Joint Scheduling and Resource Allocation in Uplink OFDM Systems for Broadband Wireless Access Networks

226 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 Jont Schedulng and Resource Allocaton n Uplnk OFDM Systems for Broadband Wreless Access Networks Janwe Huang, Vay G. Subramanan, Raeev Agrawal, and Randall Berry Abstract Orthogonal Frequency Dvson Multplexng OFDM) wth dynamc schedulng and resource allocaton s a key component of most emergng broadband wreless access networks such as WMAX and LTE Long Term Evoluton) for 3GPP. However, schedulng and resource allocaton n an OFDM system s complcated, especally n the uplnk due to two reasons: ) the dscrete nature of subchannel assgnments, and ) the heterogenety of the users subchannel condtons, ndvdual resource constrants and applcaton requrements. We approach ths problem usng a gradent-based schedulng framework. Physcal layer resources bandwdth and power) are allocated to mze the proecton onto the gradent of a total system utlty functon whch models applcaton-layer Qualty of Servce QoS). Ths s formulated as a convex optmzaton problem and solved usng a dual decomposton approach. Ths optmal soluton has prohbtvely hgh computatonal complexty but reveals gudng prncples that we use to generate lower complexty sub-optmal algorthms. We analyze the complexty and compare the performance of these algorthms va extensve smulatons. Index Terms Orthogonal Frequency Dvson Multplexng OFDM), schedulng, resource allocaton, optmzaton, dual decomposton, uplnk communcatons I. INTRODUCTION ORTHOGONAL Frequency Dvson Multplexng OFDM) s the core technology for most recent wreless data systems, ncludng IEEE 802.16 WMAX), IEEE 802.11a/g Wreless LANs), and LTE for 3GPP. In ths paper, we analyze an uplnk schedulng and resource Manuscrpt receved 15 January 2008; revsed 15 August 2008. Part of ths work was done whle J. Huang and V. G. Subramanan were at Motorola. J. Huang s supported by the Compettve Earmarked Research Grants Proect Number 412308) establshed under the Unversty Grant Commttee of the Hong Kong Specal Admnstratve Regon, Chna, the Drect Grant Proect Number C001-2050398) of The Chnese Unversty of Hong Kong, and the Natonal Key Technology R&D Program Proect Number 2007BAH17B04) establshed by the Mnstry of Scence and Technology of the People s Republc of Chna. V. Subramanan s supported by SFI grant 03/IN3/I346. R. Berry was supported n part by the Motorola-Northwestern Center for Seamless Communcatons and NSF CAREER award CCR-0238382. The work was partally presented at the 2007 Aslomar Conference on Sgnals, Systems and Computers. J. Huang s wth the Department of Informaton Engneerng, The Chnese Unversty of Hong Kong e-mal: whuang@e.cuhk.edu.hk). V. G. Subramanan s wth the Hamlton Insttute, Natonal Unversty of Ireland e-mal: vay.subramanan@num.e). R. Agrawal s wth the Advanced Networks and Performance Dept., Motorola Inc., e-mal: Raeev.Agrawal@motorola.com). R. Berry s wth the Dept. of EECS, Northwestern Unversty e-mal: rberry@ece.northwestern.edu). Dgtal Obect Identfer 10.1109/JSAC.2009.090213. 0733-8716/08/$25.00 c 2008 IEEE allocaton problem for OFDM wreless access networks. The specfc problem s motvated by the WMAX/802.16e standard, where there s a centralzed scheduler that knows the QoS classes, and can estmate the queue-lengths on each moble devce. The WMAX/802.16e standard specfes mechansms for communcatng ths nformaton to the scheduler and for conveyng the schedulng decsons to the mobles, both wth low delays. 1 Our approach s motvated by our prevous work on downlnk schedulng n CDMA systems [3] and OFDM systems [4]. As n [3], [4], we consder a gradent-based schedulng framework, whch s descrbed n detal n Secton II along wth our system model. In ths framework, the tme-varyng gradent of a utlty functon s used to gude the resource allocaton decsons and provde long-term Qualty of Servce QoS) guarantees. In partcular, we mze a weghted sumrate durng each schedulng nterval, where the weghts are tme-varyng. The optmzaton varables are the assgnment of OFDM subchannels to the users and the allocaton of each user s power across the assgned subchannels. We hghlght two challengng aspects of ths problem n the OFDM uplnk context. Frst, the dscrete nature of subchannel assgnments n OFDM systems usually leads to dffcult nteger programmng problems. Second, the per-user power constrants n the uplnk make the problem even less tractable. We ntally relax the nteger constrants and allow multple users to share one subchannel usng orthogonalzaton e.g. va tmesharng 2 ). In Secton III we derve an optmal soluton to ths relaxed problem va a dual decomposton. Due to the peruser power constrants, the resultng algorthm has very hgh computatonal complexty. However, ths provdes nsghts nto the structure of an optmal soluton. In Secton V we use the nsghts ganed from the optmal soluton to propose a famly of sub-optmal algorthms that also take nto account the nteger constrants on subchannel allocatons. Fnally, n Secton VI we these algorthms usng smulaton. Most ntal work on OFDM schedulng and resource allocaton focused on the downlnk case. The optmalty condtons and algorthms derved for the downlnk, however, can not be drectly appled to the uplnk due to dfferences n the resource constrants see Secton IV for a detaled dscusson). 1 Our model s also approprate for LTE [29] for 3GPP, UMB [30] for 3GPP2 and the FLASH OFDM system [19] from Qualcomm Flaron. 2 Whle super-poston codng would yeld an even larger and more tractable) capacty regon, we do not consder t as t s stll not practcal.

HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 227 Recently, uplnk OFDM resource allocaton has receved some attenton, ncludng [21] [28]. In [21], an teratve OFDM resource allocaton was proposed to fnd a Nash Barganng soluton. A heurstc algorthm that tres to mnmze each user s transmsson power whle satsfyng the ndvdual rate constrants s gven n [22]. In [23], an algorthm for mzng the sum-rate assumng Raylegh fadng s gven; ths s a specal case of the problem consdered here wth equal weghts. In our case no assumpton on the fadng dstrbuton s made. In [24], an uplnk problem wth multple antennas at the base staton s consdered; here, we focus on sngle antenna systems. The work n [25] [28] s closer to ours. In [25], a weghted rate mzaton problem s also consdered for the uplnk, but wth statc weghts. The results n [25] are generalzed n [26] to account for per-tme slot farness va a utlty functon of the nstantaneous rate. Per tme-slot farness s also consdered n [27]. Our work dffers from these n that by usng a gradent-based scheduler, we can consder longterm farness, whch depends on the average rate or queue szes. For elastc data applcatons, long-term QoS evaluaton s more reasonable than the short-term QoS evaluaton durng each tme slot. It not only more fathfully reflects users actual perceved performance, but also gves the system more flexblty n terms of explotng mult-user dversty. Fnally, [28] proposed a heurstc algorthm based on Lagrangan relaxaton, whch has hgh complexty due to a subgradent search of the dual varables. Here we use Lagrangan relaxaton to gve an optmal soluton of the uplnk problem when tme-sharng s allowed. Solvng ths problem provdes both an upperbound on the actual system performance as well as the ntuton we use to desgn good heurstc algorthms. II. PROBLEM STATEMENT We consder the problem of schedulng and resource allocaton for the uplnk of a OFDM cell, where a set M = {1,..., M} of users transmt to the same base staton. The total frequency band s dvded nto a set N = {1,..., N} of subchannels e.g., frequency bands). Let p be the power user allocates to subchannel, whch s subect to a per-user power constrant: p P, M. 1) Let x be the fracton of subchannel allocated to user, where the total allocaton across all users should be no larger than 1,.e., x 1, N. 2) We use bold symbols to denote vectors and matrces of these quanttes, e.g., P = {P, }, e = {e,, }, p = {p,, }, and x = {x,, }. Tme s dvded nto equal length slots. At the begnnng of every tme slot, the scheduler seeks to mze a tmevaryng) weghted sum of the users rates over a tme-varyng) rate-regon. We descrbe ths rate-regon next. The scheduler s assumed to have knowledge of the normalzed receved sgnal-to-nterference plus nose rato SINR) per unt transmt power, e, for each user and subchannel. 3 The tme-varyng subchannel qualty vector at tme t s denoted by e t. As n [4], ths model ncorporates varous subchannelzaton schemes where the resource allocaton s performed n terms of subchannels.e., sets of tones); e represents a collectve qualty ndcator for the subchannel, e.g., the harmonc/geometrc/arthmetc) average across the tones n the subchannel. Ths model also apples f resource allocaton s done wth a granularty of multple symbols n the tme doman. We model the feasble rate regon at tme t by { Re t )= r : r = } fx,p e t)), 3) N where x, p) X are chosen subect to 1) and 2) and the set { } X := x, p) 0 :0 x 1, p xs e, t). 4) Here, fa, b) =a log1 + b a ) so that r s the achevable rate of user n a Gaussan Multple Access Channel usng tmesharng cf. [17, Secton 15.3.6, pg. 547]). By contnuty, we assume that f0,b) = 0. The value of s s a mum SINR constrant on subchannel for user, whch can model scenaros when users have lmted choces of modulaton and codng schemes. In practcal OFDM systems, x s constraned to be an nteger,.e., we have the addtonal constrant x {0, 1} for all,. Intally, we gnore ths constrant, and consder a system n whch users can share each tone. If resource allocaton s done on blocks of OFDM symbols, then fractonal values of x can be mplemented by tme-sharng the symbols n a block. Alternatvely, ths can also be mplemented by frequency sharng e.g., [31]), f there are a large number of subchannels wth roughly equal gans. We wll re-ntroduce these nteger constrants n Sectons IV and V. Next we formulate the schedulng and resource allocaton problem. Our approach s based on the gradent-based schedulng framework n [2], [10], [12]. Each user s assgned a utlty functon U W,t,Q,t ) dependng on ther average throughput W,t up to tme t and ther queue-length Q,t at tme t. Ths s used to quantfy farness and ensure stablty of the queues. At the begnnng of each tme slot t, the scheduler chooses a r t Re t ) that mzes a weghted sum of the users rates, where the weghts are determned by the gradent of the sum utlty across all users,.e., t solves r t Re t) r t Re t) w UW t, Q t ) q UW t, Q t )) T rt, 5) where UW t, Q t )= K =1 U W,t,Q,t ). Further assumng that for each user, U W,t,Q,t )=u W,t ) d p Q,t) p, then 5) s equvalent to ) u W,t ) + d Q,t ) p 1 r,t, 6) W,t 3 In both FDD and TDD systems ths can be obtaned usng a combnaton of measurements made on the uplnk plots as well as past transmssons from the mobles.

228 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 where, u W,t ) s a ncreasng concave functon used to represent elastc data applcatons e.g., [1], [8], [13], [18]), d 0 s a QoS weght for user s queue length, and p>1 s a farness parameter assocated wth the queue length. The broad class of polces n 6) can be tuned to yeld good operatng ponts by a proper choce of parameters. If d =0forall M, the resultng polcy has been shown to yeld utlty mzng solutons see [2], [10], [12]). If u ) 0 wth d > 0 for all M, then the polcy has been shown to be stablzng n a varety of settngs see [5] [7]). The weghts can also be adapted so as to mze sum utlty subect to stablty [9] or feasble) mnmum throughput constrants see [11]). More generally, the optmzaton n 6) can be wrtten as r t Re t) w,t r,t, 7) where w,t 0 s a tme-varyng weght assgned to the th user at tme t. Our focus s on solvng such a problem for an uplnk OFDM system,.e., when Re t ) s gven by 3). Note that 7) must be re-solved at each schedulng nstant because of changes n both the subchannel state, e t, and the weghts. Whle n the above examples, the weght w,t s gven by the gradent of an utlty functon, our algorthms also apply to other methods for generatng these weghts. III. OPTIMAL SOLUTION WITH FRACTIONAL ALLOCATIONS We now consder the optmal soluton to 7) when Re t ) s gven by 3). Suppressng the tme ndex, the problem s w x log 1+ p ) e UL) x,p) X x M N subect to the per subchannel assgnment constrants n 2) and the per user power constrants n 1), where X s gven n 4). It can be shown that Problem UL has no dualty gap and so we solve t va a dual formulaton. We assocate dual varables λ =λ ) M wth constrants 1) and µ =µ ) N wth constrants 2), resultng n the Lagrangan, Lλ, µ, x, p) := w x log 1+ p ) e x, + λ P ) ) p + µ 1 x. 8) Therefore, the optmal soluton to Problem UL s gven by mn λ,µ) 0 x,p) X Lλ, µ, x, p). 9) We solve ths problem by the followng steps. Frst, we analytcally fnd the optmal p and x gven fxed values of the dual varables. We then show that the optmal µ s gven by a search for the mum value of a per-user metrc on each subchannel. The fnal step s to numercally search for the optmal value of λ. The value of p whch mzes Lλ, µ, x, p) gven x, µ and λ s gven bu p = x e mn { ) + w e 1,s }, 10) λ where ) + =, 0). Substtutng p nto L,,, ) yelds Lλ, µ, x) = x w h λ,w e,s ) µ ) + µ + λ P, 11) where we have used the functon h,, ) from [3]; namely, 0 f a b; a ha, b, c) = b 1 log a b b f 1+c a<b; 12) log1 + c) a b b c f a< 1+c, where a 0, b>0 and c 0. Optmzng 11) over x such that x [0, 1] yelds Lλ, µ) = µ + + λ P w h λ,w e,s ) µ ) +, 13) where the optmal subchannel allocaton has the followng structure 1, f w h λ,w e,s ) >µ ; x µ )= [0, 1], f w h λ,w e,s )=µ ; 14) 0, f w h λ,w e,s ) <µ. Snce the cost functon n 13) s separable, by defnng µ ) := w h,w e,s ) as n [3], we can mnmze Lλ, µ) over µ for a gven λ by settng µ = µ λ) gven by µ λ) = µ λ ). 15) From 14) and 15), t s clear that x µ λ)) 0 f arg M µ λ ),.e., users not mzng a specfc subchannel metrc are not allocated the subchannel. There wll be tes when multple users acheve the value µ on subchannel. These can be broken arbtrarly for optmzng the dual functon. Substtutng µ nto Lλ, µ), and notcng that µ, x, p are all functons of λ, we have Lλ) := µ λ )+ λ P. The soluton to 9) s gven by numercally mnmzng Lλ) over λ 0. For ths we use a subgradent-based search and update λ by [ λ t + 1) = λ t) κt) P )] + p t), M. where p s gven by 10) and x are gven by 14) and addtonally satsfy the feasblty constrant 2) n case of tes. The algorthm wll converge when κt) s chosen approprately, e.g., [20, Exercse 6.3.2]. Gven an optmal λ, by dualty, Lλ ) s the optmal obectve value to Problem UL. However, to mplement ths algorthm, the scheduler must specfy the correspondng optmal prmal values of x, p ). Here, as n [4], more care s requred. Specfcally, when tes occur n 15), t s often needed to splt the subchannel among several users.e., allowng fractonal values of x ). Followng a smlar approach as n [3] for the downlnk), the optmal

HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 229 fractonal values can be found by solvng a lnear program whose sze ncreases wth the number of users and tones nvolved n each te. As dscussed below, ths number can be qute large n the uplnk settng. Moreover, as noted earler, one s typcally nterested n an nteger allocaton n practce. We consder ths problem next. IV. INTEGER SUBCHANNEL ALLOCATION BASED ON OPTIMAL ALGORITHM We now address the problem: w x log x,p) X, x {0,1},, M N 1+ p ) e, UL-Int) x subect to per user power constrants 1). Intally, consder the followng heurstc for Problem UL-Int: ) Solve Problem UL as n the prevous secton; and ) break any tes on all subchannels,.e., whenever there s a fractonal x value, choose one user n the te and allocate subchannel to that user only. Clearly, f there are no tes, ths algorthm gves the optmal soluton to Problem UL-Int. After all tes are broken, we can then re-optmze the power allocaton for each user usng a fnte-tme water-fllng algorthm as n [4]. In [4] a smlar procedure s used for a downlnk OFDM schedulng problem. However, there are several maor dfferences between the uplnk and downlnk settngs that make ths approach less appealng for the uplnk. Frst, n the downlnk case there s a sngle power constrant, p P for the base staton nstead of the per-user constrants n 1). Hence, n the downlnk Lλ) s a functon of only a sngle dual varable λ, whch smplfes the numercal search for the optmzer. In the uplnk settng, the convergence of the subgradent search s too slow to be useful for schedulng on a fast tme-scale. Second, even f the optmal λ can be found, breakng tes s more dffcult than n the downlnk case. Scalar subgradents of Lλ) n the downlnk case can be used to devse smple te-breakng rules [4], whle n the uplnk case, the subgradents are vectors, complcatng such an approach. Also, the uplnk case can be more senstve to how tes are resolved. For example, f two users, and l, have the same weghts w = w l ) and the same gans on subchannel e = e l ), then allocatng subchannel to ether yelds the same total weghted rate and the same total power usage n the downlnk case. On the other hand, dfferent allocatons lead to dfferent ndvdual power consumptons n the uplnk case, and thus may lead to totally dfferent solutons. Fnally, the number of tes s typcally much larger n the uplnk case than n the downlnk case. Consder a smple scenaro wth two users and two subchannels. Each user has the same gan over both subchannels,.e., e 1 = e 2 = e for =1, 2, and P = P 1 = P 2, where P s the total power constrant n the downlnk case. Assume user 2 has a much better subchannel than user 1 so that n the downlnk case, the unque optmal soluton s to allocate both subchannels to user 2, and there s no te. However, n the uplnk case, t can be shown that at the optmal dual soluton, λ 1 and λ 2 wll satsfy µ 1 λ 1 )=µ 2 λ 2 ) for =1and 2,.e., there s a te n each subchannel and four possble subchannel allocatons must be consdered to determne how to break the te. Ths can be easly extended to M users and N subchannels, wth each user havng the same gan over all ts subchannels, resultng n M N tes even n ths smplstc settng. V. LOW COMPLEXITY SUBOPTIMAL ALGORITHMS Due to the ssues dscussed n the prevous secton, fndng the optmal dual soluton to Problem UL and breakng any tes to get an nteger allocaton s computatonally dffcult, even for a moderately szed system. Thus, we now present a famly of sub-optmal algorthms SOAs), whch try to reduce ths complexty usng the structure revealed by the optmal algorthm whle enforcng an nteger-tone allocaton and exhbtng good performance. In the optmal algorthm, gven the optmal λ, the optmal subchannel allocaton up to any tes s determned by sortng the users on each tone accordng to the metrc µ λ ) as n 14). Gven an optmal subchannel allocaton, the optmal power allocaton s gven by a per-user water-fllng allocaton as n 10). In each SOA, we use the same two phases but modfy them to reduce the complexty. Specfcally, we begn wth a subchannel Allocaton CA) phase whch assgns each subchannel to at most one user. Two dfferent mplementatons of the CA phase are gven. In SOA1, nstead of usng µ λ ), we consder metrcs based on a constant power allocaton over all subchannels assgned to a user. In SOA2, we agan use a dual based approach, but here we frst determne the number of subchannels assgned to each user and then match specfc subchannels and users. After the CA phase n both SOAs, we execute the Power Allocaton PA) phase n whch each user s power s allocated across the assgned subchannels as n the optmal algorthm. A. CA n SOA1: Progressve Subchannel Allocaton based on Metrc Sortng In ths famly of SOAs, subchannels are assgned sequentally n one pass based on a per user metrc for each subchannel. Let K n) denote the set of subchannels assgned to user after the nth teraton. Let g n) denote user s metrc durng the nth teraton and let l n) be the subchannel ndex that user would lke to be assgned f he/she s assgned the nth subchannel. The resultng CA algorthm s gven n Algorthm 1. Note that all the user metrcs are updated after each subchannel s assgned. Algorthm 1 CA Phase for SOA1 1: Intalzaton: set n =0and K n) = for each user. 2: whle n < N do 3: n = n +1. 4: Update subchannel ndex l n) for each user. 5: Update metrc g n) for each user. 6: Fnd n) = arg g n) break tes arbtrarly). 7: Assgn the nth subchannel to user n): { K n) = K n 1) {l n)}, f = n); K n 1), otherwse. 8: end whle

230 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 We consder several varatons of Algorthm 1 whch correspond to dfferent choces for steps 4 and 5. The choces for step 4 are: 4A): Sort the subchannels based on the best channel condton among all users. Ths nvolves two steps. Frst, for each subchannel, fnd the best channel condton among all users and denote t by µ := e. Second, fnd a subchannel permutaton {α } N such that µ α1 µ α2 µ αn, and set l n) =α n for each user at the nth teraton. Each operaton has complexty of OM), and the sortng operaton has a complexty of ON logn)). The total complexty s O NM + N log N). We note that ths s a one-tme preprocessng that needs to done before the CA phase starts. Durng the subchannel allocaton teratons, the users ust choose the subchannel ndex from the sorted lst. 4B): Sort the subchannels based on the channel condtons for each ndvdual user. For each user at the nth teraton, set l n) to be the subchannel ndex wth the largest gan among all unassgned subchannels,.e., l n) = arg N\ K n 1) e. Ths requres M sorts one per user) that only need to be performed once, snce each subchannel assgnment does not change a user s orderng of the remanng subchannels. The total complexty of the M sortng operatons s O MN log N), whch s hgher than that n 4A). Let k n) = K n). The choces for step 5 are: 5A): Set g n) to be the total ncrease n user s utlty f assgned subchannel l n), assumng constant power allocaton over all assgned subchannels,.e., [ ) P e g n) =w log 1+ K n 1) {l n)} K n 1) log k n 1) + 1 1+ P )] e. 16) k n 1) 5B): Set g n) to be user s gan from only subchannel l n), assumng constant power allocaton,.e., ) P g n) =w log 1+ k n 1) + 1 e,l n). Compared wth 5A), ths metrc gnores the change n user s utlty due to the decrease n power allocated to any subchannels n K n 1). The complexty of ether of these choces over N teratons s ONM), and so the total complexty for the CA phase 4 s O NM + N log N) f 4A) s chosen) or O MN log N) f 4B) s chosen). B. CA n SOA2: subchannel Number Assgnment & subchannel User Matchng As summarzed n Algorthm 2, SOA2 mplements the CA phase through two steps: subchannel number assgnment CNA) and subchannel user matchng CUM). CNA Step: The CNA step determnes the number of subchannels n assgned to each user M based on the 4 We note that SOA1 wth 4B) and 5B) s smlar to the algorthms proposed n [25]. In Secton VI we show that other varatons of SOA1 4B) and 5A)) and SOA2 can acheve better performance wth smlar or slghtly hgher complexty. Algorthm 2 CA Phase of SOA2 1: CNA step: determne the number of subchannels n allocated to each user such that M n N. 2: CUM step: determne the subchannel assgnment x {0, 1} for all and, such that N x = n, for each. approxmaton that each user sees a flat wde-band fadng subchannel. Ths step does not specfy whch subchannel s allocated to whch user; such a mappng s left to the CUM step. The CNA step s further dvded nto two stages: a basc assgnment stage and an assgnment mprovement stage. Stage 1, Basc Assgnment: Here, we model each user as havng a normalzed SINR e = 1 N N e over all subchannels, and then determne the number of subchannels assgned n to user by solvng: w n log {n 0, M} M subect to: n N. M 1+ P ) e n SOA2-CNA) N e s typcally a It can be shown that Problem SOA2-CNA s a standard concave mzaton problem over a convex set wth a unque and possbly non-nteger soluton; we use a dual relaxaton method to fnd ths soluton. The optmal Lagrange multpler for the subchannel constrant and any ntermedate optmal n allocaton can be found by a lne-search, over ranges [0, w log1 + P e ))] and [0,N], respectvely. Hence, the worst case complexty of solvng each subproblem wth a fxed dual varable s ndependent of M or N. Snce we need to determne the value of n for every user, the complexty of the basc assgnment step s OM). If the resultant channel allocatons contan non-nteger values, we wll approxmate wth an nteger soluton that satsfes M n = N. Snce each user s allocated only a subset of the subchannels, the normalzed SINR e = 1 N pessmstc estmate of the averaged subchannel condtons over the allocated subset. Ths motvates the followng assgnment mprovement stage of CNA. Stage 2, Assgnment Improvement: Here, we teratvely solve the followng varaton of Problem SOA2-CNA: {n t) 0, M} M subect to: w n t) log n t) N, M 1+ P n t) e t) ) SOA2-CNA-t) for t =1, 2,... Durng the t-th teraton, e t) s a refned estmate of the normalzed SINR based on the best n t 1) subchannels of user n 0) = N). The teratve procedure stops when the subchannel allocaton converges or the mum number of teratons allowed s reached. At the end an nteger approxmaton wll be performed, f needed. The complete algorthm for the CNA phase of SOA2 s gven n Algorthm 3. In order to perform the assgnment mprovement, we need to perform M sortng operatons once, wth a total complexty OMN logn)). Step 4 of each teraton has complexty of OM) due to solvng M subproblems

HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 231 for a fxed dual varable. The mum number of teratons s fxed and thus s ndependent of N or M. The nteger approxmaton stage typcally) requres a sortng wth the complexty of OM logm)). So the total complexty for the CNA phase of SOA2 s OMN logn)+m logm)). Algorthm 3 CNA Phase of SOA2 1: Intalzaton: nteger MaxIte> 0, t =0, n 0) = N and n 1) = N/2 for each user. 2: whle n t + 1) n t) for some ) & t <MaxIte) do 3: t = t +1. 4: For each user, e t) =average gan of user s best n t 1) subchannels. 5: Solve Problem SOA2-CNA-t) to determne the optmal n t) for each user. 6: end whle 7: let n = n t) for each user. CUM Step: The CNA step determnes how many subchannels are to be allocated to each user but not the exact subchannel assgnment. Ths s refned n the CUM step by fndng a subchannel assgnment that mzes the weghtedsum rate assumng each user employs a flat power allocaton,.e., we solve the problem: x w log 1+ P ) x {0,1} n e M N subect to: x = n, M, N x =1, N, M SOA2-CUM) where n =n, M) s the nteger subchannel allocaton obtaned n the CNA step. Problem SOA2-CUM s an nteger Assgnment Problem whose optmal soluton can be found by usng the Hungaran Algorthm [15]. 5 To use the Hungaran algorthm here, we need to perform the followng vrtual user splttng : For user, let r = w log 1+ P n e ), and let r =[r 1,r 2,,r N ] be user s achevable rates over all possble subchannels. We can then form an M N matrx R = [ ] r1 T, r2 T,, rm T T 6. Next, we splt each user nto n vrtual users by addng n 1 copes of the row vector r to the matrx R gvng a N N square matrx. Solvng Problem SOA2-CUM s then equvalent to fndng a permutaton matrx C =[c ] N N such that C = arg mn C R := arg mn N N c r. 17) C C C C =1 =1 Ths problem can be solved by the standard Hungaran algorthm whch has a computatonal complexty of O N 3). After 5 A smlar dea has been used to solve other OFDM resource allocaton problems, e.g., [16], [21]. 6 Here we assume that each user s allocated at least one subchannel. If only M < M users are allocated postve amount of channels, we can replace M by M n the dscussons. TABLE I WORST CASE COMPUTATIONAL COMPLEXITY OF SUBOPTIMAL ALGORITHMS Suboptmal Algorthms Worst Case Complexty 4A 5A O NM + N log N) subchannel 4A 5B O NM + N log N) Allocaton 4B 5A O MN log N) SOA1 4B 5B O MN log N) Power Allocaton O MN) Total O MN log N) subchannel CNA O MN log N + M log M) Allocaton CUM O `N 3 SOA2 Power Allocaton O MN) Total O `N 3 + MN log N + M log M obtanng C, we can calculate the correspondng subchannel allocaton x, e.g., f c k =1and vrtual user k corresponds to the actual user, then x =1. C. Power Allocaton PA) phase wth Sngle-user Water-fllng In ths phase each user optmally allocates ts power across the subchannel assgned to t n the CA phase. For user, ths can be formulated as the followng problem p P x log 1 + p e ), PA ) { where P = p 0:p s e, N p P }. The soluton to Problem PA wll be a water-fllng type of power allocaton, takng nto account the per subchannel SINR constrant. Specfcally, f N x s e P, we let p = s e. Otherwse, the optmal power allocaton s determned by ), p = mn x ν 1 e ) +, s e where the constant ν s chosen so that N p = P. Note that t s possble that some subchannel s allocated to user but gets no power due to ts relatvely) poor subchannel gan. The optmal value of ν can be found through a smple lne search. However, t s possble to devse a fnte-tme algorthm wth a mum of 2n steps) to calculate the exact value of v as n [3], [4] wth the dfference here beng that ths procedure needs to be executed for every user who s allocated a subchannel. Snce n = N, t follows that the total worst case computatonal complexty for ths approach s OMN). D. Complexty Summary of Suboptmal Algorthms The worst case computatonal complextes of SOA1 and SOA2 are summarzed n Table I. VI. SIMULATION RESULTS We report smulaton results for the followng four algorthms: 1) Integer-Dual) nteger subchannel allocaton wth te breakng) based on optmal algorthm as n Secton IV and power control as n Secton V-C, nspectng up to 128 ways of breakng the tes wth an nteger allocaton and selectng the allocaton among these wth the largest weghted sum rate before reallocatng the power);

232 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 # F:G191./=7?HI # F:G191./=7?HI!&+!&+!&*!&*!&)!&)!&!&!&"!&'!&%!&$!&#!! " #! #" $! $" %!,-./01-2!3/4567859:7-;7<-9:/=1>567?@/225=7A/12757B!6CDE Fg. 1. Emprcal CDF of the locaton-based term n users normalzed channel condton e s for all user and subchannel!&"!&'!&%!&$!&#!!"!!'!!%!!$!!#!! #! $! %! '! <-9:/=1>567?@/225=7A/12757B!6CDE Fg. 2. Emprcal CDF of users normalzed channel condton e s for all user and subchannel 2) SOA1) all four versons of subchannel allocaton procedure as n Secton V-A, and power control as n Secton V-C; 3) SOA2) subchannel allocaton as n Secton V-B wth up to 10 teratons) and power control as n Secton V-C; 4) Base-lne) each subchannel s allocated to the user wth the hghest e, wthout consderng the weghts w s and the power constrants, whle each user s power s stll allocated as n Secton V-C. We report smulaton results for the followng four algorthms: 1) Integer-Dual) nteger subchannel allocaton wth te breakng) based on optmal algorthm as n Secton IV and power control as n Secton V-C, nspectng up to 128 ways of breakng the tes wth an nteger allocaton and selectng the allocaton among these wth the largest weghted sum rate before reallocatng the power); 2) SOA1) all four versons of subchannel allocaton procedure as n Secton V-A, and power control as n Secton V-C; 3) SOA2) subchannel allocaton as n Secton V-B wth up to 10 teratons) and power control as n Secton V-C; and 4) Base-lne) each subchannel s allocated to the user wth the hghest e, wthout consderng the weghts w s and the power constrants, whle each user s power s stll allocated as n Secton V-C. We consder a system bandwdth of 5MHz consstng of 512 OFDM tones, grouped nto 64 subchannels 8 adacent tones per subchannel,.e., correspondng to the Band AMC mode of 802.16 d/e.). Each user s subchannel gans are the product of a constant locaton-based term, pcked usng an emprcally obtaned dstrbuton, and a fast fadng term, generated usng a block-fadng model and a standard moble delay-spread model wth a delay spread of 10µsec. The fast-fadng component for each mult-path component s held fxed for 2msec and an ndependent value s generated for the next block, whch corresponds to a 250Hz Doppler. The emprcal cumulatve dstrbuton functon CDF) of the locaton-based term of users locaton-based term of the e s s gven n Fg. 1, and the emprcal CDF of the net e s.e., ncludng the fastfadng component) s gven n Fg. 2. Snce the unts of e are n 1/watt, the x-axs n both fgures s measured n -dbw. The symbol duraton s 100µsec wth a cyclc prefx of 10µsec, whch roughly corresponds to 20 OFDM symbols per fadng block.e., 2msec). Ths s one of the allowed confguratons n the IEEE 802.16 standards [14]. Resource allocaton solvng Problem UL) s done once per fadng block. All the results are averaged over the last 1000 fadng blocks. For the sake of llustraton and to keep thngs smple, we assume that all users are nfntely back-logged, d =0, and have the same soelastc utlty functon u W,t ) = W,t ) α /α, where W,t s the long term average throughput of user up to tme t. All users also have the same mum power constrant of P =2W. We calculate the achevable rate of user on subchannel as ) p e r = Bx log 1+ x, where B s the subchannel bandwdth. In all tables, the Utlty column denotes U W,t ) n scentfc notatons when approprate), where t s the end of smulaton tme. The Log U column denotes the logarthmc logw,t), whch provdes an alternate utlty functon, characterzaton of farness among users. The Rate column denotes the total rate acheved by users n Mbps. The User # column denotes the average number of users who receve postve rates wthn one schedulng nterval. The Opt. Rato for the Integer-Dual algorthm denotes the rato between the obectve value of Problem UL-Int acheved wth the algorthm and the mum obectve value of Problem UL averaged over all schedulng nstances. Ths gves us an dea of how good the performance of the Integer-Dual algorthm s. We emphasze that ths value s based on the average performance of solvng problem UL n each schedulng nstance; ths does not translate drectly nto a bound on the average utlty under a optmal schedulng rule and under the gven polcy as the traectory of the schedulng weghts wll be dfferent under the two polces. Snce other suboptmal algorthms do not fnd the optmal dual value of Problem UL at each schedulng nstance, the Optmalty Rato does not apply to them. Table II shows results for all the algorthms summed over all users) when schedulng decsons are made every 20 OFDM symbols.e., a fadng block of 2msec). The utlty parameter

HUANG et al.: JOINT SCHEDULING AND RESOURCE ALLOCATION IN UPLINK OFDM SYSTEMS FOR BROADBAND WIRELESS ACCESS NETWORKS 233 TABLE II ALGORITHM PERFORMANCE FOR SCHEDULING EVERY 20 OFDM SYMBOLS, α =0.5 Algorthms Utlty Log U Rate User # Opt. Rato Integer-Dual 53922 514.0 21.56 37.5 0.9412 4A 5A 52494 510.7 22.86 34.6 N/A SOA 1 4A 5B 51697 509.2 20.22 28.1 N/A 4B 5A 54165 513.3 22.25 35.0 N/A 4B 5B 53156 511.4 21.43 28.6 N/A SOA 2 54316 513.6 22.33 35.1 N/A Base Lne 21406-1960.5 16.13 2.66 N/A TABLE IV ALGORITHM PERFORMANCE FOR SCHEDULING EVERY 20 OFDM SYMBOLS, α =1 Algorthm Utlty Log U Rate User # Opt. Rato Integer-Dual 23.24e6 472.56 23.37 20.43 0.82541 4A 5A 23.19e6 448.99 22.28 22.6 N/A SOA 1 4A 5B 23.11e6-136.03 23.20 15.6 N/A 4B 5A 24.31e6 444.02 24.42 32.7 N/A 4B 5B 23.95e6-195.10 24.05 15.6 N/A SOA 2 24.46e6 372.95 24.57 21.8 N/A Base Lne 16.08e6-1961 16.13 2.66 N/A TABLE III ALGORITHM PERFORMANCE FOR SCHEDULING EVERY 20 OFDM SYMBOLS, α =0 Algorthm Utlty Log U Rate User # Opt. Rato Integer-Dual 515 515 19.80 39.0 0.9715 4A 5A 511 511 18.47 36.9 N/A SOA 1 4A 5B 509 509 21.56 28.6 N/A 4B 5A 515 515 20.04 37.4 N/A 4B 5B 512 512 17.80 29.3 N/A SOA 2 515 515 19.91 37.4 N/A Base Lne -1961-1961 16.13 2.66 N/A s α =0.5. In Table II, SOA1 wth 4B & 5A) and SOA2 acheve the best performance n terms of total utlty. Ther performance s even better than the Integer-Dual approach, whch was obtaned based on the optmal value of the relaxed problem. Ths s lkely because only 128 ways to break tes are consdered, whch s typcally not suffcent. Snce the Integer- Dual algorthm acheves an optmalty rato of 0.9412, ths suggests that SOA1 and SOA2 acheve very close to optmal performance as well. The base-lne algorthm always has poor performance. Tables III and IV gve the results of the algorthms when the utlty parameter α s equal to 0 proportonal far allocaton) and 1 mum rate allocaton), respectvely. It s clear that there s a trade-off between farness measured by Log U) and effcency measured by total rate). The value of α =0 gves a throughput allocaton that s the farest and the least effcent, whle α =1s the most effcent and least far. Once agan we note that all of the heurstcs have good performance wth SOA1 wth 4B & 5A) and SOA2 achevng the best performances n terms of total utlty. In each case smulated, all of the SOA s have good performance wth SOA1 wth 4B & 5A) and SOA2 consstently achevng the best performance n terms of total utlty. From the analyss n Secton V-D, we note that these have slghtly hgher complexty than some of the other SOA s. Hence f lower complexty s desred, ths can be provded wth only a slght loss n performance. We also note that n each of the algorthms except the base lne one) a large number of users are scheduled n each tme-slot, whch may lead to a hgh sgnalng overhead. Ths can be addressed by addng a penalty term to our obectve whch ncreases wth the number of users scheduled. VII. CONCLUSIONS We presented an optmzaton-based formulaton for schedulng and resource allocaton n the uplnk OFDM access network. Compared to the downlnk, we showed that the uplnk was computatonally more challengng due n part to the per-user power constrants. A hgh complexty) optmal algorthm was gven as well as a famly of low complexty heurstcs. The heurstcs were shown to have good performance va smulatons for a range of dfferent user utltes and schedulng tme-scales. Two algorthms from ths famly consstently acheved the best performance, but had a slghtly hgher complexty than some of the other algorthms, enablng complexty to be traded off wth performance wthn ths famly of algorthms. Fnally, we want to menton that although we focus our dscussons on the uplnk transmssons n a sngle OFDM cell, the proposed algorthms can also be appled to downlnk transmssons n a mult-cell OFDM system. A dscusson of a more general soluton framework whch encompasses such cases s provded n [32]. REFERENCES [1] R. Agrawal, A. Bedekar, R. La, V. Subramanan, A Class and Channel- Condton based Weghted Proportonally Far Scheduler, n Proc. ITC, Salvador, Brazl, Dec. 2001. [2] R. Agrawal and V. Subramanan, Optmalty of Certan Channel Aware Schedulng Polces, n Proc. Allerton Conference, Oct. 2002. [3] R. Agrawal, V. Subramanan and R. Berry, Jont Schedulng and Resource Allocaton n CDMA Systems, n Proc. of WOpt 04, Cambrdge, UK, March 2004. [4] J. Huang, V. G. Subramanan, R. Agrawal and R. Berry, Downlnk schedulng and resource allocaton for OFDM systems, IEEE Trans. on Wreless, to appear. [5] L. Tassulas and A. 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234 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 27, NO. 2, FEBRUARY 2009 [14] IEEE 802.16e-2005 and IEEE Std 802.16-2004/Cor1-2005, 2005, Avalable:!""#$%%&&&')))*+,'-./%01%. [15] E. D. Nerng and A. W. Tucker, Lnear Programs and Related Problems. Academc Press Inc., 1993. [16] H. Yn and H. Lu, An effcent multuser loadng algorthm for OFDMbased broadband wreless systems, n Proc. IEEE Globecom, vol. 1, pp. 103 107, Dec. 2000. [17] T. M. Cover and J. A. Thomas, Elements of nformaton theory, Second edton, Wley-Interscence, Hoboken, NJ, 2006. [18] S. Shenker, Fundamental desgn ssues for the future Internet, IEEE J. Select. Areas Commun., vol. 13, no. 7, pp. 1176 1188, 1995. [19] FLASH-OFDM Technology, http://www.qualcomm.com/technology/ flash-ofdm [20] D. Bertsekas, Nonlnear Programmng, 2nd ed. Belmont, Massachusetts: Athena Scentfc, 1999. [21] Z. Han, Z. J, and K. Lu, Far Multuser Channel Allocaton for OFDMA Networks Usng Nash Barganng Solutons and Coaltons, IEEE Trans. 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Km, Effcent Subcarrer and Power Allocaton Algorthm n OFDMA Uplnk System, IEICE Trans. Commun., vol. 90, no. 2, pp. 368 371, 2007. [29] 3GPP Long Term Evoluton, 2008, Avalable: http://www.3gpp.org/ Hghlghts/LTE/LTE.htm. [30] 3GPP2 Ultra Moble Broadband, 2007, Avalable: http://www.3gpp2. org/. [31] W. Yu, R. Lu, and R. Cendrllon, Dual optmzaton methods for multuser orthogonal frequency dvson multplex systems, n Proc. IEEE Globecom, vol. 1, 2004, pp. 225 229. [32] J. Huang, V. G. Subramanan, R. Berry, and R. Agrawal, Schedulng and Resource Allocaton n OFDMA Wreless Systems, book chapter, submtted. Janwe Huang S 01-M 06) s an Assstant Professor n Informaton Engg. Department at the Chnese Unversty of Hong Kong. He receved the M.S. and Ph.D. degrees n Electrcal and Computer Engg. from Northwestern Unversty n 2003 and 2005, respectvely. From 2005 to 2007, He worked as a Postdoctoral Research Assocate n the Department of Electrcal Engg. at Prnceton Unversty. Hs man research nterests le n the area of modelng and performance analyss of communcaton networks, ncludng cogntve rado networks, OFDM and CDMA systems, wreless medum access control, multmeda communcatons, network economcs, and applcatons of optmzaton theory and game theory. Dr. Huang s an Assocate Edtor of Journal of Computer & Electrcal Engg., the Lead Guest Edtor of the IEEE Journal of Selected Areas n Communcatons specal ssue on Game Theory n Communcaton Systems, the Lead Guest Edtor of the Journal of Advances n Multmeda specal ssue on Collaboraton and Optmzaton n Multmeda Communcatons, and a Guest Edtor of the Journal of Advances n Multmeda specal ssue on Cross-layer Optmzed Wreless Multmeda Communcatons, the Techncal Program Commttee Co-Char of the Internatonal Conference on Game Theory for Networks GameNets 09). Vay G. Subramanan M 01) receved hs Ph.D. degree n Electrcal Engg. from the Unversty of Illnos at Urbana-Champagn, n 1999. From 1999 to 2006, he was wth the Networks Busness, Motorola, Arlngton Heghts, IL, USA. Snce May 2006 he s a Research Fellow at the Hamlton Insttute, NUIM, Ireland. Hs research nterests nclude nformaton theory, communcaton networks, queueng theory, and appled probablty and stochastc processes. Raeev Agrawal s a Fellow of the Techncal Staff at Motorola where hs responsbltes nclude the archtecture, desgn and optmzaton of Motorola s next generaton wreless systems. Pror to onng Motorola n 1999, Raeev was Professor of Electrcal and Computer Engg. and Computer Scence departments at the Unversty of Wsconsn - Madson. He also spent a sabbatcal year at IBM TJ Watson Research, Brtsh Telecom Labs, and INRIA-Sopha Antpols. Raeev receved hs M.S. 1987) and Ph.D. 1988) degrees n Electrcal Engg.-systems from the Unversty of Mchgan, Ann Arbor and hs B.Tech. 1985) degree n Electrcal Engg. from the Indan Insttute of Technology, Kanpur. Randall A. Berry S 93-M 00) receved the M.S. and PhD degrees n Electrcal Engg. and Computer Scence from the Massachusetts Insttute of Technology n 1996 and 2000, respectvely. In September 2000, he oned the faculty of Northwestern Unversty, where he s currently an Assocate Professor of Electrcal Engg. and Computer Scence. Dr. Berry s the recpent of a 2003 NSF CAREER award. He s currently servng on the edtoral boards of the IEEE Transactons on Informaton Theory and the IEEE Transactons on Wreless Communcatons.