O the Throughut Scalig of Cogitive Radio Ad Hoc Networks Chegzhi Li ad Huaiyu Dai Deartmet of Electrical ad Comuter Egieerig North Carolia State Uiversity, Raleigh, NC email: {cli3, hdai}@csu.edu Abstract Due to the emergece of Cogitive Radio, a secial tye of heterogeeous etworks attracts icreasig iterest recetly, i which a secodary etwork comosed of cogitive users shares the same resources oortuistically with a rimary etwork of licesed users. Network throughut i this settig is of essetial imortace. Some ioeer works i this area showed that this tye of heterogeeous etworks erforms as well as two stad-aloe etworks, uder the dese etwork model where the size of a etwork grows with the ode desity i a fixed area. A key assumtio behid this coclusio is that the desity of the secodary etwork is higher tha that of the rimary oe i the order sese, which essetially decoules the two overlaid etworks, as the secodary etwork domiates asymtotically. I this aer we edeavor to ivestigate this roblem with a weaker coditio that the dimesios of the two overlaid etworks are o the same order, ad cosider the exteded etwork model where the size of a etwork scales with the area with the ode desity fixed. Surrisigly, our aalysis shows that this weaker (ad arguably more ractical) coditio does ot degrade either etwork throughut i terms of scalig law. Our result further reveals the otetials of CR techology i real alicatios. I. INTRODUCTION The coflict betwee icreasig demads for badwidth ad scarcity of sectrum i wireless commuicatio strogly roels the study of Cogitive Radio (CR) techology i recet years [1]. This techology aims at rovidig a flexible way of sectrum maagemet, ermittig CR (secodary) users to temorally access sectrum that is ot curretly used by legacy (rimary) users. I may cases, it is referable that the oeratio of the secodary etwork is trasaret to the rimary etwork. Due to its secodary role, the CR etwork should revet ay uaccetable iterferece to the rimary etwork, while tolerate the iterferece from the rimary trasmissios. The caacity of a sigle ad hoc etwork has bee extesively exlored sice the semial work of Guta ad Kumar [2]. The caacity study of CR ad-hoc etworks is more challegig i ature due to coexistece of multile (tyically two) etworks. Here is a brief overview of some imortat works i this area. The cogitive chael, a iterferece chael comosed of oe S-D air from each of the two etworks, was studied i [3, 4] from a iformatio-theoretic ersective. The throughut of a oe-ho cogitive etwork This work was suorted i art by the US Natioal Sciece Foudatio uder Grat CNS-0721825 ad CCF-0830462. was ivestigated i [5]. Recetly the throughut of multiho CR etworks was cosidered i [6], where the authors showed that a -ode rimary etwork ad a m-ode secodary etwork, while coexistig, ca achieve the erode throughut Θ( 1 log ) ad Θ( 1 m log m ), resectively. The same results were achieved i [7] with a more ractical assumtio, requirig oly the kowledge of the locatios of rimary trasmitters, rather tha the locatios of all the rimary odes as i [6]. Note that the same results could be trivially achieved if cooeratios betwee two etworks are ermitted. Therefore, as [6, 7] we reclude such a otio i our study. I literature, both the dese etwork model ad exteded etwork model are widely used i the study of wireless ad hoc etworks. I the dese etwork model, the umber of odes grows with ode desity i a fixed area, while i the exteded etwork model, the umber of odes grows with the area with a fixed desity. I may scearios, the (scalig law) results obtaied i these two models coicide, but it is felt that the exteded etwork model is more realistic, where oe is free from the cocer of the ear field effects of electromagetic roagatio [8]. Both [6] ad [7] focus o the dese etwork model ad assume that m = β (β > 1), i.e., the desity of the secodary etwork is higher tha that of the rimary oe i the order sese. Uder this assumtio, the secodary etwork domiates asymtotically, ad the ifluece of the rimary etwork o the secodary etwork becomes icreasigly egligible. I additio, this assumtio is oly valid i the dese etwork model ad does ot hold i the exteded etwork model. These facts motivate us to study a more ractical sceario where m = costat i this aer, i.e., the dimesios of the two overlaid etworks are o the same order. We maily cosider a exteded etwork ad show that eve with this weaker coditio the throughut of both etworks ca be further boosted to Θ( 1 ) ad Θ( 1 m ), resectively, with the hel of itelliget desig of highway systems origially roosed i [9]. Differet from [9] where the etwork is tessellated ito ideedet cells, cells i the tesselated CR etwork are deedet, which comlicates the aalysis. The remaider of this aer is orgaized as follows. The system model ad roblem formulatio are give i Sectio II. The rimary ad secodary rotocols are reseted i Sectio
III, together with our mai results. The throughut aalysis o the secodary etwork is give i Sectio IV. Ad Sectio V cocludes the aer. II. SYSTEM MODEL AND PROBLEM FORMULATION We cosider two time-sychroized overlaid exteded etworks. Suose that a rimary etwork is deloyed i a square S with dimesio [0, ] [0, ], accordig to a Poisso Poit Process (P.P.P) with uit desity, i.e., λ = 1, ad the secodary etwork is distributed i the same square, accordig to a P.P.P with a costat desity λ s = m/, where m >, sharig the same resources with the rimary etwork. It is easy to check, accordig to the roerties of P.P.P., that the umber of rimary ad secodary odes lie i ((1 ϵ), (1 + ϵ)) ad ((1 ϵ)m, (1 + ϵ)m), ϵ > 0, with high robability (w.h..) 1. Therefore, their ratio is aroximately equal to /m. Each rimary ode is aired with aother oe uiformly at radom to form a source-destiatio air so that each ode is the destiatio of exactly oe source. The secodary sourcedestiatio airs are radomly groued similarly. The rimary etwork erforms as if it stads aloe, while the secodary etwork accesses the sectrum oortuistically to revet uaccetable iterferece to the rimary etwork. Idetical trasmissio ower P for all the rimary odes, ad P s for all the secodary odes, are assumed. For simlicity, we oly cosider ath loss for the hysical chaels (as i the majority literature), i.e., the ower atteuatio fuctio is give by: l(d) = mi{1, d α }, (1) where d is the Euclidea distace betwee a trasmitter ad a receiver, ad α > 2 is the ath loss exoet. The trasmissio rate R from a trasmitter X i to its corresodig receiver X D(i) is a cotiuous fuctio of the Sigal to Iterferece lus Noise Ratio (SINR) at X D(i), i.e., R(X i, X D(i) ) = log(1 + SINR). (2) Deote by {X,k ; k T 1 } the subset of cocurret rimary trasmitters, ad {X s,k ; k T 2 } the subset of cocurret secodary trasmitters. For a rimary trasmitter X,i the SINR at its receiver X,D(i) is give by: SINR = P l( X,i X,D(i) ) N 0 + I + I s, (3) where N 0 is the oise ower at the receiver; I is the iterferece ower from all the other rimary trasmitters to the receiver X,D(i), give by: I = P l( X,k X,D(i) ), (4) k T 1,k i ad I s is the iterferece ower from all the secodary trasmitters to the receiver X,D(i), give by: I s = P s l( X s,k X,D(i) ). (5) k T 2 1 with robability aroachig 1 as the the umber of odes i a etwork goes to ifiity. The SINR at a secodary receiver is defied similarly as: SINR s = P sl( X s,i X s,d(i) ) N 0 + I s + I s, (6) where I s is the iterferece ower from all the other secodary trasmitters to the receiver X s,d(i), ad I s is the iterferece ower from all the rimary trasmitters to the receiver X s,d(i), defied similarly as Eq. (4) ad (5). We study the throughut of both etworks, which is the average umber of bits er secod that all source odes ca simultaeously trasmit to their destiatios w.h... Formally the throughut of a etwork of size is defied as: Defiitio 1: The throughut er S-D air T Π () i a etwork of size uder some schedulig scheme Π is defied as the maximal quatity satisfyig ( Pr mi lim if 1 ) i t t B i,π(t) T Π () 1, (7) as, where B i,π (t) is the umber of bits that S-D air i ca trasfer i t time slots. Note that the above defiitio of throughut is a asymtotical roerty, therefore we require the umber of odes i both etworks go to ifiity (with fixed ratio) i our aalysis. III. NETWORK PROTOCOLS AND MAIN RESULTS I this sectio we itroduce the rotocols for both the rimary ad secodary etwork, followed by their achieved throughut. A. Primary rotocol The rimary etwork adots the highway system roosed i [9]. Rooted from ercolatio theory, the highway system is comosed of multile horizotal ad vertical aths, ad every rimary ode i the lae ca access at least oe horizotal ad vertical ath through oe ho ( see Fig. 1). The rimary rotocol is summarized below : Tessellatio: the square S is divided ito 2a log 2a horizotal corridors, each with dimesio 2a log 2a, where a is some costat, ideedet of. The corridors are the tessellated ito diamod cells with side legth a (see Fig. 2); a highway withi each corridor, comosed of Θ(log ) horizotal aths, is built accordig to [9], usig ercolatio theory (ote that oly oe ath is show i Fig. 1 for simlicity). Similarly we divide the square S ito 2a log 2a vertical corridors, each with dimesio 2a log 2a ad Θ(log ) vertical aths. Routig: three hases are ivolved i the routig scheme: access hase: the source drives its acket to oe of the multile horizotal aths i the horizotal corridor it is located, through oe sigle ho; exress relay: the acket traverses horizotally ad the vertically o the highway through multile hos;
a 2a log log( / ( )) Fig. 2: Tessellatio of a rimary horizotal corridor rimary etwork access Time slot highway delivery secodary etwork silece access highway delivery silece 2a log( / ( )) Fig. 1: The highway system delivery hase: the acket is fially delivered to the destiatio by a ode o a vertical ath, through oe sigle ho. Trasmissio ower: the trasmissio ower of a rimary trasmitter is P = P a α, where P is a costat 2, for all three routig stages. Schedulig: each time slot is further divided ito three sub-slots. For the rimary etwork, the three sub-slots corresod to the three routig hases, show i Fig. 3. The Θ(log )-TDMA scheme is used durig the first ad third routig hase ad the 9-TDMA scheme is adoted durig the highway trasmissio. B. Secodary rotocol The desig of the secodary rotocol is challegig. It is required that the secodary etwork kee its iterferece to the rimary etwork at a accetable level, which may limit its oeratio ad erformace greatly. Isired by the highway system desig i [9], we roose a o-trivial secodary rotocol, which erforms as well as the highway system i a stad-aloe etwork, while satisfies the above requiremet. The secodary rotocol is summarized as follows: Tessellatio: the square S is divided ito log 2a s horizotal corridors, each with dimesio log, where a s = c s m for some costat c s. The corridors are the tessellated ito diamod cells with side legth 3 a s (see Fig. (4)). Corresodigly we divide the square ito log 2a s dimesio 2a s log vertical corridors, each with. A highway system is costructed i each corridor ad it is show i Theorem 3 that there are Θ(log m) horizotal (vertical) disjoit athes i each of the horizotal (vertical) corridors. 2 The same P is used i the secodary rotocol to maitai the ower ratio betwee two etworks. 3 a s is a costat sice /m is. Fig. 3: Time slot for both etworks A reservatio regio (see Fig.5) is set aroud each rimary ode, which is defied as a square comosed of 9 secodary cells. The urose of reservatio regios is to limit the iterferece from the secodary trasmitters to rimary receivers. All the secodary odes located i the reservatio regios must kee silet, i.e., they ca ot serve as trasmitters or relays. Routig: the routig scheme of the secodary etwork is similar to that of the rimary etwork, i.e., the routig is comosed of three hases: access hase, exress relay ad deliver hase. Each secodary trasmissio is reeated for 3 times to avoid excess iterferece from the rimary trasmissio. Trasmissio ower: the trasmissio ower of a secodary trasmitter is P s = P a α s for all three routig stages. Therefore the ower ratio betwee the rimary ad secodary etwork is (a /a s ) α. Schedulig: the secodary etwork shares the same time frame structure with the rimary etwork. The time slot is also divided ito three sub-slots (Fig. 3). Durig the first ad third sub-slot all the secodary odes kee silet. The secod sub-slot is further divided ito three miislots, which corresod to three routig hases of the secodary etwork. The Θ(log m)-tdma scheme is used durig the first ad third routig hase ad the 9-TMDA scheme is adoted durig the highway trasmissio. Accordig to the rimary ad secodary rotocol above, our a s log Fig. 4: Tessellated secodary horizotal corridor ad grid costructio s
oe cell a s Preservatio regio PU secodary ode rimary ode Fig. 5: Preservatio regios of the secodary etwork mai results are summarized below. Theorem 1: Uder the rimary ad secodary rotocol give i this sectio, the rimary etwork achieves erode throughut Θ(1/ ), i the resece of the secodary etwork. Theorem 2: Uder the rimary ad secodary rotocol give i this sectio, with a costat outage robability 4, the er-ode throughut of the secodary etwork is Θ( 1 m ). Remark 1: As will be see i our aalysis below, the outage robability of the secodary etwork deeds o the ratio m/. The higher the ratio, the lower the outage is. The two theorems above imly that, at least i terms of the scalig law, there is o erformace loss for either of the two coexistig etworks; this is articularly iterestig as to the CR etwork desite its secodary role. I the iterest of sace, we maily focus o the aalysis of the secodary etwork i this aer with all the roof omitted. The reader is referred to [10] for more details. IV. THROUGHPUT ANALYSIS OF THE SECONDARY NETWORK I this sectio, we focus o the aalysis of the er-ode throughut of the secodary rotocol, which is our rimary cocer. We first show how to costruct a highway with cosideratio of reservatio regios. The we evaluate the oe-ho data rate achievable i each of the three routig hases, based o which we obtai the er-ode throughut. A. Highway costructio We describe the aroach to build the highway i a horizotal corridor, which alies almost verbatim to a vertical corridor as well. As we metioed, each (horizotal) corridor is artitioed ito diamod cells of side legth a s. We call such a cell oe (see Fig. (5)) if both of the followig two evets hae: 1) E 1 : there is at least oe secodary ode i the cell; 2) E 2 : there is o rimary ode i either this cell or its 8 eighborig cells; this evet differetiates the CR etwork with a stad-aloe etwork studied i [3]. Otherwise we call it closed. Accordig to the roerties of P.P.P., the robabilities of the two evets are give by: 1 P (E 1 ) = 1 e c2 s, 2 P (E 2 ) = e 9 m c2 s. 4 The outage robability is defied as the ercetage of the secodary S-D airs which ca ot be served. Fig. 6: A routig ath Due to the ideedece of E 1 ad E 2, a cell is oe with robability 1 2 = (1 e c2 s )e 9 m c2 s. (8) Some iterestig observatios o are i order: 1) The value of deeds o c s ad the secodary desity m/, both of which are costats ad irrelevat to the scalig law study. 2) If the assumtio m = β (β > 1) i [6] ad [7] is imosed here the robability of E roaches to 1 asymtotically, ad the two overlaid etworks essetially decoule. 3) The state of a secodary cell (oe or closed) is deedet o the states of its eighborig cells, sice oe rimary ode ca cause 9 secodary cells closed. I cotrast, the cell states i a stad-aloe etwork such as the oe cosidered i [9] are ideedet from each other. A key ste i highway costructio is to fid (or esure the existece of) disjoit aths across the etwork. For this urose we ma each of the tesselated secodary corridor ito the bod ercolatio model o a r log r grid G r, where r = = m 2cs. Edges of the grid are comosed of horizotal diagoals of some cells ad vertical diagoals of the other cells. The grid G r is give i Fig. 4, where the dashed lies rereset the tessellatio of the corridor ad the solid lies rereset the grid. The states (oe or closed) of the edges are the same as their corresodig cells, thus they are ot ideedet either. For a horizotal corridor, a L(eft)R(ight) oe ath corresods to a sequece of coected oe edges o the grid, formally defied below. Defiitio 2: I a 2-D grid G [0, ] [0, log ], let C v = (x v, y v ) be the coordiates of a vertex v. If there exist a series of vertices v 1, v 2,..., v m, such that there is a oe edge coectig two cosecutive vertices ad 1 i < m 0 x vi+1 x vi 1, x v1 = 1 ad x vm =, the the ath cosistig of these edges is called a LR oe ath. For each oe edge, there is at least oe secodary ode located i its corresodig cell which stays outside reservatio regios. Therefore, a LR oe ath ca be maed back to a routig ath i the etwork. A LR oe ath ad its corresodig routig ath are deicted i Fig. 6, where the dark solid lie is a LR oe ath ad the dark dashed lie is a actual routig ath. We show i the ext theorem that with deedet edges, there are Θ(log m) aths i a horizotal corridor. The same
routig ath secodary ode Fig. 7: Slabs i a horizotal corridor coclusio holds for a vertical corridor. Theorem 3: With large eough, there are Θ(log m) disjoit LR oe aths i each horizotal corridor. The roof follows the same lie of [9]. Cautio should be take to deal with the situatio that the states of cells are deedet (due to the itroductio of reservatio regios). A horizotal (vertical) corridor is further evely divided ito multile slabs with size h (h ), where h = log( /( 2a s )) Θ(log m) is a costat. The value of h guaratees that the umber of slabs are as may as the oe aths i a corridor so that the odes i each of the slabs share exactly oe ath. The slabs i a horizotal corridor are show i Fig. 7, where there are 5 slabs corresodig to 5 routig aths. All the odes i a cell share oe access oit, which is the closest ode located o their corresodig horizotal ath. Ad i the access hase, source odes load data to their access oits through a sigle ho with legth at most log( /( 2a s )) (the width of the corridor). The oeratios i the delivery hase follows a similar way. Oe ho i a highway has to suort the traffic of all the odes i a slab sice all the odes i a slab drive their data to the same ath. B. Throughut aalysis We first calculate the umber of odes i a cell ad a slab, ad the evaluate the throughut of oe ho durig each hase. Lemma 1: The umber of odes i a cell is at most log m, ad the umber of odes i a slab is at most 2h m/ m, both w.h... The followig two lemmas show that oe ho o the highway ca suort a costat rate ad the throughut for a access (delivery) lik scales as Ω(1/(log m) α ). Lemma 2: O the highway of the secodary etwork, each secodary cell ca suort traffic with a costat rate K s, where K s > 0 is ideedet of m. Lemma 3: The throughut for a access (delivery) lik scales as Ω(1/(log m) α ). Durig the access ad delivery hase, each lik suorts Ω(1/(log m) α ) data rate (Lemma 3) ad log m odes (Lemma 1) share it. Therefore, the er-ode throughut durig these two hases scales as Ω(1/(log m) α+1 ). Ad o the highway oe ho ca suort a costat data rate (Lemma 2) ad there are at most 2h m/ m odes (Lemma 1) sharig the same ho. Therefore the er-ode throughut o the highway scales as Θ(1/ m). Cosiderig the throughut of all the hases we reach the coclusio of Theorem 2. h C. Outage aalysis Accordig to the secodary rotocol, outage occurs ievitably sice ay S-D airs comosed of the secodary odes i the reservatio regios ca ot be served. Deote by m u the umber of secodary odes i all the reservatio regios, which is give by m u 2(1 + ϵ)9 m c2 s m, w.h.., due to the fact that the total area of reservatio regios is bouded by (1 + ϵ)9 m c2 s. Therefore, the outage robability P o ca be calculated as follows: P o = m u the umber of secodary odes 2(1 + ϵ)9 m c2 s m (1 ϵ)m = 18(1 + ϵ)c2 s (1 ϵ)m δ, where δ < 1 is a costat deedig o /m. Therefore, higher secodary desity m/ leads to lower outage robability. Note that if the higher order coditio m = β (β > 1) is alied, the outage robability is vaishig, w.h.., which coicides with the coclusio i [6]. V. CONCLUSIONS I this aer, we have studied the throughut of a tye of heterogeeous etworks cosistig of a rimary etwork of size ad a cogitive radio ad hoc etwork of size m uder a more ractical model. We show that the er-ode throughut of the rimary etwork scales as Θ(1/ ) ad there is ideed o erformace loss, i terms of scalig law, with the coexistece of the CR etwork. The CR etwork ca also achieve erode throughut Θ(1/ m), erformig as well as a stadaloe etwork excet sufferig from a o-vaishig outage robability. REFERENCES [1] I. F. Akyildiz, W. Lee, ad K. R. Chowdhury, CRAHNs: Cogitive radio ad hoc etworks, Ad Hoc Networks, vol. 7,. 810 836, July 2009. [2] P. Guta ad P. Kumar, The caacity of wireless etworks, IEEE Tras. Iform. Theory, vol. 46, o. 2, 2000. [3] N. Devroye, P. Mitra, ad V. Tarokh, Achievable rates i cogitive radio chaels, IEEE Tras. Ifor. Theory, vol. 52,. 1813 1827, May 2006. [4] A. Jovicic ad P. Viswaath, Cogitive radio: A iformatio-theoretic ersective, IEEE Tras. Ifor. Theory, vol. 55,. 3945 3958, Se. 2006. [5] M. Vu, N. Devroye, M. Sharif, ad V. Tarokh, Scalig laws of cogitive etworks, i Proc. Crowcom, Aug. 2007. [6] S. Jeo, N. Devroye, M. Vu, S. Chug, ad V. Tarokh, Cogitive etworks achieve throughut scalig of a homogeeous etwork, IEEE Tras. Ifo. Theory, 2009, submitted. Available at htt://arxiv.org/abs/0801.0938. [7] C. Yi, L. Gao, ad S. Cui, Scalig laws for overlaid wireless etworks: A cogitive radio etwork vs. a rimary etwork, IEEE Tras. Networkig, vol. 18, o. 4, Agu. 2010. [8] O. Dousse ad P. Thira, Coectivity vs caacity i dese ad hoc etworks, i Proc. IEEE INFOCOM, 2004. [9] M. Fraceschetti, O. Dousse, D. N. C. Tse, ad P. Thira, Closig the ga i the caacity of wireless etworks via ercolatio theory, IEEE Tras. Ifo. Theory,. 1009 1018, Mar. 2007. [10] C. Li ad H. Dai, O the throughut scalig of cogitive radio ad hoc etworks, Techical Reort, 2011.