Table Of Content

Energy-Efficient Spectrum Sharing in Relay-Assisted Cognitive Radio Systems (Invited Paper) Mariem Mhiri∗, Karim Cheikhrouhou∗, Abdelaziz Samet∗, François Me´riaux† and Samson Lasaulce† ∗Tunisia Polytechnic School P.B. 743-2078, University of Carthage, La Marsa, Tunisia {mariem.mhiri, karim.cheikhrouhou}@gmail.com, [email protected] †L2S - CNRS - SUPELEC, F-91192 Gif-sur-Yvette, University of Paris-Sud, France {meriaux, lasaulce}@lss.supelec.fr 3 1 0 2 Abstract—Thiswork characterizes an important solution con- anothersolution concept for this spectrum allocation game of cept of a relevant spectrum game. Two energy-efficient sources interest namely, the Nash bargaining solution (NBS), moti- n communicating with their respective destination compete for an vated by the need to design efficient solutions in distributed a extra channel brought by a relay charging the used bandwidth J wireless networks. Remarkably, such a solution exists for the through a pricing mechanism. This game is shown to possess 8 a unique Nash bargaining solution, exploiting a time-sharing considered scenario, is unique, and can be implemented in 2 argument. This Pareto-efficient solution can be implemented a decentralized manner according to the conjugate gradient by using a distributed optimization algorithm for which each algorithm. Thisconfirmstherelevanceofthisapproachwhich ] transmitterusesasimplegradient-typealgorithmandalternately T has also been adopted in other contexts such as [3] (NBS for updates its spectrum sharing policy. Typical numerical results I show to what extent spectral efficiency can be improved in a powerallocationgameswheretransmissionratesareoptimized . s system involving selfish energy-efficient sources. with no relay and pricing), [4] (wireless sensors are energy- c Keywords: resource allocation, cognitive radio, cooperative efficiently coordinated by the Raiffa-Kalai-Smorodinsky so- [ transmission, Nash bargaining solution. lution to communicate with a unique fusion center), or [5] 1 (multiple access channels without pricing are considered). v I. INTRODUCTION Compared to these references, the present work makes a step 7 Designing spectrally efficient communication systems has towards implementing an efficient solution in a decentralized 8 always been, and still is, a critical issue in wireless networks. manner, which is known to be a challenging task [6][7]. The 7 6 Theneedforenergy-efficientterminalsatboththemobileand algorithm proposed in this paper is decentralized in the sense . fixed infrastructure sides is more recent but becomes stronger of the decision but not in terms of channel state information 1 0 and stronger. This paper precisely considers both aspects. (CSI),leavingthisissueasanon-trivialextensionofthiswork. 3 More specifically, the main goal is to determine an energy- This paper is structured as follows. In the next section, we 1 efficientoperatingpointof a givendistributed communication introducethe systemmodelaswellasthespectrumallocation : v system at which the spectrum is efficiently used. We do game. In section III, we analyze the NBS and present the i not pretend to solve this tough issue for general distributed decentralized algorithm. In section IV, numerical results are X multiuserchannels. Rather,weshowthatitispossibletofully presentedand discussed. Concludingremarksare proposedin r a determine such an operating point in one possible scenario section V. which has already been considered in the literature [1]. This II. PROBLEM STATEMENT scenario is as follows. We consider an initial communication A. System model system comprising two point-to-point communications which useorthogonalchannels(sayinthefrequencydomain);ahalf- The communication system under study is represented in duplexrelay using a dedicated bandis added to the system in Fig. 1. Source/transmitter i 1,2 sends a signal √pixi ∈ { } order to help the two transmitters to improve their energy- with power p over two quasi-static (block fading) links: the i efficiency; the relay implements a pricing mechanism which link from source i to destination i whose channel gain is is directly related to the amount of band used for relaying. h C and the one from the source i to the relay r whose ii Theenergy-efficiencymetricunderconsiderationisa quantity chan∈nel gain is h C. The total band associated with ir ∈ inbitcorrectlydecodedperJouleandisdefinedasin[2]. The those two links is ω and the extra band allocated by source situation where each transmitter aims at selfishly maximizing i to communicate with the relay is denoted by ω [0,ω]. i ∈ its individual energy-efficiency (with pricing) by allocating The extrabandavailable is precisely thatofferedbythe relay. bandwidth on the extra channel on which the relay operates The relay operates in a half-duplex mode and is assumed to has been considered in [1] ; the solution concept considered implement an amplify-and-forward (AF) protocol. Time is therein is the Nash equilibrium (NE), which is shown to dividedintoblocksonwhichallchannelgainsareassumedto be unique but not Pareto efficient. Our goal is to consider be fixed. Each block is divided into two sub-blocks[8]. Over the first sub-block (first phase), only the source can transmit B. Strategic form of the spectrum allocation game andthesignalsreceivedbythedestinationandrelaynodesare As motivated in [1], the spectrum allocation problem can given by: be modeled by a strategic form game (see e.g., [7]). yii = hii√pixi+nii, (1) Definition 1: The game is defined by the ordered triplet (cid:26) yir = hir√pixi+nir, = ,( i)i∈K,(ui)i∈K where G K S where nii and nir are (complex) additive white Gaussian • (cid:0) isthesetofplayer(cid:1)s. Here, theplayersofthegameare noises (AWGN) with mean 0 and variance σ2. Following the tKhe two sources/transmitters, = 1,2 ; relevant choice of [8], only the relay is assumed to transmit • i is the set of actions/strateKgies.{Here}, the strategy of over the second sub-block (second phase): Ssource/transmittericonsistsinchoosingω initsstrategy i set =[0,ω]; yri =hri√prxri+nri, (2) • ui iSsithe utility function of each user. It is given by: where h C is the channel gain between the relay and ri ∈ (ω ω ) ω destinationi,nri ∼N(0,σ2),andxriisthetransmittedsignal ui(ω1,ω2) = α −p i f(γi,i)+α(p +ip )f(γiA,iF) fromtherelaytodestinationiand,undertheassumptionmade i i r in terms of relaying protocol, expresses as: 2 y b ωjωi, (6) xri = ir . (3) − Xj=1 yir   | | where α defines the spectral efficiency (in bit/s per Hz), f : Inthefirsttransmissionphase,thesignal-to-noiseratio(SNR) [0,+ ) [0,1] isa sigmoidalefficiencyfunctionwhichcan associatedwiththesource-destinationchannelimerelywrites ∞ → correspondto the packet success rate or probabilityof having as: p h 2 nooutage(emphasizingthelinkbetweenenergy-efficiencyand γ = i| ii| . (4) i,i σ2 the outage analysis conducted in [8]). The parameter b is a (linear) pricing factor. The SNR associated with the second transmission phase is As explained in [1], the presence of the factor b 0 amounts given by [1], [8]: ≥ toimposingacosttothesourcesforusingtherelay;thiscostis p p h 2 h 2 assumedtobeproportionaltotherelayingbandused. Thefirst γ = i r| ir| | ri| . (5) r,i σ2(p h 2+p h 2+σ2) term of the utility function corresponds to the ratio between i ir r ri | | | | the goodput (net rate in bit/s) to the cost in terms of power Interestingly,asprovenby[8],whenusingmaximal-ratiocom- (in J/s) for the direct link alone (whose band width equals bining, the equivalent SNR corresponding to the AF protocol ω ω ),whereasthesecondtermcorrespondstotheaggregated can be written in a simple form if the outage probability is − i effectsof the directtransmissionand the relayedtransmission the metric of interest. This writes as: γAF =γ +γ . i,i i,i r,i (whose bandwidth equals ω ). This game is concave in the i sense of Rosen and has a unique pure NE (see e.g., [7]). The main problem is that the NE can be very inefficient as there exist some operating points at which both transmitters havebetterutilities. Thismotivatesthestudyofmoreefficient solutions such as the NBS. The NBS analysis, the design of a simple distributed optimization algorithm to implement it, and proving its relevance in terms of performance constitute the main results of this paper. (a)Directtransmission. III. NASH BARGAINING SOLUTIONANALYSIS The objective of this section is to characterize the NBS of the game and to propose a simple distributed optimization G algorithm for implementation since the function of interest to optimize can be checked to be strictly concave under certain operation conditions explicated in Sec. III-D. A. Achievable utility region: Pareto boundary and convexity First, we study the properties of the achievable or feasible utility region, which is denoted by . It is defined as the (b)Cooperative transmission. R regionformedbyallthepointswhosecoordinatesare (u ,u ) 1 2 Figure 1: System model. that is: = (u ,u ) (ω ,ω ) [0,ω]2 . (7) 1 2 1 2 R { | ∈ } Foragivenchannelconfigurationorblockofdata(i.e.,theh Proof: The point uNE,uNE defines a threat point ij 1 2 are given), the region is compact [3], which follows from and can always be reach(cid:0)ed, which(cid:1)ensures the existence of R thecompactnessof andthecontinuityofu . However,itis a solution to the above maximization problem. i i S not always convex. This prevents one from using bargaining Regarding to the uniqueness of the NBS, Nash proved that theorywhichisbasedontheconvexityoftheachievableutility it holds under certain axioms due to the existence of the region. Itturnsoutthat,intheproblemunderconsideration,it convex hull of the achievable region and the threat point, as isrelevanttoexploittime-sharing(asdonein[3]forShannon- mentioned in [3]. As we have shown that the utility region rate efficient allocation games on the interference channel), can be convexified, this solution is also unique. which convexifies the utility region. Indeed, the main idea Finally, the cooperative outcome (NBS) must be invariant is to assume that coordination in time is part of the sought to equivalent utility representations, symmetric, independent solution. The new utility region is: of irrelevant alternatives and Pareto efficient [9]. The NBS is thereforetheuniquesolutionresultingfromtheintersectionof R¯ ={(µu1+(1−µ)u′1,µu2+(1−µ)u′2) (8) thePareto boundary ¯∗ with the Nashcurvewhichis defined ′ ′ R |0≤µ≤1, (u1,u2)∈R, (u1,u2)∈R}. as (Fig. 2): Duringa fractionµofthetime,theusersuse (ω ,ω )tohave (u ,u )=arg max π(u ,u ) (11) 1 2 1 2 1 2 (u ,u ). Duringafraction(1 µ)ofthetime,theyuseanother (u1,u2) 1 2 combination of bandwidths (−ω′,ω′) to obtain (u′,u′). Note whereπ(u ,u )=(u uNE)(u uNE)istheNashproduct 1 2 1 2 1 2 1− 1 2− 2 that this region includes several points of interest. First, it function. includes uNE,uNE which is the pointcorrespondingto the Since the NBS determination is on the subregion ¯+, we 1 2 R unique pu(cid:0)re NE of (cid:1). Second, it includes the two points for stress that the utilities arising from the NBS are higher than G which the sources or transmitters do not exploit the relay at those deduced with NE (see Fig. 2). all: ω =0 or ω =0. Let ¯∗ be the Pareto boundaryof the conve1x hull ¯. F2ig. 2 illustRrates different operating points as C. Decentralized algorithm for the NBS determination wellastheacRhievableutilityregion andtheParetoboundary Theproposedalgorithmisbasedontheideaofdetermining ¯∗. The otherelementsshown in thRis figure are defined next. analyticallythe uniquemaximum,which is the NBS fromthe R resolution of the following system of equations: ∂π = 0, (I) ∂ω1 (12)  ∂π = 0. ∂ω 2 Mathematical resolution of such a system leads to solve two second degree polynomials in ω (for i 1,2 ), the i ∈ { } discriminants of which are fourth degree polynomials in ω j (for j 1,2 i ). The study of signs of the discriminants ∈{ }\{ } show that expressing the NBS analytically is a difficult task even by exploiting Ferrari and Cardan methods for high degree polynomials resolution. This study shows the interest in: (1) finding decentralized algorithms to compute the NBS; (2) designing distributed procedures to converge towards the NBS. The scope of this paper is about (1) and (2) but with Figure 2: The achievable utility region plus some key the restriction that distribution is only performed in terms operating points. of decision and not in terms of channel state information. Instead of determining the maximum of π, we propose to find the minimum of π, denoted after as π , by focusing m − on the conjugate gradient algorithm. One of the steps of this B. Existence and uniqueness analysis of the NBS algorithm consists in determining a parameter denoted as The NBS can be characterized as follows: β (which is defined next). Accordingly, many methods k+1 Proposition 1: In the spectrum allocation game , there have been introducedsuch as: Fletcher-Reeves, Polak-Ribie`re G exists a unique NBS given by: andHestenes-Stiefel. Duetotheefficiencyofitsconvergence, we focus here on the second method based on calculating the (cid:0)uN1 BS,uN2 BS(cid:1)=(u1,mu2a)∈xR¯+(u1−uN1 E)(u2−uN2 E), (9) Pisoluapkd-Ratiebdie`rien paanraamlteetrenrat[i1n0g].mTanhneers,pejuctsrtumlikeshathriengiteproaltiicvye where sequential iterative water-filling algorithm [11]. However, in contrast with the latter, only the decision is distributed here R¯+ ={(u1,u2)|u1 ≥uN1 E, u2 ≥uN2 E}. (10) and global channel state information is needed (through the Hessian matrix). with ϕ = αf(γ )/p and ψ = αf(γAF)/(p + p ) for i i,i i i i,i i r i 1,2 . Therefore, the eigenvalues are the zeros of the Algorithm 1: Decentralized determination of the NBS ∈ { } following polynomial: (1) Set the position of the relay (2) ω0 =(ω0,ω0) (frequency initialization) P :λ2 λ tr(A)+det(A). (14) 1 2 − (3) v = π (ω0) (initialization gradient) 0 −∇ m If we denote∆ the discriminantof the polynomialP, we can (4) k=0; while v >ǫ k kk verify merely that ∆ is always positive. Indeed, we have: gtv a. t = k k (optimal parameter with Newton method k −vktAmvk ∆=(a11−a22)2+4a12a21. (15) where g = π (ωk), A is the Hessian matrix of π and ωk =(ωkk,ω∇k) ims the freqmuency bands at the kth iteratimon) Since a12 =a21, equation (15) is equivalent to: 1 2 b. ωk+1 =ωk+t v (new frequency bands) ∆=(a a )2+4(a )2 0. (16) k k 11 22 12 − ≥ c. ωk+1 =ωk+1(1) and ωk+1 =ωk(2) (alternated updates) 1 2 Therefore,theeigenvaluesofA,denotedasλ andλ arereal d. g = π (ωk+1) (new gradient) 1 2 k+1 ∇ m and are as follows: gt (g g ) e. βk+1 = k+1 gkkt+g1k− k (Polak-Ribie`re parameter) λ1 = tr(A)−√∆, f. v = g +β v (new descent direction)  2 (17) k =kk++11 − k+1 k+1 k  λ = tr(A)+√∆. 2 end 2 (5) (ωNBS,ωNBS)=(ωFI(1),ωFI(2)) where FI denotes From these expressions, we study the the strict negativity of 1 2 Final Iteration the eigenvalues depending on the relay position in a space region [0,700] [0,700] m2. The corresponding simulations × D. Convergenceofthe algorithm(Strict-concavityanalysis of (for the same settings considered in section IV) are given in the π function) Fig. 3 in which we represent in black the region where the eigenvaluesarestrictlynegative. Weassumeastandardchoice The proposed algorithm is ensured to converge to a NBS forf forallthenumericalresultsprovidedinthispaperwhich if π is strictly concave. But this property is not always true. isf(x)=(1 e−x/2)M [2]whereM isthenumberofsymbols According to the previous study, the function π is defined − perpacket. Therefore,we can deducethatthe strict concavity on the subregion ¯+ which is formed by all the utilities (u1,u2) verifying Rui ≥ uNi E for all i ∈ {1,2}. Such a set [o4f0f0u,n5c5t0io]nmπ2.is ensured in a region (xr,yr) ∈ [400,550]× can be determined when the NE point is fixed. Though, for each channels values, a NE can be identified. Consequently, fthoelloswubinrgeg,ifoonr gR¯iv+endleopceantidosnsonoftshoeurccheasnannedlsdveastliuneast.ionIns,twhee 70 D1 D2 show that there exists a region in which the π function is 60 strictly concave. 50 Rsc Proving the strict-concavity of π amounts to proving the strict-negativity of the eigenvalues of its corresponding Hes- /1040 sian matrix, which is given by: yr 30 S1 A= a11 a12 , (13) 20 S2 (cid:18) a21 a22 (cid:19) 10 where: 10 20 30 40 50 60 70 ∂2π xr/10 a = 11 ∂ω2 Figure 3: Strict-concavity of the π function on the disk sc 1 R = 2b(u uNE) 2bω ( ϕ +ψ b(2ω +ω )), when both eigenvalues are strictly negative. − 2− 2 − 2 − 1 1− 1 2 ∂2π a = 22 ∂ω2 = 2b2(u uNE) 2bω ( ϕ +ψ b(2ω +ω )), IV. NUMERICAL RESULTS −∂2π1− 1 − 1 − 2 2− 2 1 Here, we implement the NBS and compare it to the NE a = 12 [1]. We consider a scenario where the coordinates (in me- ∂ω ∂ω 1 2 = −b(u2−uN2 E)−b(u1−uN1 E)+b2ω1ω2+ ter) of each source/destination nodes Si/Di are as follows: (ϕ1 ψ1+b(2ω1+ω2))(ϕ2 ψ2+b(2ω2+ω1)), S1(300,300), D1(500,645), S2(390,257) and D2(590,603). ∂2−π − The channel gains hij 2 are given by 0.097/d4 where d is a = | | 21 ∂ω ∂ω the distance between the transmitter and the receiver. The 1 2 = a , noise power and transmission powers of the users and relay 12 are 10−13 Watt, 0.1 Watt and 0.08 Watt respectively, and α 14 is set to 0.8 bit/s per Hz. The constants b and M are set to yr=400m 10−5 and 80 respectively,while the bandwidthω is fixed to 1 12 yr=500m MHz. [%] yr=550m E 10 yr=600m N W yr=650m S 60 yr=400m NEW)/ 8 yr=500m −S 6 50 yr=550m NB 4 NEω[%]140 yyrr==660500mm (SW 2 NBω)/130 105 0 200 250 300 350 400 450 500 550 600 650 − x-coordinateoftherelay(xr)[m] NE(ω120 Figure 6: Gains in terms of sum energy-efficiencywith 10 pricing (social welfare) by operating at the NBS instead of the NE. 0 150 200 250 300 350 400 450 500 550 600 650 x-coordinateoftherelay(xr)[m] Figure 4: Gains in terms of individual bandwidth for user 1 In Fig. 5, we plot the gain in terms of total bandwidth when operating at the NBS instead of the NE. demand. Thus, we deduce that a maximum gain with NBS is reachedwhentherelayispositionedat(x ,y ) [400,550] r r ∈ × [400,550] m2. In this region, the total bandwidth demand is reduced to 20 25%. The study of the social welfare in Fig. − 30 6 confirmsthat the maximumenergy-efficiency(with pricing) yr=400m gain,whichisaround10 12%,isreachedinthesameregion. yr=500m − 25 Consequently, the results obtained according to the strict- yr=550m %] yr=600m concavityanalysisare well confirmedwhenimplementingthe [20 E yr=650m conjugate gradient algorithm. N ω B)/15 Nω V. CONCLUSION − NE 10 This paper studies an efficient solution for a relevant game ω ( introducedin[1]byreferringtotheNBS.Remarkably,uptoa 5 time-sharingargument,the correspondingspectrumallocation game can be checked to possess all the properties to have a 0 150 200 250 300 350 400 450 500 550 600 650 unique NBS. Through implementing a conjugate gradient al- x-coordinateoftherelay(xr)[m] gorithminadecentralizedway,considerablegainsof20 25% Figure 5: Gains in terms of system/total bandwidth by − can be obtained in terms of used bandwidth. The results operating at the NBS instead of the NE. reached for the two-user case are very encouraging to extend the case study to larger multi-user systems. Interestingly, Our results highlight that the NBS requires less bandwidth our analysis gives some insights into how to deploy some thantheNE.Additionally,theenergy-efficiency(withpricing) relaysforimprovingadistributednetworkbothfromaspectral at the NBS is higher than the one at the NE. In Fig. 4, and energy standpoint. This paper is a first step towards we represent the relative bandwidth gain (NBS vs NE) in designing fully distributed algorithms (in terms of channel % of user 1 w.r.t. the coordinates of the relay (the relative stateinformation)orlearningtechniqueswhichconvergetoef- gain of user 2 shows a similar behavior). Simulations show ficientsolutionssuchastheNBSorRaiffa-Kalai-Smorodinsky that maximum gains are obtained when y [400,550] m. solution ; this task is known to be challenging and this paper r ∈ Moreover,fordifferenty ,therearesomeregionsofx where shows the existence of relevant wireless scenarios where this r r the gains vanish. In these regions, the optimum bandwidth objective might be reachable. with NBS is equalto thatwith NE. Since user1 cannotprofit from the presence of the relay (when this latter is far from REFERENCES the source), we have ωNE = ωNBS = 0. However, user 2 1 1 [1] L. Cong, L. Zhao, K. Yang, H. Zhang, and G. Zhang, “A Stackelberg maximizesitsutilityattheNE(whentherelayismuchcloser game for resource allocation in multiuser cooperative transmission to its location) and we have ωNE = ωNBS = 0. This shows networks,” Wireless Communications and Mobile Computing, vol. 11, 2 2 6 pp.129–141, Jan.2010. that our analysis provides some insights to an operator who [2] D.J.GoodmanandN.B.Mandayam,“Powercontrolforwirelessdata,” would like to optimize the location of a relay. IEEEPersonalCommunications, vol.7,pp.48–54,Apr.2000. [3] E. G. Larsson and E. A. Jorswieck, “Competition versus cooperation [7] S. Lasaulce and H. Tembine, Game Theory and Learning for Wireless ontheMISOinterferencechannel,”IEEEJournalonSelectedAreasin Networks: FundamentalsandApplications. AcademicPress,Elsevier, Communications, vol.26,pp.1059–1069, Sep.2008. Jul.2011. [4] K.Ma,X.Guan,B.Zhao,andJ.Wang,“Acooperation strategybased [8] J.N.Laneman,D.N.C.Tse,andG.W.Wornell,“Cooperativediversity onbargainingsolutioninwirelesssensornetworks,” JournalofControl in wireless networks: Efficient protocols and outage behavior,” IEEE TheoryandApplications, vol.9,no.1,pp.121–126,2011. TransactionsonInformationTheory,vol.50,pp.3062–3080,Dec.2004. [9] W. Thomson and T. Lensberg, Axiomatic theory of bargaining with a [5] Z. Zhang, J. Shi, H.-H. Chen, M. Guizani, and P. Qiu, “A cooper- variable numberofagents. CambridgeUniversity Press,1989. ation strategy based on nash bargaining solution in cooperative relay [10] E. Polak, Computational Methods in Optimization. Academic Press, networks,” IEEETransactions onVehicular Technology, vol.57,no.4, 1971. pp.2570–2577, Jul.2008. [11] W.Yu,G.Ginis,andJ.Cioffi,“Distributedmultiuserpowercontrolfor [6] S.Lasaulce,M.Debbah,andE.Altman,“Methodologies foranalyzing digital subscriberlines,” IEEEJournal onSelected AreasinCommuni- equilibriainwirelessgames,”IEEESignalProcessingMagazine,Special cations, vol.20,no.5,pp.1105–1115, Jun.2002. issueonGameTheoryforSignalProcessing,vol.26,no.5,pp.41–52, Sep.2009. Prob[λ1 <0] Prob[λ2 <0] 100 100 70 70 D1 D1 90 90 D2 D2 60 60 80 80 50 70 50 70 R R 1 2 60 60 40 40 0 0 1 1 / 50 / 50 r r y y 30 30 40 40 S1 S1 S2 30 S2 30 20 20 20 20 10 10 10 10 0 0 20 40 60 20 40 60 xr/10 xr/10