Downlink Scheduling and Resource Allocation for OFDM Systems

Transcription

1 288 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 1, JANUARY 2009 Downlnk Schedulng and Resource Allocaton for OFDM Systems Janwe Huang, Member, IEEE, Vjay G. Subramanan, Member, IEEE, Rajeev Agrawal, and Randall A. Berry, Member, IEEE Abstract We consder schedulng and resource allocaton for the downlnk of a cellular OFDM system, wth varous practcal consderatons ncludng nteger tone allocatons, dfferent subchannelzaton schemes, maxmum SNR constrant per tone, and self-nose due to channel estmaton errors and phase nose. Durng each tme-slot a subset of users must be scheduled, and the avalable tones and transmsson power must be allocated among them. Employng a gradent-based schedulng scheme presented n earler papers reduces ths to an optmzaton problem to be solved n each tme-slot. Usng a dual formulaton, we gve an optmal algorthm for ths problem when multple users can tme-share each tone. We then gve several low complexty heurstcs that enforce nteger tone allocatons. Smulatons are used to compare the performance of dfferent algorthms. Index Terms Orthogonal frequency dvson multplexng OFDM, WMax, cellular downlnk, schedulng, resource allocaton, nonlnear optmzaton, wreless communcatons. I. INTRODUCTION MOST recent hgh-speed wreless data systems dynamcally schedule users and allocate physcal layer resources among them based on the users channel condtons and qualty of servce QoS requrements. Many of the schedulng algorthms consdered can be vewed as gradentbased algorthms, whch select the transmsson rate vector that maxmzes the projecton onto the tme-varyng gradent of the system s total utlty [1] [4]. Several such algorthms have been studed for tme-dvson multplexed TDM systems, ncludng the proportonally far rule [4], [6] whch s based on a logarthmc utlty functon of each user s throughput. A larger class of throughput-based utltes s consdered n [2], [5], where effcency and farness are Manuscrpt receved November 13, 2007; revsed Aprl 12, 2008 and July 31, 2008; accepted September 17, The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was D. I. Km. J. Huang s wth the Dept. of Informaton Engneerng, The Chnese Unversty of Hong Kong e-mal: [email protected]. V. G Subramanan s wth the Hamlton Insttute, Natonal Unversty of Ireland emal: [email protected]. R. Agrawal s wth Motorola Inc. e-mal: [email protected]. R. A. Berry s wth the Dept. of EECS, Northwestern Unversty e-mal: [email protected]. Part of ths work was done whle J. Huang and V. G. Subramanan were at Motorola. Ths work has been supported by the Compettve Earmarked Research Grants Project Number establshed under the Unversty Grant Commttee of the Hong Kong Specal Admnstratve Regon, Chna, the Drect Grant Project Number C of The Chnese Unversty of Hong Kong, SFI grant IN3/03/I346, the Motorola-Northwestern Center for Seamless Communcatons, NSF CAREER award CCR , and the Natonal Key Technology R&D Program Project Number 2007BAH17B04 establshed by the Mnstry of Scence and Technology of the People s Republc of Chna. Dgtal Object Identfer /T-WC /09$25.00 c 2009 IEEE allowed to be traded-off. The Max Weght polcy e.g. [7] [9] can also be vewed as a gradent-based polcy, where the utlty s also a functon of the user s queue-sze or delay. In TDM systems, one only needs to schedule one user n a tme-slot and choose the modulaton and codng scheme for that user. However, n many current systems, multple users may be multplexed wthn a tme-slot usng Orthogonal Frequency Dvson Multplexng OFDM e.g. IEEE /WMAX [11] and 3GPP LTE [12]. Ths paper addresses gradent-based schedulng and resource allocaton for the downlnk of such a system where n addton to determnng whch users are scheduled, the allocaton of physcal layer resources e.g. transmsson power and subcarrers must also be specfed. Our approach s motvated by [10], where a gradent-based schedulng algorthm s used for a system whch multplexes users n a tme-slot va code dvson multple access CDMA. Compared to CDMA, OFDM offers more degrees of freedom to allocate resources across.e., tone allocaton n the frequency doman. Ths enables explotng both mult-user dversty and frequency dversty at a fner granularty, but also sgnfcantly ncreases the complexty of the optmzaton. At the begnnng of each schedulng nterval, the gradentbased schedulng algorthm maxmzes the weghted throughput sum over the current set of feasble rates. In Secton II, we gve a model for ths rate regon, takng nto account the followng mportant practcal consderatons for OFDM systems: 1 dfferent subchannelzaton technques n whch resource allocaton s performed at a larger granularty.e, groups of tones or symbols to reduce the channel measurement and feedback overhead; 2 constrants that each subchannel/tone can be allocated to at most one user; 3 constrants on the maxmum rate per tone to model a lmtaton on the avalable modulaton and codng schemes; and 4 self-nose due to channel estmaton errors e.g., [13] or phase nose [23]. In Secton III, we consder a dual formulaton for the resultng optmzaton problem, whch enables us to explot the problem s structure and develop both optmal and smple suboptmal algorthms wth low complexty. Smulaton results are gven n Secton IV for these algorthms wth dynamcally varyng weghts under dfferent choces of utlty functons, subchannelzaton schemes, self-nose and per tone rate constrants. We conclude n Secton V. A number of related formulatons wthout self-nose and per tone rate constrants for downlnk OFDM resource allocaton have been studed ncludng [14] [20]. In [15], the goal s

2 HUANG et al.: DOWNLINK SCHEDULING AND RESOURCE ALLOCATION FOR OFDM SYSTEMS 289 to mnmze the total transmt power gven target bt-rates for each user. Sum-rate maxmzaton s consdered n [16], [18], [19], where [18], [19] also enforce a mnmum bt-rate per user. Weghted sum-rate maxmzaton for a fxed set of weghts s studed n [14], [20]. In [14], a suboptmal algorthm wth constant power per tone was shown n smulatons to have lttle performance loss. Other heurstcs that use a constant power per tone are gven n [16] [18]. We also consder such a heurstc n Secton III-D. In [20], a smlar dual-based algorthm to ours s consdered and smulatons are gven whch show that the dualty gap of ths problem quckly goes to zero as the number of tones ncreases; we wll revst ths n Secton III-B. Fnally, n [21], the capacty regon of a downlnk broadcast channel wth frequency-selectve fadng usng a TDM scheme s gven that covers our rate regon wthout any maxmum rate constrants or self-nose. The prevous papers optmze a statc objectve functon whle we are nterested n the case where the objectve changes accordng to a gradent-based algorthm. It s not a pror clear f a good heurstc for a statc problem appled to each tme-step, wll be a good heurstc for the dynamc case, snce the optmalty result n [1] [4], [7] [9] s predcated on solvng the optmzaton problem exactly n each tme-slot. Our smulaton results show that the heurstcs contnue to perform well, at least for the scenaros consdered n ths paper. In a companon paper [25], we use smlar methods to solve the correspondng uplnk problem. A more general soluton framework that encompasses both the uplnk and downlnk cases s provded n [29]. II. PROBLEM FORMULATION We consder downlnk transmssons n an OFDM cell from a base staton to a set K = {1,...,K} of moble users. In each tme-slot, the schedulng and resource allocaton decson can be vewed as selectng a rate vector r t =r 1,t,...,r K,t from the current feasble rate regon Re t R K +,wheree t ndcates the tme-varyng channel state nformaton avalable at the scheduler at tme t. Followng the gradent-based schedulng framework n [1] [4], an r t Re t s selected that has the maxmum projecton onto the gradent of a system utlty functon UW t := K =1 U W,t, where U W,t s an ncreasng concave utlty functon of user s average throughput, W,t, up to tme t. In other words, the schedulng and resource allocaton decson s the soluton to max UW t T r t = r t Re t max r t Re t U W,tr,t, 1 where U s the dervatve of U. For example, one class of utlty functons gven n [2], [5] s { c U W,t = α W,t α, α 1, α 0, 2 c logw,t, α =0, where α 1 s a farness parameter and c s a QoS weght. Wth equal class weghts, α =1results n the schedulng rule that maxmzes the sum-rate durng each slot; α =0results n the proportonally far rule. In general, we consder the problem of w,t r,t, 3 max r t Re t where w,t 0 s a tme-varyng weght assgned to the th user at tme t ted to the QoS requrements of the user [1] [4], [7] [9]. We note that 3 must be re-solved at each schedulng nstance because of changes n both the channel state and the weghts e.g., the gradents of the utltes. Whle the former changes are due to the tme-varyng nature of wreless channels, the latter changes are due to new arrvals and past servce decsons. A. OFDM capacty regons The soluton to 3 depends on the channel state dependent rate regon Re, where for smplcty we suppress the dependence on tme. We consder a model approprate for downlnk OFDM systems; related models have been consdered n [14], [21]. In ths model, Re s parameterzed by the allocaton of tones to users and the allocaton of power across tones. In a tradtonal OFDM system, at most one user may be assgned to any tone. Intally, as n [15], we make the smplfyng assumpton that multple users can share one tone usng some orthogonalzaton technque e.g. TDM. 1 In practce, f a schedulng nterval contaned multple OFDM symbols, we can mplement such sharng by gvng a fracton of the symbols to each user. We dscuss the case where only one user can use a tone n Secton III-C. Let N = {1,...,N} denote the set of tones. For each j N and user K, let e j be the receved sgnalto-nose rato SNR per unt power. We denote the power allocated to user on tone j by p j and the fracton of that tone allocated to user by x j. The total power allocaton must satsfy,j p j P, and the total allocaton for each tone j must satsfy x j 1. For a gven allocaton, wth perfect channel estmaton, user s feasble rate on tone pj ej j s r j = x j B log1 + x j, whch corresponds to the Shannon capacty of a Gaussan nose channel wth bandwdth x j B and receved SNR p j e j /x j. Ths SNR arses from vewng p j as the energy per tme-slot user uses on tone j; the correspondng transmsson power becomes p j /x j when only a fracton x j of the tone s allocated. Wthout loss of generalty we set B =1n the followng. In a realstc OFDM system, mperfect carrer synchronzaton and channel estmaton may result n self-nose e.g. [23], [13]. We model ths n a smlar way as [13]. Let the receved sgnal on the jth tone of user be gven by y j = h j s j +n j, where h j, s j and n j are the complex channel gan, transmtted sgnal and addtve nose, respectvely, wth n j CN0,σ 2. Assume that h j = h j + h j,δ,where h j s recever s estmate of h j and h j,δ CN0,δj 2.After matched-flterng, the receved sgnal wll be z j resultng n an effectve SNR of Eff-SNR = h j 4 p j σ 2 j h j 2 + δ 2 j p j h j 2 = = h j y j p j ẽ j 1+β j p j ẽ j, 4 1 We focus on systems that do not use superposton codng and successve nterference cancellaton wthn a tone, as such technques are generally consdered too complex for practcal systems.

3 290 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 1, JANUARY 2009 where p j = E s j 2, β j = δ2 j and ẽ h j 2 j = h j 2. 2 Here, σj 2 β j p j ẽ j s the self-nose term. As n the case wthout selfnose β j =0, the effectve SNR s stll ncreasng n p j. However, t now has a maxmum of 1/β j.forthesakeof presentaton, we assume that β = β j for all and j. The analyss s almost dentcal f users have dfferent β j s. Wth self-nose, user s feasble rate on tone j becomes pj ẽj r j = x j log1 + x j+βp jẽ j, where agan x j models tmesharng of a tone. Under these assumptons, we have { Re = r : r = x j log 1+ pjẽj x j+βp jẽ j, j p j P, } x j 1 j, x, p X, 5,j where X := N j=1 X j, and for all j N, { } X j := x j, p j xj sj 0 : x j 1,p j ẽ j, 6 wth x j := x j, K and p j := p j, K. Here, s j = Γj 1 Γ,whereΓ jβ j < 1/β s a maxmum SNR constrant on tone j for user, e.g., to model a constrant on the maxmum rate per tone due to lmted avalablty of modulaton and codng schemes. At the cost of addtonal complexty, we could also nclude mnmum rate constrants to model nelastc traffc, and maxmum rate constrants to ncorporate buffer szes. We assume that ẽ j s known by the scheduler for all and j as s β or δj 2. In a frequency dvson duplex FDD system, ths knowledge can be acqured by havng the base staton transmt plot sgnals, from whch the users can estmate ther channel gans and feed them back to the base staton. In a tme dvson duplex TDD system, these gans can also be acqured by havng the users transmt uplnk plots; the base staton can then explot recprocty to measure the channel gans. In both cases, ths feedback nformaton would need to be provded wthn the channel s coherence tme. B. Subchannelzaton Wth many tones and users, provdng plots and/or feedback per tone can requre excessve overhead; e.g., n IEEE e [11], a channel wth bandwdth 1.25Mhz to 20Mhz s dvded from 128 to 2048 tones. One way to reduce ths overhead s for feedback and resource allocaton to be done at the granularty of subchannels of dsjont sets of tones,.e., constant power s used and codng s done across the tones n the same subchannel. Our model can be adapted to ths settng by vewng N as the set of subchannels and ẽ j as the effectve SNR per unt power for user on the jth subchannel. Specfcally, assumng that k tones are bundled nto subchannel j, ẽ j s chosen so that the total rate gven by x j j l N j log1 + pj ẽj l x j+βp jẽ jl where N j s the set of tones 2 Ths s slghtly dfferent from the Eff-SNR n [13] n whch the sgnal power s nstead gven by h j 4 p j ; the followng analyss works for such a model as well by a smple change of varables. For the problem at hand, 4 seems more reasonable n that the resource allocaton wll depend only on h j and not on h j. We also note that 4 s shown n [22] to gve an achevable lower bound on the capacty of ths channel. n the jth subchannel and ẽ jl s the SNR per unt power for tone j l for user n ths subchannel s approxmately kx j log1 + pjẽj x j+βp jẽ j.sncelog1 + pe x+βpe s a concave functon of e, usng Jensen s nequalty the rate acheved over a subchannel s upper bounded by takng ẽ j to be the arthmetc average of the channel gans of tones n subchannel j. The rate can be lower bounded usng the strct convexty of logl +expy for y R wth l > 0 and Jensen s nequalty. If β = 0,takngy = log pe x and l = 1 we lower bound the rate by settng ẽ j equal to the geometrc average of the subchannel gans. When β > 0 we take y = log 1+ x βpe and l = β, apply Jensen s nequalty followed by the arthmetc-mean geometrc-mean nequalty to lower bound the rate by settng ẽ j equal to the harmonc average of the subchannel gans. The gap between the upper and lower bounds s qute small for reasonable values of pe; for the SNRs acheved by scheduled users n our smulatons, we do not see much dfference. 3 From here onwards we wll use the terms tone/carrer/subchannel to mean the basc allocaton unt; the specfc dstnctons wll be clear from the context. We consder the followng subchannelzatons: adjacent channelzaton, where adjacent tones are grouped together as n the optonal band AMC mode n IEEE d/e [11]; nterleaved channelzaton, where tones are perfectly nterleaved as n the nterleaved channelzaton n IEEE d/e [11]; and random channelzaton, where tones are randomly assgned as n systems that employ frequency hoppng as n the Flash OFDM system [24]. Adjacent channelzaton enables the resource allocaton to better explot frequency dversty. Interleaved or random channelzaton reduces the varance of the effectve SNR across subchannels for each user; when the varance s small, user can smply feed back a sngle e value. Random channelzatons also ad n managng nter-cell nterference. III. OPTIMAL AND SUBOPTIMAL ALGORITHMS From 3 and 5, the schedulng and resource allocaton problem can be stated as: max V x, p := x,p X,j subject to:,j w x j log 1+ pjẽj x j+βp jẽ j p j P, and x j 1, j N, where we stll assume that users can tme-share subchannels. Next we show how to solve 7 va a dual formulaton. 3 For example, n our smulatons of the optmal algorthm wth β =0.01, the dfferences between acheved utltes under arthmetc average and harmonc average approxmatons are 0.005%, 0.1%, and 0.4% under adjacent, nterleaved and random subchannelzatons, respectvely. 7

4 HUANG et al.: DOWNLINK SCHEDULING AND RESOURCE ALLOCATION FOR OFDM SYSTEMS A. Optmal Dual Soluton Consder the Lagrangan, Lx, p, λ, µ := λp + N j=1 L jx j, p j,λ,μ j, where K L j x j, p j p j ẽ j,λ,μ j := w x j log 1+ x =1 j + βp j ẽ j K K + μ j 1 x j λ p j, 8 =1 =1 and µ = μ j N j=1. The correspondng dual functon Lλ, µ :=max p,x X Lx, p,λ,µ can then be wrtten as Lλ, µ = λp + N max L j x j, p j,λ,μ j. 9 p j,x j X j j=1 By drectly evaluatng the Hessan of x log1 + x+βp t can be seen that ths s jontly concave n x, p. It follows that Problem 7 s convex and satsfes Slater s condton. Hence, there s no dualty gap and so V := mn λ 0,µ 0 Lλ, µ s the optmal objectve value [26]. Next we gve a closed-form representaton of Lλ, µ n 9. We then show that mnmzng Lλ, µ over µ only requres searchng for the maxmum of user dependent metrcs for each tone j. The only numercal search needed s for the mnmzaton over λ, whch s a one-dmensonal search. 1 Computng the Dual Functon: For a gven x j, μ j and λ, thep j whch obtans the maxmum n 9 s gven by p jx,λ,µ = x j p j λ wth ] + p j λ := 1 wẽ ẽ j [q β, j λ 1 s j,10 where x + =maxx, 0, a b =mna, b, and { z, f β =0; qβ,z := 2β+1 2ββ ββ+1 2β+1 z 1, f β>0. 2 Fgure 1 shows p j n 10 as a functon of ẽ j for β =0, 0.01, and 0.1. Whenβ =0, 10 becomes a waterfllng soluton n whch p j x,λ,µ s non-decreasng n ẽ j.forafxed β > 0, due to self-nose, less power may be allocated to better subchannels. The constant β case s applcable when the self-nose s due to phase nose as n [23]. On the other hand, when self-nose arses prmarly from estmaton errors, β may not be constant but could depend on the channel qualty. The exact dependence wll depend on the detals of channel estmaton. As an example, we also show a curve for when βe = 10/e, whch s motvated by the analyss n [22, Secton IV] for the estmaton error of a Gauss-Markov channel from a plot wth known power. For that model, when the plot power s ether constant or nversely proportonal to channel qualty subject to maxmum and mnmum power constrants modelng power control, β wll be nversely proportonal to e. It can be seen that the curve has a dfferent shape and ampltude compared to the β =0 case. For smplcty of presentaton, we assume constant β s n the remander of the paper. p Optmal power p j * x, λ,μ β=0 β=10/e β=0.1 β= Channel condton e j db Fg. 1. Optmal power p j 15 wth w =1versus channel condton e j. Notce from 10 that the optmal value of p j s always a lnear functon of x j. Substtutng 10 nto L j x j, p j,λ,μ j also results n a lnear functon of x j, namely, L j x j, p j,,λ,μ j = where μ j λ :=w h β, wẽj λ h β,ω, s j :=log x j μ j λ μ j +μ j,, s j, and 1+ qβ,ω 1+ s j 1+βqβ,ω 1 + s j 1 ω q β,ω 1 + s j. From ths t follows that any choce {1}, f μ j λ >μ j ; x jλ, µ [0, 1], f μ j λ =μ j ; {0}, f μ j λ <μ j, 11 wll maxmze L j x j, p j,,λ,μ j. Hence, Lλ, µ :=λp + N j=1 L jλ, μ j,where L j λ, μ j = + μj λ μ j + μj Optmzng the Dual Functon over λ and µ: Lemma 1 characterzes the optmzaton of Lλ, µ over µ. Lemma 1: For all λ 0, Lλ :=mn Lλ, µ =λp + μ j λ, 13 µ 0 j where for every tone j, the mnmzng value of μ j s μ j λ =maxμ j λ. 14 Proof of Lemma 1 s smlar to the proof n [10]. For each tone j, 14 computes the maxmum of user metrc μ j. Snce Lλ s the mnmum of a convex functon over a convex set, t s a convex functon of λ; hence, t can be mnmzed usng an terated one dmensonal search e.g., the Golden Secton method [26] for whch the computaton complexty s Olog1/ɛ, whereɛ s the target relatve error bound. Snce there s no dualty gap, ths mnmzaton gves the optmal objectve value n 7.

5 292 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 1, JANUARY 2009 B. Optmal prmal varables wth tme-sharng Now we fnd optmal values of the prmal varables x, p. For every λ 0, wth µ λ as n 14, let x, p :=arg max L x, x,p X p,λ,µ λ ; 15 note that these satsfy 10 and 11. Gven that λ = λ, t follows from dualty theory, that f the x, p satsfyng 15 are prmal feasble and satsfy complmentary slackness, then they are prmal optmal prmal. In partcular, f for each tone j there exsts a unque user that acheves the maxmum n 14, then snce there s no dualty gap, allocatng tone j only to that user must be prmal optmal. In general, gven λ 0, leta j := { μ j λ =max î μ îj λ} be the set of users who acheve the maxmum on tone j, and A j be the sze of A j. From 11 t follows that all x that solve 15 are those that satsfy the followng propertes: for A j, x j =0; f A j =1,thenx j =1for A j; and f A j > 1, then for all A j, x j [0, 1] and A x j j =1. In case, not all tone allocatons satsfyng A j x j =1may be prmal feasble e.g., j p j maybe largerthan P. Breakng these tes s necessary to fnd a prmal optmal soluton. A key pont s that when tes occur at a gven λ, Lλ may not be not dfferentable at that λ. However, snce Lλ s a convex functon, subgradents exst [27]. Proposton 1: For any λ 0, d s a subgradent of Lλ f and only f there exsts x, p satsfyng 15, x j 1 for all j, μ j λ 1 j x = 0 for each j, andp,j p j = d. The proof of Proposton 1 can be found n [28] and follows by observng that that dual functon s the maxmum of a set of Lagrangan functons whch are lnear n λ and that the gradent of each of the Lagrangan functons wth respect to λ sgvenbyp,j p j. At any gven λ, we need to restrct attenton to the maxmzng x, p to obtan the set of subgradents of Lλ. The rest follows by observng that the resultng subgradent P,j p jλx j s lnear n x j, whch takes values n a convex set product of smplexes. Thus, n order to fnd the dual optmal, we need to search for λ whch has a zero subgradent f λ > 0; and nonnegatve f λ =0. From Proposton 1, ths wll also be the check for prmal feasblty and complmentary slackness for the power constrant. Next we provde a soluton for ths check. We refer to an allocaton as an extreme pont f t satsfes - andx j {0, 1} for all and j; suchan allocaton can be represented by a functon f : N K,so that fj A j ndcates the user who s allocated channel j,.e., x fjj =1.LetB = {j : A j =1} and B c = N\B. For each j B, there are no tes, and so fj s unque. For each tone j B c,thereare A j users n the te, and so the total number of extreme ponts s j B A j. Each extreme c pont satsfes Proposton 1 and so provdes a subgradent for Lλ. From Proposton 1 t follows that all the subgradents of Lλ can be obtaned as a convex combnaton of the values at the extreme ponts. Gven an extreme pont f, from 10, t follows that the correspondng subgradent df s gven by df =P p fjj p fjj. 16 j B j B c Choosng dfferent extreme ponts only effects the last term on the rght of 16. It follows that the maxmum subgradent of Lλ corresponds to the extreme ponts gven by ˆfj :=arg mn p j, j. 17 Aj The mnmum subgradent corresponds to the extreme ponts fj :=arg max p j, j. 18 Aj At λ, the maxmum subgradent usng 17 s always nonnegatve, and the mnmum subgradent usng 18 s always non-postve. If ether s zero, an nteger prmal optmal soluton s found. In general, we have the followng: Proposton 2: There exsts an optmal prmal soluton x λ, p λ, where x λ s gven by tme-sharng between the two extreme ponts n 17 and 18 so that the convex combnaton of the correspondng subgradents s equal to zero, and p λ s gven by 10. Proposton 2 mples that each tme-shared tone s shared n the same proporton. The above steps gve an algorthm for fndng the optmal soluton to 7 n two stages. Frst, fnd λ that mnmzes Lλ as n Secton III-A. Ths nvolves evaluatng Lλ for a fxed value of λ as an nner loop, and a one-dmensonal search over λ as an outer loop. The outer loop has a complexty that s ndependent of N and K. The nner loop has a complexty of ONK due to searchng for the maxmum of K metrcs 14 on each of the N tones. Thus the total complexty of ths stage s ONK. Second, gven λ, we compute the maxmum and mnmum extreme ponts and fnd the optmal prmal varables as n Proposton 2 whch also has a complexty of ONK. Hence, the overall complexty of the optmal algorthm s ONK. In our smulatons, the actual complexty of the second stage s typcally much smaller than ONK because typcally only a few tes occur. 4 However, the number of extreme ponts can be very large under nterleaved channelzaton. Ths s because f two users are ted on one subchannel, t s very lkely that they wll also be ted on other subchannels snce all subchannels have roughly the same channel gan for the same user. However, f all the tes are due to the same two users, we can just allocate all subchannels wth a te to the same user and ths wll lead to ether the largest or smallest subgradent. These observatons are consstent wth [20], whch argues that an OFDM system wth β =0n whch no tme-sharng s allowed wll have a certan dualty gap that s small for a reasonable number of sub-channels. Problem 7 can be vewed as the dual of the dual problem n [20, eqn. 9] and the dualty gap n [20] can be vewed as a measure of the accuracy of approxmatng the OFDMA schedulng problem by the tme-sharng verson of t from 7. When there s exactly one extreme pont, the dualty gap s clearly zero snce we have an nteger soluton. The arguments n [20] for a vanshng dualty gap roughly correspond to showng 4 For example, extensve smulaton results show that for a system of 64 subchannels grouped from 512 tones and 40 users n a hgh moblty envronment, there are on average only two extreme ponts typcally on one subchannel nvolvng two users, at each schedulng nterval averaged over 3000 schedulng ntervals under ether adjacent or random channelzatons.

6 HUANG et al.: DOWNLINK SCHEDULING AND RESOURCE ALLOCATION FOR OFDM SYSTEMS 293 that the spread n the power consumpton of dfferent extreme ponts.e., the maxmum dfference n subgradent values s typcally small for a reasonable number of carrers. When ths spread s small, one expects that fewer tes occur whch s consstent wth the above dscusson. Dscussons above also argue that the conclusons n [20] extend to the β>0 case. C. Sngle user per tone We now consder the case where no tme-sharng s allowed,.e., x j {0, 1} for all and j. Suppose we stll fnd the optmal λ as n Secton III-A. If there are no tes on any of the tones or f there s an extreme pont wth j N p fjj = P, the optmal prmal soluton gven n Secton III-B only has one user per tone, and we are done. If not, Proposton 2 wll no longer gve a soluton that satsfes the nteger constrants. In ths case, a reasonable heurstc s to smply choose one extreme pont allocaton. In our smulatons, we choose the extreme pont correspondng to the subgradent wth the smallest non-negatve value;.e., the extreme pont f, forwhch j N p fjj s closest to P, wthout exceedng t. Other rules for choosng an extreme pont can also be used. Note that ths requres searchng over all extreme ponts, whch has a worstcase complexty of OK N f all users were ted on every tone. However, as dscussed above, typcally there are only two users ted on one tone and so ths has almost constant complexty. If nstead the largest or smallest subgradent was used, the worst-case complexty would agan be ONK. For a gven extreme pont f, the total transmt power j N p fjj wll be ether greater or less than the constrant P unless ths pont s optmal. We then need to re-optmze the powerallocatonfor the gvenfxed feasble tone allocaton x.e., x j =1f = fj, otherwsex j =0,.e., solve max V x, p s.t. p:p,x X,j p j P. 19 Let L x λ be the dual functon for ths problem. Gven λ =argmn λ 0 L x λ, the optmal power allocaton to 19 s gven by 10 wth λ = λ and the gven tone allocaton x. A smple one-dmensonal search once agan yelds the optmal λ. Ths wll have a complexty of ON to get wthn ɛ of the optmal snce each tone has at most one user. When the self-nose term β =0, we can actually fnd the optmal λ n fnte steps based on the followng alternatve characterzaton of λ, the proof of whch s based on a smlar argument as n [10]. Proposton 3: For β =0agvenˆλ s the unque optmal soluton to the dual problem mn λ 0 L x λ f and only f,j ˆλ = x jw 1 {ˆλ Wj} P Γ j,j e j 1 +, 20 1 {ˆλ Yj},j e j 1 {ˆλ Wj} [ [ xjw where W j = e j 1+Γ j,x j w e j,andy j = 0, xjwej 1+Γ j. Proposton 3 suggests the followng algorthm [28] for fndng λ. Frst check f the power constrant s volated when all users use maxmum power on the allocated tones,.e., f > P. If ths s false, the problem s solved.,j x j e j Γ j Otherwse, we need to search for λ by startng from the largest λ, and calculatng the rght sde of 20. If the result s less than the chosen value of λ, then we decrease λ and recalculate, untl a fxed-pont s found. It can be shown that the algorthm wll stop [28] n at most 2N steps at the correct λ. Ths algorthm sorts 2N values and thus, has a complexty of ON log N whch s larger than the ON complexty of the one-dmensonal search, but yelds the exact optmal soluton n fnte tme as opposed to an ɛ-optmal soluton. However, regardless of how the power s allocated, we frst need to fnd the optmal λ. It follows that f the largest or smallest subgradents are used to break tes, the overall algorthm wll have a complexty of ONK or ONK + N log N dependng on how the power s re-optmzed. D. Sngle sort suboptmal algorthm Now we ntroduce two sub-optmal algorthms that do not requre fndng the optmal λ teratvely. Instead, a carrer allocaton s determned by a sngle sort on each tone based on some easly calculated metrc. These heurstc algorthms are much faster than the prevous algorthms, although ther complexty s agan ONK. 1 HEURISTIC 1: Each subchannel j s allocated to the user wth the largest value of w Rj,where [ ] ẽj P/N R j =log 1+ s j 1+βẽ j P/N s the rate user could acheve on subchannel j under power allocaton P/N. Any tes are broken arbtrarly, and power allocaton P/N s used. Ths metrc was motvated n part by work n [14], [16] where a unform power allocaton not necessarly over all tones was shown to be nearly optmal. 2 HEURISTIC 2: Here subchannels are allocated as n HEURISTIC 1. However, after ths procedure, an optmal power allocaton s performed as n Secton III-C nstead of power allocaton P/N. It may turn out that no power s assgned to some subchannels. IV. SIMULATION STUDY We report smulaton results based on a realstc OFDM smulator wth assumptons and parameters commonly used n IEEE standards [11]. We focus on the followng algorthms: the OPTIMAL algorthm whch fnds the optmal λ and then chooses a tone-allocaton wth one user per tone as descrbed n Secton III-C 5, and HEURISTIC 1 and HEURISTIC 2 from Secton III-D. We smulate a sngle OFDM cell wth M = 40 users and a total transmsson power of P = 6W at the base staton. The channel gans e j s are the product of a fxed locaton-based term for each user and a frequency-selectve fast-fadng term. The locaton-based components are pcked usng an emprcally obtaned dstrbuton for many users n a large system. The fast-fadng term s generated usng a block-fadng model based upon the Doppler frequency for 5 We smulated both the algorthms n Secton III-B and III-C, and found that they have dentcal performance under all parameter choces. Ths could be due to the fact that the gap n makng the tme-sharng assumpton s small owng to there beng very few sgnfcantly dfferent extreme ponts at each schedulng nterval as dscussed at the end of Secton III-B. We thus refer to the algorthm n Secton III-C smply as the OPTIMAL algorthm.

7 294 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 1, JANUARY 2009 the block-length n tme and a standard reference moble delay-spread model for varaton n frequency. For the fastfadng terms, each mult-path component s held fxed for 2msec.e., a fadng block length, whch corresponds to a 250Hz Doppler frequency. The delay-spread s set to 1μsec. The users channel condtons are averaged over the applcable subchannelzaton scheme and fed back to the scheduler. We consder a system bandwdth of 5MHz consstng of 512 OFDM tones, grouped nto 64 subchannels 8 tones per subchannel. The symbol duraton s 100μsec wth a cyclc prefx of10μsec, whch roughly corresponds to 20 OFDM symbols per fadng block.e., 2msec. Ths s one of the allowed confguratons n the IEEE standards [11]. Resource allocaton.e. solvng 1 s done once per fadng block. All the results are averaged over the last 2000 OFDM symbols out of OFDM symbols.e., 3000 fadng blocks by whch tme we can be reasonably confdent that the system has reached statonarty. All users are nfntely back-logged and assgned a throughput-based utlty as n 2 wth parameter c =1and the same farness parameters α across users. The rate of user on subchannel j s calculated as r j =0.28Bx j log p jẽ j x j + βp j ẽ j, where B s the subchannel bandwdth. Here 0.56 accounts for the SNR gap due to lmted modulaton and codng choces and 0.28 accounts for varous factors such as hybrd ARQ transmsson scheme and the overhead due to guard tones and control symbols, etc. Whle the schedulng s based on the geometrc average for β =0and harmonc average for β>0, the decoded rate s based on per tone channel condtons. The frst set of smulaton results are for a system wth adjacent channelzaton, no self-nose β =0,andnoper-user SNR constrants.e., Γ j = for all and j. Table I shows the results for all three algorthms under dfferent choces of the utlty parameter α. The column Utlty gves the average utlty per user for each algorthm. The column log U shows the log utlty per user; ths gves an alternate ndcaton of the farness of the resultng allocaton same as utlty for α =0. The column Rate s the average throughput per user n Kbps, and the fnal column s the average number of users scheduled per schedulng nterval. For each choce of α, the three algorthms perform close to each other for each of these metrcs. HEURISTIC 2 performs better than HEURISTIC 1, snce the former re-optmzes the power allocaton after tone allocaton, and the latter just uses constant power allocaton. When α =1maxmum throughput, all three algorthms have almost dentcal performance. Next we consder the effect of dfferent subchannelzaton schemes. Table II shows the performance of the three algorthms for the adjacent, random, and nterleaved channelzaton schemes from Secton II-A. We set α = 0.5, β = 0, and Γ j = for all and j. Agan, both HEURISTIC algorthms perform close to the OPTIMAL algorthm. In all cases, nterleaved and random channelzatons result n lower utlty than the adjacent channelzaton. Ths s lkely due to hgher frequency dversty wth adjacent channelzaton. TABLE I PERFORMANCE FOR DIFFERENT CHOICES OF α ADJACENT CHANNELIZATION, NO-SELF-NOISE, NO SNR CONSTRAINTS. α Algorthm Utlty Log U Rate Num. 0 OPTIMAL HEURISTIC HEURISTIC OPTIMAL HEURISTIC HEURISTIC OPTIMAL HEURISTIC HEURISTIC TABLE II PERFORMANCE OF DIFFERENT SUBCHANNELIZATION SCHEMES α =0.5, NO SELF-NOISE, NO SNR CONSTRAINTS. Channelzaton Algorthm Utlty Log U Rate Num. Adjacent OPTIMAL Adjacent HEURISTIC Adjacent HEURISTIC Interleaved OPTIMAL Interleaved HEURISTIC Interleaved HEURISTIC Random OPTIMAL Random HEURISTIC Random HEURISTIC Indeed, for the channel model used here, n the nterleaved case all subchannels can be shown to be almost dentcal, explanng why t typcally schedules only one or two users. Next we consder the case when the self-nose coeffcent β = n Table III. Here we assume α =0.5, and no peruser SNR constrant. The performance gap between the three algorthms s slghtly larger compared to the case wthout selfnose n Table II. Fgure 2 shows the throughput CDFs for all three algorthms, wth β = and β = 0. Here adjacent channelzaton s used, α =0.5, and s j = for all and j. It s clear that users acheve better throughput when there s no self-nose β =0. For each β the OPTIMAL algorthm always acheves better rates compared to the HEURISTIC ones. Table IV llustrates the effect of SNR constrants. In partcular, we choose the SNR constrant to be, 32.5dB, and 22.5dB, respectvely, and the same across all users and all tones. We choose adjacent channelzaton wth utlty parameter α =0.5 and no self-nose. Compared to the no SNR constrants case, a constrant of 32.5dB does not change the results sgnfcantly, whle a constrant of 22.5dB substantally decreases the achevable rates 13% for the OPTIMAL algorthm and 27% for HEURISTIC 1 algorthm.

8 HUANG et al.: DOWNLINK SCHEDULING AND RESOURCE ALLOCATION FOR OFDM SYSTEMS 295 TABLE III PERFORMANCE OF DIFFERENT SUBCHANNELIZATION SCHEMES α =0.5, β =0.0056, NO SNR CONSTRAINTS. TABLE IV PERFORMANCE OF DIFFERENT SNR CONSTRAINTS ADJACENT CHANNELIZATION, α =0.5, NO SELF-NOISE. Channelzaton Algorthm Utlty Log U Rate Num. Adjacent OPTIMAL Adjacent HEURISTIC Adjacent HEURISTIC Interleaved OPTIMAL Interleaved HEURISTIC Interleaved HEURISTIC Random OPTIMAL Random HEURISTIC Random HEURISTIC SNR Max Algorthm Utlty Log U Rate Num. OPTIMAL HEURISTIC HEURISTIC dB OPTIMAL dB HEURISTIC dB HEURISTIC dB OPTIMAL dB HEURISTIC dB HEURISTIC HEURISTIC 1 β=0 OPTIMAL β= HEURISTIC 2 β= HEURISTIC 1 β= OPTIMAL β=0 HEURISTIC 2 β= Throughput n Kbps Fg. 2. Emprcal CDF of users throughputs adjacent channelzaton, α = 0.5, no per-user SNR constrants. V. CONCLUSIONS We have consdered the problem of gradent-based schedulng and resource allocaton for a downlnk OFDM system, whch essentally reduces to solvng a convex optmzaton problem n each tme-slot. We studed ths problem for a model that accommodates varous choces for user utlty functons, dfferent subchannelzaton technques, and selfnose due to mperfect channel estmates or phase nose. Usng dualty theory we frst gave an optmal algorthm for solvng a relaxed verson of ths problem n whch users can tmeshare each subchannel. Ths nvolves fndng a maxmum of a per user closed-form metrc for each subchannel and a one-dmensonal search of an optmal dual varable. More nterestngly, ths algorthm typcally automatcally yelds an nteger carrer allocaton except on one or two tones. To enforce such a constrant on all tones, we further proposed an algorthm that pcks an nteger carrer allocaton and reoptmzes the power allocaton accordngly. The numercal performance of ths algorthm s almost dentcal to the optmal soluton of the relaxed problem. Fnally, we proposed two even smpler suboptmal algorthms that only perform a sngle sort on each of the tones and avod any teratve calculatons. Smulatons show that the suboptmal algorthms acheve close to optmal performance under a wde range of scenaros, and the performance gap wdens when per user SNR constrants or channel estmaton errors are consdered. REFERENCES [1] R. Agrawal and V. Subramanan, Optmalty of certan channel aware schedulng polces, n Proc Allerton Conference, [2] R. Agrawal, A. Bedekar, R. La, and V. Subramanan, A class and channel-condton based weghted proportonally far scheduler, n Proc. ITC 2001, Salvador, Brazl, Sept [3] A. L. Stolyar, On the asymptotc optmalty of the gradent schedulng algorthm for multuser throughput allocaton, Operatons Research, vol. 53, no. 1, pp , [4] H. Kushner and P. Whtng, Asymptotc propertes of proportonal-far sharng algorthms, n Proc. 40th Annual Allerton Conference, [5] J. Mo and J. Walrand, Far end-to-end wndow-based congeston control, IEEE/ACM Trans. Networkng, vol. 8, no. 5, pp , Oct [6] A. Jalal, R. Padovan, and R. Pankaj, Data throughput of CDMA- HDR a hgh effcency-hgh data rate personal communcaton wreless system, n Proc. IEEE VTC, Sprng, [7] L. Tassulas and A. Ephremdes, Dynamc server allocaton to parallel queue wth randomly varyng connectvty, IEEE Trans. Inform. Theory, vol. 39, pp , [8] M. Andrews, et al., Provdng qualty of servce over a shared wreless lnk, IEEE Commun. Mag., vol. 39, no. 2, pp , Feb [9] A. L. Stolyar, Maxmzng queueng network utlty subject to stablty: greedy prmal-dual algorthm, Queueng Syst., vol. 50, no. 4, pp , [10] R. Agrawal, V. Subramanan, and R. Berry, Jont schedulng and resource allocaton n CDMA systems, n Proc. WOpt, Cambrdge, UK, Mar 2004 journal verson under submsson. [11] IEEE e-2005 and IEEE Std /Cor [12] Long Term Evoluton of the 3GPP rado technology, [13] H. Jn, R. Laroa, and T. Rchardson, Superposton by poston, n Proc. IEEE ITW 2006, Mar [14] L. Hoo, B. Halder, J. Tellado, and J. Coff, Multuser transmt optmzaton for multcarrer broadcast channels: asymptotc FDMA capacty regon and algorthms, IEEE Trans. Commun., vol. 52, no. 6, pp , [15] C. Y. Wong, R. S. Cheng, K. B. Letaef, and R. D. Murch, Multuser OFDM wth adaptve subcarrer, bt, and power allocaton, IEEE J. Select. Areas Commun., vol. 17, no. 10, pp , [16] J. Jang and K. Lee, Transmt power adaptaton for multuser OFDM systems, IEEE J. Select. Areas Commun., vol. 21, no. 2, pp , [17] Y. Zhang and K. Letaef, Multuser adaptve subcarrer-and-bt allocaton wth adaptve cell selecton for OFDM systems, IEEE Trans. Wreless Commun., vol. 3, no. 5, pp , [18] T. Chee, C. Lm, and J. Cho, Adaptve power allocaton wth user prortzaton for downlnk orthogonal frequency dvson multple access systems, n Proc. ICCS 2004, pp , [19] H. Yn and H. Lu, An effcent multuser loadng algorthm for OFDMbased broadband wreless systems, n Proc. IEEE Globecom, [20] K. Seong, M. Mohsen, and J. M. Coff, Optmal resource allocaton for OFDMA downlnk systems, n Proc. IEEE ISIT, pp , 2006.

9 296 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 8, NO. 1, JANUARY 2009 [21] L. L and A. Goldsmth, Capacty and optmal resource allocaton for fadng broadcast channels part I: ergodc capacty, IEEE Trans. Inform. Theory, vol. 47, no. 3, pp , [22] M. Medard, The effect upon channel capacty n wreless communcatons of perfect and mperfect knowledge of the channel, IEEE Trans. Inform. Theory, vol. 46, no. 3, pp , May [23] J. Lee, H. Lou and D. Toumpakars, Analyss of phase nose effects on tme-drecton dfferental OFDM recevers, n Proc. IEEE GLOBE- COM, [24] QUALCOMM Flaron Technologes, [25] J. Huang, V. Subramanan, R. Berry, and R. Agrawal, Jont schedulng and resource allocaton n uplnk OFDM systems for broadband wreless access networks, IEEE J. Select. Areas Commun., to appear, [26] D. Bertsekas, Nonlnear Programmng, 2nd ed. Belmont, MA: Athena Scentfc, [27] R. Rockafellar, Convex Analyss. Prnceton Unversty Press, [28] J. Huang, V. Subramanan, R. Berry, and R. Agrawal, Downlnk schedulng and resource allocaton for OFDM systems, tech. report. Avalable: jwhuang/publcaton/ofdm DL Techncal Report.pdf. [29] 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 Engneerng Department at the Chnese Unversty of Hong Kong. He receved the M.S. and Ph.D. degrees n Electrcal and Computer Engneerng from Northwestern Unversty n 2003 and 2005, respectvely. From 2005 to 2007, He worked as a Postdoctoral Research Assocate 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 &ELECTRICAL ENGI- NEERING, a Guest Edtor for IEEE JOURNAL OF SELECTED AREAS IN COMMUNICATIONS and JOURNAL OF ADVANCES IN MULTIMEDIA, and a TPC Co-Char of the Internatonal Conference on Game Theory for Networks GameNets 09. Vjay G. Subramanan M 01 receved hs Ph.D. degree n Electrcal Engg. from the Unversty of Illnos at Urbana-Champagn, n 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. Rajeev 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 jonng Motorola n 1999, Rajeev 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. Rajeev receved hs M.S and Ph.D degrees n Electrcal Engg.-systems from the Unversty of Mchgan, Ann Arbor and hs B.Tech degree n Electrcal Engg. from the Indan Insttute of Technology, Kanpur. COMMUNICATIONS. 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 joned 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 TRANSACTIONS ON INFORMATION THE- ORY and the IEEE TRANSACTIONS ON WIRELESS