RECENTLY, cooperative communications [1] have gained

Transcription

1 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL Dstrbuted Relay Selecton and Power Control for Multuser Cooeratve Communcaton Networks Usng Stackelberg Game Bebe Wang, Student Member, IEEE, Zhu Han, Member, IEEE, and K.J. Ray Lu, Fellow, IEEE Abstract The erformance n cooeratve communcaton deends on careful resource allocaton such as relay selecton and ower control, but the tradtonal centralzed resource allocaton requres recse measurements of channel state nformaton (CSI). In ths aer, we roose a dstrbuted game-theoretcal framework over multuser cooeratve communcaton networks to acheve otmal relay selecton and ower allocaton wthout knowledge of CSI. A two-level Stackelberg game s emloyed to jontly consder the benefts of the source node and the relay nodes n whch the source node s modeled as a buyer and the relay nodes are modeled as sellers, resectvely. The roosed aroach not only hels the source fnd the relays at relatvely better locatons and buy an otmal amount of ower from the relays, but also hels the cometng relays maxmze ther own utltes by askng the otmal rces. The game s roved to converge to a unque otmal equlbrum. Moreover, the roosed resource allocaton scheme wth the dstrbuted game can acheve comarable erformance to that emloyng centralzed schemes. Index Terms Cooeratve communcaton networks, relay selecton, dstrbuted ower allocaton, game theory. Ç 1 INTRODUCTION RECENTL, cooeratve communcatons [1] have ganed much attenton as an emergng transmt strategy for future wreless networks. The basc dea s that relay nodes can act as a vrtual antenna array to hel the source node forward ts nformaton to the destnaton. In ths way, cooeratve communcaton effcently takes advantage of the broadcastng nature of wreless networks. Besdes, t exlots the nherent satal and multuser dverstes. The erformance n cooeratve communcaton deends on careful resource allocatons such as relay lacement, relay selecton, and ower control [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17]. In [2], the ower allocaton s otmzed to satsfy the outage robablty crteron. The authors n [3] rovde the analyss on symbol error rates and otmum ower allocatons for the decodeand-forward cooeraton rotocol n wreless networks. The energy-effcent broadcast roblem n wreless networks s consdered n [4]. The work n [5] evaluates the cooeratve dversty erformance when the best relay s chosen accordng to the average sgnal-to-nose rato (SNR) and the outage robablty of relay selecton based on nstantaneous SNRs. In [6], the authors roose a dstrbuted relay selecton scheme that requres lmted network knowledge wth nstantaneous SNRs. In [7], the relay. B. Wang and K.J.R. Lu are wth the Deartment of Electrcal and Comuter Engneerng, Unversty of Maryland, College Park, MD E-mal {bebewang, kjrlu}@umd.edu.. Z. Han s wth the Electrcal and Comuter Engneerng Deartment, Unversty of Houston, N324 Engneerng Buldng 1, Houston, TX E-mal zhan2@mal.uh.edu. Manuscrt receved 25 Ar. 2008; revsed 1 Aug. 2008; acceted 11 Set. 2008; ublshed onlne 16 Oct For nformaton on obtanng rernts of ths artcle, lease send e-mal to tmc@comuter.org, and reference IEEECS Log Number TMC Dgtal Object Identfer no /TMC assgnment roblem s solved for the multuser cooeratve communcatons. In [8], the cooeratve resource allocaton for OFDM s studed. The authors of [9] and [10] nvestgate the relay selecton roblem wth focus on when to cooerate and whch relay to cooerate wth, whch requres channel state nformaton (CSI). In [11], centralzed ower allocaton schemes are resented by assumng that all the relay nodes hel. In order to further mnmze the system outage behavors and mrove the average throughut, a selecton forward rotocol s roosed to choose only one best relay node to assst the transmsson. A centralzed resource allocaton algorthm for ower control, bandwdth allocaton, relay selecton, and relay strategy choce n an OFDMA-based relay network s roosed n [12]. The work n [13] develos dstrbuted ower control strateges for multho cooeratve transmsson schemes. Lfetme extenson for wreless sensor networks wth the ad of relay selecton and ower management schemes s nvestgated n [14]. The authors of [15] study the otmal ower allocaton roblem n the hgh-snr regme for dfferent relayng rotocols. Relay staton lacement and relay tme allocaton n IEEE j networks s nvestgated n [16]. However, most exstng work focuses on resource allocaton n cooeratve communcatons by means of a centralzed fashon. Such schemes requre that comlete and recse CSI be avalable n order to otmze the system erformance, whch are generally nether scalable nor robust to channel estmaton errors. Ths fact motvates the research on dstrbuted resource allocaton wthout requrng CSI. For dstrbuted resource allocaton, there are two man questons over multuser cooeratve wreless networks 1) among all the dstrbuted nodes, who can hel relay and mrove the source node s lnk qualty better and 2) for the selected relay nodes, how much ower do they need to transmt? /09/$25.00 ß 2009 IEEE Publshed by the IEEE CS, CASS, ComSoc, IES, & SPS Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

2 976 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 To answer these two questons, game theory s a natural and flexble tool that studes how the autonomous nodes nteract and cooerate wth each other. In game theory lterature of wreless networkng, n [18], the behavors of selfsh nodes n the case of random access and ower control are examned. In [19], statc rcng olces for multle-servce networks are roosed. Such olces can offer ncentves for each node to choose the servce that best matches ts needs so as to dscourage overallocaton of resources and mrove socal welfare. The work n [20] resents a ower control soluton for wreless data n the analytcal settng of a game-theoretcal framework. Prcng of transmt owers s ntroduced to mrove user utltes that reflect the qualty of servce a wreless termnal receves. A rcng game that stmulates cooeraton va rembursements to the relay s roosed n [21], but there was no detaled analyss on how to select the best relays and how to acheve the equlbrum dstrbutvely. In [22], the authors emloy a cooeratve game for the sngle-cell OFDMA resource allocaton. In general, n multuser cooeratve wreless networks wth selfsh nodes, nodes may not serve a common goal or belong to a sngle authorty. Therefore, a mechansm of rembursement to relay nodes s needed such that relay nodes can earn benefts from sendng ther own transmsson ower n helng the source node forward ts nformaton. On the other hand, f the source node remburses relay nodes for ther hel, t needs to choose the most benefcal relay nodes. Accordng to such characterstcs, n ths aer, we emloy a Stackelberg game [25] to jontly consder the benefts of the source node and relay nodes n cooeratve communcatons. The game s dvded nto two levels. The source node lays the buyerlevel game, snce t ams to acheve the best erformance wth the relay nodes hel wth the least rembursements to them. We analyze how many and whch relay nodes are selected by the source node to artcate n relayng after they announce ther otmal rces. In addton, we otmze how much servce (such as ower) the source node wll buy from each relay node. On the other hand, each relay node lays the seller-level game, n whch t ams to earn the ayment that not only covers ts forwardng cost but also gans as many extra rofts as ossble. Therefore, the relay node needs to set the otmal rce er unt for the servce so as to maxmze ts own beneft. To study the game outcomes, we analyze several roertes of the roosed game. Then, we develo a dstrbuted algorthm that can converge to the otmal game equlbrum. From the smulatons, the relay nodes close to the source node lay an mortant role n ncreasng the source node s utlty, so the source node lkes to buy ower from these referred relay nodes. On the other hand, n order to attract more buyng from the source, the relay adots a low-rce, hgh-market olcy to further ncrease ts beneft. If the total number of avalable relay nodes ncreases, the source node wll obtan a larger utlty value whle the average ayment to the relay nodes wll decrease. We fnally show that the roosed resource allocaton scheme wth dstrbuted game acheves comarable erformance to those of the centralzed scheme [11]. Fg. 1. System dagrams. The rest of the aer s organzed as follows In Secton 2, we descrbe the system model and formulate the cooeratve otmzaton as a Stackelberg game. We construct the dstrbuted mlementaton of multuser cooeraton transmssons and rovde the solutons n Secton 3. Smulaton results are shown n Secton 4. Fnally, conclusons are drawn n Secton 5. 2 SSTEM MODEL AND PROBLEM FORMULATION In ths secton, we frst derve the exresson of the maxmal achevable rate n cooeratve transmsson wth the relay nodes hel. Then, we formulate the otmzaton roblem of relay selecton and ower control usng a Stackelberg game. 2.1 System Model In the sequel, we emloy the amlfy-and-forward (AF) cooeraton rotocol [1] as our system model; other cooeraton rotocols [1] can be consdered n a smlar way. The system dagrams are shown n Fg. 1, n whch there are n total N relay nodes, one source node s, and one destnaton node d. The cooeratve transmsson conssts of two hases. In Phase 1, source node s broadcasts ts nformaton to both destnaton node d and each relay node r. The receved sgnals y s;d and y s;r at node d and node r can be exressed as y s;d ¼ ffffffffffffffffffffffffff P s G s;d x þ s;d ð1þ and y s;r ¼ ffffffffffffffffffffffffffff P s G s;r x þ s;r ; ð2þ Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

3 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS where P s reresents the transmt ower at node s, x s the broadcast nformaton symbol wth unt energy from node s to node d and node r, G s;d and G s;r are the channel gans from node s to node d and node r, resectvely, and s;d and s;r are the addtve whte Gaussan noses (AWGNs). Wthout loss of generalty, we assume that the nose ower s the same for all the lnks, denoted by 2. We also assume that the channels are stable over each transmsson frame. Wthout the relay nodes hel, the SNR that results from the drect transmsson from node s to node d can be exressed by s;d ¼ P sg s;d 2 ; ð3þ and the rate of the drect transmsson s R s;d ¼ W log 2 1 þ s;d ; ð4þ where W s the bandwdth for transmsson, and s a constant reresentng the caacty ga. In Phase 2, relay node r amlfes y s;r and forwards t to destnaton d wth transmtted ower P r. The receved sgnal at destnaton node d s y r;d ¼ ffffffffffffffffffffffffffffff P r G r;dx r;d þ r;d; ð5þ where x r;d ¼ y s;r jy s;r j ð6þ s the transmtted sgnal from node r to node d that s normalzed to have unt energy, G r;d s the channel gan from node r to node d, and r ;d s the receved nose. Substtutng (2) nto (6), we can rewrte (5) as ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff P r G r;d P s G s;r X þ s;r r ;d ¼ ffffffffffffffffffffffffffffffffffffffffffffffff þ P s G s;r þ 2 r ;d ð7þ Usng (7), the relayed SNR for source node s, whch s heled by relay node r, s gven by P r P s G r ;dg s;r s;r;d ¼ 2 ðp r G r ;d þ P s G s;r þ 2 Þ ð8þ Therefore, by (4) and (8), we have the rate at the outut of the maxmal-rato combnng (MRC) detector wth one relay node r helng as R s;r ;d ¼ W 2 log 2 1 þ s;d þ s;r;d ð9þ If the relay nodes avalable to hel source node s at a certan tme consttute a set, denoted by L ¼fr 1 ;...;r N g, then we have 0 s;d þ P 1 s;r;d B r R s;r;d ¼ L W log 2 1 þ 2L A; ð10þ where L denotes a bandwdth factor. Accordng to dfferent network alcatons, L can have dfferent defntons. For the energy-constraned networks, L s set to 1. For the network wth a lmted bandwdth, the bandwdth should be dvded for the source node and relay nodes, and L deends on the number of relay nodes that actually hel forwardng, snce not all the relay nodes wll contrbute to a better erformance for the source node. If N 0 out of N relay nodes are selected by the source node, N 0 N, then L ¼ 1 N 0 þ1.1 We wll study the energy constraned scenaro frst, then we show the effects of the varyng L n the smulaton art. 2.2 Problem Formulaton To exlot the cooeratve dversty for multuser systems, from (10), two fundamental questons on resource allocaton need to be answered 1) whch relay nodes wll be ncluded, and 2) what s the otmal ower P r? However, solvng these ssues n a centralzed manner requres accurate and comlete CSI, brngng consderable overheads and sgnalng of nformaton about channel estmatons. In contrast, the dstrbuted resource allocaton only needs local knowledge about channel nformaton. Moreover, n general, nodes n multuser cooeratve wreless networks may belong to dfferent authortes and act selfshly. Incentves need to be rovded by the source node to the relay nodes for relayng the nformaton. Consequently, the source node needs to choose the most benefcal relay nodes. Accordng to the behavors of the source node and the relay nodes, we emloy a dstrbuted resource allocaton usng a Stackelberg-game-based scheme as the followng formulated roblem 1. Source node/buyer. The source node s can be modeled as a buyer and ams to obtan the most benefts at least ossble ayments. The utlty functon of source node s can be defned as U s ¼ ar s;r;d M; ð11þ where R s;r;d denotes the achevable rate wth the relay nodes hel, a denotes the gan er unt of rate at the MRC outut, and M ¼ X r 2L P r ¼ 1 P r1 þ 2 P r2 þþ N P rn ð12þ reresents the total ayments ad by source node s to the relay nodes. In (12), reresents the rce er unt of ower sellng from relay node r to source node s, and P r denotes how much ower node s wll buy from node r. The relay nodes helng source node s consttute a set, stll denoted by L; then, the otmzaton roblem for source node s or the buyer-level game can be formulated as max U s ¼ ar s;r;d M; st P r 0;r 2 L ð13þ fp r g 2. Relay node/seller. Each relay node r can be seen as a seller and ams to not only earn the ayment 1. The source node can know the number of avalable relay nodes by broadcastng ts sgnal and lstenng to the relay nodes feedback on whether to hel forward the source node s nformaton. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

4 978 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 that covers ts forwardng cost but also gan as many extra rofts as ossble. We ntroduce a arameter c, the cost of ower for relayng data, n our formulaton. Then, relay node r s utlty functon can be defned as U r ¼ P r c P r ¼ð c ÞP r ; ð14þ where P r s the source node s ower consumton by otmzng U s descrbed n (13). The otmzaton roblem for relay node r or the seller-level game s max >0 U r ¼ð c ÞP r ; 8 ð15þ The choce of the otmal rce s affected not only by each relay node s own channel condtons to the source node and the destnaton node but also by the other relay nodes rces. Ths s because the seller-level game s noncooeratve, and the relay nodes comete to get selected by source node s. If a certan relay node r j asks such a hgh rce that makes t less benefcal than the other relay nodes to source node s, then source node s wll buy less from relay node r j or even dscard t. On the other hand, f the rce s too low, the roft obtaned by (14) wll be unnecessarly low. Overall, there s a trade-off for settng the rce. If under the otmal rce, denoted by, the resultng utlty of relay node r s negatve,.e., Ur 0, then node r wll qut the seller-level game snce t cannot cover the basc cost by sellng ower to the source node. It s worth notcng that the only sgnalng requred to exchange between the source node and the relay nodes are the rce and the nformaton about how much ower P r to buy. Consequently, the roosed two-level gametheoretcal aroach can be mlemented n a dstrbuted way. The outcome of the roosed games wll be shown n detal n the followng secton. 3 ANALSIS OF THE PROPOSED GAMES Frst, we obtan closed-form solutons to the outcomes of the roosed games. Then, we rove that these solutons are the global otma. Furthermore, we show that the set of the solutons s a unque fxed ont and the roosed dstrbuted game converges to that ont. Fnally, we comare the erformance of the roosed dstrbuted scheme to that of a centralzed scheme. 3.1 Analyss of the Buyer-Level Game for the Source Node Relay Selecton by the Source Node As relay nodes are located n dfferent laces and ask dfferent rces for helng the source node, t may not be good for source node s to choose all relay nodes, esecally those wth bad channel condtons but askng a hgh rce. Moreover, f the source node wll exclude the less benefcal relay nodes sooner or later durng the buyer-level game, t s better to reject them at the begnnng so as to reduce the sgnalng overhead. Because source node s ams at maxmzng utlty U s through buyng an otmal amount of ower P r, then a natural way of relay selecton for source node s s to observe how U s vares wth P r,.e., observe the r. Snce source node s gradually ncreases the amount of ower bought from the relay nodes to aroach the otmum, by observng the sgn r when P r ¼ 0, node s can exclude (or select) those less (or more) benefcal relay nodes. From the defnton n (11), we know s ¼ s;r;d ; ¼ 1;...;N r When P rj ¼ 0, j ¼ 1;...;N, f r for relay node r, then r > 0, meanng that the source node wll obtan a larger U s by ncreasng P r. Otherwse, relay node r should be excluded. Then, a queston s how each relay node r asks ts rce at the begnnng. Snce n a dstrbuted mlementaton, each relay node does not know the other relay nodes rces, t s natural to frst tentatvely set ¼ c.if the ntal rce s lower than c, utlty U r wll be negatve and, hence, mractcal; on the other hand, f the ntal rce s greater than c, relay node r may be at the rsk of beng excluded by the source node. If under these lowest ntal rces, the source node would choose not to buy any ower from some relay node r, then r wll not artcate n the seller-level game because U r ¼ 0. To summarze the analyss above, the relay rejecton crtera of the source node are descrbed as follows Assume that the total number of the relay nodes s N. At frst, the source node tentatvely chooses P r ¼ 0, ¼ 1;...;N, and all the relay nodes set ther ntal rces as ¼ c, 8. For relay node r j, f c j rj Þ, then r j s rejected by the source node wth corresondngly P rj ¼ 0. It wll be shown later that ths rejecton s fxed and wll not change after the game s layed. Wth the roosed relay rejecton crtera, source node s can exclude the least benefcal relay nodes at the very begnnng. In ths way, the sgnalng overhead can be further reduced, because the source node and the rejected relay nodes no longer need to exchange ther nformaton about the urchased ower and rces Otmal Power Allocaton for the Selected Relay Nodes After the selecton, for the selected relay nodes that consttute a set L h ¼fr 1 ;...;r N 0g, we can solve the otmal ower P r by takng the dervatve of U s n (11) wth resect to P r r ¼ r ¼ 0; r 2 L h ð17þ For smlcty, defne C ¼ 1 þ s;d and W 0 ¼ aw ln 2. By (10), we get the frst term of U s as ar s;r;d ¼ aw log 2 C þ 1 X s;r;d r 2L h ð18þ ¼ W 0 ln 1 þ SNR 0 tot þ W 0 ln C; where SNR 0 tot ¼ X 0 s;r ¼ 1 X ;d C r 2L h r 2L h s;r ;d; ð19þ Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

5 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS and 0 s;r ¼ s;r ;d ;d C ¼ A ¼ A P r ; 1 þ B P P r þ B r wth A ¼ P sg s;r ð 2 þp and B sg s;dþ ¼ P sg s;r þ 2 G r. ;d Substtutng (12) and (18) nto (17), we have W 0 1 þ P A kp rk P rk þb k r k2l h ¼ A B ðp r þ B Þ 2 ð20þ ð21þ Snce the left-hand sde (LHS) of (21) s the same for any relay node on the rght-hand sde (RHS), by equatng the RHS of (21) for relay nodes r and r j, we get sffffffffffffffffffffffffffff P rj ¼ A j B j ðp r þ B ÞB j j A B ð22þ Substtutng the above P rj nto (20) and smlfyng, we have sffffffffffffffffffffffffffff A j j A B A j B j 0 s;r j;d ¼ 1 þ Bj P rj ¼ A j A j B j ðp r þ B Þ ð23þ Then, (19) can be reorganzed as " sffffffffffffffffffffffffffffff # SNR 0 tot ¼ A 1 A B A 1 B 1 1 þþ A A B A 1 B 1 ðp r þb Þ P r þ B " sffffffffffffffffffffffffffffffffffffff # N 0A B A N 0B N 0 þþ A N 0 A N 0B N 0 ðp r þ B Þ ¼ X sffffffffffffffffffff A B 1 X ffffffffffffffffffffffffffffff A j j A j B j r j 2L P r þ B h r j 2L h ð24þ Substtutng (24) nto (21), after some manulatons, we can have a quadratc equaton of P r. The otmal ower consumton s sffffffffffffffffffff Pr A B þ ffffffffffffffffffffffffffffffffffffffffffffffff ¼ 2 þ 4XW 0 B ; ð25þ 2X where X ¼ 1 þ P r j2l h A j, and ¼ P ffffffffffffffffffffffffffffff r j2l h j A j B j. The soluton n (25) can also be verfed by the Karush- Kuhn-Tucker (KKT) condton [27] to be the global otmum to roblem (13), snce the U s functon s concave n fp r g N and the suortng set fp r jp r 0;¼ 1;...;Ng s convex. 3.2 Analyss of the Seller-Level Game for the Relay Nodes Substtutng (25) nto (15), we have max U r ¼ð c ÞPr f ð 1 ;...; ;...; N 0Þ ð26þ g>0 We can note that (26) s a noncooeratve game by the relay nodes, and there exsts a trade-off between the rce and the relay node s utlty U r. If relay node r n good channel condtons asks for a relatvely low rce at frst, source node s wll buy more ower from relay node r, and U r wll ncrease as grows. When kees growng and exceeds a certan value, t s no longer benefcal for source s to buy ower from relay r, even though relay r may be n very good channel condtons. In ths way, P r wll shrnk and hence results n a decrement of U r. Therefore, there s an otmal rce for each relay node to ask for, deendng on the relay node s channel condtons. Besdes, the otmal rce s also affected by the other relay nodes rces snce the source node only chooses the most benefcal relay nodes. From the analyss above, by takng the dervatve of U r to and equatng t to zero, we ¼ P r þð c ¼ 0; r 2 L h ð27þ Solvng the above equatons of, we denote the otmal rces as ¼ 2 ; fg s;r g; fg r;dg ; r 2 L h ð28þ In Secton 3.1, we assume that the source node transmts wth a constant ower. However, f the source node has a lower transmsson ower, t s wllng to buy more ower from the relay nodes n order to obtan a hgh data rate, and hence, the relay nodes can ask hgher rces for helng the source node. On the other hand, f the source has a hgher transmsson ower, t wll buy less ower from the relay nodes and also ay less to them. 3.3 Exstence of the Equlbrum for the Proosed Game In ths secton, we rove that the solutons Pr n (25) and n (28) are the Stackelberg Equlbrum (SE) for the roosed game and show the condtons for the SE to be otmal by the followng roertes, rooston, and theorem. We frst defne the SE of the roosed game as follows Defnton 1. Pr SE and SE are the SE of the roosed game f for every r 2 L, when s fxed U s n P SE r o ¼ and for every r 2 L h, when P r s fxed U r SE su U s ðfp r gþ; 8r 2 L; ð29þ fp r g0 ¼ su >c U r ð Þ; 8r 2 L h ð30þ Then, we show that the otmzer Pr of (13) can be solved by r to zero by the followng roerty. Proerty 1. The utlty functon U s of the source node s jontly concave n fp r g N, wth P r 0, and s fxed, 8. Proof. See Aendx A.1. tu Due to Proerty 1, Pr n (25) s the global otmum that maxmzes the source node s utlty U s. Therefore, Pr satsfes (29) and s the SE Pr SE. Moreover, n the ractcal mlementaton of the game, the source node can fnd the otmal ower amount by gradually ncreasng the urchased ower from each relay node untl U s reaches ts maxmum wthout knowng CSI. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

6 980 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 In the followng two roertes, we show that the relay nodes cannot nfntely ncrease U r by askng arbtrarly hgh rces Proerty 2. The otmal ower consumton Pr for relay node r s decreasng wth ts rce when other relay nodes rces are fxed. Proof. See Aendx A.2. tu Consequently, there s a trade-off for each relay node to ask a roer rce, and we can solve the otmal rce by ¼ 0, the reason of whch s shown as follows Proerty 3. The utlty functon U r of each relay node s concave n ts own rce when ts ower consumton s the otmzed urchase amount from the source node as calculated n (25) and the other relay nodes rces are fxed. Proof. See Aendx A.3. tu Based on Proertes 1, 2, and 3, we can show that the relay rejecton crtera stated n Secton hel the source node reject the least benefcal relay nodes n the followng rooston Prooston 1. The relay rejecton crtera descrbed n Secton are necessary and suffcent to exclude the least benefcal relay nodes to the source node. By necessary, t means that any r n L h cannot get further dscarded n the followng U r maxmzaton rocess; Whle by suffcent, t means that even f we kee r j that satsfes the rejecton crtera n L h,ts stll dscarded n the followng U rj maxmzaton rocess. Proof. We frst rove the suffcent art. Assume that the relay rejecton crtera aly to some relay node r rj Þ < 0, when P r ¼ 0 and ¼ c, 8. Snce U s s concave n fp r g N, r j s otmal ower allocaton Pr j < 0. Suose source s does not exclude relay r j and n the followng rce udate rocess, all remanng relay nodes gradually ncrease ther rces to get more utltes. To rove that the new resultng Pr new j < 0, t suffces to rove that Pr j < 0, where Pr j denotes the ncrease of Pr j when each relay node r ncreases by a very small ostve amount from the cost c. Ths can be verfed equvalent by rovng X r r j f ¼c ;8g < 0 ð31þ We know that j A j B j þ ffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffff ¼ 2 þ 4XW 0 1 A B ffffffffffffffffffffffffffffffffffffffffffffff >0 j 2X 2 2 þ4xw 0 and j A j B j þ ffffffffffffffffffffffffffffffffffffffffffffffff ¼ 2 þ 4XW j j 2X 1 ffffffffffffffffffffffffffffff ð33þ j A j B j 1 ffffffffffffffffffffffffffffffffffffffffffffffff < 0; 2 j 2 þ 4XW 0 so t suffces to rove (31) by rovng the followng X ffffffffffffffffffffffffffff 1 A B ffffffffffffffffffffffffffffffffffffffffffffffff < 1 2 6¼j 2 þ 4XW 0 2 j ffffffffffffffffffffffffffffff j A j B ð34þ j 1 ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 f ¼c ;8g Wthout loss of generalty, assumng that the selected relay nodes generally share smlar roertes,.e., c ¼ c j ¼ c, 8 6¼ j, we can rove (34) by the followng nequalty X ffffffffffffffffffffffffffff 1 A B ffffffffffffffffffffffffffffffffffffffffffffffff < 1 X ffffffffffffffffffffffffffff A B 2 2 þ 4XW 0 2c 6¼j ¼ 1 2 j ffffffffffffffffffffffffffffff j A j B j 1 6¼j < 1 2 j ffffffffffffffffffffffffffffff j A j B j 1 ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 ð35þ Therefore, n the followng rce ncreasng rocess, r j s stll dscarded by the source node by observng Pr new j < 0. Next, we rove the necessary art. In each round, any two relay nodes r k and r udate ther rces n two consecutve stes. Frst, r k ncreases ts rce k to the new otmal new k, and then, by (32), the resultng Pr new s larger than Pr, where Pr > 0. Thus, Pr new > 0, whch means that r wll not be dscarded f r k ncreases k. Second, after k s ncreased, r ncreases ts own rce. In (54), assumng that s the rce for r such that Pr ¼ 0 when the other relay nodes rces are fxed, we have r A B þ ffffffffffffffffffffffffffffffffffffffffffffffff < B þ 2 þ 4XW 0 0 2X By Proerty 3, the otmal rce ¼ 0 must satsfy c < <. Ths means that to maxmze U r, r asks a lower rce than to avod beng rejected by the source node. tu If relay node r gets selected by the source node, due to the concavty of U r roved n Proerty 3, r can always fnd ts otmal rce 2ðc ; 1Þ, and thus, U r ð ÞU r ð Þ, 8r 2 L h. Together wth Proerty 1, we conclude the followng theorem Theorem 1. The ar of fpr g N n (25) and f gn0 n (28) s the SE for the roosed game, where the SE s defned n (29) and (30). In the next secton, we wll show that the SE s unque, and the roosed game converges to the unque SE when each relay node udates ts rce accordng to a smle functon. 3.4 Convergence of the Dstrbuted Prce Udatng Functon From the revous secton, one relay node needs to modfy ts own rce after the other relay nodes change ther rces. Consequently, for every r 2 L h, relay node r udates so that ts utlty U r satsfes the followng @ h ð c ÞPr ¼ P r þð c ¼ 0; ð37þ Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

7 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS wth the equalty holdng f and only f reaches the otmum. After rearrangng (37), we have ¼ I ðþ ¼ ð38þ =@ In order to calculate n (38), each relay node r lstens to the nstantaneous feedback nformaton about Pr =@ from the source node, whch s smlar to the needed nformaton exchange n teratve ower control [28]. Then, the udatng of the relay nodes rces can be descrbed by a vector equalty of the form ¼ IðÞ; ð39þ where ¼ð 1 ;...; N 0Þ, wth denotng relay node r s rce, and IðÞ ¼ðI 1 ðþ;...;i N 0ðÞÞ, wth I ðþ reresentng the rce cometton constrant to r from the other relay nodes. Therefore, for the N 0 relay nodes n set L h wth the cometton constrants n (39), the teratons of the rce udatng can be exressed as follows P r ðt þ 1Þ ¼IðtÞ ð Þ ð40þ Remark. If K source nodes, denoted by S¼fs 1 ;s 2 ;...;s K g, exst n the network, assumng that the rce of relay node r when t hels source node s k s s k wth corresondng ower P s k r, then the buyer-level game for each source node s k s essentally the same as the sngle-buyer case. However, the seller-level game becomes more comlcated, because now, relay node r needs to choose K rces, f sk g s k2s, n order to maxmze ts utlty U r ¼ X sk c P s k r ð41þ s k 2S If the relay nodes treat all source nodes equally wth s k ¼, 8s k 2S,.e., relay node r asks a unform rce no matter whch source node t hels, then utlty U r s smlfed as U r ¼ð c Þ X s k2s P sk r ; ð42þ and the roosed algorthm s stll alcable, wth the followng modfed rce udatng functon ¼ I ðþ ¼ c P s P s k k2s r P s k r =@ ð43þ However, f the relay nodes treat the source nodes dfferently, then each relay node r needs to udate K rces, f sk g s k2s, usng the followng udatng functon sk ¼ I ð sk Þ¼ c P s k r s k r =@ k Therefore, f there are multle source nodes n the network, the roosed algorthm s stll alcable the buyer-level game of each source node s essentally the same as the sngle-source case; the only change s n the seller-level game of the relay nodes, where the rce udatng functon s modfed as n (43) or (44). We show next the convergence of the teratons n (40) by rovng that the rce udatng functon IðÞ s a standard functon [28]. Defnton 2. A functon IðÞ s standard f for all 0, the followng roertes are satsfed [28]. Postvty. IðÞ > 0.. Monotoncty. If 0, then IðÞ Ið 0 Þ.. Scalablty. For all >1, IðÞ >IðÞ. Prooston 2. The rce udatng functon IðÞ s standard. Proof. See Aendx A.4. tu In [28], a roof has been gven that startng from any feasble ntal ower vector, the ower vector I n ðþ roduced after n teratons of the standard ower control algorthm gradually converges to a unque fxed ont. As we have dscussed n Secton 3.1.1, t s natural for the relay nodes to ntalze the rces as ¼ c, because lowerng below c wll result n a negatve utlty U r, whle by settng above c, relay node r may be at the rsk of beng excluded by the source node at the very begnnng. Therefore, we assume that the ntal rce vector s c ¼ðc 1 ;...;c N 0Þ, where c s the cost er unt of ower for relay node r, as ntroduced n (14). Therefore, we can conclude that startng from the feasble ntal rce vector c ¼ðc 1 ;...;c N 0Þ, the teraton of the standard rce udatng roduces a nondecreasng sequence of rce vectors I n ðcþ that converges to a unque fxed ont. From (37), we know that for relay node r 2L h, ts utlty U ¼ 0 every tme after r udates ts rce gven the feedback from the source. After the vector I n ðþ converges to, no relay can gan a hgher utlty by further varyng ts rce, ¼ 0 8r 2 L h. From (27) and (28), we know that s exactly the otmal rce vector. As Proerty 1 shows, U s s concave n P r, so the source node can gradually ncrease the ower from 0 and fnd the otmal Pr. Thus, f the rces of all the selected relay nodes converge to ther otma, then the source node wll corresondngly buy the otmal ower. Therefore, once I n ðþ converges to, P r and converge to the SE. It s worth mentonng that although the closed-form solutons fpr g N n (25) and f gn0 n (28) are functons of the CSI, n the ractcal mlementaton of the game, the source node can fnd the otmal ower amount by gradually ncreasng the urchased ower from each relay node untl U s reaches ts maxmum due to Proerty 1. Actually, the reason why we exress the closed-form soluton fpr g N as a functon of CSI s just to show that the relay node s utlty U r s concave n (Proerty 3) and hence to rove that the relay nodes can utlze the roosed rce udatng algorthm and gradually converge to the otmal rce f gn0 (Prooston 2). Hence, the only sgnalngs between an ndvdual relay node and the source node are the nstant rce and corresondng ower, and no CSI s needed. Moreover, there s no rce nformaton exchange between the relay nodes. Therefore, the roosed game acheves ts equlbrum n a dstrbuted way wth local nformaton. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

8 982 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL Comarson wth the Centralzed Otmal Scheme In order to demonstrate the erformance of our roosed game-theoretcal scheme, we frst nvestgate a centralzed otmal ower allocaton roblem wth closed-form solutons. Then, we llustrate the numercal comarson of the erformance n Secton 4. Suose the system resources are shared by all avalable N relay nodes. From [11], we can model the centralzed otmal ower allocaton roblem as follows max P r st W N þ 1 log 2 X N P r 1 þ P s;d þ N s;r ;d P tot r ; 0 P r P max r 8; ð45þ where s;d and s;r ;d are defned n (3) and (8), resectvely. Because log 2 ð1 þ xþ s a strctly ncreasng functon of x, reorganzng the objectve functon of (45), we can get an equvalent otmzaton roblem as n [11] mn st X N X N P r P 2 s a2 þ P sa P s a þ P r b þ 1 P tot r ; 0 P r P max r 8; where a ¼ Gs;r, and b 2 ¼ Gr ;d The soluton of (46) can be solved as 0sffffffffffffffffffffffffffffffffffffffffffffffffff 1 Ps P r 2a2 þ P sa P sa þ 1 A b b 2. Pr max 0 ð46þ ; ð47þ where s a constant chosen to meet the total ower constrant, and ðxþ u l s defned as 8 < l; x < l; ðxþ u l ¼ x; l x u; ð48þ u; u < x In order to make a far comarson, n the roosed game-theoretcal scheme, we can change a, the gan er unt of the rate, to equvalently reflect dfferent Pr tot constrants as n the centralzed scheme. The reason s exlaned as follows When a s so large that the total ayment M n U s s neglgble, U s ar s;r;d, then the otmal ower consumton of the roblem n (13) wll be Pr 1. It s equvalent to have Pr tot 1 n the centralzed scheme. On the contrary, when a s so small that the total gan of the rate ar s;r;d n U s s neglgble, U s M ¼ P P r, then n ths case, we get Pr ¼ 0. It s equvalent to have Pr tot ¼ 0 n the centralzed scheme. Therefore, by varyng a n a large range, we can get the otmal achevable rates corresondng to dfferent total ower consumtons and farly comare the erformance wth that of the centralzed scheme. 2 For more detaled dscussons, lease see Aendx A We do not nclude exlctly the constrants on the relay nodes ower n the roosed game for ease of analyss. From the smulaton n the next secton and the analytcal roof n Aendx A.5, t wll be shown that the game wll acheve comarable erformance when we consder the constrants on relay nodes ower. However, the centralzed otmal ower allocaton scheme needs consderable overheads and sgnalng, because t requres that the comlete CSI,.e., G s;d, G s;r, and G r ;d, s avalable. In Secton 4, we show that our roosed dstrbuted scheme can acheve comarable erformance whle the needed sgnalng between the source node and the relay nodes s only the nformaton about the rces and the ower consumtons. 4 SIMULATION RESULTS AND ANALSIS To evaluate the erformance of the roosed scheme, n what follows, the smulaton results for a one-relay case, for a two-relay case, and for a multle-relay case are to be shown. Then, we rovde the erformance comarsons of the roosed aroach wth the centralzed otmal scheme. Fnally, we dscuss the effect of the bandwdth factor. 4.1 One-Relay Case There are one source-destnaton node ar ðs; dþ and one relay node r n the network. Destnaton node d s located at coordnate (0 m, 0 m), and source node s s located at coordnate (100 m, 0 m). We fx the y-coordnate of relay node r at 25 m, and ts x-coordnate vares wthn the range of [250 m, 300 m]. The roagaton loss factor s set to two. The transmt ower P s ¼ 10 mw, the nose level s 2 ¼ 10 8 W, and we select the caacty ga ¼ 1, W ¼ 1 MHz, the gan er unt of rate a ¼ 001, and the cost er unt of ower c ¼ 02. In Fg. 2a, we show the otmal rce for relay node r and the otmal ower bought by source node s, resectvely. In ths smulaton, relay node r moves along a lne. We observe that when relay node r s close to source node s at (100 m, 0 m), the source can gan a hgher U s n the game, so the relay can more effcently hel source node s. However, the relay cannot arbtrarly select ts rce n order to mrove ts utlty. As we have shown n Proerty 2 and Proerty 3, the otmal ower Pr the source buys from relay node r s decreasng wth, and node r s utlty U r s concave n. Snce the objectve of the relay node s to maxmze ts utlty U r, the rce should be carefully selected nstead of an arbtrarly large value. As decreasng rce can attract more buyng from the source, relay node r reduces ts rce to enhance ts utlty U r. When relay node r moves close to destnaton node d at (0 m, 0 m), relay node r can use a very small amount of ower to relay source node s s data, so relay node r sets a very hgh rce n order to get more rofts by sellng ths small amount of ower. However, even the rce s hgher than that when r s closer to the source, the utlty U r s stll lower when the relay s close to the destnaton. When relay node r kees movng away from destnaton node d, source node s stos buyng servces because askng for relay node r s hel s no longer benefcal to source node s. Smlarly, when relay node r moves n the ooste drecton and locates far away from source node s, s would not buy servces ether. In Fg. 2b, we show, resectvely, the otmal utltes relay node r and source node s can obtan usng the roosed game. When relay node r s close to source node s, both r and s can get ther maxmal utltes. The reason s Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

9 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS Fg. 2. One-relay case wth the relay node at dfferent locatons. (a) Otmal rce and ower of the relay node. (b) Otmal utlty of the relay and the source node. that around ths locaton, relay node r can most effcently hel source node s ncrease ts utlty, and the otmal rce of relay node r s lower than that when r s at other locatons. Therefore, source node s buys more ower, resultng n a hgher utlty to relay node r. 4.2 Two-Relay Case We also set u two-relay smulatons to test the roosed game. In the smulatons, the coordnates of s and d are (100 m, 0 m) and (0 m, 0 m), resectvely. Relay node r 1 s fxed at the coordnate (50 m, 25 m), and relay node r 2 moves along the lne from (250 m, 25 m) to (300 m, 25 m). For each r, we set c ¼ 01. Other settngs are the same as those of the one-relay case. In Fg. 3, we can observe that even though only r 2 moves, the rces of both the relay nodes change accordngly, and s buys dfferent amounts of ower from them. Ths fact s because the relay nodes nfluence and comete wth each other n the roosed game. When relay node r 2 s close to d at (0 m, 0 m), t sets a very hgh rce as exlaned n the one-relay case. Accordngly, r 1 ncreases ts rce and Pr 1 slghtly decreases. When r 2 s close to s at (100 m, 0 m), r 2 s more sutable to hel s than r 1, and Ur 2 s very hgh. Hence, n order to attract source s to buy ts servce, r 1 reduces ts rce a lot, but Ur 1 stll dros. Because r 2 close to s results n the most effcent hel to s from the relay nodes, both U s and M reach ther maxma around ths locaton. As r 2 moves far away from s or d, r 2 s rce dros because r 2 s less comettve than r 1. When ts utlty s less than 0, r 2 quts the cometton, and Pr 2 ¼ 0mW. At that moment, r 1 can slghtly ncrease ts rce snce there s no cometton. However, source node s wll buy slghtly less ower from r 1. Ths fact suresses the ncentve of r 1 to ask an arbtrarly hgh rce n the absence of cometton; otherwse, r 2 wll rejon the cometton. At the transton ont when r 2 quts, U r1 s smooth. Note that when r 2 moves to (50 m, 25 m), whch s the same locaton as r 1, the ower consumtons, the rces, and the utltes of both relay nodes are the same. Ths s because the source node s ndfferent for the two relay nodes located together and treats them equally. 4.3 Multle-Relay Case We then set u multle-relay smulatons to test the roosed game. The coordnates of the source node and the destnaton node are (100 m, 0 m) and (0 m, 0 m), resectvely, and the relay nodes are unformly located wthn the range of [50 m, 150 m] n the x-axs and [0 m, 20 m] n the y-axs. In Fg. 4, we can observe that as the total number of the avalable relay nodes ncreases, the comettons among the relay nodes become more severe, so the average rce er relay node decreases. The source node ncreases the amount of average ower urchase when the number of the relay nodes s not so large (less than three), because the average rce s decreased. When the number of the relay nodes becomes larger (greater than three), the source node decreases the amount of average ower urchase, because t buys ower from more relay nodes. Corresondngly, the total ayments are shared by more relay nodes, whch leads to less average ayment from the source node. Thus, the source node obtans an ncreasng utlty. 4.4 Convergence Seed of the Proosed Game As descrbed n Secton 3.4, the relay nodes start ncreasng ther rce from c after the N 0 more benefcal relay nodes have been selected by the source node. Denote the rce vector at tme t as ðtþ ¼ð 1 ðtþ; 2 ðtþ;...; N 0ðtÞÞ. From (25), the otmal ower urchased by the source node at tme t can be denoted as Pr ðtþ ¼ Pr ððtþþ ¼ Pr ð 1 ðtþ; 2 ðtþ;...; N 0ðtÞÞ ð49þ In order to =@ and udate ther rces by (38), the selected relay nodes wll smultaneously ncrease each ðtþ by a small amount. The source node receves ths rce udatng and =@ usng the followng P r ð 1 ðtþ;...; ðtþþ ;...; N0ðtÞÞPr ððtþþ Substtutng the above aroxmaton sgnaled from the source node nto (38), where the numerator Pr ðtþ s as defned n (49), the relay nodes can obtan ðt þ 1Þ¼IððtÞÞ. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

10 984 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 Fg. 3. Two-relay case wth relay node r 2 at dfferent locatons. (a) Otmal rces of the relay nodes. (b) Otmal ower consumtons of the relay nodes. (c) Otmal utltes of the relay nodes. (d) U s and M of the source node. In the above udatng rocess, the source node can sgnal the aroxmated dervatves calculated by (50) to all the relay nodes at one tme and need not nteract wth them one by one. Therefore, ths rocess can be vewed as one teraton and does not deend on the number of relay nodes. Then, we conducted smulatons when two to four relay nodes are avalable to hel the source node and observe the convergence behavor of the roosed game. In Fg. 5a, t s seen that the roosed scheme has fast convergence to the SE. It takes less than 15 teratons untl the rce vector converges to the otmum when there are two relay nodes n the system for a ¼ 1, where a denotes the gan er unt of rate as defned n (11), and less than 10 teratons for a ¼ 02. In addton, n Fg. 5b, the convergence behavor of R s;r;d to the otmzed transmsson rate usng P r and aears to be exonentally fast. Fnally, we kee a ¼ 1, ncrease the number of relay nodes to three and four, and show the convergence behavor of Fg. 4. Multle-relay case wth dfferent numbers of relay nodes. (a) Average rce and ower versus number of relay nodes. (b) Average U s and M versus number of relay nodes. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

11 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS Fg. 5. Observaton of convergence seed. (a) Prces of relay nodes versus teraton ndex. (b) Dfference n R s;r;d versus teraton ndex. (c) Prces of relay nodes versus teraton ndex (three relays). (d) Prces of relay nodes versus teraton ndex (four relays). the rces n Fgs. 5c and 5d, resectvely. We can see that the number of teratons untl convergence haens almost kees the same as there are more relay nodes exstng n the system. 4.5 Comarson wth the Centralzed Otmal Scheme To comare the erformance of the roosed game wth the centralzed scheme, we set u two smulatons as follows There are two relay nodes and one ðs; dþ ar. One of the relay nodes s fxed at coordnate (50 m, 25 m) and the other node s fxed at (60 m, 25 m) and (40 m, 25 m) n the two smulatons, resectvely. For the centralzed scheme defned n (45), we set Pr max ¼ 10 mw and let Pr tot vary wthn the range of [10, 20] mw. Then, we can obtan a curve of the maxmal rates versus dfferent total ower consumton constrants. For the dstrbuted scheme, as exlaned n Secton 3.5, by varyng a and ncludng the same constrant P max r ¼ 10 mw on P r, we can also get dfferent total ower consumtons and corresondng maxmal rates. In Fgs. 6a and 6b, we observe that the roosed game acheves almost equal rates as the centralzed scheme under the same total ower consumtons. 4.6 Effect of the Bandwdth Factor As exlaned n Secton 2.1, for the network wth a lmted bandwdth, the bandwdth should be dvded for the source node and the relay nodes. If N 0 out of the total N avalable relay nodes are selected by the source node, where N 0 N, then L ¼ 1 N 0 þ1 n (10), ndcatng that the bandwdth factor decreases as more relay nodes hel the source node. Thus, usng fewer relay nodes among the selected N 0 relay nodes may further ncrease U s for the source node. Therefore, for the networks wth a lmted bandwdth, t s not suffcent for the source node to mlement only one round of relay selecton. Instead, after source node s selects N 0 relay nodes usng the relay rejecton crtera, s contnues to try dfferent subsets of the N 0 selected relay nodes, get the corresondng otmal utlty Us for each tral, and choose the subset of relay nodes that results n the largest Us. In ths secton, we set u smulatons to observe the effect of the varyng bandwdth factor. We set a ¼ 085, relay node r 1 s at (100 m, 5 m), and r 2 moves along the lne between onts (250 m, 5 m) and (300 m, 5 m). In Fg. 7, we show the otmal Us obtaned by the source node under four scenaros,.e., when no relay node, only r 1, only r 2, and both relay nodes are avalable to hel, resectvely. We see that when r 2 moves close to r 1 and the source node s,.e., the x-coordnate of r 2 les n the nterval of (85 m, 115 m), both r 2 and r 1 are benefcal to node s. Moreover, as exlaned n the multle-relay case n Secton 4.3, snce there s cometton between two relay nodes, the average ower bought from the relay node s much greater, whle the average ayment s lower, comared wth the one-relay case. Hence, although L s only 1/3, U s ðr 1 ;r 2 Þ s stll greater than U s (r only), for Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

12 986 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 Fg. 6. Otmal rate n dstrbuted and centralzed schemes. (a) x-coordnate of r 2 ¼ 60 m. (b) x-coordnate of r 2 ¼ 40 m. ¼ 1; 2, and both relay nodes are selected. When r 2 moves farther away from s, r 2 s less benefcal and asks a hgher rce, and r 1 s also nfluenced to ask a hgher rce. Therefore, U s ðr 1 ;r 2 Þ decreases and becomes smaller than U s (r 1 only) where L s 1/2. Thus, choosng r 1 only s better than choosng both relay nodes. When r 2 kees movng away from s, t s no longer benefcal for the source node s to select t to hel. Hence, r 2 wll be rejected, and the bandwdth factor jums from 1/3 to 1/2. Therefore, there are two bums of U s ðr 1 ;r 2 Þ when the x-coordnate of r 2 s about 70 m and 140 m. 5 CONCLUSIONS In ths aer, we roose a game-theoretcal aroach for the dstrbuted resource allocaton over multuser cooeratve communcaton networks. We target to answer two questons Who wll be the relays, and how much ower for the relays to transmt for the cooeratve transmsson? We emloy a Stackelberg (buyer/seller) game to jontly consder the benefts of the source node and the relay nodes. The roosed scheme not only hels the source node otmally choose the relay nodes at better locatons but also hels the cometng relay nodes ask otmal rces to maxmze ther utltes. From the smulaton results, relay nodes closer to the source node can lay a more mortant role n ncreasng the source node s utlty, so the source node buys more ower from these referred relay nodes. If the total number of the avalable relay nodes ncreases, the source node can obtan a larger utlty value, and the average ayment to the relay nodes shrnks, due to more severe comettons among the relay nodes. It s also shown that the dstrbuted resource allocaton can acheve a comarable erformance to that of the centralzed scheme, wthout requrng knowledge of CSI. The roosed Stackelberg-game-based framework can be extended as a buldng block n large-scale wreless ad hoc networks to stmulate cooeraton among dstrbuted nodes. APPENDIX A A.1 Proof of Proerty 1 Takng the second-order dervatves of the source node s utlty U s, we can get 2 U 2 r " # 2 W 0 A B ¼ 2 1 þ PN A k P rk ðp r þ B Þ 2 P rk þb k k¼1 W 0 2 A B 1 þ PN A kp rk ðp r þ B Þ 3 P rk þb k k¼1 ð51þ Fg. 7. Otmal U s ncludng the bandwdth-factor effect, wth dfferent relay nodes hel, a ¼ 2 U s 2 rj 1 þ PN A k P rk W 0 P rk þb k k¼1 A B A j B j ðp r þ B Þ 2 2 P rj þ B j ð52þ Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

13 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS For each relay, by defnton, W 0 >0, A >0, B >0, and P r 0. As a U < 0, U s r rj < 0. It s straghtforward to verfy U 2 U U s r j rj Þ 2 > 0, 8 6¼ j. Moreover, U s s contnuous n P r, so when P r 0, U s s strctly concave n each P r, 8, and jontly concave over fp r g N as well. A.2 Proof of Proerty 2 Takng the frst-order dervatve of the otmal ower consumton Pr, we have A B þ ffffffffffffffffffffffffffffffffffffffffffffffff ¼ 2 þ 4XW 2X 1 2 ffffffffffffffffffffffffffff A B 1 ffffffffffffffffffffffffffffffffffffffffffffffff < 0 2 þ 4XW 0 ð53þ Therefore, Pr s decreasng wth. Ths s because when some relay node ndvdually ncreases ts rce whle the others kee the same rces as before, the source node wll buy less from that relay node. A.3 Proof of Proerty 3 Pr s a contnuous functon of,sou r s contnuous n too. Takng the dervatves of the relay node s utlty U r results n r A B þ ffffffffffffffffffffffffffffffffffffffffffffffff ¼B þ 2 þ 4XW 2X 1 ffffffffffffffffffffffffffff ð54þ c A B 1 ffffffffffffffffffffffffffffffffffffffffffffffff 2 2 þ 4XW 0 and, further, 2 ffffffffffffffffffffffffffff U r A 2 ¼ 2X 1 A B ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 qffffffffffffff A B 3c 4 2 þ ffffffffffffffffffffffffffffffffffffffffffffffff 8X þ 4XW 0 h ffffffffffffffffffffffffffff 2 2 þ 2 A B þ 4XW 0 ð 3c Þ ffffffffffffffffffffffffffff þ A B 2 þ 2 A B ð 3c Þ þ A B 4XW 0 ð4c Þ ; ð55þ where ¼ ffffffffffffffffffffffffffff A B. Snce A, B,,, c, X, and W 0 > 0, we U r < 0. Therefore, 2 r s concave wth resect to. A.4 Proof of Prooston 2 Postvty. By Proerty < 0. Moreover, f c > 0 and P r 0, then by the defnton of (38), I ðþ c > 0. Therefore, n a real rce udatng rocess, each relay node starts ncreasng ts rce from c. Scalablty. Comarng IðÞ and IðÞ n an elementwse manner, we have I ðþ I ðþ ¼ð 1Þc P r ðþ þ P r ðþ r r ðþ=@ Snce >1, ð 1Þc > 0. Then, the roblem reduces to rovng that the second term n the RHS of (56) s ostve. If we defne F ðw 0 Þ as follows F ðw 0 Þ¼ Then, we can get P r r ðþ=@ ¼ 1 2 P r r ðþ=@ ¼ 1 qffffffffffffff B ffffffffffffffffffffffffffffffff A B þ 2 þ4xw 0 2X ffffffffffffffffffffffffffff 1 A B 1 ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 B 1 qffffffffffffff ffffffffffffffffffffffffffffffffffffffff þ 1 2 A B 2 þ4xw 0 = 2X " ffffffffffffffffffffffffffff # 1 A B 1 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 = ¼ F ðw 0 =Þ ð57þ ð58þ Therefore, to rove the ostvty of the second term of the RHS of (56) s equvalent to rove F ð W 0 Þ >F ðw 0 Þ, where W 0 <W0. Snce F ðw 0 Þ s contnuous and dfferentable n W 0, we only need to < 0. we get 0 0 ¼ 8X A B ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 ffffffffffffffffffffffffffff 2 ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 A B 2 þ ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 " ffffffffffffffffffffffffffff XB 2 þ 4XW 0 þ A B þ 1 2 A B þ ffffffffffffffffffffffffffffffffffffffffffffffff # 2 2 þ 4XW 0 ð59þ The frst four terms of the RHS of (59) are all ostve. After extensve numercal tests for a wde range of arameters when the nodes are randomly located, we observe that the last term n the square brackets s negatve. 0 n (59) s less than zero. Therefore, we can clam that IðÞ >IðÞ. Monotoncty. Suose and 0 are dfferent rce vectors, and the vector nequalty 0 means that 0, 8 2f1;...;N 0 g.if86¼ j,, j 2f1;...;N 0 g, I j ð½ 1 ;...; ;...; j ;...; N 0ŠÞ I j ð½ 1 ;...; 0 ;...; j;...; N 0ŠÞ; and I ð½ 1 ;...; ;...; j ;...; N 0ŠÞ I ð½ 1 ;...; 0 ;...; j;...; N 0ŠÞ; then monotoncty can be shown to hold. Therefore, the roblem reduces to j ðþ=@ 0 ðþ=@ 0. Exandng and j ðþ=@ to Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

14 988 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 Fg. 8. Comarson of otmal relay ower of the game and the centralzed scheme. (a) Otmal ower n game. (b) Otmal ower n centralzed scheme. (c) Otmal ower n game ðp r P max Þ. (d) Otmal ower n centralzed scheme ðp r r P max Þ. r exress t as a roduct of a ostve term and a second term, we get ffffffffffffffffffff 1 A j ðþ ffffffffffffffffffffffffffffffff ¼ 2 þ4xw ffffffffffffffffffffff 1 j A j B j j 1 ffffffffffffffffffffffffffffffff 2 þ4xw 0 " qffffffffffffffff ffffffffffffffffffffffffffffffff A j B j þ 2 þ4xw 0 j 2X B j 1 qffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff A j B j ð60þ þ 2 þ4xw 0 j 2X ffffffffffffffffffff ffffffffffffffffffffff ja jb j 1 ffffffffffffffffffffffffffffffff þ ja jb j # 2 þ4xw 0 2 þ4xw 0 ffffffffffffffffffffff ja jb j 1 ffffffffffffffffffffffffffffffff 2 þ4xw 0 The frst term of the RHS of (60) s ostve; to decde the sgn of the second term, t suffces to comare the dfference of the denomnator and numerator of the fracton nsde qffffffffffffff the square brackets, whch are both ostve. A By usng B < XB, roved n the scalablty roerty, we can fnally show that sffffffffffffffffffffff A j B j þ ffffffffffffffffffffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffff 2 þ 4XW 0 j A j B j 1 ffffffffffffffffffffffffffffffffffffffffffffffff j 2X 2 þ 4XW 0 sffffffffffffffffffffff A j B j þ ffffffffffffffffffffffffffffffffffffffffffffffff 2 þ 4XW 0 B j j 2X ffffffffffffffffffffffffffffff j A j B j 1 ffffffffffffffffffffffffffffffffffffffffffffffff þ ffffffffffffffffffffffffffffff j A j B j 2 þ 4XW 0 2 þ 4XW 0 >B j 1 2 ffffffffffffffffffffffffffffff ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff j A j B j j A j B j j A j B j ffffffffffffffffffffffffffffffffffffffffffffffff þ 2 þ 4XW 0 2 þ 4XW 0 ffffffffffffffffffffffffffffff 2 j A j B j ¼ B j 1 ffffffffffffffffffffffffffffffffffffffffffffffff > 0; 2 þ 4XW ð61þ > 0. Smlarly, we can also > 0, so monotoncty holds for the rce udatng functon. Fnally, from the above three arts, we rove that the rce udatng functon s standard. Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

15 WANG ET AL. DISTRIBUTED RELA SELECTION AND POWER CONTROL FOR MULTIUSER COOPERATIVE COMMUNICATION NETWORKS A.5 Analytcal Comarson between the Centralzed Scheme and the Proosed Game In ths Aendx, we sketch the analytcal comarson between the centralzed otmzaton scheme n Secton 3.5 and the roosed dstrbuted game. Frst, accordng to (45), we can reresent the Lagrangan of the centralzed otmal scheme as follows L cen ðp r ;;Þ¼R s;r;d þ XN þ Nþ1 X N ðp r Þþ XN P r P tot r ; P r P max r ð62þ where the Lagrangan multlers are ¼ð 1 ;...; Nþ1 Þ and ¼ð 1 ;...; N Þ, wth, 0. In the roosed game, each node maxmzes ts own utlty, defned n (13) and (15), so we can equvalently vew the objectve as a vector otmzaton, and the scalarzaton can be reresented n the followng max U s þ XN w U r ð63þ st 0 P r P max r ; ¼ 1;...;N; ð64þ 0; ¼ 1;...;N; ð65þ where w ¼ðw 1 ;...;w N Þ s any weght vector, and w > 0, 8. Smlarly, we can exress the Lagrangan for the scalarzed otmzaton as ~L game ðpr; ; ~ ; ~; ~Þ ¼U s þ XN þ XN w U r þ XN ~ ðp r Þþ XN ~ ð Þ ~ P r Pr max ; ð66þ where the Lagrangan multlers are ~ ¼ð ~ 1 ;...; ~ N Þ, ~ ¼ð~ 1 ;...; ~ N Þ, and ~ ¼ð~ 1 ;...; ~ N Þ, wth ~, ~, ~ 0, 8. Substtutng (13) and (15) nto (66), after some manulaton, ~L game ðpr; ; ~ ; ~; ~Þ becomes ~L game ðpr; ; ; ~ ~; ~Þ ¼aR s;r;d þ XN ½w ð c Þ XN þ XN ~ þ XN ~ ðp r Þ ~ P r Pr max ŠP r ð67þ Snce a>0 and for smlcty, the above Lagrangan can be further converted to ~L 0 game ðpr; ;;;Þ¼R s;r;d þ XN XN þ XN ~ a þ XN ~ a ½w ð c Þ Š a P r P max r ~ a ðp r Þ P r ð68þ Comarng (62) and (68), we can fnd that they have smlar terms, whch can be vewed as one-to-one mangs,.e., $ ~ a, $ ~ a, and X N P N Nþ1 P r Pr tot $ ½w ð c Þ ŠP r P N ~ a Wthout loss of generalty, let us vew a as a arameter n the roosed game and, corresondngly, Pr tot as a arameter n the centralzed otmal scheme. When a ncreases, P N ½w ð c Þ ŠP r P N ~ a decreases. In order to ma Nþ1 ð P N P r Pr tot Þ to t, Pr tot should ncrease. That s the reason why varyng the arameter a n the roosed game s equvalent to varyng Pr tot n the centralzed otmzaton. To justfy our clam, we show the otmal owers versus Pr tot and a of the two schemes n Fg. 8, wth or wthout the Pr max constrants, resectvely. From both the smulaton and the above analyss, we can see that due to the equvalence of the Lagrangan n the two aroaches, the roosed game can acheve comarable erformance to that n the centralzed otmal scheme. REFERENCES [1] J.N. Laneman, D.N.C. Tse, and G.W. Wornell, Cooeratve Dversty n Wreless Networks Effcent Protocols and Outage Behavor, IEEE Trans. Informaton Theory, vol. 50, no. 12, , Dec [2] M.O. Hasna and M.-S. Aloun, Otmal Power Allocaton for Relayed Transmssons over Raylegh Fadng Channels, Proc. 57th IEEE Vehcular Technology Conf. (VTC Srng 03), vol. 4, , Ar [3] W. Su, A.K. Sadek, and K.J.R. Lu, Cooeratve Communcatons n Wreless Networks Performance Analyss and Otmum Power Allocaton, Wreless Personal Comm., vol. 44, no. 2, , Jan [4] I. Marc and R.D. ates, Cooeratve Multho Broadcast for Wreless Networks, IEEE J. Selected Areas n Comm., vol. 22, no. 6, , Aug [5] J. Luo, R.S. Blum, L.J. Greensten, L.J. Cmn, and A.M. Hamovch, New Aroaches for Cooeratve Use of Multle Antennas n Ad Hoc Wreless Networks, Proc. 60th IEEE Vehcular Technology Conf. (VTC Fall 04), vol. 4, , Set [6] A. Bletsas, A. Lman, and D.P. Reed, A Smle Dstrbuted Method for Relay Selecton n Cooeratve Dversty Wreless Networks, Based on Recrocty and Channel Measurements, Proc. 61st IEEE Vehcular Technology Conf. (VTC Srng 05), vol. 3, , May [7] A.K. Sadek, Z. Han, and K.J.R. Lu, An Effcent Cooeraton Protocol to Extend Coverage Area n Cellular Networks, Proc. IEEE Wreless Comm. and Networkng Conf. (WCNC 06), vol. 3, , Ar [8] Z. Han, T. Hmsoon, W. Srwongarat, and K.J.R. Lu, Energy Effcent Cooeratve Transmsson over Multuser OFDM Networks Who Hels Whom and How to Cooerate, Proc. IEEE Wreless Comm. and Networkng Conf. (WCNC 05), vol. 2, , Mar [9] A. Ibrahm, A.K. Sadek, W. Su, and K.J.R. Lu, Relay Selecton n Mult-Node Cooeratve Communcatons When to Cooerate and Whom to Cooerate Wth? Proc. IEEE Global Telecomm. Conf. (GLOBECOM 06),. 1-5, Nov [10] A. Ibrahm, A.K. Sadek, W. Su, and K.J.R. Lu, Cooeratve Communcatons wth Relay Selecton When to Cooerate and Whom to Cooerate wth? IEEE Trans. Wreless Comm., vol. 7, no. 7, , July Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.

16 990 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 8, NO. 7, JUL 2009 [11]. Zhao, R.S. Adve, and T.J. Lm, Imrovng Amlfy-and- Forward Relay Networks Otmal Power Allocaton Versus Selecton, Proc. IEEE Int l Sym. Informaton Theory (ISIT 06), , July [12] T. Ng and W. u, Jont Otmzaton of Relay Strateges and Resource Allocatons n Cooeratve Cellular Networks, IEEE J. Selected Areas n Comm., vol. 25, no. 2, , Feb [13] S. Savazz and U. Sagnoln, Energy Aware Power Allocaton Strateges for Multho-Cooeratve Transmsson Schemes, IEEE J. Selected Areas n Comm., vol. 25, no. 2, , Feb [14] T. Hmsoon, W. Srwongarat, Z. Han, and K.J.R. Lu, Lfetme Maxmzaton Framework by Cooeratve Nodes and Relay Deloyment n Wreless Networks, IEEE J. Selected Areas n Comm., vol. 25, no. 2, , Feb [15] R. Annavajjala, C. Cosman, and B. Mlsten, Statstcal Channel Knowledge-Based Otmum Power Allocaton for Relayng Protocols n the Hgh SNR Regme, IEEE J. Selected Areas n Comm., vol. 25, no. 2, , Feb [16] B. Ln, P. Ho, L. Xe, and X. Shen, Otmal Relay Staton Placement n IEEE j Networks, Proc. Int l Conf. Wreless Comm. and Moble Comutng (IWCMC 07), , Aug [17] A.K. Sadek, W. Su, and K.J.R. Lu, Mult-Node Cooeratve Communcatons n Wreless Networks, IEEE Trans. Sgnal Processng, vol. 55, no. 1, , Jan [18] A.B. MacKenze and S.B. Wcker, Game Theory and the Desgn of Self-Confgurng, Adatve Wreless Networks, IEEE Comm. Magazne, vol. 39, no. 11, , Nov [19] L.A. DaSlva, D.W. Petr, and N. Akar, Statc Prcng and Qualty of Servce n Multle Servce Networks, Proc. Ffth Jont Conf. Informaton Scences (JCIS 00), vol. 1, , Feb [20] C.U. Saraydar, N.B. Mandayam, and D.J. Goodman, Effcent Power Control va Prcng n Wreless Data Networks, IEEE Trans. Comm., vol. 50, no. 2, , Feb [21] N. Shastry and R.S. Adve, Stmulatng Cooeratve Dversty n Wreless Ad Hoc Networks through Prcng, Proc. IEEE Int l Conf. Comm., , June [22] Z. Han, Z. J, and K.J.R. Lu, Far Multuser Channel Allocaton for OFDMA Networks Usng Nash Barganng and Coaltons, IEEE Trans. Comm., vol. 53, no. 8, , Aug [23] B. Wang, Z. Han, and K.J.R. Lu, Stackelberg Game for Dstrbuted Resource Allocaton over Multuser Cooeratve Communcaton Networks, Proc. IEEE Global Telecomm. Conf. (GLOBECOM 06),. 1-5, Nov [24] B. Wang, Z. Han, and K.J.R. Lu, Dstrbuted Relay Selecton and Power Control for Multuser Cooeratve Communcaton Networks Usng Buyer/Seller Game, Proc. IEEE INFOCOM 07, , May [25] D. Fudenberg and J. Trole, Game Theory. MIT Press, [26] Z. Han and K.J.R. Lu, Resource Allocaton for Wreless Networks Bascs, Technques, and Alcatons. Cambrdge Unv. Press, [27] M.S. Barzaraa, Nonlnear Programmng Theory and Algorthms, second ed. John Wley & Sons, [28] R. ates, A Framework for Ulnk Power Control n Cellular Rado Systems, IEEE J. Selected Areas n Comm., vol. 13, no. 7, , Set Bebe Wang receved the BS degree n electrcal engneerng (wth hghest honors) from the Unversty of Scence and Technology of Chna, Hefe, n July She s currently a PhD canddate n the Deartment of Electrcal and Comuter Engneerng, Unversty of Maryland, College Park. Her research nterests nclude resource allocaton and management n cogntve rado systems, wreless communcatons and networkng, game theory, wreless multmeda, and network securty. She was the recent of both the Graduate School Fellowsh and the Future Faculty Fellowsh from the Unversty of Maryland. She s a student member of the IEEE. Zhu Han receved the BS degree n electronc engneerng from Tsnghua Unversty n 1997 and the MS and PhD degrees n electrcal engneerng from the Unversty of Maryland, College Park, n 1999 and 2003, resectvely. From 2000 to 2002, he was an R&D engneer of JDSU, Germantown, Maryland. From 2002 to 2003, he was a graduate research assstant, and from 2003 to 2006, a research assocate, at the Unversty of Maryland. From 2006 to 2008, he was an assstant rofessor at Bose State Unversty, Idaho. Currently, he s an assstant rofessor n the Electrcal and Comuter Engneerng Deartment, Unversty of Houston, Texas. From June to August 2006, he was a vstng scholar at Prnceton Unversty. From May to August 2007, he was a vstng rofessor at Stanford Unversty. From May to August 2008, he was a vstng rofessor at the Unversty of Oslo, Norway, and Suelec, Pars, France. Hs research nterests nclude wreless resource allocaton and management, wreless communcatons and networkng, game theory, wreless multmeda, and securty. He s the MAC Symosum vce char of the IEEE Wreless Communcatons and Networkng Conference n He s the guest edtor for the secal ssue on farness of rado resource management technques n wreless networks of the EURASIP Journal on Wreless Communcatons and Networkng and the secal ssue on game theory of the EURASIP Journal on Advances n Sgnal Processng. He s a member of the techncal rogrammng commttee for the IEEE Internatonal Conference on Communcatons, the IEEE Vehcular Technology Conference, the IEEE Consumer Communcatons and Networkng Conference, the IEEE Wreless Communcatons and Networkng Conference, and the IEEE Globe Communcaton Conference. He s a member of the IEEE. K.J. Ray Lu s a Dstngushed Scholar-Teacher at the Unversty of Maryland, College Park. He s the assocate char of graduate studes and research n the Electrcal and Comuter Engneerng Deartment. He also leads the Maryland Sgnals and Informaton Grou, conductng research encomassng broad asects of nformaton technology ncludng communcatons and networkng, nformaton forenscs and securty, multmeda sgnal rocessng, and bomedcal technology. He s the recent of numerous honors and awards, ncludng the IEEE Sgnal Processng Socety Dstngushed Lecturer Award, the EURASIP Mertorous Servce Award, the Natonal Scence Foundaton oung Investgator Award, and best aer awards from the IEEE Sgnal Processng Socety (twce), the IEEE Vehcular Technology Socety, and EURASIP. He has also receved varous teachng and research recogntons from the Unversty of Maryland, ncludng the unversty-level Inventon of the ear Award, the Outstandng Faculty Research Award, and the Poole and Kent Senor Faculty Teachng Award from the A. James Clark School of Engneerng. He s the vce resdent for ublcatons and s on the Board of Governors of the IEEE Sgnal Processng Socety. He was the edtor-n-chef of the IEEE Sgnal Processng Magazne and the foundng edtor-n-chef of the EURASIP Journal on Aled Sgnal Processng. Hs recent books nclude Cooeratve Communcatons and Networkng (Cambrdge Unversty Press, 2008), Resource Allocaton for Wreless Networks Bascs, Technques, and Alcatons (Cambrdge Unversty Press, 2008), Ultra-Wdeband Communcaton Systems The Multband OFDM Aroach (IEEE-John Wley & Sons, 2007), Network-Aware Securty for Grou Communcatons (Srnger, 2007), Multmeda Fngerrntng Forenscs for Trator Tracng (Hndaw, 2005), and Handbook on Array Processng and Sensor Networks (IEEE-John Wley & Sons, 2009). He s a fellow of the IEEE and AAAS.. For more nformaton on ths or any other comutng toc, lease vst our Dgtal Lbrary at Authorzed lcensed use lmted to Unversty of Maryland College Park. Downloaded on May 30, 2009 at 1407 from IEEE Xlore. Restrctons aly.