Throughput Capacity of Hybrid Radio-Frequency and Free-Space-Optical (RF/FSO) Multi-Hop Networks

Transcription

1 Throughput Capacity of Hybrid Radio-Frequecy ad Free-Space-Optical (RF/FSO) Multi-Hop Networks Di Wag ad Alhussei A. Abouzeid Departet of Electrical, Coputer ad Systes Egieerig Resselaer Polytechic Istitute Abstract The per-ode throughput capacity of hybrid radio frequecy ad free space optics (RF/FSO) etworks is studied ad the beefit of usig this hybrid etwork architecture over the pure RF wireless etworks is evaluated. The hybrid RF/FSO etwork cosists of a RF ad hoc etwork of odes, of the (so called super odes) are equipped with a additioal FSO trasceiver. Every RF ad FSO trasceiver is able to trasit at a axiu data rate of ad bits/sec, respectively. All the super ode are coected by the FSO liks ad thus ca for a stad-aloe FSO etwork. With a iiu trasit power objective, a upper boud o the per ode capacity of c + log c2w2 is derived. I order to prove that this upper boud is achievable, we desig a hybrid routig schee i which the data traffic is divided ito two classes ad use differet routig strategies: a portio of data will be forwarded with the (partial) support of super odes i a hierarchical routig fashio, ad the rest will be purely routed through RF liks i a ulti-hop fashio. By properly balacig the load betwee these two classes of traffic, it is show that this upper boud is tight whe the axiu data rate ratio of W FSO ad RF trasceivers, 2, grows slower tha. Uder such circustaces, the capacity iproveet with the support of FSO odes, as copared with the results for RF wireless etworks i [], is evaluated. A sigificat capacity gai will be achieved if = Ω(). The results characterize the uber of super odes ad/or the FSO data rate ecessary i order to cause a o-trivial icrease i the per-ode throughput. I. INTRODUCTION The capacity of Radio Frequecy (RF) wireless etworks is costraied by provable liits ad does ot scale well with the icreasig uber of odes i the syste due to the iterferece betwee cocurret trasissio fro eighborig odes []. While the RF wireless etworks cotiue to develop ad grow i deads, the techology of free space optics (FSO) starts to draw attetio fro both acadeia ad idustry. FSO ca provide high data rate ad highly directioal trasissios usig free-space laser beas, whose bea divergece agle is o the order of illi-radias.this perfect directioality i trasissio akes the FSO couicatios early iue fro iterferece. However, oe of the ajor liitatios of FSO techology is the eed for optical liks to aitai lieof-sight (LOS), ad FSO lik availability ca be further liited by adverse weather coditios like fogs ad heavy sowfalls. The copleetary property of RF ad FSO otivates us to desig hybrid RF/FSO etworks, i which the weakesses of each lik type are expected to be utually itigated. It is quite atural to view this RF/FSO cobiatio as a way to solve the capacity scarcity proble i RF wireless etworks, or at least tur it aroud to soe extet. Several practical desigs of such hybrid etworks have bee proposed ad/or ipleeted [2], [3], [4], [5] ad [6], which ay iprove the etwork perforace by providig higher throughput ad/or better reliability. However, there is still o theoretical work that ay give isight o how uch the perforace ca be obtaied ad how uch these iproveets ca be possibly achieved. Istead of tryig to provide ew practical solutios to these hybrid RF/FSO etworks or to copare the perforaces of differet protocols or algoriths available, this work ais to derive the fudaetal liit of the capacity of hybrid RF/FSO etworks ad show exactly how uch this ay deviate fro the capacity results i the ladark paper by Gupta ad Kuar []. I this paper we cosider a rado etwork sceario, where odes are radoly located, i.e., idepedetly ad uiforly distributed o a uit area. Each ode is equipped with a RF trasceiver. Oly of the, which are called super odes, are equipped with a additioal FSO trasceiver each. Each RF trasceiver ca oi-directioally trasit or receive at bits/sec withi the couicatio rage r. Each FSO trasceiver ca directioally trasit withi a ifiitesially sall agle ad oi-directioally receive, all at bits/sec. The couicatio rage of every FSO trasceiver is s. For each ode, its destiatio is radoly chose ad let λ (i bits/sec) deote the axiu data rate at which each source-destiatio pair is required to trasit ad receive. By followig the siilar defiitio i [], we defie this λ as the throughput capacity of this hybrid RF/FSO etwork. Here we let this rado etwork satisfy two asyptotic coectivity assuptios: a) All the odes are asyptotically coected by RF liks; b) All the super odes are asyptotically coected by FSO liks. This allows us to have two stadaloe etworks cosistig of odes coected oly by RF liks ad odes coected oly by FSO liks, respectively. We eed to otice that weather coditios ay have differet effects o RF ad FSO liks, as FSO liks are highly susceptible to dese fog, soke ad dust particles but relatively less vulerable to rai coditios ad the opposite is true for RF systes. Thus uder these asyptotical coectivity assuptios, the rado hybrid etwork ca accoodate

2 for adverse weather coditios such as heavy rais or heavy fogs [5]. Although i this paper we do ot study the case whe RF or FSO liks are ureliable, the odellig applied here ay reserve possible aveues for the future work. We aalyze the throughput capacity of the hybrid RF/FSO etworks uder the costrait of iiu FSO trasit power cosuptio. We iiize the FSO trasit power cosuptio by choosig the FSO couicatio rage s to be the iiu i aitaiig the asyptotic coectivity aog super odes. Although our aalysis idicates that the larger the FSO couicatio rage is, the higher the throughput ay be achieved, we still have to ote that it is ipossible to realize a couicatio syste with a arbitrarily large couicatio rage. The it is reasoable to add certai kid of FSO trasit power costraits whe tryig to obtai the throughput capacity λ. The costrait of iiu FSO trasit power cosuptio cosidered i this paper is certaily a coservative oe, ad it ay be ore suitable for the case whe eergy savig is give high priority i the etwork desig. Sice our aalysis also show that the RF couicatio rage r is autoatically iiized whe axiizig the throughput capacity λ, the this power costrait ca be rewritte as the costrait of iiu trasit power cosuptio with o abiguity. Our derivatio of the throughput capacity ca be divided ito two parts. First we derive a upper boud o the throughput capacity λ over all routig ad trasissio strategies. The we costruct a hybrid routig schee i which the data traffic is divided ito two classes ad use differet routig strategies: a portio of data will be forwarded with the (partial) support of super odes i a hierarchical routig fashio, ad the rest will be purely routed through RF liks i a ulti-hop fashio. It is show that uder certai coditios, the etwork ca actually achieve at a throughput o the sae order of ad as the upper boud. We copare the results obtaied here with the results o throughput capacity of RF wireless etworks evaluated i [] i order to characterize the capacity iproveet cotributed by those super odes. It is observed that a oticeable capacity iproveet will be achieved whe the uber of super odes ad/or the FSO data rate are high eough. The ai cotributios of this paper are suarized as follows: a) Derive the scalig laws for the throughput capacity of hybrid RF/FSO etworks as a fuctio of the total uber of odes of which are super odes; b) Preset a routig ad trasissio schee that achieves the derived throughput capacity, which ay guide the practical routig protocol desig for the hybrid RF/FSO etworks; c) Copare with the results agaist pure RF wireless etworks, aalyze the capacity iproveet of hybrid etworks, ad characterize the uber of super odes ad/or the FSO data rate ecessary i order to cause a o-trivial icrease i the per ode throughput. The rest of this paper is orgaized as follows. A brief overview of related work is preseted i Sectio II. Sectio III outlies the syste odel that is cosidered throughout the paper. Sectio IV derives a upper boud o throughput capacity of RF/FSO etworks with iiu trasit power cosuptio. A costructive lower boud o throughput capacity of RF/FSO etworks is preseted i sectio V. Sectio VI discusses the capacity results obtaied, ad provides soe practical iplicatios. Sectio VII cocludes the paper. II. RELATED WORK To the best of our kowledge, there is o prior work i the literature o the capacity aalysis of RF/FSO ultihop etworks. However, several attepts have bee ade to provide capacity iproveet by utilizig directioal ateas or itroducig ifrastructure support. The authors i [7] aalyzed the capacity iproveet of RF wireless etworks usig directioal ateas i three 2π α cases. The capacity gai is show to be a) whe usig directioal trasissio with divergece agle α ad oi receptio, b) 2π β whe usig oi trasissio ad 2π directioal receptio with divergece agle β, ad c) αβ whe both trasissio ad receptio are directioal. These capacity gais ca oly be achievable whe the divergece agle is ot too sall. Thus the results derived i [7] are ot applicable to our work sice we deal with FSO liks with ifiitesially sall divergece agle. [8], [9] ad [0] try to iprove the capacity by the itroductio of ifrastructure support (access poits or base statios). Authors i [8] depict the ifrastructure etwork as a cellular etwork, where the base statios are wired by a broadbad etwork ad placed at the ceter of hexagoal cells. They ivestigate how the uber of base statios should scale with the uber of ad hoc odes to achieve sigificat capacity iproveet over the pure RF ad hoc wireless etworks. They apply differet routig strategies i which the ad hoc odes are divided ito two groups depedig o whether they use the cellular etwork to reach the destiatio or ot. The decisio criteria i forig the groups rely o heuristic arguets ad ay ot be the optial routig strategies. They show that the uber of base statios should at least scale with to achieve a oticeable gai. The ifrastructure etwork used i [9] cosists of radoly located access poits which are pre-wired ad allocated ifiite capacity. The authors i [9] odel the ode distributio ad traffic patter i the sae aer as the rado etwork odel used i []. They assue that the uber of ad hoc odes per access poit is bouded above, ad each wireless ode is able to trasit at W bits/sec usig a fixed trasissio rage. Uder this rado etwork sceario, they show that a per ode capacity of Θ( W log ) ca be achieved. To do this, they specify the upper boud of throughput capacity over all routig ad trasissio strategies, ad the desig a specific routig ad trasissio schee to achieve this upper boud. The results i [0] exted the work of [9] by allowig odes to perfor power cotrol ad properly choosig the uber of access poits, ad further show that it is possible to provide a throughput of Θ() to ay fractio f, 0 < f <, of odes. Although overlaps ay exist betwee our work ad the 2

3 prior work such as [8], [9] ad [0], there are also ajor differeces that uderlie the cotributio of our work: a) It is ore realistic to deploy the super odes radoly rather tha eployig a hexagoal cell structure, as copared to [8]; b) We do ot ipose strog assuptios such as the uber of odes per super ode should be bouded or properly chose, as copared to [9] ad [0]; c) We let the axiu data rate of each FSO trasceiver,, be soe fiite value, which is ore realistic as copared to the ifiite capacity assuptios used i [9] ad [0]. III. SYSTEM MODEL The syste odel icludes the settigs of ode distributios, traffic patters, ad RF/FSO couicatio odels. A. Node Distributios ad Traffic Patter I a rado sceario, odes each equipped with a RF trasceiver are radoly located, i.e., idepedetly ad uiforly distributed o the surface S 2 of a three-diesioal sphere of area 2. Oly of the, which are called super odes, are equipped with a additioal FSO trasceiver each. Our purpose i studyig S 2 is to separate edge effects fro other pheoea. Each ode has a radoly chose destiatio to which it wishes to sed λ bits/sec. The destiatio for each ode is idepedetly chose as the ode earest to a radoly located poit, i.e., uiforly ad idepedetly distributed. Thus destiatios are o the order of away o average. This odel, except for the FSO capability, is siilar to the odel i []. B. RF Couicatio Model Each RF trasceiver ca oi-directioally trasit or receive at bits/sec. We use the Protocol Model itroduced i [] as the RF iterferece odel here. All odes eploy a coo rage r for all their trasissios. Note that all the distaces are easured o the surface S 2 of the sphere by segets of great circles coectig two poits. Let X i deote the locatio of a ode; we will also use X i to refer to the ode itself. Whe ode X i trasits to a ode X j over the RF chael, the data will be successfully received by X j if ) The distace betwee X i ad X j is o ore tha r, i.e., X i X j r () 2) For every other ode X k siultaeously trasittig over the RF chael X k X j ( + )r (2) The quatity > 0 odels situatios where a guard zoe is specified by the protocol to prevet a eighborig ode fro trasittig o the RF chael at the sae tie. It also allows for iprecisio i the achieved rage of trasissios. C. FSO Couicatio Model Every super ode is equipped with a FSO trasceiver. Each FSO trasceiver ca directioally trasit at bits/sec withi a ifiitesially sall agle ad a coo rage s. Each FSO trasceiver ca receive at bits/sec oidirectioally. We assue that the orietatio of each FSO trasitter ca be steered to ay possible directio withi S 2. The every super ode ca trasit data to ay other super ode withi the distace s. I practice, the bea steerig fuctioality has bee realized i several ways. Miler et.al. [2] ipleeted the bea steerig usig echaical devices, while Kha et.al. [] desiged a 3-diesioal wide-agle o-ovig-parts laser bea steerig ethod. May other desig choices are also available ad ca be foud fro [2], [3], [4] ad [5]. The oi-directioality assuptio for the FSO receiver is also reasoable, as oi-directioal FSO receivers are also ipleeted i [4]. IV. AN UPPER BOUND ON THROUGHPUT CAPACITY OF RF/FSO NETWORKS WITH MINIMUM POWER CONSUMPTION I this sectio, we aalyze the throughput capacity of hybrid RF/FSO etworks with iiu total power cosuptio. I RF wireless etworks, it ay be desirable to reduce the trasit power level, or equivaletly, to reduce the trasissio rage r i order to icrease etwork capacity. Kawadia ad Kuar [6] argue that the area of the iterferece is proportioal to r 2 whereas the relayig burde, i.e., the uber of hops, is iversely proportioal to r. The the area cosued by a packet is thus proportioal to r, iplyig that reducig the trasit power level icreases the capacity. I FSO etworks, however, reducig the trasit power level ad icreasig the capacity are essetially cotradictig with each other. Sice the area of the FSO iterferece is actually egligible, the relayig burde is alleviated by siply icreasig the FSO trasissio rage s, which iplies that icreasig the trasit power level icreases the capacity. Therefore, the objective of iiizig the total power cosuptio i hybrid RF/FSO etworks sees to be cotradictory i achievig a higher capacity. However, the capacity results derived here ay still be valuable, sice reducig the power cosuptio is always cosidered to be iportat whe deployig large-scale etworks where the resource of eergy is liited. Theore 4.: The throughput capacity of hybrid RF/FSO etworks with iiu power cosuptio is bouded fro above by λ c log + c 2 where c ad c 2 are costat. Proof: Each packet i this hybrid etwork ay traverse a uber of RF hops ad a uber of FSO hops. Let d deote the uber of RF hops traversed by a sigle packet averaged over (3) 3

4 all packets i the etwork, ad let d2 deote the uber of FSO hops traversed by a sigle packet, also averaged over all packets i the etwork. Fro Lea 5.4 i [], the uber of siultaeous trasissios o a RF chael is o ore tha c3 r. (All the c 2 i s used above ad throughout are costats.) Sice each source geerates λ bits/sec, there are sources, ad each packet eeds to be relayed o the average by d RF hops, it follows that the total uber of bits per secod served by the RF part of the etire etwork is λ d. To esure that all the required RF traffic is carried, we therefore eed λ d c 3 r 2 (4) O the other had, sice FSO ca operate with egligible iterferece, the uber of siultaeous trasissio o FSO chael is o ore tha. To esure that all the required FSO traffic is carried, we therefore eed λ d 2 (5) Now let L deote the ea legth of a lie coectig two idepedetly ad uiforly distributed poits o S 2. The we have the followig iequality d r + d 2 s L (6) Recall that we ake the assuptio that all the odes are asyptotically coected by RF liks, ad all the super odes are asyptotically coected by FSO liks. To satisfy these two asyptotic coectivity assuptios, [7] suggests that the trasissio rages, r ad s, should satisfy r c 4 log s c 4 The lik equatio of a FSO couicatio syste is give by [3] A P r = P t exp( βs) (9) (sθ) 2 where P t is the laser output power, θ is the bea divergece agle (i radias), A is the receiver area, P r is the received power, ad β is the atospheric atteuatio factor. Copared with the cooly used iverse α th law path loss odels, this lik equatio suggests α > 2. It has bee show i [8] that, sice α > 2, the trasissio rage should be kept to iiu to iiize the total power cosuptio i the etire etwork. Thus we set the FSO trasissio rage s to be s = c 4 (7) (8) (0) Fro (4), (5), (6), (7) ad (0), we have λ L λ d r + λ d s c 3 r 2 = c 3 r c 3 c 4 r + s + s log + c 4 The λ L ( ) c3 c 4 log + c 4 which yields the result. V. A CONSTRUCTIVE LOWER BOUND ON THROUGHPUT CAPACITY OF RF/FSO NETWORKS WITH MINIMUM POWER CONSUMPTION () To obtai a costructive lower boud o the throughput capacity λ, we eed to derive the throughput capacity of rado FSO etworks with iiu power costrait. This result is give by the followig theore. Theore 5.: Cosider a rado FSO etwork with odes radoly ad idetically distributed o S 2. Every ode is equipped with a FSO trasceiver. Each FSO trasceiver ca directioally trasit at bits/sec withi a ifiitesially sall agle ad a coo rage s. Each FSO trasceiver ca receive at bits/sec oi-directioally. The the throughput capacity, λ, of this rado FSO etwork with iiu power cosuptio is give by λ = Θ ( ) (2) Proof: We derive the throughput capacity of rado FSO etworks by providig a upper boud o λ, ad the show by costructio that a throughput of the sae order i ca actually be achieved. To derive the upper boud o λ, we otice that the uber of siultaeous trasissios is o ore tha, thus the total data rate served by the etire FSO etwork is o ore tha. Now let L deote the ea legth of a lie coectig two idepedetly ad uiforly distributed poits o S 2. The the ea legth of the path of packets is at least L o(). Thus the ea uber of hops take by a packet is at least L o() s. The the total uber of bits per secod served by the etire etwork eeds to be at least ( L o())λ s. The we have the followig iequality: ( L o())λ (3) s I order to iiize the total trasit power cosuptio, the FSO couicatio rage s is chose to be s = 4

5 c 4, the we have the followig upper boud o λ : λ c 5 (4) To show that this upper boud is achievable, we desig a routig schee usig siilar techiques as i []. Let ρ () := radius of a disk of area 00 / (5) We ca costruct a Vorooi tessellatio V i relatio to the uber of odes ad the locatios of odes, i which every Vorooi cell cotais a disk of radius ρ () ad is cotaied i a disk of radius 2ρ (). We choose the rage s of each trasissio such that s = 8ρ () (6) This rage allows direct couicatio withi a cell ad betwee adjacet cells. Every ode i a cell is withi this distace s fro every other ode i its ow cell or adjacet cell. The routig strategy is to choose the routes of packets to approxiate the straight-lie which is coectig the source ad destiatio. So the routes actually are the cells that the straight-lie itersects. Siilar to Lea 4.4 ad Lea 4.8 i [], it ca be show that the traffic to be served by each cell is bouded fro above, ad the uber of odes i each cell is bouded fro below. Forally, for ay cell V, there is a δ () 0 such that Prob{ sup V V (traffic eedig to be carried by cell V ) c 6 λ } δ () (7) Ad for every disk D of area 00 / i S 2, there is sequece δ() 0 such that Prob{uber of odes i D 50 } δ() (8) It ca be thus oted that each cell is able to trasit at least (50 ) bits/sec. The with high probability, the rate c 6 λ ca be accoodated by all cell if c 6 λ (50 ) (9) The the per ode achievable throughput is give by λ = c 6 which proves Theore 5.. (20) The lower boud o λ is give by the followig theore. Theore 5.2: The throughput capacity of hybrid RF/FSO etworks with iiu power cosuptio, λ, is bouded fro below by ) Case : λ c 7 log ( ) + c 8 c 9 (2) whe,, ad satisfy ( ) ( ) = o log W 2) Case 2: λ c 0 log whe,, ad satisfy ( ) ( ) = ω log W (22) (23) (24) Proof: I order to show that the lower bouds o λ preseted above are achievable, we desig a hybrid routig schee i which the data traffic is divided ito two classes ad use differet routig strategies: a portio of data will be forwarded with the (partial) support of super odes i a hierarchical routig fashio, ad the rest will be purely routed through RF liks i a ulti-hop fashio. The lower bouds o λ ca thus be derived by properly balacig the load betwee these two classes of traffic. The proof is orgaized as follows: The hybrid routig schee is proposed i Sectio V.A, i which we desig a RF ulti-hop routig strategy ad a hierarchical routig strategy for the two classes of traffic. I Sectio V.B, the axiu throughput achieved by the hybrid routig schee is derived. A. A Hybrid Routig Schee We eploy a hybrid routig schee by dividig the per ode throughput λ ito two classes. Each source-destiatio pair trasits packets with costat data rate λ by purely usig RF liks, ad trasits packets with data rate λ 2 by (fully or partly) usig FSO liks. Hece we have λ = λ + λ 2 (25) The detailed routig strategies for these two classes of traffic are as follows. ) A RF ulti-hop routig strategy for the traffic correspodig to λ : The routig strategy for the traffic correspodig to λ is the sae as the oe i []. Let ρ() := radius of a disk of area 00 log / (26) We ca costruct a Vorooi tessellatio V such that every Vorooi cell cotais a disk of radius ρ() ad is cotaied i a disk of radius 2ρ(). We choose the rage r of each trasissio such that r = 8ρ() (27) This rage allows direct couicatio withi a cell ad betwee adjacet cells. The routig strategy is to choose the routes of packets to approxiate the straight-lie which is coectig the source ad destiatio. So the routes actually are the cells that the straight-lie itersects. Accordig to [], we have the followig leas. 5

6 Lea : Every cell i V has o ore tha c iterferig eighbors. Furtherore, there is a schedule for trasittig packets such that i every ( + c ) slots, each cell i the V gets oe slot i which to trasit, ad such that all trasissios are successfully received withi the trasissio ad receptio coverage. Lea 2: There is a δ () 0 such that Prob{ sup V V (RF traffic correspodig to λ eedig to be carried by cell V ) c 6 λ log } δ () (28) 2) A hierarchical routig strategy for the traffic correspodig to λ 2 : The routig strategy for the packets correspodig to λ 2 is described by the followig steps: Step : The packets origiated by a source ode are routed through RF liks to the super ode earest to the source ode, if eeded; Step 2: Packets are the routed through FSO liks to the super ode earest to the destiatio ode; Step 3: Packets are the routed through RF ode to the destiatio ode, if eeded. Here the RF routig schee used i steps ad 3 is the sae as the oe used i []. The FSO routig schee used i Step 2 is described i the proof of Theore 5.. Note that the routig schee used here for the the traffic correspodig to λ 2 is siilar to hierarchical state routig e.g.[5]. Siilar to Lea 4.8 i [], it ca be proved that every disk D of area 00 / cotais at least oe super ode with high probability. Forally, for every disk D of area 00 / i S 2, there is δ() 0 such that Prob{D cotais a super ode} δ() (29) Sice every disk of area 00 / cotais at least oe super ode with high probability, ad each Vorooi cell i V cotais a disk of area 00 log /, the the ea uber of RF hops traversed by a packet durig its first ad thirdstep i the routig schee described above is at ost c / 2 log /. The the total data rate correspodig to λ 2 served by theetire RF wireless etwork is o ore tha c 2 λ 2 ( ) / log /. Thus the average data rate correspodig to λ 2 served by each Vorooi cell V is o ore tha / c 3 λ 2 ( ) log / log log = c 3 λ 2. (30) The by akig use of the property that the sequece of straight-lie segets {L i } i= is i.i.d, we ca exploit uifor covergece i the law of large ubers usig the sae techique itroduced i ([], Sectio IV.I), ad thus obtai the followig lea: Lea 3: There is a δ () 0 such that Prob{ sup (RF traffic correspodig to λ 2 eedig to be V V log carried by cell V ) c 4 λ 2 } δ () (3) B. Maxiu Throughput Achieved by the Hybrid Routig Schee Fro Lea, Lea 2 ad Lea 3, we ca coclude that the data trasissio utilizig RF liks ca be accoodated by all cells if c 6 λ log + c4 λ 2 log + c (32) Sice the per ode throughput capacity of the stad-aloe FSO etwork is give by Theore 5., the data trasissio (partially) utilizig FSO liks ca be accoodated by the FSO etwork if λ 2 c 6 (33) The we ca derive the axiu throughput achieved by the proposed routig schee by solvig the followig optiizatio proble: subject to axiize λ λ + λ 2 = λ log c 6 λ log + c4 λ 2 + c λ 2 c 6 The optial solutio falls ito the followig two utually exclusive cases: Case : Whe,, ad satisfy ( ) < ( + c )c 6 c 4 log W (34) the axiu throughput λ ad the data rates of the two classes of traffic, λ ad λ 2, are give by λ = ( + c )c 6 log ( ) + c 6 c 4 (35) λ = c 4 ( + c )c 6 log 2 (36) λ 2 = c 6 (37) Case 2: Whe,, ad satisfy ( ) ( + c )c 6 c 4 log W (38) 6

7 the axiu throughput λ ad the data rates of the two classes of traffic, λ ad λ 2, are give by λ = (39) ( + c )c 4 log λ = 0 (40) λ 2 = (4) ( + c )c 4 log Hece we have proved Theore 5.2. VI. DISCUSSIONS A. The Tightess of the Bouds o λ ad The Routig Schee Selectio Criteria We evaluate the tightess of the capacity bouds o λ by coparig the upper boud results i Theore 4. with the lower boud results uder two differet cases derived i Sectio V. It is clear that the lower boud result for Case i (35) will asyptotically becoe the for of the upper boud o λ i Theore 4., as grows. Thus i Case, Theore 4. ad Theore 5.2 provide a tight boud o λ whe ( ) < c 5 log W (42) For Case 2, which requires,, ad satisfyig ( ) c 5 log W, (43) the lower boud result i (39) does ot atch the for of the upper boud result i Theore 4.. Although we caot get a geeral result like i Case, we ca evaluate the tightess by cosiderig the extree case whe. Uder these circustaces, the upper boud result becoes ifiity, which is certaily ot tight. The lower boud result atches the results provided by [9], where the support of a ifiite capacity ifrastructure etwork is cosidered. This iplies that the lower boud is tighter tha the upper boud uder such circustaces. We ca ote that for Case the proposed routig schee works as a cobiatio of two idepedet schees, while uder Case 2 it copletely degeerates to the hierarchical routig schee as described i Sectio V.2. I order to guide the practical routig protocol desig, we eed to derive clear coditios o,, ad for Case ad 2. Although there is geerally o closed for solutios whe (42) or (43) hold with equality, we are still able to exploit soe practical iplicatios by presetig the followig protocol desig priciples: ) Whe W2 = o( 2 ), the coditio for Case is satisfied. The it is better to eploy the hybrid routig schee to achieve a higher per ode throughput. 2) Whe W2 = Ω( 3 2 ), the coditio for Case 2 holds. Thus it is better to use the hierarchical routig schee to achieve a higher per ode throughput. 3) Whe W2 = Θ( k ), where 0.5 k <.5, there exists two critical ubers M (k) ad M 2 (k), M (k) < M 2 (k) <, satisfyig f() := ( ) = c 5 log W, for = M, M 2. (44) Sice f() is strictly cocave over [, ], hece ( ) < c 5 log W, for < M (k) or > M 2 (k) (45) ( ) > c 5 log W, for M (k) < < M 2 (k) (46) Thus the hybrid routig schee is ore desirable to achieve a higher per ode throughput whe < M (k) or > M 2 (k). O the other had, the hierarchical routig schee is preferred whe M (k) < < M 2 (k). B. Capacity Iproveet We defie the capacity gai G, as the capacity ratio betwee hybrid RF/FSO etworks ad RF wireless etworks. We further defie the capacity iproveet ratio, R, as R = G. First we take Case ito cosideratio. By coparig with the capacity results i [], we ca show that R ay diiish to zero as grows whe = o() (47) O the other had, we ca possibly achieve a o-egligible capacity iproveet by addig FSO trasceivers to odes if ad oly if = Ω() (48) For Case 2, (39) iplies that the capacity iproveet ratio R will ever diiish to zero. Thus a sigificat capacity iproveet ca be achieved with high probability uder Case 2. Thus we ca geeralize the results for Case ad 2 by claiig that we ca achieve a o-egligible capacity iproveet with high probability if = Ω() (49) This result characterizes the uber of super odes ad/or the FSO data rate eeded i order to guaratee a sigificat capacity iproveet as copared to the case of pure RF wireless etworks. 7

8 VII. CONCLUSION AND FUTURE WORK I this paper, we have studied the per ode throughput capacity of hybrid RF/FSO etworks. A hybrid RF/FSO etwork cosists of a RF wireless etwork with rado deployet ad coected by RF liks, ad oly a portio of odes (so called super odes) are equipped with a additioal FSO trasceiver. All the super ode are coected by the FSO liks ad thus for a stad-aloe FSO etwork. The objective of this paper is to derive the asyptotic capacity of such hybrid etworks, ad evaluate the beefit of usig this hybrid RF/FSO etwork architecture over the pure RF wireless etworks. We cosider a hybrid RF/FSO etwork with a total uber of odes, ad of the are super odes. Every RF ad FSO trasceiver is able to trasit at a axiu data rate of ad bits/sec, respectively. We show that the throughput capacity of hybrid RF/FSO etworks with the costrait of iiu power cosuptio, λ, is bouded fro above by λ c log + c 2. I order to evaluate the tightess of this upper boud, we desig a hybrid routig schee as we divide the data traffic ito two classes ad use differet routig strategies: a portio of data will be forwarded with the (partial) support of super odes i a hierarchical routig fashio, ad the rest will be purely routed through RF liks i a ulti-hop fashio. Uder such routig strategies, we have show that the upper boud is a tight oe whe the W axiu data rate ratio of FSO ad RF trasceivers, 2, grows slower tha. The aalysis regardig our proposed routig strategy also yields soe practical iplicatios which ay guide the routig protocol desig for the hybrid RF/FSO etworks. Our aalysis have show that if W2 grows slower tha, high per ode throughput ca be achieved by properly balacig the load betwee the two classes of traffic. If W2 grows o slower tha 3, the a pure hierarchical routig strategy will suffice to achieve high per ode throughput, i.e., all the data traffic will be forwarded with the (partial) support of super odes. Furtherore, we have also evaluated the capacity iproveet with the support of FSO odes, as copared with the results for RF wireless etworks i []. We have show that a sigificat capacity gai will be achieved if W2 = Ω(). Possible aveues for future research still exist, as we have ot derived a tight boud o throughput capacity for the case whe W2 grows o slower tha. I additio, we are also iterested i extedig the capacity results to the case whe RF ad FSO trasceivers ay becoe ureliable (e.g., uder adverse weather coditios). [2] S.D. Miler, C.C. Davis, Hybrid free space optical/rf etworks for tactical operatios, Military Couicatios Coferece(MILCOM), [3] S. Bloo, W. S. Hartley, The last ile solutio: Hybrid FSO Radio, Airfiber, May [4] J. Akella, Chag. Liu, D. Partyka, M. Yuksel, S. Kalyaaraa, P. Dutta, Buildig blocks for obile free-space-optical etworks, Secod IFIP Iteratioal Coferece o Wireless ad Optical Couicatios Networks (WOCN), [5] J. Dereick, C. Thore, J. Spletzer, O the deployet of a hybrid freespace optic/radio frequecy (FSO/RF) obile ad-hoc etwork, IEEE/RSJ Iteratioal Coferece o Itelliget Robots ad Systes (IROS), [6] A. Kashyap, M. Shaya, Routig ad traffic egieerig i hybrid RF/FSO etworks, IEEE Iteratioal Coferece o Couicatios (ICC), [7] S. Yi, Y. Pei, ad S. Kalyaaraa, O the capacity iproveet of ad hoc wireless etworks usig directioal ateas, i Proc. of the 4th ACM iteratioal syposiu o Mobile ad hoc etworkig ad coputig(mobihoc), pp. 08-6, [8] B. Liu ad Z. Liu ad D. Towsley, O the capacity of hybrid wireless etworks, I IEEE INFOCOM 03, March [9] U.C.Kozat, L.Tassiulas, Throughput Capacity of Rado Ad Hoc Networks with Ifrastructure Support, Proceedigs of ACM/MOBICOM 2003, [0] Ashish Agarwal ad P. R. 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