels, i the sequel we will derive the bouds o capacity ad delay of these trasissios separately. Oe ay also ote that the fial capacity will be just Λ a (, )+Λ b (, ). A. Lower Boud o Capacity ad Delay for Purely ad hoc trasissios This sectio costructs a schee that achieves Λ a (, ) bits per secod for every ode i the etwork to its chose destiatio, with high probability (whp). Cosider the ad hoc layer of the hybrid wireless etwork ad tessellate the uit area log regio by subcells of area a() =Ω( ). We the choose the trasissio rage of each ode as r() = a() such that a ode i a subcell ca trasit to soe other ode lyig withi its four eighborig subcells. As show i Fig., we further lay out a virtual layer, cellular layer, fored by cells each of size at the top of this ad hoc layer. To be ore precise, such a costructio will result i each cell of area to cosist of a() subcells, each of area a(). Oce the etwork is costructed, our ext step is to preset a schee that allows each ode to route the data to its chose The throughput capacity of hybrid wireless etworks is said to be of order Θ(f(, )) bits per secod, if there exist deteriistic positive costats c ad c such that li Prob (Λ(, ) = c f(, ) is feasible) = ; ad li Prob (Λ(, ) =c f(, ) is feasible) <. 465

5 Cell Flow- Ad hoc Layer Flow- Flow-5 Flow-5 Flow-4 Flow 3 Cellular Layer Flow- Fig.. Sae Cell Routig Policy. Flows,3 ad 4 are forwarded i purely ad hoc ode ad Flows ad 5 are forwarded i hybrid ode. destiatio. For this purpose, draw a straight lie, that passes through soe subcells, coectig each source-destiatio (S- D) pair. As etioed before, if a S-D lie lies copletely iside (outside) a cell the packets are trasitted fro source to destiatio (base statio) i purely ad hoc ode by hops alog the adjacet subcells lyig o its S-D lie. Therefore to trasit the packet alog the S-D lie, we eed to choose a ode fro each of these subcells which i tur requires at least oe ode per subcell. We ow state the Lea fro [5] that bouds the uber of odes preset i a subcell of area a() =Ω( log ). Lea. (Ref. [5]) If a() is greater tha subcell has Θ(a()) odes per subcell, whp. 50 log, each Fro Lea, it follows that each subcell with area log a() > will have at least oe ode whp, thus esurig successful trasissio alog each S-D lie. Our ext step is to schedule trasissios such that each ode i a subcell ca trasit to odes i its adjacet subcells, without causig iterferece to siultaeous seders, at regularly scheduled tie slots. We propose a tie divisio ulti-access schee (TDMA) to schedule the trasissios which is i fact based o the followig two Leas. Lea shows that there is a schedule for trasittig packets such that oce i every +c 3 tie slots, each subcell i the tessellatio gets oe slot to trasit to its adjacet subcells. Oce a subcell gets a opportuity to trasit, the odes withi it are required to relay the traffic for each of the S-D routes passig through it. I Lea 4, we deterie the aout of relayig traffic per subcell by boudig the axiu uber of routes passig through it. Before statig Lea, we use the followig defiitio [],[5]. A subcell X is said to iterfere with aother subcell Y, if there is a seder i subcell X which is withi a distace ( + Δ)r() of soe seder i subcell Y. Lea. The uber of subcells that iterfere with ay give subcell is bouded by a costat c 3 = O(( + Δ) ), i.e, idepedet of, ad a(). Proof: Accordig to the defiitio, ay two odes trasittig siultaeously are separated by a distace of at least ( + Δ)r() ad hece disks of radius ( + Δ )r() cetered aroud each trasitter are essetially disjoit. Thus, usig siple geoetric arguets we get the uber of iterferig ( (+Δ) a() subcells, c 3,asatost 4 a() = O(( + Δ) ), which is a costat idepedet of, ad a(). This iplies that i every +c 3 slots, each subcell i the tessellatio gets oe slot to trasit, thus guarateeig a successful trasissio to odes withi a distace of r() fro their trasitters. Also, ote that the fact +c 3 follows fro the vertex colorig of a graph with bouded degree c 3 (see [] for ore details o vertex colorig). Next, we calculate the axiu uber of routes passig through ay subcell i a cell. To prove Lea 4, we eed Lea 3 that bouds the uber of source-destiatio (S-D) pairs couicatig usig purely ad hoc ode ad hybrid ode i a give cell. Lea 3. For a give cell k of size, the uber of S-D pairs couicatig usig purely ad hoc ode ad hybrid ode are ad ( ) respectively, uder sae cell routig policy. Proof: Cosider a tagged cell k. LetXi k be a idicator rado variable that represets whether the source ode i ad its destiatio are i the sae cell, k. Thus, we have { Xi k If both source i ad destiatio are i cell k = 0 Otherwise Recall that i the cellular layer, base statios divide the uit area torus ito cells each with a area of, ad i the ad hoc layer, we have odes radoly distributed. The probability that a source ode i is located i cell k is /; the probability that the destiatio of ode i is also located i cell k is /. Therefore, E[X k i ]=/.LetX k = i= Xk i deterie the uber of source-destiatio pairs couicatig usig purely ad hoc ode withi cell k. Now applyig the liearity of expected value ad the fact that all E[X k i ] s are equal, we have E[X k ]= i= E[Xk i ]=E[Xk i ]=. Our ext step is to deterie the probability that ay cell ca have at ost / odes couicatig i purely ad hoc ode. Fro the applicatio of the Cheroff boud; see Appedix B ad settig δ =, we obtai that for ay cell Pr [X k > ] exp( 3 ). Sice there are total of cells, by the applicatio of the uio boud (see pg.38 []), it follows that the above boud holds for all cells with probability.exp( 3 ). Siilarly, we will also calculate the uber of sourcedestiatio pairs Y k = k i= Yi couicatig usig hybrid ode withi cell k, where Yi k is defied as follows: { Yi k If source is i cell k ad destiatio i cell x k = 0 Otherwise Agai, applyig the liearity of expected value ad the fact that all E[Yi k] s are equal, we have E[Y k ] = E[Yi k] = ( ), where the factor ( ) is the probability that source ad destiatio lie i two differet cells. Followig the above techiques, we get the Pr [ay cell has > ( ) odes couicatig i hybrid ode].exp( 3 ( )). ) 466

6 Lea 4. The uber of S-D routes passig through ay a() subcell i a cell is O( ), whp. Proof: Cosider a arbitrary cell k. Fro Lea 3, we have S-D pairs couicatig usig purely ad hoc ode i cell k. Letd i be the distace betwee the S-D pair i ad h i be the ea uber of hops per packet for each S-D pair i. The, h i = d i / a(). LetH = i= h i be the total uber of hops required to sed a packet fro each seder S to its correspodig destiatio D i purely ad hoc ode. Cosider a tagged subcell i a cell k ad defie the Beroulli rado variables Zj i for S-D pairs i ad hops j h i as follows: If hop j of S-D pair i origiate fro a Zj i = ode i the subcell of a give cell (3) 0 Otherwise As a result, the total uber of S-D lies passig through a h i subcell due to the purely ad hoc ode is Z = Zj. i i= j= Now cosider the rado variable H = i= h i= i= d i/ a(). Recall that i sae cell routig policy, each ode trasits the data to its destiatio i purely ad hoc ode if both the ode ad its destiatio are located i the sae cell. Sice each cell has a size of, for all i, we have d i =[0, ] ad H = O( 5/ ). Usig this a() result, we fid the boud o E[Z] as follows [5]: E[Z] = E H [E[Z H]] = E H [HE[Z]] ( ) = 5/ a() = a() (4) a() 3/ where the last equality follows fro the fact that ay hop is equally likely to origiate fro ay subcell of the /a() subcells. Next, we also eed to boud the uber of lies passig through a subcell due to the ulti-hop traffic (i.e., the traffic fro source to the base statio) i a hybrid ode. Let Z be the total uber of S-D lies passig through a subcell due to the ulti-hop traffic i a hybrid ode. Followig the above techiques ad replacig S-D pairs by ( ) we get E[Z ] = ( ) a(). Sice E[Z] 3/ ad E[Z ] are idepedet, we have the total uber of lies passig through a subcell i a cell as E[Z]+E[Z a() ]=. Due to space costraits, we oit the proof for coputig the a() probability that ay subcell has at ost lies passig through it, usig cheroff bouds. However followig the sae techiques i [5] ad settig δ = log /(E[Z]+E[Z ]), we obtai that for ay cell the uber of lies passig through it are bouded about by a() with probability /. Throughput Capacity: We are ow ready to calculate Λ a (, ), the throughput capacity cotributed by purely ad hoc trasissios. It follows fro Lea that each subcell will receive a opportuity to trasit oce i every +c 3 tie slots. I other words, i every oe secod tie period each subcell ca be active for a costat fractio of tie period of legth Ω( +c 3 ) secods. Lea 4 suggests that if each tie period correspodig to a subcell, that is Ω( +c 3 ) secods, a() is further divided ito Ω( ) tie slots, each S-D pair hoppig passig through it ca use oe slot. Equivaletly, each ode ca successfully trasit for Ω( a() ) fractio of tie at the rate of W a bps. Thus, we have Λ a (, ) = Ω( Wa a() ), where a() = log, as the throughput capacity correspodig to purely ad hoc trasissios. Average Packet Delay. We will also copute the average packet delay, D(, ), of all the packets i the etwork. Fro Lea 4, it follows that each S-D pair i, couicatig i purely ad hoc ode or hybrid ode, has a legth of at ost /, i.e., d i = O(/ ). Besides, it also follows fro [5], that the delay per packet is the su of the aout of tie spet at each hop. Sice each hop covers a distace of Θ( a()), the uber of hops per packet for S-D pair i is Θ(d i / a()). Thus the average uber of hops take by a packet averaged over all S-D pairs caot be ore tha i= d i/ a()=. Fro the above discussio, we a() ca coclude the delay D(, ) as Ω( log ). I additio, oe ay ote that as the base statios grows faster tha / log, the uber of hops take by each packet for S-D pair i is at ost Θ(). Ituitively as icreases, the size of each cell served by a base statio also decreases which i tur leads to decreasig uber of hops take by each packet i purely ad hoc ode. As a result, we ca boud the delay by Θ() for =Ω(/ log ) ad by Ω( log ) for = O(/ log ). B. Upper Boud o Capacity ad Delay for Purely ad hoc trasissios Now, we tur to the upper bouds o the per-ode throughput ad delay of purely ad hoc trasissios. Before derivig the boud, we state the followig Lea fro [] that deteries the uber of siultaeous trasissios possible o ay particular chael. Lea 5. Ref []: The uber of siultaeous trasissios 4 o ay particular chael is o ore tha c 4πΔ r () i the Protocol Model. Therefore, observig that each purely ad hoc trasissio o a give chael is of W a bits/secod, by suig all the trasissios takig place at the sae tie, we ote that they 4W a c 4πΔ r () caot be ore tha bits per secod i the protocol odel. Cosider a particular cell of size ad let L be the ea legth of the distace take by a packet i purely ad hoc ode. This also iplies that the ea uber of hops L take by a packet i a give cell is at least r(). Fro previous discussios we also kow that whether i a purely ad hoc ode or a hybrid ode, each source ode i a cell eeds to trasit 467

7 the packet at a distace of at ost. Thus we have L as at ost. [Oe ay ote that i [6],[8]-[], authors do ot L accout for the ipact of r() o the capacity aalysis which i tur leads to iaccurate results.] Sice each source geerates Λ a (, ) bits per secod, there are sources i each cell, ad each bit eeds to be retrasitted o the average by at least odes, it turs out that the total uber of bits per secod L r() that eeds to be served by a give cell as at least LΛ a(,) r(). Agai otig that there are total of cells, the total uber of bits per secod served by the etire etwork eeds to be at least LΛ a(,) r(). To guaratee that all the purely ad hoc traffic is carried, we eed LΛ a (, ) r() 4W a c 4 πδ r () Thus we have, Λ a (, ) LΔ c5wa. Fro a precursor result r() i [], it also follows that r log () > π is ecessary to esure coectivity i a idepedetly ad uiforly distributed ad hoc etwork. [Oe ay agai ote that we cosider a stadaloe ad hoc etwork i.e., eve i the breakdow of ay ifrastructure odes, the etwork ca still fuctio properly. However, i [6], the liited trasissio rage of odes ay soeties allow odes to couicate oly with a sall uber of earby eighbors ad hece, ca lead to several discoected copoets i the etwork.] Together with the fact that L =, we have the per-ode throughput of each ode i purely ad hoc ode as Λ a (, ) =O(W a log ) bits per secod. Next, we will copute the upper boud o the delay of each packet i the hybrid etwork. Recall fro previous sectio that the delay per packet is the su of the aout of tie spet at each hop. Notig that the ea uber of hops take by a packet i purely ad hoc ode is about O( log L r() (5), we ca coclude the delay D(, ) as ). This copletes the upper boud proof. C. Capacity of Base Statio Relative Trasissios Our last step is to deterie the throughput capacity cotributed by the trasissios relative to a base statio. Oe ay ote that each packet trasitted fro a source to destiatio i hybrid ode will use oe uplik trasissio ad oe dowlik trasissio ad as a result each trasissio should be couted oly oce i the coputatio of the throughput capacity. It follows fro Sectio II-D that the badwidth allocated for a uplik trasissio is W u bps. Hece, the throughput capacity cotributed by per cell caot be ore tha W u which i tur is the upper boud. We will ext tur to the lower boud o the capacity. Fro sectio II-B, it follows that each cell eploys a frequecy reuse policy (siilar to a cellular etwork) to avoid utual iterferece fro adjacet cells. Equivaletly, each cell will use a differet frequecy fro its adjacet or iterferig cells ad if a set of say c 6 differet frequecies are used for a group of c 6 adjacet cells, the badwidth occupied by each cell (trasissio rate) will be lower bouded by Wu c 6. Sice the upper ad lower bouds are tight, we have the throughput capacity per cell as Θ(W u ). Fro earlier discussios we kow that each cell has a() = log subcells, each with Θ(a()) odes. This iplies that withi each cell there are total of Θ(/) odes. Therefore, ote that if the throughput available to a cell is Θ(W u ), the each ode withi a cell gets a throughput of Λ b (, ) =Θ( Wu ). Sice both the upper ad lower bouds aps to each other, we have the tight bouds deoted by Θ( ) i Theores ad respectively. This cocludes the proof. Oe ay also ote that the capacity figures, Λ a (, ) ad Λ b (, ), i Theore i fact depeds o differet chael allocatio schees. Therefore to get the axiu throughput capacity, oe has to axiize the throughput over all possible cobiatios of W a ad W u. We the have the followig cases: (a) whe = O( log ), we achieve the axiu throughput capacity by allocatig ost of the badwidth for ad hoc ode trasissios ad oly allocatig a iial aout of badwidth for the base statio relative trasissios i the hybrid ode. I other words, whe W u /W 0 or whe W a = W, we get the throughput available for each ode as Λ(, ) = Θ(W log whe =Ω( ) (See Appedix B); (b) log ), we observe that each ode realizes the axiu throughput by allocatig ost of the badwidth for carryig base statio relative trasissios. Alteratively, whe W a /W 0 or W u = W/, we get the throughput available to each ode as Λ(, ) =Θ(W ). Ituitively, we see that whe > log, the uber of odes couicatig i ad hoc ode withi a cell decreases ad hece, ost of the traffic has to be carried through the base statio which i tur requires larger badwidth. IV. COMPARISON WITH EXISTING SOLUTIONS I this sectio, we copare our desig with several existig schees. For the coveiece of elucidatio, i the sequel we ter the pure ad hoc etworks i [], obile ad hoc etworks i [4], k-earest cell routig policy i [6] ad L-Maxiu hop routig strategy i [7] as PANs, MOBILITY, kearest ad L-Hop respectively. To study the perforace of our desig, two paraeters are cosidered, capacity gai ad delay gai deoted by G c ad G d respectively. G c ad CAPACITY OF OUR DESIGN G d are coputed as follows: G c = CAPACITY OF RELATED WORKS DELAY OF RELATED WORKS ad G d =. DELAY OF OUR DESIGN ) Copariso with Pure Ad Hoc Wireless Networks: I [], Gupta et.al studied the capacity of pure ad hoc wireless etworks as grows to a large level. Their results priarily idicate that whe a source ode radoly chooses its destiatio ode placed at the axiu distace of O() apart, the W log ). throughput capacity available to each ode is of Θ( Our paper aily studied the beefits that ca be realized by regularly placig base statios i these pure ad hoc etworks. Specifically, we observe that whe = O( log ) ad =Ω( log ), sae cell routig policy achieves a gai of log ad respectively i copariso to PANs. This iproveet ca be iterpreted by lookig at the ipact 468

8 Gai= Gai= for our desig. I [7], authors studied the capacity of hybrid TABLE I G c UNDER THE SAME CELL ROUTING POLICY OVER L-HOP [7] Gai= sae cell routig policy k-earest Pure ad hoc Networks Regio (a) Regio (b) Regio (c) Fig.. Network Capacity Λ(, ) vs. for sae cell routig policy ad k-earest cell [6]. the uber of hops has o the capacity of each ode. It follows fro [] that each source ode couicates with a radoly chose destiatio at a distace of O() apart. Sice the average trasissio rage is of log /, each packet fro the source ode has to be retrasitted by at least / log / = / log relayig odes before reachig the fial destiatio. This i tur icreases the traffic burde at each relayig ode ad subsequetly leads to a per-ode throughput of / log. However, uder sae cell routig policy we allow odes to couicate i purely ad hoc ode oly withi a distace of /. As a result, the average uber of hops betwee each S-D pair is reduced to O( / log ) (O()) for = O(/ log ) (Ω(/ log )). Sice it leads to a decreased aout of relayig traffic per ode, a higher capacity gai is observed for our desig. ) Copariso with Hybrid Network Solutios: I [6], authors propose a k-earest cell routig policy for hybrid wireless etworks. Uder this policy, a source ode uses ad hoc ode to sed data oly whe the destiatio is located withi its k earest eighborig cells. Particularly, their results idicate that whe = O( ) the etwork capacity is Θ(W log / ), iplyig that the beefit of addig base statios to pure ad hoc etwork o capacity is isigificat. However, whe =Ω( ) the etwork capacity is Θ(W ); iplyig that the capacity icreases liearly with the uber of base statios. Fig. copares the capacity achieved uder sae cell routig policy with k-earest as the uber of base statios icreases. Oe ca observe a higher gai for our desig whe copared to k-earest which is represeted by the followig three gai regies (a), (b) ad (c). (i) Regio a: whe = O( log ), G c = log ; (ii) Regio b: whe =Ω( ) ad O( log ), G c = log ; ad (iii) Regio c: whe =Ω( log ), G c =. As etioed earlier, we idetified ad solved the liitig factors, ISSUE I, ISSUE II ad ISSUE III, i [6], which i tur provides a higher gai L G c Ω( /3 ) (Ω( log /3 L log ),O( log )) log Ω( log ) O( /3 ) log /3 O( L log ) L log O(L log ) L log 3/ Ω( log ) Ω(L log ),O( log ) log wireless etworks uder L-Maxiu hop routig strategy. I this policy, a ode seds the data i ad hoc ode oly if the destiatio ca be reached withi L hops. Otherwise, the data is trasitted through the base statios. Agai, as i [6], authors assue a oe-hop wireless uplik betwee a ode ad its base statio, which sees to be a ipractical solutio (ISSUE II). Their results aily idicate that (a) whe L =Ω( /3 ) ad is Ω( log /3 L log ), the etwork capacity is give by Θ(W ). Otherwise, whe = O( etwork capacity is Θ( W L log L log ),the ); (b) whe L = O( /3 log /3 ) ad grows faster tha L log, the etwork capacity is give by Θ(W ). Otherwise, whe grows slower tha L log, the etwork capacity is give by W (L log ). I Table I, we illustrate the gai of sae cell routig policy over L-Hop for several cobiatios of ad L. For exaple, we see that whe L = /3 ad = /3, sae cell routig policy log /3 log /3 achieves a gai of /6 over L-Hop. That is for a 0, 000- log /3 ode etwork, our desig realizes a gai of 3 over L-Hop routig policy. This iproveet stes fro the accuracy of our desig. 3) Delay of Sae Cell Routig Policy: I [5], Gaal et. al studied the optial capacity-delay tradeoff for pure ad hoc etworks as well as obile etworks. For obile etworks, they prove that oe ca achieve a per-ode capacity of Θ() with obility but at the cost of Θ() delay. Whereas for pure ad hoc etworks, they show that the average edto-ed packet delay ca be bouded by Θ( log ) which is i tur depedet o the uber of odes. I both cases, we observe that as icreases, delay also icreases largely. Table II illustrates G d of sae cell routig policy over existig schees i [],[4]. Agai, we observe a iproved perforace for our desig as opposed to those schees i [][4]. TABLE II G d UNDER THE SAME CELL ROUTING POLICY OVER EXISTING SOLUTIONS Existig Works PANs MOBILITY = O( ) =Ω( log log ) log log 469

9 V. CONCLUSION I this paper, we idetify three critical factors, uber of hops, ulti-hop upliks ad failure tolerat, which is overlooked by existig schees ad propose a desig that accurately resolves these issues. Specifically, we observe that whether = O( log ) or Ω( log ), our desig achieves a per ode capacity that scales with. This is i fact a sigificat result as opposed to previous efforts which states that if grows slower tha soe threshold, the beefit of augetig those base statios to the origial ad hoc etwork is isigificat. Besides, i copariso to existig works we clearly show the gai oe could obtai o delay as well as o capacity i executig our desig. APPENDIX A. D legth Routig Policy I this sectio, due to space costraits we oly provide the upper bouds o the capacity for D legth routig policy. However, the lower bouds ca be obtaied i a siilar aer give by Sectio III-A. Oe ay also ote that oly the capacity figures related to the purely ad hoc trasissios differs fro the sae cell routig policy ad the figures related to the base statio relative trasissios reai the sae. Next to quatify the capacity bouds, we first deterie the uber of odes couicatig i purely ad hoc ode as well as i hybrid ode. We have the followig Lea. Lea 6. The uber of S-D pairs couicatig usig purely ad hoc ode ad hybrid ode are πd ad ( πd ) respectively, uder D legth routig policy. Proof: The probability that a ode couicates with its destiatio i purely ad hoc ode i.e., withi D legth is πd. Likewise, the probability that a ode couicates with its destiatio ode i hybrid ode is πd.now, followig the sae techiques i Lea 3 ad replacig the correspodig probabilities with πd ad πd, we get the uber of odes couicatig i ad hoc ad hybrid ode as πd ad ( πd ) respectively. Let La ad L h respectively be the ea legth of the distace take by a packet i purely ad hoc ode ad i hybrid ode. Sice each ode couicates with its chose destiatio i ad hoc ode oly if it is withi D legth fro the source, we have L a as at ost D. Fro earlier discussios, we also kow that i a hybrid ode source couicates with its base statio i a ulti-hop fashio. Hece i a cell of size, Lh is at ost. Replacig the left-had side( of the expressio i eq. (5) i Sectio III-B by Λa(,) r() πd 3 + ( πd ) ),we have Λ a (, ) ( W a πd 3 + πd ) log. We ca thus write the capacity of D legth routig policy as follows Λ(, ) = W Θ( a ) + log W u). Oe ay ote that whe ( πd 3 + πd = O(/ log ), settig D =0ad aps the capacity of D legth routig policy to capacity of sae cell routig policy ad pure ad hoc etworks respectively. Thus, the axiu capacity realizable uder this policy is o ore tha W log for = O(/ log ). Ituitively, as D decreases the probability of odes couicatig i hybrid ode icreases; this i tur iplies that ost of the odes couicate with the base statio i purely ad hoc ode at a distace of O( / ). 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