DUE to the global aspiration for energy saving and reduction. Relay Power Allocation in Auction-based Game Approach

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1 Globecom 0 - Communcatons QoS, Relablty and Modellng Symposum Relay Power Allocaton n Aucton-based Game Approach Dan Wu, Yuemng Ca, Lang Zhou, and Joel Rodrgues Insttute of Communcatons Engneerng, PLA Unversty of Scence and Technology, Nanjng, Chna. The Key Lab of Broadband Wreless Communcaton and Sensor Network Technology Nanjng Unversty of Posts and Telecommuncatons, Mnstry of Educaton, Nanjng, Chna. Insttuto de Telecomuncaçőes, Unversty of Bera Interor, Portugal. Abstract In ths work, wth respect to the uncertanty about the ndvdual nformaton, we nvestgate the relay power allocaton problem from the energy-effcent, Pareto optmal, and compettve farness perspectve. At frst, we desgn an easymplementaton energy effcency metrc, whch ams at strkng a balance between the QoS provsonng and the energy consumpton. Then, an aucton mechansm s proposed for relay power allocaton. By transferrng the aucton mechansm nto a game, we prove the exstence, unqueness, and Pareto optmalty of the Nash equlbrum NE for our aucton game, and show that the allocaton strateges from the NE can acheve the energy effcency n terms of the proposed metrc. Next, we develop a dstrbuted relay power allocaton algorthm based on our bestresponse functons to reach the Pareto optmal NE. Importantly, we not only certfy the convergence of the proposed algorthm, but also provde quanttatve analyss on t. Extensve smulatons results are conducted to confrm the valdty of the analytcal results. Index Terms Wreless cooperatve networks, aucton theory, energy effcency, power allocaton, outage probablty. I. INTRODUCTION A. Motvaton and Background DUE to the global aspraton for energy savng and reducton of carbon footprnt, the energy-effcent communcaton s becomng a crucal trend []. In partcular, cooperatve communcaton technque has been wdely consdered as a promsng approach to acheve energy effcency []. In order to take full advantage of the energy savng potental, some ssues deserve further consderaton: Energy effcency metrcs play an mportant role n comparng and assessng the energy consumpton of varous components and overall networks. Snce future wreless networks wll support a varety of servces wth heterogeneous QoS requrements, t s expected to seek a refned energy effcency metrc to provde a comprehensve evaluaton of energy savngs and general QoS performance. Resource allocaton and coordnaton among the cooperatve members become an mportant guarantee for realzng the energy-effcent advantages. Gven a fxed relay, the paramount ssue s how to dstrbute the energy resource of the relay to the nodes n need, even n a dstrbuted and ncompletenformaton manner. For a multple node-pars relay network model, where each relay s delegated to assst one or more source-destnaton pars, multple source-destnaton pars share rado resources from the same relay, and thus, effcent resource utlzaton seems partcularly mportant. A typcal example of such scenaros s the deployment of few relays n ad hoc networks or the transmssons between the hot-spot nodes located at the edge of the cell [3]. Recently, aucton-theoretc approaches to resource allocaton problems have been explored n wreless cooperatve networks [4], [5]. The aucton mechansm s benefcal to allocatng avalable resources among multple bdders n a dstrbuted way, wth uncertanty about the ndvdual nformaton, and achevng compettve farness. In ths work, we propose a relay power allocaton scheme for multple nodepars relay wreless networks by usng aucton mechansm. The man contrbutons are three-fold: We derve a refned energy effcency metrc, whch s characterzed by easy mplementaton and adequately shows the relatonshp between the QoS provsonng and the energy consumpton. We desgn an aucton mechansm for relay power allocaton by modelng the source-destnaton pars as bdders, the relay as an auctoneer, and the avalable energy of relay as a traded resource. Then, we treat the proposed aucton mechansm as an aucton game, and prove the exstence, unqueness and Pareto optmalty of the Nash equlbrum NE. Accordng to the proposed energy effcency metrc, the resultng NE can yeld the energy-effcent relay power allocaton strateges. 3 We propose a dstrbuted algorthm based on bestresponse functons to reach the Pareto optmal NE of our aucton game. Moreover, we analyze the convergence and provde the upper bound of the convergence. Interestngly, the convergence s only dependent on the number of sourcedestnaton pars and ndependent of the ntal bddng strateges and the avalable energy of the relay. II. NETWORK MODEL AND PROBLEM FORMULATION A. Network Model We consder a multple node-pars relay wreless network, where the source node s communcates wth ts correspondng destnaton node d, =,..., N, and one relay r s avalable to assst the data transmssons from N source-destnaton 830

2 pars. Orthogonal transmssons are assumed among dfferent nodes for smultaneous communcatons of all sourcedestnaton pars. Takng the node-par for example, the cooperatve communcaton scheme s carred out by two phases, namely a local transmsson and a long-haul cooperatve transmsson. For both phases, the channels are modeled by a path loss exponent δ and frequency flat Raylegh fadng. More precsely, h s,r CN 0,, h s,d CN 0,, and h r,d CN 0, are untary power, Raylegh fadng channel coeffcents. Wthout loss of generalty, each node has the same Gaussan nose varance σ. Here, we adopt the outage performance as a QoS measure. In order to guarantee the basc QoS requrement, the outage probablty P out, should not be larger than the threshold value Pout, thr, under the fxed outage capacty C out,. Durng Phase, there exsts the channel recprocty between s and r, and the relay r s expected to cooperate durng Phase. Hence, settng the rate of Phase to C out,, the capacty regon of Phase s constraned to: log + p s,r σ l local δ hs,r C out,, where p s,r s the transmt power, and l local represents the local dstance between the source node s and the relay r, respectvely. Moreover, we normalze a unt of tme scale to, and derve the energy consumpton for Phase as E = p s,r + p ct + p cr /, where p ct and p cr denote the transmtter and the recever crcut power, respectvely. For smplcty, all nodes have the same p ct and p cr. Specfcally, the part of s s p s,r + p ct /, and the part of r s p cr /. Durng Phase, there s no channel recprocty, and thus, s and r jontly transmt data to d based on the Alamout codes. Then, each transmttng member has the same transmt power [6], namely p s,d = p r,d =. Hence, the energy consumpton for Phase s formulated as E = + p ct, where the part of s s p s,d + p ct /, and the part of r s p r,d + p ct /. From [6], the outage probablty s C out, σ l δ P out, = Γ,, where Γ, m = m 0 xe x dx, and l s the dstance of the th long-haul cooperatve transmsson lnk. B. Desgn of Energy Effcency Metrc As we know, the bt-per-energy effcency s a wdely-used metrc n terms of the energy effcency desgn [], and well captures the tradeoff between the throughput and the energy consumpton. However, throughput s not an approprate measure for all the communcaton networks, n partcular for multmeda communcatons [7]. Ths motves us to propose a refned energy effcency metrc to precsely capture the characterstcs of the applcaton-drven transmsson system. In ths regard, we formulate the rato of QoS provsonng to the energy costs as our energy effcency metrc. As outage performance s related to large scale channel fadng and statstcal small scale fadng, t s approprate for the model descrpton whch lacks nstantaneous channel knowledge. Therefore, t s adopted as a QoS representatve n ths work. To ths end, for the node-par, we have α P thr out, Type I : P out, E +, 3 E where α s a postve constant. However, the use of Type I may ncrease the computatonal complexty, whch motvates us to rewrte 3 as Type II : β P thr out, P out, E + E, 4 where β s also a postve constant. In partcular, both α and β are tunable parameters, whch s used to adjust the steepness, and acheve the farness and heterogeneous servce requrements. Proposton : There exsts β = ρ α where ρ > 0 s a scalng factor enablng Type I and Type II to acheve ther optmal values at the same allocated power. That s, Type I and Type II are equvalent n ths case. Indeed, Proposton can be proved by frst settng the dervatves of 3 and 4 wth respect to as zero, then fndng the relatonshp between α and β when 3 and 4 acheve the optmal solutons at the dentcal. Hence, 4 can be treated as an energy effcency metrc. Partcularly, Type II can smplfy the dervaton analyss, and facltate resource allocaton strategy updates. Moreover, n order to consume energy as lttle as possble, p s,r can be obtaned by transformng nto the equaton,.e., p s,r = C out, l local δ σ / h s,r. For Type II, achevng the energy effcency of the node-par s equvalent to maxmzng 4 n terms of. III. AUCTION-BASED RELAY POWER ALLOCATION FOR A. Aucton Mechansm Desgn COOPERATIVE NETWORKS In our model, each source-destnaton par competes for relayng assstance,.e., relay transmt power, and s only nterested n ts own acton and ts resultng beneft. Hence, we resort to the noton of compettve farness [8] to acheve the far allocaton desgn for compettve scenaros. As a notable example of compettve farness, the aucton approach s exploted to determne relay power allocaton n ths work. Specfcally, n the aucton mechansm for relay power allocaton, the source-destnaton pars consttute a set of rsk-neutral bdders, denoted by N = {,..., N}, and the relay r acts as the auctoneer. The object for sale s the avalable energy of the relay r. Moreover, the prvate value of the bdder, V, whch represents the value of the sale object to tself, should exhbt two measures of nterest, namely, the amount of cooperaton benefts and costs. Here, the two measures for bdder are characterzed by ts outage performance mprovement and the sum of energy consumpton for Phase and Phase, respectvely. To ths end, the prvate value of bdder can be expressed as V = η P thr out, P out, p s,r + p s,d + p ct, 5 83

3 3 where η s a postve constant. The bdder submts ts bd b to r, and then, obtans a porton of the relay transmt power p r,d = b j N b j + θ p r, 6 where p r s the avalable transmt power of the relay r, and θ s the reserved bd whch s a postve bd placed by the relay r. The relay r adjusts ts reserved bd accordng to the varaton of energy effcency demand. Moreover, the bdder pays for the help of the relay r. The payment should be proportonal to the amount of energy that r dedcates to servng the bdder, that s, C = μ p r,d + p cr + p ct, 7 where μ s the unt prce whch the relay r announces to all bdders at the begnnng of the aucton. B. Nash Equlbrum Under Aucton Game The essence of an aucton s a game, where the players are the bdders, the strateges are the bds, and both allocatons and payments are functons of the bds. Accordngly, the proposed aucton mechansm can be compactly represented n strategc form as G = N, {B } N, {u } N, whch specfes for each bdder a set of bddng strateges B b B, and a utlty functon u b,..., b N assocated wth outcome of the aucton arsng from a bddng strategy profle b = b,..., b N. Moreover, the jont strategy space s B = B... B N. For b B, we can rewrte b = b, b, where b = b,..., b, b +..., b N denotes the bddng strateges for all N others excludng bdder. Hence, the utlty for the bddng strategy profle b s u b, b = V b, b C b, b, 8 Formally, our aucton game G s defned as max u b, b, for all N. 9 b B Our work manly concentrates on the relay power allocaton, hence, we take the sngle-relay case for example. The proposed aucton-based relay power allocaton scheme can be extended to networks wth multple relays. The key dfference between them s to add the step of each node-par choosng ts approprate relay. Consequently, f there exst N r relays, the mult-relay network can be dvded nto N r + clusters of nodes, and each cluster s ndependently analyzed as a snglerelay case. The readers can be referred to [5] for more detals. Proposton : The outcome of the proposed aucton game G can result n the energy-effcent relay power allocaton. Proof: Note that the utlty functon n 8 s almost dentcal to the energy effcency metrc n Type II, and the addtonal parameters, η and μ, do not affect the relatonshp between the energy effcency and relay transmt power The reason s smlar to Proposton.. Moreover, each bdder of our aucton game G s to maxmze the utlty functon n terms of b. s a monotoncally ncreasng functon wth b when b s fxed, and thus, the soluton whch maxmzes the utlty functon n terms of concdes wth the outcome of G. Snce achevng the energy effcency n Type II s equvalent to maxmzng 4 n terms of, the proof s complete. Consderng that the NE s treated as a desrable outcome of any mult-player game, we are nterested n the relay power allocaton whch results from the NE strateges b. Theorem : A NE exsts for the proposed aucton game G. Proof: We rewrte 8 as u b, b = Y b, b X b, b, where Y b, b = η Pout, thr Γ, C out, σ l δ, and X b, b = b,b [ + μ pco b, b + p s,r + μp cr + + μ p ct ]. By takng the second dervatve of Y b, b wth respect to b, we fnd that f b, b C out, σ l δ, Y b, b / b 0. Notng that Cout, σ l δ s tny e.g., t asymptotcally acheves to zero, the above constrant on b, b can be satsfed. Then, Y b, b s concave. Also, X b, b s convex. As a result, the utlty functon u s concave, whch n turn mples quas-concavty. Moreover, B s a non-empty, compact and convex set. Hence, the condtons of the exstence of NE, as n [9], are satsfed, and accordngly G has a NE at least. Next, we dscuss the propertes of NE. Frst, we derve the dervatve of u b, b wth respect to b by [ u b, b η m 3 = e m ] j = b j + θ p r + μ b M,, j N b j + θ 0 where M, = C out, σ l δ, and m = C out, σ l δ b,b. Notce that the second term on the rght-hand sde of 0 s strctly postve. In order to fnd the best-response strateges, 3 we have M, /3 m = e m, where M, = +μm, η. By usng the prncpal branch of the Lambert W functon whch s denoted by W 0, we obtan = M, 3W 0 3 M, /3. Hence, the best-response functon can be expressed as j = b j + θ BR b, b = p r. Then, we can defne the best-response vector functon BR b as BR b=br b,..., BR N b, and ts components are gven by. Defnton [0]: If a best-response vector functon BR satsfes the followng condton: for all ψ >, /ψ b b ψb mples /ψ BR b < BR b < ψbr b, t s sad to be two-sded scalable. Theorem [0]: If a best-response vector functon BR s two-sded scalable and a fxed pont exsts, then that fxed pont s unque. Theorem 3: The proposed aucton game G has a unque NE. Proof: Accordng to Theorem, our aucton game G has a NE b, and then, b s a fxed pont of the best-response vector functon,.e., b = BR b. For N and all ψ >, f /ψ b b ψb, by monotoncty, we can 83

4 get BR /ψ b BR b BR ψb. From, we obtan ψbr /ψ b > BR ψ /ψ b = BR b. These above nequaltes jontly result n /ψ BR b < BR b < ψbr b. Accordngly, BR s two-sded scalable. By usng Theorem, we conclude that the NE b s unque. Theorem 4: The relay power allocaton soluton dervng from the unque NE of the proposed aucton game G s Pareto optmal. Proof: For the proposed aucton game G, we assume that the bddng strategy profle b s the outcome of the utltaran socal welfare problem whch s defned as max b N u b, 3 = but not Pareto optmal. Then, there exsts a certan strategy profle b such that u b u b, for all N and u b > u b for some N. Ths mples that b N = u > N = u b. As a result, b cannot be a soluton to the problem n 3, whch contradcts the orgnal assumpton. Therefore, t must be a Pareto optmal pont. Furthermore, the soluton of 3 should satsfy the followng N equatons: N / = u b b = 0, for all N. The result of NE satsfes ths case Please refer to the analyss of best-response functon for the detals.. Therefore, the resultng NE b s a soluton to the problem n 3, that s, t s Pareto optmal. C. Dstrbuted Algorthm for Aucton Game In ths subsecton, we propose a relay power allocaton algorthm to reach the NE of G n a dstrbuted way, and the detals are shown n Table I. Specfcally, the bdders teratvely update ther bddng strateges based on best-response functons, and the teraton vector functon can be expressed as b n+ = BR b n, 4 = pco j = bn j p r +θ whose component s gven by b n+, for N, where s gven by. In partcular, the proposed algorthm favors dstrbutng the avalable energy of the relay, wth uncertanty about the ndvdual nformaton e.g., the utlty value, the chosen strateges, etc.. In fact, [0, Theorem 0] shows that f the best-response functon s two-sded scalable and a fxed pont exsts, ths fxed pont can be reached va usng ts correspondng teraton vector functon. Hence, we conclude that the proposed algorthm can converge to the NE b. Moreover, we nvestgate the convergence tme of the proposed algorthm, whch s necessary and mportant for communcaton qualty. Note that f b n s a NE, then b n+ = b n and the allocated power stops changng. In order to characterze the dfference between each teraton durng the aucton process, we defne the potental functon as Φ b = N = b b, where b = N N = b. Table I: Dstrbuted Relay Power Allocaton Algorthm begn ntalzaton n = ; r announces the nformaton,.e., the reserved bd θ, the prce μ, and ts total transmt power p r to all bdders; determnes the transmt power p s,r by transformng nto the equaton, chooses a feasble bd b arbtrarly, and submts b to r; whle n > 0 r broadcasts the sum of bds of all bdders N bn ; whle N calculates ts updated bddng strategy b n+ accordng to 4, and submts t to r; end whle f b n+ = b n n = ; else n = n + ; end f end whle Lemma : Let b n = b n,..., bn N be the bddng strategy profle at teraton n by usng the proposed algorthm, and we have E [ Φ b n+ b n] N + N Φ b n. Proof: Let z b = N N = = b b, and d b = N N = b b. By usng the upper-bound of z b and d b, we have [ E Φ b n+ b n] N + N N N b n = = 4 b n. 5 Moreover, the rght-hand sde of 5 s N + N = b n b at most. By Cauchy-Schwarz, we get E [ Φ b n+ b n] N + N Φ b n. Lemma : Suppose that the bddng strategy satsfes b b ε ε > 0 and b b b N b, for all N. Let w = w,..., w N wth w = b for [, N ]. Then, Φ b N z b Φ w N z w. Proof: Let f g = N = b g. Note that the dervaton of f g s: f g g = N g N w, 6 where w s the average value of all the elements of w. Moreover, the second dervatve s f g / g = N 0. Hence, f g s mnmzed at g = w, and then, we acheve Φ b Φ w = k + N = b b N = b w k. 7 Next, we assume that b N = b + k and S b = N = b b, then S b S w Nk. Let M = b N k and y j = { b = j}. Thus, we have b b N k, y j = 0 for j > b N, and y j = 0 for j < M. Smlarly, let B b = b N j=m y j y j y j+, and we can further get B b B w y bn k N y bn k/

5 5 Snce the rght-hand sde of 8 s at most N, we can obtan: N z b S b S w B b B w z w N N N k. 9 In the case of b N = b k, the proof s smlar to the above. Therefore, the proof s complete. Theorem 5: Let N be the number of the source-destnaton pars n the wreless cooperatve network, and T be the number of rounds taken by the proposed algorthm to reach a NE for the frst tme. Then, E [T ] O N + N. Proof: Note that the relatonshp between b n+ and b n satsfes: E [ Φ b n+] N + N E [ Φ b n]. Then, let τ = O N, Υ n = Φ b n+τ d b n, and Ω n = mn Υ n, z b n. From Lemma, we know that Υ n s a super-martngale. Hence, and E E [ Ω n+ Ω n < z b n] Υ n = Ω n, 0 [ Ω n+ Ω n = z b n] z b n = Ω n. If b n s not a NE, P [ Φ b n+ =Φ b n Φ b n] Φ / w n zwn N Φ w n. Based on Lemma, we have for x > 0, P Υ n+ = Υ n Υ n = x / b max b, for 0 < x < z b n, N P Ω n+ = Ω n Ω n = x / b max b. Consequently, N we can show [ E Ω n+ Ω n 0 < Ω n < z b n] b. In fact, T can be vewed a stoppng tme. Defne Ξ n = z b n Ω n b T, 3 The Maxmum Utltaran Socal Welfare CPAS Proposed Aucton Number of Bdders Fg.. Performance comparson n terms of dfferent number of bdders The avalable transmt power of the relay s W.. The Maxmum Utltaran Socal Welfare CPAS Proposed Aucton The Avalable Relay Power W and we fnd that Ξ n T s a sub-martngale. Let P be the probablty for Ω n = 0, that s, b n+τ s a NE. Usng Optonal Stoppng Theorem [], we get P z b n [ be [T ] = E Ξ T ] Ξ > 0. 4 By means of the Estmaton Theorem [], we have E [T ] O N + P z b n / b. For Ω n = 0, accordng to Lemma, we have E [ T Ω T = 0 ] O N + z b n / b O N + N. Smlarly, for Ω n = z b n, we also have E [ T Ω T = z b n] O N + N. Therefore, we can get E [T ] O N + N. IV. NUMERICAL RESULTS The smulaton setup s as follow: several source-destnaton nodes are randomly located around the relay. For both the local broadcastng and the long-haul cooperatve channels, Raylegh fadng coeffcents are modeled as untary power, complex Gaussan random varables. The constant κ s set to, and Fg.. Performance comparson n terms of dfferent avalable relay power values The number of the bdders s 5.. the path loss exponent δ s set to 3. The Gaussan nose varance σ s 0 W. For smplcty, we take the outage capacty C out, and the threshold value of outage probablty as the same values for all source-destnaton nodes,.e., C out, =.4bps/Hz and Pout, thr = 0 4 [6]. Besdes, we choose θ = for all the smulatons as n [5]. For llustratve purpose, we compare our scheme wth the centralzed power allocaton scheme CPAS whch acheves the optmal soluton []. Fg. depcts the maxmum utltaran socal welfare wth dfferent number of the bdders, and Fg. shows the case of dfferent avalable relay power values. Note that these results are obtaned usng 50 runs n order to obtan statstcally meanngful average values. From the gven fgures, we can fnd that our scheme almost acheves the same performance as that of CPAS, rrespectve of the number of bdders and avalable relay power values. In other words, P thr out, 834

6 6 Convergence Tme Iteraton Number Experment Theoretcal Analyss Number of Bdders Fg. 3. The convergence tme as the number of bdders vares from 6 to 0. the resultng NE s a soluton of maxmzng the utltaran socal welfare. Hence, the resultng NE va usng the proposed dstrbuted algorthm s Pareto optmal. Moreover, Fg. 3 shows the convergence tme as the number of bdders vares from 6 to 0 wth random relay transmt power. The more the bdders, the slower the convergence speed. That s because the ncrease n the number of bdders may result n the growth of opportunty of nteracton among the bdders. Also, we verfy the convergence performance when the avalable transmt power of the relay s a dynamc value. We fnd that the convergence tme s strongly dependent on the number of the bdders, and ndependent of the ntal bddng strateges and the avalable transmt power of the relay node. Most notably, when N s small, Theorem 5 provdes an accurate estmaton, whle as N goes large, Theorem 5 tends to become a bt of lose. For example, when N = 6, the theoretcal upper bound s 4, and the practcal convergence round s 39. These numbers change to 40 and 347 when N = 0. Ths s due to the fact that we use a conservatve estmaton of Φb n Lemma, and we refer the readers to [, Theorems ] for more detals on the performance devaton by usng ths estmaton method. ACKNOWLEDGEMENT Ths work s supported by the NSF of Chna Grant No , 60007, and the Major Natonal Scence & Technology Specfc Projects Grant No. 00ZX REFERENCES [] Y. Chen, S. Zhang, S. Xu, and G. Y. L, Fundamental trade-offs on green wreless networks, IEEE Communcatons Magazne, vol. 49, no. 6, pp , 0. [] C. Han, T. Harrold, Green rado: Rado technques to enable energyeffcent wreless networks, IEEE Communcatons Magazne, vol. 49, no. 6, pp , 0. [3] K. Phan, L. Le, S. Vorobyov, and T. Le-Ngoc, Power allocaton and admsson control n mult-user relay networks va convex programmng: centralzed and dstrbuted schemes, EURASIP Journal on Wreless Communcatons and Networkng, vol. 009, artcle ID 90965, 009. [4] A. Mukherjeen and H. M. Kwon, General aucton-theoretc strateges for dstrbuted partner selecton n cooperatve wreless networks, IEEE Transactons on Communcatons, vol. 58, no. 0, pp , 00. [5] J. Huang, Z. Han, M. Chang, and H. V. Poor, Aucaton-based resource allocaton for cooperatve communcatons, IEEE Journal on Selected Areas n Communcatons, vol. 6, no. 7, pp. 6-37, 008. [6] A. D. Coso, U. Spagnoln and C. Ibars, Cooperatve dstrbuted MIMO channels n wreless sensor networks, IEEE Journal on Selected Areas n Communcatons, vol. 5, no., pp , 007. [7] L. Zhou, X. Wang, W. Tu, G. M. Muntean, and B. Geller, Dstrbuted schedulng scheme for vdeo streamng over multchannel multrado multhop wreless networks, IEEE Journal on Selected Areas n Communcatons, vol. 8, no. 3, pp , 00. [8] J. Sun, E. Modano, and L. Zheng, Wreless channel allocaton usng an aucton algorthm, IEEE Journal on Selected Areas n Communcatons, vol. 4, no. 5, pp , 006. [9] A. B. Mackenze and L. DaSlva, Game Theory for Wreless Engneers. Morgan & Claypool, 006. [0] C. W. Sung and K.-K. Leung, A generalzed framework for dstrbuted power control n wreless networks, IEEE Transactons on Informaton Theory, vol. 5, no. 7, pp , 005. [] S. Boyd and L. Vandenberghe, Convex Optmzaton. Cambrdge, U.K.: Cambrdge Unv. Press, 004. V. CONCLUSIONS In ths work, we develop an aucton-based relay power allocaton scheme for multple node-pars relay wreless networks. Frstly, we desgn a refned energy effcency metrc to descrbe the relatonshp between the QoS provsonng and the energy consumpton. Then, we propose an aucton mechansm for relay power allocaton. From the game theoretc perspectve, we prove the exstence, unqueness, and Pareto optmalty of the NE for our aucton game. From the resultng NE, we can determne the energy-effcent allocaton strateges. Moreover, we develop a dstrbuted algorthm to converge to the Pareto optmal NE wth the ncomplete ndvdual nformaton. Interestngly, we fnd that the convergence s only dependent on the number of source-destnaton pars. 835