Auction-based Resource Allocation for Multi-relay Asynchronous Cooperative Networks

A UCTION -B ASED RESOUR CE ALLOC A TION FOR MUL TI-RELA Y ASYNCH R ONOUS COOPE RA TIVE NE TWORKS Jianwei Huang, Z hu Han , Mung Chiang, and H. V incent P oor ABSTRA CT Resource allocation is co nsidered for cooperative transmissions in multiple-rela y wirele ss networks. T wo auction mechanisms, SNR auctions and power auctions, are propo sed to distrib utively co- ordinate th e alloca tion of power among multiple relays. In th e SNR a uction, a user chooses th e re lay with th e lowest weighted price. In th e power auction, a user may choose to use multiple re- lays simultaneo usly , depen ding o n the network topo logy an d the relays’ prices. Sufficient cond itions for the e xistence (in both auc- tions) and u niquen ess ( in th e SNR au ction) of the Nash equ ilib- rium ar e given. The fairness of the SNR au ction and efficiency of the power auction are further discussed. It is also p roven that users ca n achieve the unique Nash equ ilibrium distributiv ely via best response updates in a completely asynchro nous ma nner . Keywords : W ireless Networks, Relay Network s, Auction The- ory , Po wer Control, Resource Allocation 1. INTRODUCTION Cooperative communication (e.g., [1 ]) takes advantage of the b road- cast natu re of wireless channels, u ses relay nodes as v irtual an- tennas, and thus realizes the be nefits of multiple- input-m ultiple- output (M IMO) comm unications in situatio ns wher e phy sical mul- tiple an tennas are difficult to install (e.g., on small sen sor n odes). Although the phy sical la yer performa nce of coope rativ e co mmu- nication h as been extensi vely studied in the context o f small net- works, there are still many o pen prob lems of h ow to realize its full b enefit in large-scale networks. For example, to o ptimize co - operative com municatio n in large networks, we need to consider global chann el information (including that fo r source- destination, source-re lay , and relay-destina tion channels), heterogen eous re- source constra ints a mong users, an d various u pper laye r issues (e.g., rou ting and traffic demand). Recently some ce ntralized net- work control algorithm s (e.g., [2 , 3]) have b een proposed for co- operative communication s, but they require considerab le overhead for signalin g and measur ement a nd do not scale well with network size. This motivates ou r stud y of distributed resour ce allocation algorithm s for coo perative commun ications in this p aper . In this paper, we design two distributed auctio n-based resourc e allocation algo rithms that achiev e fairness and efficiency for mu ltiple- relay coo perative communication networks. Here fairness means an allocation tha t equalizes the (weighted) marginal rate increase among users who use the r elay , and efficiency means an alloca- tion that maximizes the total r ate increase realize d by use of the J. Huang is with the Department of Information Engineering , the Chinese Uni versity of Hong Kong , Shatin, NT , Hong H ong. Z. Han is with the Elec- trical and Computer Engineer ing Department, Boise State Unive rsity , Boise, ID, USA. M . Chia ng and H.V . Poor are with the Department o f Electri cal Engineering, Princet on Uni versity , Princeton, NJ, USA. The work of J. Huang is supported by Direct Grant of the Chinese Uni v . of Hong K ong under Grant 2050398. This work of H. V . Poor was supported by the U.S. National Sci ence Fo undation under Grants ANI-03-38807 and CNS-06-25637 relays. Precise definition s of fairn ess and efficiency will be given in Section 2. I n both au ctions, each user decides “when to use re- lay” based on a locally computable thr eshold po licy . Th e q uestion of “how to relay ” is answer ed by a simple weig hted pr oportio nal allocation among users who use the relay . In our previous work [4], we have propo sed similar auc tion mechanisms for a single-r elay coop erative commun ication net- work, where user s can achiev e the desired auction o utcomes if they up date their bids in a synchr onous manner . This paper co n- siders the more gener al case wher e there are mu ltiple rela ys in the network with different locations and av ailable resou rces. The existence, uniquene ss, and prop erties of the auction outcome s are very dif feren t from the single-relay case. M oreover , we show that users can achieve the desirable auction ou tcomes in a completely asynchr onous manner, which is m ore realistic in practice an d mo re difficult to prove. Due to the space lim itations, a ll th e p roofs ar e omitted in this conferenc e pap er . 2. S YSTEM MODEL AND NETWORK OBJECTIVES As a concrete example, we co nsider th e amplify-a nd-forward (AF) cooper ativ e commun ication pr otocol in this paper . The system diagram is shown in Fig. 1, wher e there is a set K = (1 , ..., K ) of relay nodes and a set I = (1 , ..., I ) of sour ce-destination pairs. W e also refer to pair i as user i , which inclu des source nod e s i and destination node d i . For each u ser i , the co operative tran smission consists o f two phases. In Phase 1 , source s i broadc asts its information with power P s i . The received signals Y s i ,d i and Y s i ,r k at destination d i and relay r k are given by Y s i ,d i = p P s i G s i ,d i X s i + n d i and Y s i ,r k = p P s i G s i ,r k X s i + n r k , where X s i is the transmitted in- formation symbo l with unit energy at Phase 1 at sour ce s i , G s i ,d i and G s i ,r k are the chann el gains fr om s i to d estination d i and re- lay r k , respectively , and n d i and n r k are additive white Gaussian noises. Without loss o f g enerality , we a ssume th at the noise level is the same for all lin ks, and is deno ted by σ 2 . W e also assume that th e transmission time o f one frame is less than the chan nel coheren ce time. The signal-to-noise ratio (SNR) t hat is realized at destination d i in Phase 1 is Γ s i ,d i = P s i G s i ,d i σ 2 . In Phase 2 , user i ca n use a subset o f (includin g all) relay nodes to he lp imp rove its throug hput. If relay r k is u sed b y u ser i , r k will amplify Y s i ,r k and forward it to destination d i with transmitted power P r k ,d i . The received signal at destinatio n d i is Y r k ,d i = p P r k ,d i G r k ,d i X r k ,d i + n ′ d i , where X r k ,d i = Y s i ,r k / | Y s i ,r k | is th e unit-en ergy transm itted sign al th at relay r k receives fr om source s i in Phase 1 , G r k ,d i is the ch annel gain from relay r k to destination d i , and n ′ d i is the receiver noise in Phase 2 . Equiv a- lently , we can write Y r k ,d i = √ P r k ,d i G r k ,d i ( √ P s i G s i ,r k X s i ,d i + n r k ) √ P s i G s i ,r k + σ 2 + n ′ d i . The addition al SNR increase due to relay r k at d i is Source 1 Relay 1 Destination 1 Destination 2 Source 2 Source i Relay r k Destination i combining Phase 1 Phase 1 Phase 2 k i r s X , k i r s Y , i i d s X , i i d s Y , i k d r X , i k d r Y , i k i d r s R , , Relay 2 Fig. 1 . System Mo del for Cooperation T ransmission △ SN R ik = P r k ,d i P s i G r k ,d i G s i ,r k σ 2 ( P r k ,d i G r k ,d i + P s i G s i ,r k + σ 2 ) . (1) The total informatio n rate user i achiev es at the output of maximal ratio combinin g i s R s i ,d i ( P r ,d i ) = W log 2 (1 + Γ s i ,d i + P k △ SNR ik ) P k ∈K 1 { P r k ,d i > 0 } + 1 . (2) Here P r , d i = ( P r k ,d i , ∀ k ∈ K ) is the tran smission power vector of all relays to destination d i , W is the total ban dwidth of th e system, and 1 {·} is the indicator function. Eq uation (2) includes a special case w here user i do es not use any relay (i.e., P r k ,d i = 0 for all k ∈ K ) , in which case th e ra te is W lo g 2 (1 + Γ s i ,d i ) . The denomin ator in (2) mod els the fact that relay transmissions occupy system resou rce ( e.g., time slots, band width, cod es). W e write R s i ,d i ( P r , d i ) to e mphasize tha t P r , d i is th e resource allo cation decision we ne ed to make, a nd it is clear that R s i ,d i depend s on other system parameter s s uch as ch annel gains. W e assume that the source transmission power P s i is fixed for each u ser i . Each relay r k has a fixed total transmission power P r k , and can choo se the tran smission power vector P r k , d , ( P r k ,d 1 , ..., P r k ,d I ) fro m the feasible set P r k , ( P r k , d      X i P r k ,d i ≤ P r k , P r k ,d i ≥ 0 , ∀ i ∈ I ) . (3) Finally , d efine P r , d = ( P r k , d , ∀ k ∈ K ) to be the transmission power of all relay s to all u sers’ destination s. Th e resource alloca- tion decision we need to make is the v alue of P r , d . From a n etwork d esigner’ s poin t of view , it is imp ortant to consider bo th efficiency and fairn ess . An efficient p ower alloca - tion P efficienc y r , d maximizes the total rate increases of all users, i.e., max { P r k , d ∈P r k , ∀ k ∈K } X i ∈I △ R i ( P r ,d i ) , (4) where △ R i ( P r , d i ) den otes the rate increase o f user i due to the use of relays △ R i ( P r ,d i ) = max { R s i ,d i ( P r , d i ) − R s i ,d i ( 0 ) , 0 } . In many cases, an efficient allocatio n discriminates against users who are far away fr om the r elay . T o avoid th is, w e also con sider a fair power allocation P fair r , d , where each re lay r k solves the fol- lowing pr oblem max P r k , d ∈P r k X i P r k ,d i , s.t. ∂ △ R i ( △ SNR ik ) ∂ ( △ SNR ik ) = c k q ik · 1 { P r k ,d i > 0 } , ∀ i ∈ I . (5) Here q ik ’ s are the priority coefficients den oting the importan ce of each user to each relay . Whe n q ik = 1 for each i , all users who use relay r k have the same marginal utility c k , which lead s to st rict fairness amon g users. In the special case where users are symmet- ric and only use the same relay r k , the fairness maximizing po wer allocation leads to a Jain’ s fairness index [5 ] equal to 1. Howev er, the d efinition of fairness here is m ore gen eral than the Jain’ s fair- ness index. Notice that a fair a llocation is Pareto o ptimal, i.e., no user’ s rate can be fur ther incr eased withou t decreasing the rate of another user . Since △ R i ( P r , d i ) is non -smooth and non- concave ( due to the max operation) , it is well k nown that Problems (4) and (5) are N P hard to solve even in a cen tralized fashion . Next, we will propo se two auction mechan isms that can solve these prob lems under certain technical condition s in a distributed fashion. 3. A UCTION MECHANISMS An auctio n is a decentr alized m arket mechanism for allocating resources without kn owing the pr iv ate valuations of ind ividual users in a market. Auction the ory has been recently used to study various wireless resourc e allocation prob lems (e.g ., time slot al- location [6] an d power c ontrol [7 ] in cellular n etworks). Here we p ropose two au ction mechan isms f or alloc ating resour ce in a mu ltiple-relay network. Th e rules of the two auction s are de - scribed belo w , with the only difference bein g in paym ent d etermi- nation. • Initialization : Each relay r k annou nces a positiv e reserve bid β k > 0 and a price π k > 0 to all users befor e the auction starts. • Bids : Each u ser i su bmits a nonnegative bid vector b i = ( b ik , ∀ k ∈ K ) , on e compone nt to each relay . • Allocation : Each relay r k allocates transmit power as P r k ,d i = b ik P j ∈I b j k + β k P r k , ∀ i ∈ I . (6) • P ayments : User i pays C i = P k π k q ik △ SNR ik in an SNR auction or C i = P k π k P r k ,d i in a power au ction. The two a uction mech anisms that we p ropo se are highly distributed, since each user only nee d to know th e public system paramete rs (i.e., W , σ 2 and P r k for all relay k ), local in formatio n ( i.e., P s i and G s i ,d i ) an d the channel g ains with relays ( G s i ,r k and G r k ,d i for each relay r k , which can be obtained thro ugh chan nel feed- back). The relays do not need to know any network inform ation. A bidding profile is defined as t he vector containing the users’ bids, b = ( b 1 , ..., b I ) . The bidding pro file of user i ’ s oppo nents is defined as b − i = ( b j , ∀ j 6 = i ) , so that b = ( b i ; b − i ) . User i chooses b i to maximize its payoff U i ( b i ; b − i , π ) = △ R i ( P r , d i ( b i ; b − i )) − C i ( b i ; b − i , π ) . (7) Here π = ( π k , ∀ k ∈ K ) is the pr ices of all relay s. It can be shown that th e values of the re serve b ids β k ’ s do not affect the resource allocation, thus we can simply choose β k = 1 for all k . The desirab le outcome of an au ction is called a Nash Equ ilib- rium (NE ), wh ich is a bidd ing profile b ∗ such th at n o u ser wants to deviate unilaterally , i.e., U i  b ∗ i ; b ∗ − i , π  ≥ U i  b i ; b ∗ − i , π  , ∀ i ∈ I , ∀ b i ≥ 0 . (8) Define user i ’ s best r esp onse (for fixed b − i and price π ) as B i ( b − i , π ) =  b i     b i = arg max ˜ b i ≥ 0 U i  ˜ b i ; b − i , π   , (9) which can be written as B i ( b − i , π ) = ( B i,k ( b − i , π ) , ∀ k ∈ K ) . An NE is also a fixed poin t solution of all users’ best resp onses. Next we will consider the existence, uniqu eness and prope rties of the NE, an d ho w to achie ve it in practice. Although in gen eral NE is not the most desirab le operatio nal po int from an overall sys- tem point of vie w , we will show later that the t wo auctio ns indeed achieve o ur d esired network o bjectives under suitable technical condition s. 3.1. SNR A uction W e first consider the SNR auction where user i ’ s paymen t is C i = P k π k q ik △ SNR ik . Theorem 1 In an SNR auction with multiple relays, a user i ei- ther does n ot u se any relay , o r u ses o nly one relay r k ( i ) with the smallest weighted price, i.e., k ( i ) = arg min k ∈K π k q ik . Theorem 1 imp lies th at we can divide a multiple- relay network into K + 1 clusters of nod es: each of the first K clusters contains one relay no de an d the users who use this relay , and the last cluster contains users that d o not use any re lay . Then we can an alyze each cluster indep enden tly as a single-r elay network as in [4] . In particular, for a user i belo nging to cluster k ( i ) ≤ K , its best response function is B i,k ` b − i,k , π k ´ = ( f s i,k ( π k ) “ P j 6 = i b j,k + β k ” , k = k ( i ) , 0 , otherwise . (10) Note that user i ’ s b est response is related only to the b ids fr om users wh o are in the sam e clu ster . T he lin ear coefficient f s i,k ( π k ) is derived as f s i,k ( π k ) = (11) 8 > > > < > > > : ∞ , π ≤ π s i , ( P s i G s i ,r k + σ 2 ) σ 2 P r k G r k ,d i P s i G s i ,r k W 2 π k q ik ln 2 − 1 − Γ s i ,d i − ( P s i G s i ,r k + P r k G r k ,d i + σ 2 ) σ 2 , π ∈ ` π s i , ˆ π s i ´ , 0 , π ≥ ˆ π s i , where π s i , W/ ( 2 q ik ln 2 ) 1 + Γ s i ,d i + P r k G r k ,d i P s i G s i ,r k ( P s i G s i ,r k + P r k G r k ,d i + σ 2 ) σ 2 , (12) and ˆ π s i is the smallest positive r o ot o f the following equation in π π q ik (1 + Γ s i ,d i ) − W 2 „ log 2 „ 2 π q ik ln 2 W (1 + Γ s i ,d i ) 2 « + 1 ln 2 « = 0 . (13) In the degenerate case wh ere ˆ π s i > π s i , we have f s i,k ( π k ) = ∞ for π k < ˆ π s i and f s i,k ( π k ) = 0 for π k ≥ ˆ π s i . Notice that the linear coefficient is d etermined based on a simple thr eshold policy , i.e., comparin g th e p rice an noun ced by the relay with the two loc ally computab le thr eshold prices. Now let us assume th at all users use th e same relay r k , then from (6) and (10) we know that the total deman d for the r elay power is P i ∈I f s i,k ( π k ) f s i,k ( π k )+1 P r k , wh ich ca n not exceed P r k . It is also clear that f s i,k ( π k ) is a non-inc reasing fu nction of π k . T hen we can find a threshold p rice π s k,th such that P i ∈I f s i,k ( π k ) f s i,k ( π k )+1 < 1 when π k > π s k,th , and P i ∈I f s i,k ( π k ) f s i,k ( π k )+1 ≥ 1 when π k ≤ π s k,th . Theorem 2 In an SNR au ction with multiple relays, a un ique NE exis ts if π k > π s k,th for each k . Finally let us consider the proper ty of the NE. For a single- relay network, we show in [4] th at the SNR auction achieves the fair resou rce allocation (i.e. it solves Problem (5)) if at least one user wants to use the relay at the thr eshold price π th . In the multiple-rela y c ase, howev er, som e re lays may never be able to achieve a Pareto op timal allocation , which is a basic r equirem ent for a fair allocation . This is because if the relay an nounc es a high price, no users will use th e relay . If the relay d ecreases th e pr ice, there mig ht be too ma ny users switching to the same relay simu l- taneously su ch that an NE d oes not exist. On th e oth er h and, we can show t he fo llowing: Theorem 3 If ther e exists a NE such that eac h r elay’s r esource is full utilized and ea ch r elay is used by at lea st one user , the corr e- sponding power allocation is fair (i.e., it solves Pr ob lem (5)). 3.2. Power A uction Here we consider th e p ower auction , where user i ’ s p ayment is C i = P k π k P r k ,d i . There are two key differences h ere com- pared with the SNR auctio n. First, a user m ay choo se to use mul- tiple relay s simultaneou sly here. User i ’ s best respo nse can be written in the following lin ear for m: B i,k ( b − i,k , π ) = f p i,k ( π )  P j 6 = i b j,k + β k  , ∀ k ∈ K . T o calculate f p i,k ( π ) , user i needs to consider a total of P K l =0  K l  cases of choosing relays. For exam- ple, when ther e are two relays in the netw or k, a user needs to con- sider fo ur cases: n ot using any relay , using relay 1 only , using relay 2 only , and u sing both r elays. For the given relay choice in case n , it calc ulates the linear coefficients f p,n i,k ( π ) for all k in closed- form (this inv olves thr eshold p olicy similar to the SNR auc tion) and the correspon ding r ate increase △ R n i . Th en it find the case that yields the largest pay off, n ∗ = arg max n △ R n i , and sets f p i,k ( π ) = f p,n ∗ i,k ( π ) ∀ k . Second, the linear coefficient f p i,k ( π ) depend s on the prices anno unced by all relay s. F or example, ei- ther a large π k or a small π k ′ ( k ′ 6 = k ) can make f p i,k ( π ) = 0 , i.e., user i will choo se not to use relay r k . Similar to in the SNR au ction, we can a lso calcu late a th resh- old price π k,th for re lay r k . In this case, we assume that all re- lays anno unce infin itely high price s excep t r k , and the n calcu- late π p k,th such that P i ∈I f s i,k ( π k ) f s i,k ( π k )+1 < 1 whe n π k > π p k,th , and P i ∈I f s i,k ( π k ) f s i,k ( π k )+1 ≥ 1 when π k ≤ π p k,th . Colloary 1 In a p ower auction with multiple relays, ther e exists an NE if π k > π p k,th for each k . On the other hand , necessary condition for existence of NE as well as conditio ns for unique ness are no t straigh tforward to specify , and are left for futur e resear ch. W e can character ize the property of the NE as follows: Theorem 4 If ther e exists a NE such that eac h r elay’s r esource is full utilized and all users use all relays, the corresponding power allocation is efficient (i.e., it solves Pr oblem (4)). 3.3. Asynchronous Best Response Updates The last question we want to answer is how the NE can be reached in a distributed fashion. Since user i does n ot kn ow th e best re- sponse fu nctions of oth er users, it is impossible for it to calculate the NE in one shot. In the context of a single-relay network [ 4], we ha ve shown that distributed b est response updates can globally conv erge to the uniq ue NE (if it exists) in a synchr o nous ma nner, i.e., all users u pdate their bids in each time slot simultaneously ac- cording ly to b i ( t ) = f s i ( π )  P l 6 = i b l ( t − 1) + β  . In p ractice, howe ver , it w ould be difficult or even undesirab le to coordinate all users to update their bid s at the same tim e, an d th e fo llowing can be used: Algorithm 1 Asynchro nous Best Respo nse Bid Updates 1: t = 0 . 2: Each user i rand omly chooses a b i (0) ∈  b i , ¯ b i  . 3: t = t + 1 . 4: for each user i ∈ I 5: if t ∈ T i then 6: b i,k ( t ) = h f s i ( π )  P l 6 = i b l ( t − 1) + β i ¯ b i,k b i,k , ∀ k . 7: end if 8: end for 9: Go to Step 3. W e show th at asynchr onou s b est respon se updates conver ges in the multiple- r elay case. The comp lete asynch ronou s be st r e- sponse u pdate alg orithm is giv en in Algor ithm 1 ( [ x ] b a = max { min { x, b } , a } .), where each user i up dates its b id only if the current time slot belongs to a set T i , which is an un boun ded set of time slots a nd co uld be d ifferent from user to user . W e make a very mild assumption that the asynchronism of the updates is bounde d, i.e., there exists a finite but sufficiently large positive constant B , and for all t 1 ∈ T i , ther e exists a t 2 ∈ T i such that t 2 − t 1 ≤ B . Each user updates its bid at least onc e during any time in terval of length B slots. The exact v alue of B is not importa nt (as l on g as it is bou nded) for the conver gen ce proo f and needs no t to be known by the users. Theorem 5 If there exists a un ique nonzer o NE in the SNR auc- tion, ther e a lways exists a lowerbou nd bid vector b =  ¯ b i , ∀ i ∈ I  and an u pperbou nd bid vector ¯ b = ( b i , ∀ i ∈ I ) , under which Al- gorithm 1 globally conver ges to the unique NE. In practice, we can cho ose b to be a sufficiently sm all po si- ti ve vector ( to a pprox imate zero bids from users) and ¯ b to be a sufficiently large finite vector . 4. SIMULA TION RESUL TS For i llustratio n purpose, we show the convergence of Algorith m 1 in a mu ltiple-relay SNR auction. W e consid er a network with three users an d two relays. The three transmitters are located at (100m,- 25m), ( -100m ,25m) and (10 0m,5m) , an d the th ree receivers ar e located at (-100m ,25m) , (1 00m,2 5m) and (- 100m, 5m). The two relays are located at (0m, -2m) an d (0m,0 m). All the prio rity co - efficients q ik = 1 . Since the first r elay annou nces a price lower than the seco nd r elay , all u sers choose to use th e first relay . In Fig. 2.a, we show th e con vergence of the users’ bids t o the first re- lay under s yn chrono us upda tes, wher e each user updates its bid i n each time slot. T he so lid line s show the evolution of the bids and the d otted lines show the optimal values of th e bid s after con ver- gence. In Fig. 2.b, we s how the conv ergence under the same setup with asyn chron ous conv ergence. Three users rand omly and inde - penden tly ch oose to up date their own bid s in each time slot with (a) Synchronous Updates (b) Asynchron y Updates Fig. 2 . Bids update in an SNR auction (the same one relay). probab ility 0 . 1 , 0 . 5 and 1 , respectively . W e can see that the al- gorithm conver ges to the same op timal values as the synch ronou s update case but in longer time (a s expected). 5. CONCLUSIONS In this pap er , a coopera ti ve comm unication network with multip le relays has b een co nsidered , and two auction mechanisms, th e SNR auction and the power auction, h av e bee n prop osed to distribu- ti vely coord inate the relay power alloca tion a mong users. Unlike the sing le-relay case studied in [4], here the users’ cho ices of re- lays d epend on the prices an noun ced by all relays. I n the SNR auction, a user will cho ose the relay with th e lowest weig hted price. I n the power auction, a u ser migh t use mu ltiple relay s si- multaneou sly , dep ending on the network topology and the relative relationship amon g the relays’ price s. A sufficient condition is shown for the existence o f the Nash equ ilibrium in b oth auctio ns, and conditio ns are derived for uniq ueness in the SNR auctio n. The fairness of the SNR auction and the efficiency o f the power auctio n are also discussed. Fin ally , if a n NE exists, users can achieve it in a distributed fashion via best respo nse upda tes in an a synchro nous manner . 6. REFERENCES [1] A. Sendon aris, E. Erkip, and B. Aa zhang, “User coo peration div ersity - Part I: System d escription, ” IEEE T rans. Commun. , vol. 51, no. 11, pp. 1927–193 8, 2003. [2] T . Ng and W . Y u, “Joint op timization of relay strategies and resou rce allocations in coo perative cellular network s, ” in IEEE J . Selected Ar eas Commun. , v ol. 25, no. 2 , pp. 328-339 , Feburary 2007. [3] S. Sa vazzi and U. Spagnolin i, “Energy aw are power allocation strate gies f or mu ltihop-co operative transmission schemes, ” in IEEE J . Selected Areas Commun . , v ol. 25, no . 2, pp. 318- 327, 2007. [4] J. Huang, Z. Han, M. Chiang, and H. V . Poor, “ Auction-b ased distributed re source allocation for cooperation tran smission in wireless networks, ” in Pr o c. of IEEE Global Communicatio ns Confer ence , W ashington D.C., November 2007. [5] R. Jain , W . Hawe, D. Chiu , “ A q uantitative measure o f fairness and discrimin ation for resource allocation in Sha red Computer Systems, ” DEC T ech nical Repo rt, September 19 84. [6] J. Sun, L. Zhen g, and E. Mo diano, “Wireless channe l alloca- tion using an auction a lgorithm, ” in IE EE J. Selected A r eas Commun. , vol. 24, no. 5, pp. 1085-109 6, May 2006 . [7] J. Huang, R. Ber ry , and M. L. Ho nig, “ Auction-ba sed spec- trum sharing , ” ACM Mo bile Netw . a nd Applic. Journal , vol. 11, no. 3, pp. 405– 418, J un e 2006.