ower Allocaton n Wreless Relay Networks: A Geometrc rogrammng-based Approach Khoa T. han, Tho Le-Ngoc, Sergy A. Vorobyov, and Chntha Telambura Department of Electrcal and Computer Engneerng, Unversty of Alberta, Edmonton, AB, CANADA Department of Electrcal and Computer Engneerng, McGll Unversty, Montreal, QC, CANADA Emal: khoa, vorobyov, chntha@ece.ualberta.ca, tho@ece.mcgll.ca Abstract 1 In ths paper, we consder an amplfy-and-forward (AF) wreless relay system where multple source nodes communcate wth ther correspondng destnaton nodes wth the help of relay nodes. Whle each user 2 s asssted by one relay, one relay can assst many users. Conventonally, each relay node s assumed to equally dstrbute the avalable bandwdth and power resources to all sources for whch t helps to relay nformaton. Realzng the sub-optmalty of ths approach, n ths paper, we present effcent power allocaton schemes to ) mze the mnmum end-to-end sgnal-to-nose rato among all users; ) mnmze the total transmt power over all sources; ) mze the system throughput. Our approach s based on geometrc programmng (G), a well-studed class of nonlnear and nonconvex optmzaton. Snce a G problem s readly transformed nto an equvalent convex optmzaton problem, optmal power allocaton can be obtaned effcently. Numercal results demonstrate the effectveness of our proposed approach. Index terms ower allocaton, geometrc programmng, relay networks. I. INTRODUCTION It has been shown that the operaton effcency and qualtyof-servce (QoS) of cellular and/or ad-hoc networks can be ncreased through the use of relay(s) [1], [2]. In such systems, the nformaton from the source to the correspondng destnaton s transmtted va a drect-lnk and also forwarded va relays. A crtcal ssue for mprovng the performance of wreless networks s effcent management of avalable rado resources. artcularly, resource allocaton va power control s commonly used to ensure the performance and stablty of the wreless network. There have been numerous works that attempt to optmze the avalable communcaton resources,.e., power and bandwdth to mprove the system performance [7]-[10]. A sngle source-destnaton par s typcally consdered n the aforementoned papers. In [7], for example, the authors derve closed-form expressons for the optmal and near-optmal relay transmsson powers for the sngle relay and the multple relays cases. Furthermore, the problem of mnmzng the transmsson power gven that a target outage probablty s acheved was tackled n [8]. In [9], the authors derve power allocaton strateges for -node amplfy-and-forward 1 Ths work was supported n parts by the Natural Scence and Engneerng Research Councl (NSERC) of Canada and Alberta Ingenuty, Alberta, Canada. 2 Hereafter, the term user refers to a source-destnaton (S-D) par or only the source node dependng on the context. (AF) relayng system based on the knowledge of channel means. Gven ether channel state nformaton (CSI) or channel statstcs, two power allocaton schemes to mnmze the outage probablty are presented n [10]. We note, however, that very few exstng works have consdered the 2-hop relay model wth multple users. The latter setup of multple users s the more practcal as compared to the prevously consdered confguratons. Therefore, the above mentoned analyss s applcable only to a specal case of the problem n hands, snce each relay s usually delegated to assst more than one users, especally when the number of relays s usually (much) smaller than the number of users. An example of such scenaro s the deployment of few relays n a cell at approprate locatons to assst moble users operatng n heavly scatterng envronment for uplnk transmsson. Resource allocaton n a mult-user system usually has to take nto account the farness ssue among users, ther relatve QoS requrements, channel qualty and avalable resources. Mathematcally, optmzaton of relay networks wth multple users s a dffcult (f tractable) problem, especally for systems wth large number of sources and relays. In ths paper, we develop effcent power allocaton schemes for mult-user wreless relay systems. artcularly, we derve optmal power allocaton schemes to ) mze the mnmum end-to-end sgnal-to-nose ratos (SNRs) among all users; ) mnmze the total transmt power of all sources; ) mze the system throughput. We show that the correspondng optmzaton problems can be formulated as geometrc programmng (G) problems. Therefore, optmal power allocaton can be obtaned effcently even for large-scale networks usng convex optmzaton technques. Note that G has been successfully appled to solve the problem of power allocaton n tradtonal cellular and ad hoc networks [] [6]. II. SYSTEM MODEL Consder a mult-user relayng model where a set of M source nodes S, {1,...M} wants to transmt data to ther correspondng destnaton nodes D, {1,...M}. Moreover, L relay nodes, denoted by R, {1,..., L} are employed for forwardng the nformaton from source to destnaton nodes. The conventonal two-stage AF relayng Ths ncludes the case of one destnaton node for all sources, for example, base staton n a cellular network, or central processng unt n a sensor network. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 2008 proceedngs.
s assumed. We also assume orthogonal transmsson usng tme dvson [10], [1], [2]. Each source S s asssted by one relay R S. The set of source nodes whch use the relay R s denoted by S (R ),.e., S(R )={S R S = R }. Let S, RS denote the power transmtted by source S and relay R S correspondng to S -R S -D lnk, respectvely. Snce unt duraton tme slots are assumed, S and RS correspond also to the average energes consumed by source S and relay R S. For smplcty, we present the sgnal model for lnk S -R S -D only. In the frst tme slot, source S transmts the sgnal x wth unt energy to the relay R S. The receved sgnal at relay R S can be wrtten as r SR S = S a SR S x + n RS where a SR S stands for the channel gan for lnk S -R S, n RS s the addtve crcularly symmetrc whte Gaussan nose (AWGN) at the relay R S wth varance N RS. The channel gan ncludes the effects of path loss, shadowng and fadng. We assume that the relay R S knows the CSI for lnk S -R S. If the AF protocol s used,.e., the sgnal receved by a relay node s normalzed and retransmts to the destnaton node D, the receved sgnal at the destnaton node D can be expressed as r SR S r D = RS a RS D E { r SR S 2} + n D = RS S S a SR S 2 a RS D + N a SR S x +ˆx D RS where E{.} denotes statstcal expectaton operator, a RS D s the channel coeffcent for lnk R S -D, n D s the AWGN at the destnaton node D wth varance N D, ˆx D s the ( modfed AWGN nose at D wth equvalent varance N D + RS a ) ( ) R S D 2 N RS / S a SR 2 S + N RS. The equvalent end-to-end SNR of the vrtual channel between the nodes S and D can be wrtten as [10] γ = RS S a RS D 2 a SR S 2 S a SR S 2 N D + RS a RS D 2 N RS + N D N RS = S RS η S + α RS + β where η = N D a 2, R S D α = N R S a 2 S R S, β = N R S N D a 2 S R S a 2 R S D. It can be seen that for fxed RS, γ s a concave ncreasng functon of S. However, no matter how large S s, the mum achevable γ can be shown to be equal to RS /η. Vce versa, when S s fxed, γ s a concave ncreasng functon of RS and the correspondng mum achevable γ s S /α. Moreover, snce γ s concave ncreasng on S, The sngle relay assgnment may be done durng the connecton setup phase, or by relay selecton process [10]. We consder the case n whch the source-to-relay lnk s (much) stronger than the source-to-destnaton lnk, that s usual scenaro n practce. the ncremental change n γ s smaller for large S, and, therefore, γ s monotone. Note that monotoncty s a useful property helpng to provde some nsghts nto optmzaton problems at optmalty. We assume a central unt (CU) to coordnate the power allocaton at the sources and at the relays. For such purpose, the CU should have CSI for all the transmsson lnks,.e., source-relay and relay-destnaton lnks. The power allocaton factors can be communcated to relays and sources va a secured channel. The sources and relays then adust ther transmt power accordngly. A slow fadng envronment s also assumed. The letter assumpton corresponds to networks wth statonary topology or low-moblty users. III. ROBLEM FORMULATIONS In ths secton, we propose power allocaton schemes for mult-user wreless relay systems. A. Max-mn SNR based allocaton ower control n wreless networks often has to take nto account the farness among dfferent users. In other words, the performance of the worst user(s),.e., the user(s) wth smallest end-to-end SNR, s often of concern to the network operator. The tradtonally used mum sum SNR power allocaton s based towards users that have the best channel qualty and s unfar to the other lnks. Here we consder the -mn far power allocaton problem whch ams at mzng the mnmum SNR over all users. 6 The problem can be mathematcally posed as mn γ ( S, RS ) (1a) S, RS =1,...,M subect to: RS R,=1,...,L (1b) where R S S(R ) S (1c) =1 0 S S, S, {1,...M} (1d) s the total power avalable at the relay node R and s the total power allocated to all sources. The rghthand sde of (1b) s the total power that the relay R allocates to the users whch t asssts. Ths power s constraned to be less than the relay s total power. Smlarly, constrants (1d) specfy the peak power lmt S for each source S. It can be seen that the set of lnear nequalty constrants wth postve varables n the optmzaton problem (1a) (1d) s compact and nonempty. Hence, the problem (1a) (1d) s always feasble. Moreover, snce the obectve functon mn =1,...,M γ s an ncreasng functon of the allocated powers S and RS, the nequalty constrants (1b), (1c) must be met wth equalty at optmalty. It can be observed that whle the performance of user depends only on the allocated powers S and RS, the performance of all users nteract 6 In ths way, the mnmum data rate among users s also mzed snce data rate s a monotonc ncreasng functon of SNR. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 2008 proceedngs.
wth each other va shared and lmted power resource at the relays and the sources. Therefore, proper power allocaton among users s necessary to mze a specfc crteron on the system performance. Resource allocaton n a mult-user network s not as smple as allocaton of resources for each user ndvdually, albet orthogonal transmssons are assumed. B. ower mnmzaton based allocaton In practcally used wreless networks, one of the targets of power allocaton s to prolong the lfetme of battery-powered devces snce nodes wth long lfetme help to ensure unnterrupted nformaton exchange. Therefore, the mnmzaton of the power consumpton of the source nodes n relay systems s partcularly mportant. The transmt power mnmzaton problem subect to constrants on the end-to-end SNR for each user can be formulated as follows mn S (2a) S, RS subect to: =1 γ γ mn,=1,...,m (2b) The constrants (1b), (1d) (2c) where γ mn s the threshold SNR for th user. 7 In the problem (2a) (2c), the relays are assumed to be power-lmted but energy-unlmted. Ths problem answers the queston of how to explot effcently the avalable power resource at the relays to mnmze the power consumpton at the battery-powered nodes. It can be seen that under the assumpton of energy-unlmted relays the optmzaton problem (2a) (2c) s always feasble. It also can be observed that at optmalty, the nequalty constrants (2b), (1b) n (2c) must be met wth equalty. Ths s because γ s an ncreasng functon of S and RS. In order to mnmze the obectve functon n (2a) (2c), S, γ and RS must attan ther mnmum and mum values, respectvely. Fnally, note that we have mplctly assumed n (2a) (2c) that none of the source needs transmts more than S at optmalty. C. Throughput mzaton based allocaton The data rate R of the th S-D par can be wrtten as a functon of γ as R = 1 T log 2(1 + Kγ ) 1 T log 2(Kγ ) ζ 1 where T s the symbol perod, K = ln(ζ 2BER), BER s the target bt error rate, and ζ 1, ζ 2 are constants dependent on the modulaton scheme []. Note that we have approxmated 1+Kγ as Kγ. For notatonal smplcty we also set K =1. Then, the aggregate throughput for the system can be wrtten as [6] R sys = R 1 [ T log M 2 γ ]. =1 =1 7 We assume that the threshold γ mn s not larger than the mum achevable SNR for user as dscussed above. The power allocaton problem to mze the system throughput can be mathematcally posed as R sys = 1 [ S, RS T log M ] 2 γ (a) =1 subect to: The constrants (1b), (1c), (1d). (b) Therefore, n the hgh SNR regon, mzng network throughput can be approxmately replaced by mzng the product of SNRs. 8 At optmalty, the nequalty constrants (1b), (1c) n (b) of the problem (a) (b) must be met wth equalty. IV. OWER ALLOCATION IN RELAY NETWORKS VIA G G s a well-nvestgated class of nonlnear, nonconvex optmzaton problems wth attractve theoretcal and computatonal propertes [], [6]. Snce equvalent convex reformulaton s possble for a G problem, only global optmum exsts. Moreover, the avalablty of large-scale software solvers makes G more appealng for practcal use. A. Max-mn SNR based allocaton Introducng a new varable t, the optmzaton problem (1a) (1c) can be equvalently rewrtten as mn S, RS,t 0 subect to: 1 (a) t S RS t, =1,...,M(b) η S + α RS + β The constrants (1b), (1c), (1d). (c) The obectve functon n the problem (a) (c) s a monomal functon. Moreover, the constrants n (b) can be easly converted nto posynomal constrants. The constrants (1b), (1c), (1d) are lnear wth respect to the power varables, and thus, are posynomal constrants. Therefore, the optmzaton problem (a) (c) s a G problem. B. ower mnmzaton based allocaton In ths case, the obectve functon s clearly a posynomal functon. The constrants can also be wrtten as posynomal ones. Therefore, the power mnmzaton based allocaton s a G problem. C. Throughput mzaton based allocaton A smple manpulaton of the optmzaton problem (a) (b) gves 1 mn S, M RS =1 γ (a) subect to: The constrants (1b), (1c), (1d). (b) The obectve functon n ths problem can be shown to be a posynomal. Moreover, the constrant can also be easly converted nto posynomal constrants as for the prevous 8 Note, however, that n the low SNR regon, the approxmaton of 1+γ by γ does not hold satsfactorly, and therefore, wll not lead to accurate results. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 2008 proceedngs.
200 m Y-Axs Source 2 Source 1 Relay Destnaton 1 Destnaton M... R Source Relay 2 6 Destnaton 2 Source M Relay 1 Destnaton 0 m X-Axs 1 m 200 m Fg. 1. A wreless mult-user relay system. 2 8 6 R problems. Therefore, the optmzaton problem (a) (b) also belongs to the G class. We have shown that the three aforementoned power allocaton schemes can be reformulated as G problems. The proposed optmzaton problems wth dstnct features of relayng model are mathematcally smlar to the ones n [6] developed for conventonal cellular networks. However, the numerator and denomnator of the SNR expresson for each user consdered n [6] are lnear functons of the power varables whch s not the case n our work. V. SIMULATION RESULTS Consder a wreless relay network shown n Fg. 1 wth ten users and three relays dstrbuted n a two-dmensonal regon 200m 200m. The relays are fxed at coordnates (100,), (100,100), and (100,1). The ten source nodes and ther correspondng destnaton nodes are deployed randomly n the area nsde the box area [(0, 0), (, 200)] and [(1, 0), (200, 200)] respectvely. In our smulaton, each source s asssted by a random (and then fxed) relay. We assume that there s no mcroscopc fadng and the gan for each transmsson lnk s computed usng the path loss model as a =1/d where d s the Eucldean dstance between two transmsson ends. The nose power s assumed to be equal to N 0 = db. Although each relay node may assst dfferent number of users, whch are assumed to have the same mum power level R. Smlarly, all users are also assumed to have the equal mnmum SNR thresholds γ mn. The software package for solvng convex programs, whch can be downloaded at [11], s used n our smulatons. Fgs. 2 and show the mnmum rate among all users and the network throughput,.e., the sum of users rates, when the mum power levels of the relays R and sources are vared. The performance of the equal power allocaton (EA) scheme s also plotted for comparson. Accordng to the EA scheme the power s allocated equally among all sources,.e., S = /10, S and each relay dstrbutes power equally among all users whch t asssts. For =(see Fg. 2), the optmal power allocaton (OA) scheme acheves about 7 6 Fg. 2. Data rate versus R, =. 2 70 60 0 0 Fg.. Data rate versus, R =. 0.8 bts performance mprovement over the EA scheme for the worst user data rate. The performance mprovement of both schemes s hgher when R s small (less than 0). The EA scheme provdes a slght performance mprovement for the worst user(s) for R. However, the OA scheme s able to take advantage from larger R. Ths demonstrates the effectveness of OA scheme n general and our proposed approach n partcular. In Fg., we fx R =. It can be seen that the OA scheme also outperforms the EA scheme. The mprovement s about 0.8 bts and ncreases when ncreases. In both scenaros, there s a loss n the network throughput snce our obectve s to mprove the performance of the worst user(s). Ths confrms the well-known fact that achevng -mn farness among users usually results n performance loss for the whole system. Fg. dsplays the total power consumed by source nodes n two scenaros. The frst scenaro s to attan a mnmum Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 2008 proceedngs.
Sum ower () Sum ower () 2 200 1 100 0 6 7 8 9 10 11 12 1 1 1 16 17 18 19 20 mn γ 0 2 20 1 10 10 1 20 2 0 0 R.. R 6 2 8 6 R Fg.. Total power of sources. Fg.. Network throughput versus R, =. SNR γ mn wth fxed R =, whle n the second scenaro R s vared when γ mn =10dB s fxed. It can be seen that, n the frst scenaro, the OA scheme allocates less requred power than that of the EA scheme when γ mn 17 db. Moreover, when γ mn 18 db, EA scheme can not fnd a feasble power allocaton (n fact, negatve power allocaton), represented by werd part n the EA curve. It s because the threshold γ mn 18 db exceeds the mum value of γ for some users. It can be seen that by proper power dstrbuton at the relays, the OA scheme can fnd power allocaton to acheve larger target SNR γ mn. Ths further demonstrates the advantage of our proposed approach over the EA scheme. For the second scenaro, the OA scheme requres less sum power than that of the EA scheme, especally when R s small. It can be observed that as there s more avalable power R, less sum power s requred to acheve a target SNR. In the last example, the OA s used to mze the sum users throughput. Fg. shows the performance of our proposed approach versus R when =.TheOA scheme outperforms the EA for all values of R.Its notceable that the OA scheme acheves better performance n terms of both worst user data rate and network throughput. Comparng wth the results n Fgs. 2 and, we can see the tradeoff between achevng farness and the sum throughput. VI. CONCLUSION In ths paper, optmal power allocaton schemes have been proposed for wreless relay networks. AF relayng model s assumed where each of the source nodes communcates wth ts correspondng destnaton node wth the help of one relay node. The proposed approach s based on G. Although G s nonconvex, t allows for an equvalent convex reformulaton whch provdes an effcent method for obtanng optmal soluton. In partcular, we presented power allocaton schemes to ) mze the mnmum end-to-end SNR for all source-destnaton pars; ) mnmze the total transmt power over all sources; ) mze the system throughput. Smulaton results demonstrate the effectveness of the proposed approachs over the EA scheme. Acknowledgements The authors are grateful to Duy H. N. Nguyen from the Unversty of Saskatchewan, Canada for helpful dscussons and comments. REFERENCES [1] M. O. Hasna, and M. S. Aloun, End-to-end performance of transmsson systems wth relays over Raylegh fadng channels, IEEE Trans. Wreless Commun., vol. 2, pp. 1126-111, Nov. 200. [2] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperatve dversty n wreless networks: Effcent protocols and outage behavor, IEEE Trans. Inform. Theory, vol., pp. 062-080, Dec. 200. [] A. Goldsmth, Wreless Communcatons, Cambrdge Unversty ress, 200. [] L. B. Le, and E. Hossan, Multhop cellular networks: otental gans, research challenges, and a resource allocaton framework, IEEE Communcatons Magazne, vol., pp. 66-7, Sept. 2007. [] D. Julan, M. Chang, D. O Nell, and S.. Boyd, QoS and farness constraned convex optmzaton of resource allocaton for wreless cellular and ad hoc networks, n roc. IEEE INFOCOM 02, New York, NY, Jun. 2002, pp. 77-86. [6] M. Chang, C. W. Tan, D. alomar, D. O Nell, and D. Julan, ower control by geometrc programmng, IEEE Trans. Wreless Commun., vol. 6, pp. 260-261, Jul. 2007. [7] Y. L, B. Vucetc, Z. Zhou, and M. Dohler, Dstrbuted adaptve power allocaton for wreless relay networks, IEEE Trans. Wreless Commun., vol. 6, pp. 98-98, Mar. 2007. [8] M. Chen, S. Serbetl, and A. Yener, Dstrbuted power allocaton for parallel relay networks, n roc. of IEEE Global Communcatons Conference (GLOBECOM 0), St. Lous, MO, pp. 1177-1181, Nov. 200. [9] X. Deng, and A. M. Hamovch, ower allocaton for cooperatve relayng n wreless networks, IEEE Commun. Letters, vol. 9, pp. 99-996, Nov. 200. [10] Y. Zhao, R.S. Adve and T.J. Lm, Improvng amplfy-and-forward relay networks: optmal power allocaton versus selecton, IEEE Trans. Wreless Commun., vol. 6, pp. 11-12, Aug. 2007. [11] M. Grant, and S. Boyd, CVX: Matlab software for dscplned convex programmng (web page and software), http://stanford.edu/ boyd/cvx, Feb. 2008. Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subect matter experts for publcaton n the IEEE "GLOBECOM" 2008 proceedngs.