Distributive Subband Allocation, Power and Rate Control for Relay-Assisted OFDMA Cellular System with Imperfect System State Knowledge

IEEE TRANSACTIONS ON WIRELESS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 1 Distrib uti v e Subband Allocat ion, P o we r and Rate Control for Relay-Assisted OFDMA Ce llular System with Imperfec t Sy stem State Kno wledge Y ing Cui, Student Member , V incent K. N. Lau, Senior Me mber and Rui W ang, Student Member , IEEE Abstract — In this paper , we consid er distributive subband, power and rate allocation for a two-hop transmission in an orthogonal frequency-division multip le-access (OFDMA) cell ular system with fixed relays wh ich operate in decode-and -for ward strategy . W e take in to account of system fairness by considerin g weighted sum goodput as our opti mization objective . Based on the clu ster -based architecture, we obtain a fast-con ve rging distributive solution with only local imperfect CSIT by using decomposition of th e optimization p roblem. T o furth er reduce the signaling ov erhead and computational complexity , we pr opose a reduced feedback distributive solution, whi ch can achiev e asymptotically optimal p erfo rmance for large number of users with arbitrarily small feedback over head p er user . W e also derive asymptotic ave rage system throughput fo r th e relay-assisted OFDMA system so as to obtain usefu l design insights. Index T erms — relay , OFDM A, resource all ocation, fairness, distributive algorithm I . I N T R O D U C T I O N The relay-assisted OFDMA ce llular system is a promising architecture for futur e wireless communicatio n systems be- cause it offers hu ge p otential fo r enhan ced system capacity , coverage as well a s reliability [1], [ 2]. Full-dup lex relay stations wer e discussed in the early literature. Howe ver , full- duplex transceiver is hard to implement in practical systems because the relay has to transmit and receiv e at that same time 1 . Since prac tical r elays op erate in a half-du plex m anner, there is a dup lexing penalty associated wh en u sing the r elay to fo rward packets. As a result, it is not alw ays advantageo us to deli ver packets via relay stations and it is very imp ortant to adapt the relay resource dy namically accor ding to the glob al channel state in formation (GCSI) 2 in o rder to captu re the advantage of the relay-assisted OFDMA a rchitecture. How- ev er , perfect knowledge of GCSI is very d ifficult to obtain in both TDD (implicit CSI feedback) and FDD ( explicit CSI feedback ) systems due to the huge s ignaling overhead in v olved in deliv ering GCSI to the controller . Manuscript recei v ed July 10, 2008; revised December 1, 2008 and March 26, 2009; acce pted May 24, 2009. The associate editor coordi nating the revie w of this paper and approv ing it for publicati on was N. Kato. The au thors are with the Department of ECE, the Hong Kong Univ ersity of Science and T ec hnologie s, Clear W at er Bay , K owl oon, Hong Kong (e-mail : cuiying , eeknl au, wray@ust.hk). 1 In practice , there are cross-couplin g between the transmit path and the recei v e path in any transcei ver circui t. Hence, when the relay transmit (with high power), there will be some leakag e into the recei ver path, which will cause strong interference to the recei ved signal (which is much weaker than the transmit signal ). 2 Global CSI refers to the CSI of the base station (BS) and relay (RS), the CSI of the RS and mobile (MS) as well as the CSI of the BS and the MS. There ar e a lot of research interests fo cused o n improvin g the sy stem throug hput by the optimal r esource allo cation of th e relay-assisted OFDMA system. For examp le, in [3], [4], o ptimal subban d allocation is co nsidered fo r d ifferent scheduling sch emes based on equal p ower allocation across all th e subband s. In [5], h euristic separate subband and power allocation sch emes are co nsidered to imp rove the system capacity . T o furth er im prove the system th roug hput, jo int subband and power allocation is propo sed in [6], [ 7]. Due to the subband allocation co nstraint in the OFDMA system, the joint optimizatio n problem is a NP-h ard integer progr amming problem an d contin uous relaxation as well as grap h theor etical approa ch a re u sed to tackle the pro blem. However , a s a simplification in both [6], [7], it is assumed that the source- relay , sour ce-destination and relay-d estination links use the same subb and. They cannot effecti vely exploit the multiuser div ersity , which is a very im portant com ponen t to system perfor mance. While these works p rovide important initial in- vestigations on th e poten tial benefit of relay -assisted OFDMA systems, centralized r esource allo cation solutions and perfect knowledge of GCSI are requir ed. In g eneral, there are still se veral important remainin g technical issues to b e resolved in order to bridg e the gap between theoretica l gain s an d practical implementatio n consider ations. They are elaborated b elow . • Distributi ve Implementation : There are two potential is- sues associated with cen tralized implemen tations, namely the comp lexity issue and th e signaling lo ading issue . For instance, the centralized joint optimization has a huge computatio nal load ing to the BS. Similarly , large signaling overhead is needed to collect the GCSI (BS- RS, BS-MS, RS-MS) from the RSs and MSs as well as to broad cast the schedu led results to the RSs an d MSs. In [8 ], the au thors p ropo sed two semi-d istributed sub-optim al subband allocation metho ds based on eq ual power allocation , which h ave offloaded certain com- putational lo ad f rom the BS, but substan tial signaling overhead is still needed to collect the GCSI fr om th e RSs. • Imperfect Knowledge of GCSI : While all the above works assume perfect GCSI knowledge at the transmitter, the CSIT me asured at the transmitter side can not be perfect due to either the CSIT estimatio n noise o r the outdatedn ess of CSIT resulting f rom dup lexing d elay . When the CSIT is imperfect, there will be systematic packet errors (despite the use of strong channel coding) as IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 2 long as the schedu led data r ate exceeds the instantaneous channel mu tual infor mation. As a result, rate adaptation (in addition to power and subband allocations) must be considered in systems with imper fect CSIT in o rder to take into account of potential packet errors due to channel outage [9], [10], [1 1], [12]. • Fairn ess Co nsideration : Most of existing works only focus on sum-thr oughp ut maximizatio n. Howe ver , fair- ness amon g u sers in the system is also an imp ortant consideratio n. In particular, it is importan t to study the potential fairness advantage of relay- assisted OFDM A systems as well. • Analytical Performance Results for Design Insights : In all the existing works above, the system perform ance is obtained by simu lations only . Ho wever , it is very d ifficult to obtain u seful design insigh ts (e.g. how the system perfor mance scales with system p arameters such as M , K as well as path loss exponents of BS-RS and RS-MS links) without analytical p erform ance results. In this paper , we shall attempt to shed some lights on the above issues. W e consider a two-hop relay-assisted OFDMA system in a single cell with one base station ( BS), M relay stations (RS) and K mobile stations (MS). In addition, we account f or the pe nalty of p acket erro rs du e to imper fect CSIT by consid ering system go odpu t (b/s/Hz successfully deliv ered to the MS) as our pe rforman ce measure in stead of trad itional sum-ergodic capac ity (wh ich only measures b/s/Hz sent by the BS or RS). W e take into acco unt o f sy stem fairness by consider ing weigh ted sum goo dput as our optimization objective (which includes pr oportio nal fair sche duling (PFS) as a specia l case). Based on the cluster-based arch itecture, we obtain a fast-conver ging distributive solution with only local CSIT using careful decomp osition o f optimization prob lem. For the d ownlink implementatio n, to furth er re duce the sig- naling overhead and comp utational co mplexity , we pro pose a redu ced feedb ack d istributi ve solution and show th at only an arbitrarily small feedb ack overhead p er MS is needed to achieve asym ptotically op timal perfor mance fo r large K . Finally , we also der i ve asy mptotic average system thro ughp ut for the r elay-assisted OFDMA system so as to obtain useful design insights. I I . S Y S T E M M O D E L In this section, we shall describe the cluster-based system architecture , the imperfect GCSI model and the system utility . A. Cluster-based Arc hitecture and Channel Model Fig.1 illustrates the system m odel of the r elay-assisted OFDMA system with one BS, M fixed RSs and K MSs. Th e coverage area is divided into M + 1 c lusters with cluster 0 served by the BS and cluster m  m ∈ { 1 , M }  served by the m th RS. The K MSs are assigned to on e of the M + 1 clusters accordin g to their large scale path lo ss 3 . The number of MSs in cluster m is K m . MSs in cluster 0 will re ceiv e downlink 3 The k th user is assigned to the cluster with the strongest recei ved pil ot strength. Cluster 1 Cluster M Cluste m Cluster 0 Spreading code 2 Spreading code 1 MS k in C m -RS m RS m -MS k in C m ( m = 1 , . . , M ) ( m = 1 , . . , M ) MS k in C 0 -BS RS m -BS BS-MS k in C 0 BS-RS m Phase 1 Phase 1 Phase 2 Phase 2 UPLINK DOWNLINK Uplink CSI estimation Use uplink CSI estimation as downlink CSIT TDD Frame Timing d i r e c t l i n k ( p h a s e 1 ) I n d i r e c t l i n k ( p h a s e 1 ) I n d i r e c t l i nk ( p h a s e 2 ) (m = 1,..,M) (m = 1,..,M)  Fig. 1. System Model packets or transmit up link packets directly fro m or to the BS, and M Ss in cluster m  m ∈ { 1 , M }  will re ly on th e m th RS forwarding the do wnlink or uplink packets in th e packet transmission process. Let K m  m ∈ { 1 , M }  denote the set of MSs in clu ster m , and K 0 denote the set o f MSs in cluster 0 and th e RSs . For n otation con venience, we assum e index k in clu ster 0 denotes the k th MS if k ∈ { 1 , K 0 } and the m th RS if k = K 0 + m  m ∈ { 1 , M }  . W e consider freq uency selective fading where there are N indep endent multipa ths. OFDMA is em ployed to c onv ert the freq uency selectiv e fading chann el into T orthogo nal subcarriers with N indepen dent subbands. The RSs operate in a half -duplex manner using decod e-and- forward (DF) strategy . In order to facilitate relay-a ssisted packet transmission, a physical frame in the downlink is divided in to tw o phases: • In ph ase one, the BS transmits data to th e MSs and the RSs belonging to cluster 0. • In phase two, the BS stops tran smitting wh ile the RSs (which have successfu lly decoded data from the BS in phase one) will forward the data to the MSs b elongin g to their o wn clu sters. Similarly , the two ph ases of a physical f rame in th e uplink are • In ph ase on e, the MSs b elongin g to the relay clusters w ill transmit data to the ir o wn RSs. • In ph ase two, the MSs and th e RSs ( which have suc- cessfully d ecoded data from their MSs in ph ase one ) belongin g to cluster 0 will transmit the data to the BS. T wo orthog onal freq uency spr eading codes are assigned to adjacent RS clusters to mitigate p otential mutual interfere nce between them as illu strated in Fig.1 4 . Since th e path loss between non -adjacent RS clusters are u sually quite large, there is pr actically negligib le mutua l interfere nce between non-ad jacent RS clusters. As a resu lt, all the RS clusters can oper ate simu ltaneously on th e entire f requen cy band with 4 T o achie v e the orthogonal signal separati on in the frequenc y domain, signals from/to differen t RSs must be synchronize d in the OFDM symbol boundary . T his require s timing synchronizatio n within the unused cycl ic prefix in the OFDMA system and is implementable in practice. As a result, it is a common assumptio n in OFDMA systems such as W iMAX. IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 3 negligible in terference 5 . In two phases ( m = 0 for the phase one of the downlink (phase two of the u plink) a nd m ∈ { 1 , M } for the p hase two of the downlink (phase one of the uplink)), the re ceiv ed symbol Y m,k,n carrying user k ’ inform ation in cluster m in the n th subband is given by Y m,k,n = p p m,k,n l m,k H m,k,n X m,k,n + Z m,k,n m ∈ { 0 , M } where X m,k,n is the transmitted symb ol from (to) BS ( m = 0 ) or the m th RS  m ∈ { 1 , M }  to (from) user k ( k ∈ K m ) in cluster m in the n th subb and, p m,k,n is the tran smit SNR, l m,k is the pa th loss, an d Z m,k,n ∼ C N (0 , 1) is the noise in th e n th sub band. W e consider Rayleigh fading and hence, H m,k,n ∼ C N (0 , 1 ) . B. Globa l Cha nnel State In formation Model In this p aper, we co nsider a relay -assisted cellular system with large coverage, and hence, both small scale fading (du e to multipath) and la rge scale fading (due to path loss) are considered . W e con sider the blo ck fading chan nel w here the small scale fading co efficient is quasi-static within each fram e and m ay be d ifferent a mong frames. W e conside r the resou rce allocation fo r the low-mobility users. Since the tim e scale for mobility is much larger than that for small scale fadin g, the path loss remains co nstant for a large nu mber of frames and can be estimated with high accuracy 6 . The CSIT of small scale fading can be o btained fro m either explicit feedback (FDD systems) or implicit feedb ack (TDD s ystems) using r eciprocity between uplink and downlink 7 . W e consider TDD sy stems, and assume perfect CSIR and im perfect CSIT due to estima tion noise on th e uplink pilots or du plexing delay as illustrated in Fig. 1 . T he estima ted CSIT in phase on e an d phase two ca n be modeled a s ˆ H m,k,n = H m,k,n + △ H m,k,n m ∈ { 0 , M } (1) where H m,k,n is the a ctual CSI, and △ H m,k,n ∼ C N (0 , σ 2 e ) is the CSIT error . W e d enote the set of local imperfect CSIT of cluster m as ˆ H m = { ˆ H m,k,n | k ∈ K m , ∀ n } , an d the glob al imperfect CSIT as ˆ H = S M m =0 ˆ H m . C. System P olicy In this paper, we con sider jo int su bband , power and rate allocation 8 . The system po licies are d efined below . 5 For tract able analysis, we assumed all RSs are separat ed eit her spatially or on the cod e domain. Y et, the effect of mutual int erferenc e is taken int o considera tion in the performan ce simulat ion. 6 For exa mple, in 802.16e system, frame duration is 5ms . Users with pedestri an mobilit y 5 km/hr will have coherence time at least ove r 20 frames. 7 In practica l systems, like IEEE 802.16e, a mechanism named ”channel sounding” is proposed to enable the BS to determine the BS-to-MS channel response under the assumption of TDD reciproc ity . 8 For both the uplink and do wnlink transmission, the resourc e alloca tion is performed at BS and RSs. 1) Subba nd Alloca tion P olicy S : Let s B m,k,n ∈ { 0 , 1 } be the subband alloca tion ind icator at the BS for MS k in clu ster m  k ∈ { 1 , K 0 } when m = 0 , k ∈ K m when m ∈ { 1 , M }  . Let s R m,k,n ∈ { 0 , 1 } be the subba nd allo cation indica tor at the m th  m ∈ { 1 , M }  RS fo r user k ( k ∈ K m ) in cluster m . The subband allocation po licy is S = n s B m,k,n , s R m,k,n ∈ { 0 , 1 } |∀ n, M X m =1 X k ∈K m s B m,k,n + K 0 X k =1 s B 0 ,k,n = 1 , X k ∈K m s R m,k,n = 1 ∀ m ∈ { 1 , M } o 2) P ower Allocatio n P olicy P : Let p m,k,n be th e scheduled transmit SNR at BS ( m = 0) a nd the m th RS  m ∈ { 1 , M }  respectively to user k ( k ∈ K m ) in cluster m in the n th subband . Le t P m be the total tran smit SNR at BS ( m = 0) and the m th RS  m ∈ { 1 , M }  . Let P m,k be th e total tran smit SNR at the u ser k ( k ∈ K m ) . The po wer allocation policy in downlink and uplink systems ( ∀ m ∈ { 0 , M } ) are P DL = n p m,k,n ≥ 0 | N X n =1 X k ∈K m p m,k,n ≤ P m , ∀ m o P U L = n p m,k,n ≥ 0 | N X n =1 p m,k,n ≤ P m,k , ∀ m, k ∈ K m o 3) Rate Alloca tion P olicy R : Let r m,k,n be the sch eduled data rate at BS ( m = 0) and the m th RS ( m ∈ { 1 , M } ) respectively to user k ( k ∈ K m ) in cluster m in the n th subband . The rate allocation p olicy is R = n r m,k,n ≥ 0 | ∀ n, m ∈ { 0 , M } , k ∈ K m o D. Maximum Achievable Date Rate and System Goo dput In this part, we sha ll introdu ce the system u tility based on the notatio ns and system policies defined in the p revious part 9 . Giv en perfect CSIR, the instan taneous mutual information (bit/s/Hz) between the m th tran smitter (receiver) a nd the k th receiver (transmitter) ( k ∈ K m ) in the n th subband is given by C m,k,n = c m log 2 (1 + p m,k,n l m,k | H m,k,n | 2 ) (2) where c 0 = 0 . 5 and c m = 0 . 25 ( m ∈ { 1 , M } ) 10 . Due to imperfec t CSIT , the transmitter d oes no t have knowledge on the instantaneo us mu tual inform ation in (2) and hence, the sche duled data r ate at the BS and th e RS might b e larger th an the correspond ing m utual in formatio n, leading to pac ket outage. T o take the p otential ou tage in to 9 Note that p 0 ,k,n , r 0 ,k,n ( k = K 0 + m ) are the po wer and rate control var iables for the m th the RS ( m = 1 , · · · , M ). s B m,k,n is the subband allocation v ariable for the direc t link BS-MS k with m = 0 , k ∈ { 1 , · · · , K 0 } . s B m,k,n , s R m,k,n are the subband al locati on varia bles for indirec t link BS-RS m -MS k with m ∈ { 1 , M } , k ∈ K m .). 10 1 2 is due to duplexing penal ty . 1 4 is due to duplexing penal ty an d the spreading c ode with c ode rate 1 2 used in RS clust ers. For si mplicity , we assume that the same amount of time resources is used for each phase. Howe ver our design can be direct ly applied on the system with unequal phase durati on by changin g the duple xing penalty ratio between phase one and phase two. IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 4 consideratio n and to guarantee the scheduling fairness, we c on- sider the average weigh ted system goo dput (average weigh ted b/s/Hz successfu lly delivered to th e MSs) as our perfor- mance measur e. The average weighted total system g oodp ut is g iv en by ¯ U wg p ( S , P , R ) = E ˆ H [ U wg p ( S , P , R , ˆ H )] , wh ere U wg p ( S , P , R , ˆ H ) is the co nditional av erage to tal weigh ted goodp ut for given ˆ H , and is g iv en b y U wg p ( S , P , R , ˆ H ) = 1 N  M X m =1 X k ∈K m α m k min n N X n =1 s B m,k,n r 0 ,K 0 + m,n (1 − Pr[ C 0 ,K 0 + m,n < r 0 ,K 0 + m,n | ˆ H ]) , N X n =1 s R m,k,n r m,k,n (1 − Pr[ C m,k,n < r m,k,n | ˆ H ]) o + K 0 X k =1 α 0 k N X n =1 s B 0 ,k,n r 0 ,k,n (1 − Pr[ C 0 ,k,n < r 0 ,k,n | ˆ H ])  (3) where α m k denotes the weig ht 11 of the MS k in Cluster m , and Pr[ C m,k,n < r m,k,n | ˆ H ] ( ∀ n , m ∈ { 0 , M } , k ∈ K m ) is the condition al packet error pr obability of one-ho p link for g iv en global ˆ H . I I I . S U B B A N D , P O W E R A N D R A T E S C H E D U L I N G P RO B L E M F O R M U L A T I O N In this sectio n, we sh all formulate the relay-assisted schedul- ing p roblem as an optimization p roblem max imizing the av er- age total weighted goodput ¯ U wg p ( S , P , R ) subject to the target outage pro bability ǫ . Note that op timization on ¯ U wg p ( S , P , R ) w .r .t. policies (set o f a ctions for all CSIT re alizations) is equiv alent to optimiz ation on U wg p ( S , P , R , ˆ H ) w .r .t. the actions S , P , R ( S = S ( ˆ H ) , P = P ( ˆ H ) , R = R ( ˆ H )) f or each g i ven CSIT realization. Hence, th e optim ization pro blem is gi ven by Pr oblem 1 : ( Sub band, P o wer a nd Rate Optimization Pr ob- lem ) max S , P , R U wg p ( S , P , R , ˆ H ) s.t. Pr[ C m,k,n < r m,k,n | ˆ H ] = ǫ ∀ n, m ∈ { 0 , M } , k ∈ K m (4) s B m,k,n , s R m,k,n ∈ { 0 , 1 } ∀ n, m ∈ { 0 , M } , k ∈ K m (5) M X m =1 X k ∈K m s B m,k,n + K 0 X k =1 s B 0 ,k,n = 1 ∀ n (6) X k ∈K m s R m,k,n = 1 ∀ n, m ∈ { 1 , M } (7) p m,k,n ≥ 0 ∀ n, m ∈ { 0 , M } , k ∈ K m (8) N X n =1 X k ∈K m p m,k,n ≤ P m , ∀ m ∈ { 0 , M } (for DL) (9a) N X n =1 p m,k,n ≤ P m,k , ∀ m ∈ { 0 , M } (for UL) (9b) 11 Proportion al fairness (PF) is a speci al case of the weighted sum goodpu t system utilit y . I V . D U A L P R O B L E M A N D D I S T R I B U T I V E S O L U T I O N Note that Problem 1 is a mixed integer real optimizatio n and hence, is n ot conve x. Brute-force optimization is N P-hard with exponential complexity in term of the nu mber of subband s. In this sectio n, we shall ap ply co ntinuo us relax ation techniq ue [6], [ 3], [13], [1 4], to obtain asymptotically optimal solution as well as discu ss the distrib utive im plementation . A. Continu ous Relaxation of the Inte ger Pr ogramming Pr ob- lem T o perfo rm contin uous relaxation, we allow time sharing between users for each subband by relaxing subban d allocation indicator to ratio nal value b etween 0 an d 1 , wh ich describes the f raction o f time a particu lar subband is occupie d b y a p articular user . Mathem atically , to a pply th e co ntinuou s relaxation, the constraint ( 5) is r eplaced by s B m,k,n , s R m,k,n ≥ 0 ∀ n, m ∈ { 0 , M } , k ∈ K m (10) After equiv alent tr ansforma tion and continuo us relaxatio n, Problem 1 is approxima ted b y the following con vex optimiza - tion problem. Pr oblem 2 (R elaxed Optimization Pr oblem): max S,P, t M X m =1 X k ∈K m α m k t m k + K 0 X k =1 α 0 k N X n =1 s B 0 ,k,n ˜ r B 0 ,k,n s.t. t m k ≤ N X n =1 s B m,k,n ˜ r B m,k,n m ∈ { 1 , M } , k ∈ K m (11) t m k ≤ N X n =1 s R m,k,n ˜ r R m,k,n m ∈ { 1 , M } , k ∈ K m (12) as well as constraints in ( 10 ) , ( 6 ) , ( 7 ) , ( 8 ) , ( 9a ) for DL or ( 9b ) for UL where 12 ˜ r B m,k,n = 1 2 log 2  1 + p m,k ′ ,n ˆ g m,k ′ ,n s B m,k,n  , m ∈ { 0 , M } ( m = 0 , k ′ = k ∈ { 1 , K 0 } ; m > 0 , k ′ = K 0 + m, k ∈ K m ) ˜ r R m,k,n = 1 4 log 2  1 + p m,k,n ˆ g m,k,n s R m,k,n  , m ∈ { 1 , M } , k ∈ K m ˆ g m,k,n = l m,k 1 2 σ 2 e F − 1 | ˆ H m,k,n | 2 / 1 2 σ 2 e ( ǫ ) , m ∈ { 0 , M } , k ∈ K m (13) Note that Problem 2 is a con vex p roblem. For the objective function , the first part P M m =1 P k ∈K m α m k t m k is linear in t m k , and the second part P K 0 k =1 α 0 k P N n =1 s B 0 ,k,n ˜ r B 0 ,k,n is a positive linear combination of functions of the type f ( x, y ) = x log(1 + y /x ) , y ≥ 0 , x ≥ 0 . Accor ding to Lemma 1 of [14], the second part is co ncave in ( s B 0 ,k,n , p 0 ,k,n ) . Theref ore, the ob jectiv e function is concave. Similarly , after movin g the R.H.S. of 12 F − 1 | ˆ H m,k,n | 2 / 1 2 σ 2 e ( · ) de notes the in v erse cdf of non-c entral chi-squa re random vari able with 2 degre es of freedom and non-centra lity paramete r | ˆ H m,k,n | 2 / 1 2 σ 2 e . (13) can be deri ve d from the condi tional PER constrai nt (4), the e xpression of instant aneous mutual information (2) and the no n-centra l chi-squar e distrib ution of | H m,k,n | 2 gi ven ˆ H m,k,n , which is omitte d due to page limit . IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 5 constraint (11) and ( 12) to the L.H.S., it ca n be proved that the inequality con straint fun ctions are conv ex. In addition , the left equ ality constraint function s are linear . Hence, Prob lem 2 is a c onv ex optimization problem. The continuo us relaxation in Problem 2 does not necessarily yield a solution where all the subban d allocation indicato rs are 0 or 1. Howe ver , they are 0 or 1 with hig h pr obability due to the prop erty of margin al b enefit of extra b andwidth defined in Appendix B. If the integer solu tion is required, we can ro und th e f ractional values to 0 or 1. In fact, under som e mild con dition ( T N → ∞ ) 13 , the solution of Problem 2 will be identical to that of Problem 1 when we do subcarrier allocation in both problems. Remark 1: Let T N be th e numb er of sub carriers in ea ch indepen dent sub band. W e assume the ch annel gain s of all the subcarriers in o ne subban d are h ighly correlated . Suppo se a user is assigned n (0 ≤ n ≤ T N ) sub carriers in a particu lar subband durin g one transmission. W e can in terpret n/ T N as the frequen cy sharing factor in this subban d, which takes discrete value f rom 0 to 1. When T N → ∞ , the fr equency sharing factor can take any ration al between 0 and 1. It can be achieved by the further subcarrier allo cation w ithin the subband. B. P artial Dual Decomposition and Distributed Solution The subb and, power and bit r ate allo cations fo r re lay- assisted OFDMA system is a complicated optimization p rob- lem. Ex isting approac h [3], [6 ], [7] con sidered centralized solutions but these solu tions hav e hu ge com putation al load ing at the BS. Furthermor e, these solutions requ ire kn owledge of GCSI wh ich induces heavy signaling overhead in th e system. In this section, we shall d eriv e a low-complexity distributi ve solution from Problem 2 using decom position tech niques. Using co n vex optim ization technique s (details in Appendix A), the dual function of Problem 2 can be simplified as follows max S,P K 0 X k =1 α 0 k N X n =1 s B 0 ,k,n ˜ r B 0 ,k,n + M X m =1 X k ∈K m λ m k N X n =1 s B m,k,n ˜ r B m,k,n + M X m =1 X k ∈K m ( α m k − λ m k ) N X n =1 s R m,k,n ˜ r R m,k,n (14) s.t. ( 10 ) , ( 6 ) , ( 7 ) , ( 8 ) , ( 9a ) fo r DL or ( 9b ) for UL In what f ollows, we apply du al decom position approach [15], [16]. The du al function can b e deco mposed into two subprob lems: Subpr oblem 1: ( Resour ce Allocations at BS in P hase One of Do wnlink (Phase two of Uplink) ) Given M vectors of Lagrang ian mu ltipliers λ m , optimize the subb and and power allocations at BS such that the weighted go odpu t is maximized subject to corr espondin g su bband constra ints at th e BS, and power constrain ts at the BS for do wnlink (at the MSs in cluster 13 The condition is quite mild and can be s atisfied in most practical systems. For example , we have N = 2048 and T ≈ 6 in WiMAX (802.16e) s ystems. 0 and RSs fo r uplink). g B S ( λ 1 , · · · , λ M ) =      max S,P P M m =1 P k ∈K m λ m k P N n =1 s B m,k,n ˜ r B m,k,n + P K 0 k =1 α 0 k P N n =1 s B 0 ,k,n ˜ r B 0 ,k,n s.t. s B m,k,n ≥ 0 , ( 6 ) , ( 8 ) , ( 9a ) or ( 9b ) ( m = 0) Subpr oblem 2: ( Resource Allocatio ns a t the m th RS in Phase T wo of Downlink (Phase one of Uplink) ) Giv en the vector of Lagr angian multiplier s λ m , op timize subb and and power allo cations at the m th RS such that the weighted go od- put is maxim ized subject to co rrespon ding subban d constraints at the m th RS, and power constraints at the m th RS for downlink (at the MSs in cluster m for uplink ). g m RS ( λ m ) =  max S,P P k ∈K m ( α m k − λ m k ) P N n =1 s R m,k,n ˜ r R m,k,n s.t. s B m,k,n ≥ 0 , ( 7 ) , p m,k,n ≥ 0 , ( 9 a ) or ( 9b ) ( m > 0) There are M subpro blems of this kind, each one corresponding to the resource alloca tions at o ne RS. Subpro blem 1 and Subpro blem 2 are similar . Please re fer to the Append ix B fo r the optimal so lution. With the du al function m entioned above, the dua l p roblem is summarized below: Pr oblem 3 (Du al Pr oblem): Find the optimal dual v ariables which maximize th e dual function min λ g B S ( λ 1 , · · · , λ M ) + M X m =1 g m RS ( λ m ) s.t. 0 4 λ m 4 α m , m ∈ { 1 , M } where α m =  α m k  K m × 1 ( m ∈ { 1 , M } ) . W e use the subgradien t m ethod [1 7] to update eac h du al variable as follows λ m k ( i + 1) (15) = h λ m k ( i ) − δ m k ( i )( N X n =1 s B m,k,n ˜ r B m,k,n − N X n =1 s R m,k,n ˜ r R m,k,n ) i X m k where δ m k ( i ) is a po siti ve step size and [ · ] X m k denotes the pr o- jection onto the feasible set X m k = { λ m k | 0 ≤ λ m k ≤ α m k } (i.e. let λ m k ( i + 1) = 0 ∀ [ · ] < 0 , λ m k ( i + 1) = α m k ∀ [ · ] > α m k and keep λ m k ( i + 1) = [ · ] if [ · ] ∈ X m k ). W e study the co n vex Prob lem 2 by solving its dual problem with deco mposition techn ique and subgrad ient method . Since the Prob lem 2 is con ve x and strictly feasible 14 , Slater’ s condition holds. Therefor e, d uality gap is zero and he nce, the prima l variables S ( λ 1 ( i ) , · · · , λ M ( i )) and P ( λ 1 ( i ) , · · · , λ M ( i )) will co n verge to the primal optim al variables S ∗ , P ∗ . Since the dual p roblem is always conve x, the convergence of the prop osed scheme is gua ranteed [17] . For any initial value in the feasible set, the d ual variable λ m k ( i ) will con verge to the dual optimal λ m ∗ k as i → ∞ . The distributi ve architectur e of the derived solu tion is il- lustrated in Fig.2. In tuitively , for user k in cluster m ( m ∈ 14 For ex ample, we can easily fi nd s B m,k,n , s R m,k,n > 0 , which satisfy (6) and (7), and p m,k,n > 0 , which satisfy (9a ) for do wnlink or (9b) for uplink . By choosing t m k = min { P N n =1 s B m,k,n ˜ r B m,k,n , P N n =1 s R m,k,n ˜ r R m,k,n } − ǫ ( ∀ ǫ > 0) , we can obtai n a strictly feasible solution to the con ve x problem 2. IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 6 Ma ster pro blem U p da te du al v ariables . Su bp rob lem 1 1 st h op resou rce allocation for ne ar u sers an d for R S s at B S Su bp rob lem 2 2 nd h op resou rce allocation for far users in cluster 1 at R S 1 Su bp rob lem 2 2 nd h op resou rce allocation for far users in cluster M at R S M Su bp rob lem 2 2 nd h op resou rce allocation for far u sers in cluster m a t R S m 1 , M λ λ  m λ 1 λ M λ 1 ( , ) M BS g λ λ  1 ( ) RS g λ ( ) m RS g λ ( ) M RS g λ Fig. 2. Decomposit ion Struct ure { 1 , · · · , M } ) , the d ual variable λ m k ( i ) can be interpreted as the equivalent weight in phase one for downlink (phase two for uplink) , while α m k − λ m k as the equiv alent weight in phase two for downlink (phase one for uplink) . Based on the weights assigned by the master pr oblem, Subpr oblem 1 can be solved at BS with its local imperf ect CSIT ( ˆ H 0 ), and Subprob lem 2 for th e m th RS ca n be solved at the m th RS with its local imperfect CSIT ( ˆ H m ). Then the du al problem updates the dual variables at BS to re duce the difference between the scheduled data rate for a particular user in phase one and phase two at each iteration. The distributi ve implementation offloads great co mputation s from the BS to M RSs. In addition, it sa ves large sig naling overhead for co llecting CSI of RS-MS links an d broad casting the sched uled results to RSs for reso urce allocatio ns of RS- MS links. These advantages are highly desirable fo r p ractical implementatio n. V . R E D U C E D F E E D B AC K D I S T R I B U T I V E S O L U T I O N F O R D O W N L I N K S Y S T E M S Compared with the centralized solution, the co mplexity and signaling overload in the above distributi ve solution are greatly reduced . Ho wev er , in downlink systems, the overhead due to the fe edback o f CSI fro m MSs to their cluster co ntrollers (the BS and RSs ) and the signaling between the BS an d RSs still gr ows linea rly with the nu mber of u sers K 15 . Selective multiuser di versity ( SMUD) has been proposed in [ 18], [19] for cellula r system to red uce feedback overhead by im posing a local threshold- based screening. W e shall extend the threshold- based mechanism to reduce the overall system feedback over - head of th e distributiv e so lution for relay-assisted cellular network with fairness con sideration. The reduce d feedb ack distributi ve solution is outlined below 16 . 15 In uplink systems, since resource alloca tion is perfo rmed at the BS and RSs, which are the recei ve rs, we do not have the issues of CSI feedback ov erhead. After performing distrib uti ve resource allocat ion, the BS and acti ve RSs inform the corresponding scheduled MSs with the sched uled subban d and transmit SNR for transmission . 16 In practice (e.g. HSDP A, W iMAX, 802.16m, L TE ), there is alw ays a pre- alloc ated dedicate d UL signaling channel associate d with high speed do wnlink pack et data access and DL resource alloca tion. So our feedb ack informatio n from MS- > RS and RS- > BS can be transmitted over these av ailabl e dedicated control channe ls. Algorithm 1 ( Distributive Alg orithm with Re duced F eedback): 1) The controller of cluster m ( m ∈ { 0 , M } ) finds out MS i with the maximum weight α m i among MSs in Cluster m , and broadcasts the feedba ck threshold γ m th = α m i l m i ˜ γ m to the MSs in its cluster . 2) Each MS in Cluster m com pares and f eedback s its CSI h m k,n in the n th subband iff α m k l m k | h m k,n | 2 ≥ γ m th . Then the lo cal r educed user set an d cor respond ing imperfect CSIT 17 is a vailable at each cluster manager . 3) BS decides and b roadcasts th e initial multipliers { λ m (0) | m ∈ { 1 , M }} in phase one. The initial m ul- tipliers { α m − λ m (0) | m ∈ { 1 , M }} in phase two are av ailable at RSs for subpro blem 2. 4) In the i th iteration , BS solves Subproblem 1 and each RS solves its own Su bprob lem 2. Each RS reports the scheduled data r ate of users in its cluster to BS. 5) BS up dates the multipliers { λ m ( i ) } to { λ m ( i + 1) } in phase one accord ing to (1 5), and b roadcasts { λ m ( i + 1) } . 6) If the difference of the sched uled data rate in two ph ases for each user in RS clusters is less than a threshold, or the number of iterations has already reached a predeter- mined value, then terminate the algorith m. Oth erwise, jump to step 4 ). In the ab ove algo rithm, the com munication overhead be- tween BS and each RS grows linearly on ly with the to tal number of users in M + 1 reduced user sets. Hence, the feed- back load of this algor ithm is much smaller than the directly distributi ve imp lementation with full feed back. Although this algorithm is in g eneral suboptim al du e to the existence of feedback outage, it can be pr oved in the following lemm a that und er some con ditions, its pe rforma nce will con verge to that of the directly distributi ve imp lementation with all MSs transmitting f eedback s. Due to the sy mmetric situation in each subband , we only con sider on e sub band and ignore the ind ex n for simplicity . Lemma 1: ( F eed back Outage, F eedback Load and Asymp- totic P erforman ce ) Assume that th e weight of each user is no t smaller than 1. W ithout loss of gen erality , assume u ser i h as the maximum weight α m i in Cluster m . Defin e T m k = α m i l m i α m k l m k . 1) Giv en the threshold γ m th = α m i l m i ˜ γ m , the outage pro b- ability P m 0 (the probability that n o one feedbacks) and the feedb ack load F m (average number of feedback s per user) are given b y P m 0 = Q K m k =1 (1 − exp( − T m k ˜ γ m )) and F m = 1 K m P K m k =1 (1 − exp( − T m k ˜ γ m )) respectively . 2) Let the upper boun d of P m 0 be P m 0 ( K m ) . If P m 0 ( K m ) satisfies P m 0 ( K m ) → 0 and P m 0 ( K m ) 1 K m → 1 as K m → ∞ , choo se ˜ γ m ( P m 0 ( K m )) = 1 max T m k log 1 1 − P m 0 ( K m ) 1 K m , so that P m 0 → 0 an d F m → 0 as K m → ∞ . There is a tradeoff between feedb ack ou tage probability and feedback loa d b y adjusting ˜ γ m . Howe ver , for sufficiently large 17 The reduce d user set is made up of users who feedback at least in one subband. T he channel coef fici ent of any user in this set will be treated as 0 in the subbands without feedba ck from this user . IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 7 K m , Algorithm 1 can achie ve asymptotically optimal weighted av erage go odput at asym ptotically zero feedb ack cost per user by choosing ˜ γ m ( P m 0 ( K m )) = 1 max T m k log 1 1 − P m 0 ( K m ) 1 K m . Pr oof: Please refer to Appen dix C. V I . A S Y M P T O T I C P E R F O R M A N C E A N A L Y S I S U N D E R P F S F O R D O W N L I N K S Y S T E M S In this section, we shall focus on stud ying how the sys- tem perfor mance of the schedulin g algorithm der iv ed in the precedin g section fo r d ownlink sy stems grow with various importan t system parameter s such as the nu mber of RSs an d the n umber of MSs. Specifically , we co nsider pro portion al fair scheduling (PFS) [20] performan ce in the limit situation t c → ∞ [21] , [ 22] for sufficiently large K in the analy sis. T o obtain insights on th e performa nce gains, we impose a set of simplifying assumption s. W e assume each RS contains K MSs with K 0 = 0. Furthermo re, we assume line-o f-sight link ( with high gain anten na) between the RSs and the BS and hen ce, the throu ghpu t is limited by the seco nd hop. Under above assumptions, we can sub stantially simplify th e system mo del without af fecting the asymptotic performance and he nce, we can der iv e closed form results wh ich would oth erwise b e impossible in general r egimes. Let D be the rad ius of a cell. Path loss takes on the form P L ( d ) = d − α , where α is the p ath loss expon ent. In the r elay-assisted system, assume ther e are M RSs and K users uniform ly distributed in each RS cluster . Assume equal power alloca tion to each sub band at RSs 18 . Accor dingly , in the system withou t RS, we assum e M K u sers in the same cell edge region as the rela y clusters, which is eq uiv alent to the total nu mber o f user s in the relay-assisted system, and BS allocates equ al power to each sub band. The asymp totic perfor mance un der PFS of the two systems is sum marized in the following lemma. Lemma 2: ( Asymptotic P erforma nce fo r the Systems with RSs and without RSs ) For the sy stem with M RSs and K MSs in each RS-cluster , th e average asymp totic through put for large K is E [ T ] = M 4  log 2 ( P m N ln K ) − α (log 2 D + log 2 t − 1 2 ln 2 )  . For the system with M K MSs and no RSs, the a verage asymptotic th rough put for large K is E [ T ( b ) ] = log 2 ( P 0 N ln M K ) − α (log 2 D + (1 − 2 t ) 2 1 − (1 − 2 t ) 2 log 2 1 1 − 2 t − 1 2 ln 2 ) , where t = sin π M 1+sin π M . Pr oof: Please refer to Appen dix D. Define the gain o f relay-assisted d esign over the non-relay design as g = E [ T ] − E [ T ( b ) ] = g f sr + g pl , where g f sr = M 4 (log 2 ( P m N ln K ) − α log 2 D ) − (log 2 ( P 0 N ln( M K )) − α log 2 D ) is th e thr oughp ut gain for fr equency and spacial reuse and g pl = α  (1 − 2 t ) 2 4 t − 4 t 2 log 2 1 1 − 2 t − M 4 log 2 t + ( M 4 − 1) 1 2 ln 2  is the thro ughpu t gain for r educing energy reductio n due to path loss. Remark 2: Since g f sr = O ( M ln ln K ) − O (ln ln( M K )) , the through put of relay-assisted system gro ws much f aster than 18 Since the number of MSs are lar ge, equal powe r allocati on is asymptot- icall y optimal due to multi-use r dive rsity . 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 Users Goodput(Mb/s) Proposed Algorithm(sum 61.247) SSA(sum 30.800) MaxPF without RS(sum 12.209) Round Robin without RS(sum 6.266) Fig. 3. A v erage goodput allocat ion for 10 users in a single cell scenar io at BS transmit po wer 36 dBm, RS transmit powe r 36 dBm at N = 16 . the system without RS, which is d ue to th e spatial r euse of relay-assisted architecture. I t can b e sho wn that g pl > 0 and increases with M . This d emonstrates the ben efits o f r elay- assisted architecture on red ucing energy reductio n. V I I . S I M U L A T I O N R E S U LT S A N D D I S C U S S I O N In this part, we shall com pare ou r distributi ve subband , power an d rate allocatio n for relay-assisted OFDMA system with several baseline referen ces. Baseline 1 re fers to the weighted to tal goodpu t maximizatio n versio n o f Sepa rate and Sequen tial Allo cation (SSA), which is a semi-d istributed scheme p roposed in [8 ]. Baseline 2 re fers to the maximu m total weig hted go odput sched uling witho ut RSs. In Baseline 3, we con sider Round Ro bin Sched uling with water-filling power allocation across the subb ands. W e apply PFS algo rithm to keep track of th e a verage goodpu t of each user , and consider its inverse as the weig ht fo r the thre e maximu m total weighted goodp ut scheduling desig n. W e use Jain’ s in dex as the fairn ess measure, which rang es from 1 /K (worst case) to 1 (best case). In the following simulation results, the average sum goodp ut is o btained from the optimal sch eduled go odpu t of each user . The c ell radiu s is 5 000m . 6 RSs are e venly loca ted on th e circle with radius 30 00m. W e set up ou r simulation scenarios accord ing to the prac tical settings in IEEE8 02.16 m systems [ 23]. The carr ier fre quency is 2GHz . BS/RS h eight is 32 m an d MS height is 1 .5m. Th e op erating bandwid th is 10MHz with 2 048 subcarriers and 8, 16 or 24 independ ent subband s. The path loss m odel of BS-MS and RS-MS is 128 . 1 + 3 7 . 6 log 10 ( R ) d B, and the path lo ss mod el BS-RS is 128 . 1 + 28 . 8 log 10 ( R ) d B ( R in km). The recei ve antenna gains o f MS is 0 d B, an d the directio nal rec eiv e antenna is used at RS with antenn a gain 20 dB. The lognor mal shadowing standard d eviation is 8 dB, and the CSIT error variance σ 2 e is 0.01. A. System P erformance of the Distributive Algorithm Fig.3 illustra tes the a verage schedu led g oodpu t allocation of the 10 users in a single cell scen ario, the positions of IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 8 0 10 20 30 40 50 75 100 125 150 175 200 225 250 275 Number of Iterations Weighted Goodput(Mb/s) Primal Value: K=10,N=24,M=6 Dual Value: K=10,N=24,M=6 Primal Value: K=10,N=8,M=8 Primal Value/2: K=20,N=8,M=6 Primal Value/3: K=30,N=8,M=6 Primal Value: K=10,N=8,M=6 Fig. 4. A ve rage total weighted goodput in a single cell scenario for primal problem and dual problem of 10 users (N=24, M=6, K=10) and primal problem of 4 other cases (N=8, M=8, K= 20,30 respecti v ely) versus the number of iterations at BS transmit power 36 dBm, RS transmit power 36 dBm. (Due to limit ed space in the graph, we hav e omitte d the dual values of the othe r 4 cases.) which are gener ated acco rding to unifor m distribution. It can be ob served that ou r distributi ve schedu ling algorithm can achieve mu ch higher average goodput and fairness comp ared with three baselines. Fig.4 illustrates the convergence pe rforma nce of our dis- tributi ve algor ithm in a single cell scenario . W e plot the av erage b est primal value curve and dual value curve within certain numb er of iterations for th e 10 users c ase ( N = 24 , M = 6 , K = 10 ) and the best prim al value c urve fo r fo ur other cases with N = 8 , M = 8 , K = 20 , 30 19 , re spectiv ely . It can be seen that our distributi ve algorithm con verges quite fast. The perf orman ce at the 5th iteration is abo ut 95 % of th e perfor mance at the 50th itera tion. Th us, good p erform ance can be achie ved with low overhead . B. System P erformance vers us T r ansmit P ower Fig.6 illustrates the a verage s um goodput perfo rmance of the 10 users in a multi-c ell scenar io with freq uency reuse factor 3 as illustrated in Fig. 5 versus the transmit power at BS and RS. It can be o bserved that our propo sed scheduling design has significan t gain, especially in the lower SNR region. T his is mainly because RS redu ces the path loss greatly and our propo sed algor ithm utilizes the limited power more efficiently . Fig.7 illustrates the fairness perfor mance of the 10 users in a multi-cell scenar io versus the transmit power . I t can b e seen tha t the d esigns with RSs keep much be tter fairness, especially in the lower SNR region. The main reason is that the differences of p ath loss of u sers in the c ell are grea tly red uced when the RS is half -way in b etween the BS an d the MS. Compared with Baseline 1, o ur p roposed design has similar fairness performan ce but a much better good put p erform ance as illustrated in Fig .6. 19 The con ver gence performance for the full feedback distributi v e algori thm with 30 MSs, is actual ly the same as the con ver gence performance for the reduced feedbac k distrib uti ve algorithm with 150 MSs under only 5% goodput performanc e penalt y . 3 1 2 1 3 1 3 1 2 1 2 1 1 1 R = 3D r = 2 D D =5000m D m = 3 0 0 0 m Fig. 5. Multi-Cell Scenario with frequenc y reuse factor 3 (The number in the cell center shows the frequency band used by that cell .). T wo orthogonal frequenc y spreading code s are assigned to adjacent RSs to achie ve the orthogona l signal separat ion in the frequenc y domain (blue and yello w parts represent two frequency sprea ding codes adopted in RS clusters.). The cell radius D is 5000m. 6 RSs are ev enly located on the circle with radius D m = 3000m. The distance betwe en the BSs of the nea rest co-cha nnel cells is R = 3 D . The distance between the RSs of the nearest relay clusters of the nearest co-cha nnel cells is r = 2 D . 16 21 26 31 36 41 46 0 10 20 30 40 50 60 70 80 Transmit Power (dBm) Sum Goodput(Mb/s) Proposed Algorithm SSA MaxPF without RS Round Robin without RS Fig. 6. A v erage sum goodput of 10 users versus BS/RS transmit power in a multi-ce ll scenario with frequenc y reuse fact or 3 as illustra ted in Fig. 5 at N = 16 . C. System P erformance versus the Numbe r of Users K Fig.8 illu strates th e average sum goodput perfo rmance ver - sus the number of users per RS cluster in a single cell scenario. The transmission in phase two of our design d irectly benefit from the in crease in th e nu mber of cell-edge u sers. When BS and RSs have the same transmit power 36 d Bm, the bottleneck of the perform ance is in ph ase on e. Hen ce, th e first curve dose not in crease much with th e nu mber of users in RS clusters. When the transmit p ower at RSs is sma ller than that at BS, i.e. 3 1 dBm, 26 dBm, th e b ottleneck shifts to phase two. T hus, the overall perfo rmance incr eases greatly with the number of users. IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 9 16 21 26 31 36 41 46 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Transmit Power (dBm) Jain‘s Index Proposed Algorithm SSA MaxPF without RS Round Robin without RS Fig. 7. Jain’ s fairness inde x of 10 users versus BS/RS transmit po wer in a multi-cell scenario with frequenc y reuse factor 3 as illustrat ed in Fig. 5 at N = 16 . 2 4 6 8 10 12 14 16 0 10 20 30 40 50 60 70 Number of Users per RS Cluster Sum Goodput(Mb/s) Proposed Algorithm (RS 36dBm) Proposed Algorithm (RS 31dBm) Proposed Algorithm (RS 26dBm) SSA (RS 36dBm) SSA (RS 31dBm) SSA (RS 26dBm) maxPF without RS Round robin without RS Fig. 8. A ve rage sum goodput versus the number of users per RS cluster (6 RSs) in a single cell scenario at BS transmit po wer 36dBm, RS transmit po wer 36dBm, 31dBm, 26dBm, and N = 8 . D. T radeo ff of P erformance a nd A ver age F eedb ack Load in the Distrib utive Reduced F eedback Algorithm Fig.9 illustrates the a verage sum g oodpu t perfo rmance fo r 60 users per cell in a sing le cell scenario versus the a verage feedback load by implementing d istributi ve re duced feedback algorithm . W e use different feedbac k thre sholds to generate different average sum g oodp ut perf ormanc e with different av erage f eedback load . Our distributi ve redu ced feed back algorithm can achieve go od perfo rmance (e. g. 95 % of the distributi ve full feedback algo rithm in Fig.8) with low feed- back load (e.g. 20 % ) in real systems. V I I I . S U M M A RY In this paper, we p ropose a cluster-based distributi ve sub - band, power and rate alloca tion fo r a two-hop transm ission in a re lay-assisted OFDMA ce llular system. W e take into account of potential packet e rrors d ue to im perfect CSIT and system fairness by co nsidering weigh ted sum good put as our o ptimization ob jectiv e. Based on the cluster-based 5 10 15 20 25 30 35 54 55 56 57 58 59 60 61 62 Normalized Average Feedback Overhead (%) Sum Goodput(Mb/s) RS Transmit Power 36dBm RS Transmit Power 31dBm Fig. 9. A vera ge sum goodput of 60 users per cell in a single cel l scenario versus the av erage feedback load at BS transmit po wer 36dBm, RS transmit po wer 36dBm, 31dBm, and N = 8 . architecture , we obtain a fast-converging distributi ve solution with only local imperfect CSIT by using careful decomp osition of optim ization problem . Our solution could be applied to both UL and DL allocation s. T o further red uce the signaling overhead and computational complexity in downlink systems, we prop ose a reduce d feedback distributi ve a lgorithm, which can achieve asy mptotically optim al pe rforman ce for large number of users with arbitrarily small feed back ou tage an d feedback lo ad. W e also derive asymptotic av erage system goodp ut f or the relay-assisted OFDMA system so as to obta in useful design insights. A P P E N D I X A : D E R I V A T I O N O F D U A L F U N C T I O N By relaxing the global coupling constraints (11) and ( 12), we hav e the fo llowing dual function max S,P, t M X m =1 X k ∈K m α m k t m k + K 0 X k =1 α 0 k N X n =1 s B 0 ,k,n ˜ r B 0 ,k,n + M X m =1 X k ∈K m λ m k ( N X n =1 s B m,k,n ˜ r B m,k,n − t m k ) + M X m =1 X k ∈K m ν m k ( N X n =1 s R m,k,n ˜ r R m,k,n − t m k ) s.t. ( 10 ) , ( 6 ) , ( 7 ) , ( 8 ) , ( 9a ) for DL or ( 9 b ) for UL where λ m , ( λ m k ) K m × 1 < 0 , ν m , ( ν m k ) K m × 1 < 0 ( m ∈ { 1 , M } are th e vectors of Lagran gian multipliers. Firstly , we optimize over t . The p art of d ual fun ction with respect to t is giv en by g 0 = max t M X m =1 X k ∈K m  α m k − ( λ m k + ν m k )  t m k =  0 , λ m k + ν m k = α m k ∞ , o therwise T o make sure that th e dual func tion is bound ed abov e, we hav e λ m k + ν m k = α m k . Hence, we can simplif y the du al function. IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 10 A P P E N D I X B : O P T I M A L S O L U T I O N T O T H E R E L A X E D P R O B L E M For downlink systems, th e subp roblem 1 and 2 share the similar form 20 as follows max S,P K X k =1 α k N X n =1 s k,n log 2  1 + g k,n p k,n s k,n  s.t. s k,n ≥ 0 , K X k =1 s k,n = 1 ∀ n, p k,n ≥ 0 , N X n =1 K X k =1 p k,n ≤ P The Lagrang ian function is gi ven by L = P K k =1 α k P N n =1 s k,n log 2  1 + g k,n p k,n s k,n  + µ  P − P N n =1 P K k =1 p k,n  + P N n =1 v n  1 − P K k =1 s k,n  . ∂ L ∂ p k,n = 0 ⇒ p k,n = s k,n  a k µ − 1 g k,n  + (16) ∂ L ∂ s k,n = 0 ⇒ X k,n , α k log 2  1 + g k,n p k,n s k,n  − α k g k,n p k,n s k,n + g k,n p k,n = v n (17) where X k,n can be interpr eted as m arginal ben efit o f extra bandwidth [14]. By substituting (16) into (17), we have X k,n = α k log  1 + g k,n  α k µ − 1 g k,n  +  − µ  α k µ − 1 g k,n  + . µ satisfies K X k =1 N X n =1 s k,n  α k µ − 1 g k,n  + = P (18) µ can be obtained b y the subgr adient metho d. For a particular µ , if there is a uniq ue k ∗ = arg ma x { X k,n } for some n , time- sharing will not h appen in th is subband. s k,n =  1 , X k,n = max k  X k,n  > 0 0 , otherwise Since for e ach g i ven µ , X k,n is a f unction o f the CSI g k,n of each user and they a re independ ent rand om variable. As a result, there is pr obability 0 for X k,n = X k ′ ,n with k 6 = k ′ . Hence, there is p robab ility 1 that th e solution o f the relaxed problem correspon ds to that of the or iginal p roblem 21 . For up link systems, we u se individual power constraint P N n =1 p k,n ≤ P k instead. Similarly , we have p k,n = s k,n  a k µ k − 1 g k,n  + , where µ k satisfies P N n =1 s k,n  a k µ k − 1 g k,n  + = P k and s k,n can be ob tained with the same m ethod as the downlink. A P P E N D I X C : P RO O F O F L E M M A 1 For n otation convenience, we only consider MSs in one cluster and omit clu ster index m in the fo llowing proof. Let g k ( = l k | h k | 2 ) be th e channel gain of user k . Defin e x k , α k g k . Giv en α i > α j ≥ 1 ∀ j 6 = i , we first show th at if ∃ j such 20 For subproblem 1, differe nt ( m, k ) pairs can be treated as dif ferent k in this form. Subproble m 2 for a give n m is equi v alent to this form by omitting inde x m . 21 If ∃ k 6 = k ′ such that X k,n = X k ′ ,n is maximum, s k,n can be obtained by solving a s et of equations P K k =1 s k,n = 1 and X k,n = X k ′ ,n for the sharing subbands and (18). that x i > x j , then su bband will not be allo cated to user j accordin g to Append ix B. x i > x j , ∀ j 6 = i ⇒      1 µ − 1 x i > 1 µ − 1 x j > 0 1 µ − 1 x i > 0 ≥ 1 µ − 1 x j 0 ≥ 1 µ − 1 x i > 1 µ − 1 x j In case 2 and 3, p j = 0 . It is obvious that u ser j can not transmit in this sub band. W e on ly need to consider ca se 1. ∀ k , if 1 µ − 1 x k > 0 , X k ( α k , x k ) = α k log  x k µ  − α k  1 − x k µ  . Now , we shall show th at X k ( α k , x k ) increase with α k and x k separately . Let t k = x k /µ , ∂ X k ∂ α k = log t k − (1 − 1 t k ) , f ( t k ) and ∂ X k ∂ x k = α k x k − µ x 2 k . T hen 1 µ − 1 x k > 0 ⇒ t k > 1 ⇒ f ′ ( t k ) = 1 t k − 1 t 2 k > 0 ⇒ f ( t k > 1) > f ( t k = 1) = 0 ⇒ ∂ X k ∂ α k > 0 . In addition, α k ≥ 1 , x k > µ ⇒ α k x k − µ x 2 k > x k − µ x 2 k > 0 ⇒ ∂ X k ∂ x k > 0 . Thu s, α i > α j , x i > x j > 0 ⇒ X i ( α i , x i ) > X j ( α j , x j ) . According to Appendix C, this su bband will not be alloca ted to user j . Giv en the threshold γ th = α i l i ˜ γ , u ser k will feedb ack its channel q uality to th e cluster manag er iff α k g k ≥ α i l i ˜ γ ⇒ γ k = | h k | 2 ≥ T k ˜ γ , where T k = α i l i α k l k . R.V . ξ k denotes the feedback action of user k in o ne subband. ξ k = 1 , if user k feedback its chann el quality; ξ k = 0 otherwise. For Ray leigh fading assumed in the co ntext, P r[ ξ k = 1] = P r [ γ k ≥ T k ˜ γ ] = e − T k ˜ γ , P r [ ξ k = 0] = P r [ γ k ≤ T k ˜ γ ] = 1 − e − T k ˜ γ . Intuitively , we can conside r l i ˜ γ as the overall ch annel gain of a potential user wh o shares the sam e weig ht α i as user i an d ser ve a s a threshold user . T he sch eduling is do ne amo ng the feedback user set an d the thresho ld user . If ˜ γ ≤ | h i | 2 , the sched uling is still optimal. I f ˜ γ > | h i | 2 and P K k =1 ξ k > 0 , the schedulin g is optimal except for the th reshold user is schedu led at last, wh ich happen s with low possibility . If P K k =1 ξ k = 0 , we dec lare a schedulin g o utage and suffer from certain p erform ance loss. The outage pro bability is P 0 = P r[ P K k =1 ξ k = 0 ] = Q K k =1 (1 − exp( − T k ˜ γ )) . The a verage feedb ack load is F = 1 K E [ P K k =1 ξ k ] = 1 K P K k =1 (1 − e ( − T k ˜ γ )) . W e have exp( − max { T k } ˜ γ ) ≤ exp( − T k ˜ γ ) ≤ exp( − min { T k } ˜ γ ) ⇒ P 0 ≤ (1 − e xp( − max { T k } ˜ γ )) K , F ≤ exp( − min { T k } ˜ γ ) . (1 − exp( − max { T k } ˜ γ )) K = P m 0 ( K ) ⇒ ˜ γ ( P m 0 ( K )) = 1 max { T k } log 1 1 − P m 0 ( K ) 1 K ⇒ exp( − min { T k } ˜ γ ) = exp( − min { T k } max { T k } log 1 1 − P m 0 ( K ) 1 K ) =  1 − P m 0 ( K ) 1 K  − min { T k } max { T k } . Since P m 0 ( K ) → 0 as K → ∞ , we ha ve P 0 → 0 . Because P m 0 ( K ) 1 K → 1 as K → ∞ , we have F → 0 . Hen ce, α i l i ˜ γ ( P m 0 ( K )) is asymptotic optimal threshold . For example, P m 0 ( K ) = 1 K satisfies all the conditions. A P P E N D I X D : P RO O F O F L E M M A 2 By applyin g the similar appr oach as in [21], throug hput (bit/s/Hz) fo r MS k ∈ K m in relay -assisted system and in the IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS, V OL. 8, NO. 10, OCTOBER 2009 11 system without RS ar e: T m,k = 1 4 K Z ∞ 0 log 2  1 + P m N l m,k xdF max , K ( x )  . = 1 4 K log 2  P m N l m,k ln K  for large K T ( b ) m,k = 1 M K Z ∞ 0 log 2  1 + P 0 N l ( b ) m,k xdF max ,M K ( x )  . = 1 M K log 2  P 0 N l ( b ) m,k ln( M K )  for large K where F max ,K ( x ) and F max ,M K ( x ) are cdf of max {| H m,k,n | 2 , ∀ k } and max {| H m,k,n | 2 , ∀ m, k } sep arately , l m,k and l ( b ) m,k are pa th loss from the RS m an d BS for the MS k ∈ K m separately . Next, we consider the a verage th roug hput over distance. T o get closed form solution, we use closed disc with radius D M to appro ximate the relay cluster s. 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A vail able: http:/ /www .iee e802.org/ 16/tgm/ Yi ng Cui recei v ed B.Eng degree (first class honor) in Electronic and Informatio n Engineeri ng, Xian Jiaotong Uni versity , Xi’an, China in 2007. She is currentl y a Ph.D candidate in the Depart- ment of Electronic and Computer Engineering, the Hong Kong Univ ersity of Science and T ec hnology (HKUST). Her current research inter ests include coopera ti ve and cogniti v e communication s, delay- sensiti v e cross-layer scheduling as well as stochastic approximat ion and Marko v Decision Process. Vi ncent K. N. Lau obtained B.Eng (Distinction 1st Hons) from the Uni ve rsity of H ong K ong in 1992 and Ph.D . from Cambridge Univ ersity in 1997. He was with PCC W as system enginee r from 1992-1995 and Bell L abs - Lucent T echnologies as member of techni cal staff from 1997-2003 . He then joined the Departmen t of ECE, HKUST as Associate Professor . His current research intere sts includ e the robust and delay-se nsiti ve cross-layer scheduli ng, cooperati v e and cognit iv e communicat ions as well as stochastic approximat ion and Marko v Decision Process. PLA CE PHO TO HERE Rui W ang recei ved B.Eng degree (first class honor) in Computer Science from the Univ ersity of Science and T echnology of China in 2004 and Ph.D de gree in the Department of ECE from HKUST in 2008. He is currently a post-doctoral researcher in HKUST . His current research interests include cross-layer optimiza tion, wireless ad-hoc network, and cogniti v e radio.