Distributed and Optimal Reduced Primal-Dual Algorithm for Uplink OFDM Resource Allocation

Transcription

1 Dstrbuted and Optmal Reduced Prmal-Dual Algorthm for Uplnk OFDM Resource Allocaton Xaoxn Zhang, Lang Chen, Janwe Huang, Mnghua Chen, and Yupng Zhao Abstract Orthogonal frequency dvson multplexng OFDM s the key component of many emergng broadband wreless access standards. The resource allocaton n OFDM uplnk, however, s challengng due to heterogenety of users qualty of servce requrements, channel condtons, and ndvdual resource constrants. We formulate the resource allocaton problem as a non-strctly convex optmzaton problem, whch typcally has multple global optmal solutons. We propose a reduced prmal-dual algorthm, whch s dstrbuted, requres smple local updates, and probably globally converges to a global optmal soluton under easly satsfed suffcent techncal condtons. The performance of the algorthm s studed through a realstc OFDM smulator based on feld measurements. Compared wth the prevously proposed standard prmal-dual algorthm, the reduced algorthm decreases the total number of teratons by 8% and the varance by 85%. I. INTRODUCTION Orthogonal Frequency Dvson Multplexng OFDM s a promsng technology for future broadband wreless networks. In OFDM, the entre frequency band s dvded nto a large number of subchannels, and network resource can be allocated flexbly over each of the subchannels. In ths paper, we consder the resource allocaton problem n a sngle cell OFDM uplnk system, where multple end users transmt data to the same base staton. Ths s motvated by several practcal wreless systems, such as WMAX/82.16e, LTE for 3GPP, and UMB for 3GPP2. Gven the channel condtons of users at a partcular tme, we need to determne whch subset of users to schedule.e., transmt wth postve rates, how to allocate subchannels to the scheduled users, and the power allocaton across these subchannels. Most prevous work on resource allocaton n OFDM systems focused on the downlnk case, where the base staton sends traffc to multple end users subect to a total power constrant. The optmzaton problem n the downlnk case s easer to solve and a centralzed algorthm s reasonable X. Zhang and Y. Zhao are wth the State Key Laboratory of Advanced Optcal Communcaton Systems & Networks, School of Electroncs Engneerng and Computer Scence, Pekng Unversty, Beng 1871, P.R.Chna {xaoxn.zhang,yupng.zhao}@pku.edu.cn L. Chen, J. Huang and M. Chen are wth the Dept. of Informaton Engneerng, Chnese Unversty of Hong Kong, Hong Kong, P.R.Chna {lchen6,whuang,mnghua}@e.cuhk.edu.hk J. Huang s supported by the General Research Fund Proect Number and establshed under the Unversty Grant Commttee of the Hong Kong Specal Admnstratve Regon, Chna, the Natonal Key Technology R&D Program Proect Number 27BAH17B4 establshed by the Mnstry of Scence and Technology of Chna, and the Mcrosoft- CUHK Jont Laboratory for Human-centrc Computng & Interface Technologes. Part of the work was done when X. Zhang vsted Chnese Unversty of Hong Kong n 28. to mplement [1]. Due to dfferent resource constrants n the uplnk case, however, the algorthms proposed for the downlnk case can not be drectly appled to the uplnk case. Uplnk OFDM resource allocaton only receves lmted attenton recently [2] [8]. In [2], the problem was formulated n the framework of Nash Barganng wth a focus of far resource allocaton. The authors of [3] proposed a heurstc algorthm that tres to mnmze each user s transmsson power whle satsfyng the ndvdual rate constrants. In [4], the author consdered the sum-rate maxmzaton problem and derved algorthms based on Raylegh fadng on each subchannel. The authors n [5] [8] proposed several heurstc algorthms to solve a problem smlar as the one consdered here wth addtonal nteger channel allocaton constrants. None of the prevous lterature focused on solvng the uplnk resource allocaton problem optmally. We formulate the resource allocaton problem as a weghted rate maxmzaton problem, whch s motvated by the gradent-based schedulng framework n [9] [11]. Ths problem, however, s qute challengng to solve due to the heterogenety of users qualty of servce requrements, channel condtons, and ndvdual resource constrants. In ths paper, we propose a dstrbuted prmal-dual algorthm that acheves the optmal resource allocaton n uplnk OFDM systems. Our key contrbutons are: Optmal algorthm wth global convergence: the proposed algorthm s provably globally convergent to one of the global optmal solutons of the resource allocaton problem, despte non-strct convexty of the problem under whch settng prmal-dual algorthms may not be able to converge [12] [14]. Dstrbuted algorthm wth low complexty: the proposed algorthm s dstrbuted, requres smple local updates, and demands only lmted message passng. Smpler algorthm wth better convergence: the proposed algorthm only needs to teratvely update a subset of all decson varables, and thus requres much fewer teratons to converge compared wth the prevously proposed standard prmal-dual algorthm. OFDM model wth self-nose: we consder an OFDM model where the achevable data rate s calculated by takng the self-nose nto consderaton. We demonstrate how ths realstc model wll affect the optmal soluton and the correspondng algorthm desgn. II. PROBLEM STATEMENT We consder a sngle OFDM cell, where there s a set M = {1,..., M} of users transmttng to the same base

2 staton. Each user M has a prorty weght w. 1 The total frequency band s dvded nto a set N = {1,..., N} of subchannels e.g., tones/carrers. A user M can transmt over a subset of the subchannels not necessarly adacent, wth transmsson power p over subchannel N satsfyng the ndvdual power constrant,.e., p P. For channel, t s allocated to user wth fracton x, and the total allocaton across all users should be no larger than 1,.e., x 1. We defne e as the receved sgnal-to-nose rato SNR per unt power for user on subchannel. We further assume that the channel condtons do not change wthn the tme of nterests,.e., we are lookng at a resource allocaton perod smaller than the channel coherence tme. 2 Wth perfect channel estmaton, user s achevable rate on subchannel s r = x B log 1 + p e x, whch corresponds to the Shannon capacty of a Gaussan nose channel wth bandwdth x B and receved SNR p e /x. Ths SNR arses from vewng p as the average power user s allowed to use on subchannel ; the correspondng nstantaneous transmsson power s p /x when only a tme fracton x of the subchannel s allocated. For notaton smplcty we normalze the bandwdth to be B = 1 n the analyss. 3 In a real OFDM system, mperfect carrer synchronzaton and naccurate channel estmaton may result n self-nose [15], [16]. We follow a smlar approach as n [15] to model self-nose and use an estmate value β to represent the level of self-nose. Wth self-nose, user s feasble rate on subchannel becomes r = x log 1 + p e x +βp e, where p e /x + βp e depcts the effectve SNR. The key notatons used throughout ths paper are lsted n Table I. We use bold symbols to denote vectors and matrces of these quanttes, e.g., p = {p,, } and x = {x,, }. Our obectve s to maxmze the weghted sum of the users rates over the feasble rate-regon defned as follows, { Re = r R+ M p e } : r = x log 1+, M, x N + βp e 1 where x, p X are chosen subect to x 1, N, 2 and the set p P, M, 3 X := { x, p : x 1, p }. 4 1 The prorty weghts w s are motvated by the gradent-based schedulng framework n [9] [11]. Assume each user has a utlty functon U W,t dependng on ts average throughput W,t up to tme t. To maxmze the total network utlty lm T 1 T Tt=1 U W,t, t s enough to solve the weghted rate maxmzaton problem durng each tme slot t wth w = U W,t / W,t. 2 Ths s partcularly sutable for fxed broadband wreless access part of the IEEE standard, where users are relatvely statc and the correspondng coherence tme s long. 3 A realstc value of B wll be consdered n the smulatons Secton V. Notaton N N M M w e p x P β TABLE I KEY NOTATIONS Physcal Meanng total number of subchannels set of all subchannels total number of users set of all users user ndex subchannel ndex user s dynamc weght normalzed SNR on subchannel for user power allocated on subchannel for user fracton of subchannel allocated to user maxmum transmt power for user self-nose coeffcent In many OFDM standards, x s constraned to be an nteger, n whch case we can add the addtonal constrant x {, 1} for all,. The nteger constrant makes the resource allocaton very dffcult to solve, and varous heurstc algorthms dealng wth such constrant are proposed n [6] [8], [17]. In ths paper, we focus on the rate regon defned by 1 to 4,.e., no nteger constrants are consdered. The correspondng optmal soluton typcally contans fractonal values of x s. There are several practcal methods of achevng these fracton allocatons. For example, f resource allocaton s done n blocks of OFDM symbols, then fractonal values of x can be mplemented by tmesharng the symbols n a block. Lkewse, f there are several tones n a subchannel, then fractonal values of x s can also be mplemented by frequency-sharng the tones n a subchannel [18]. To summarze, we want to solve the followng problem w r, 5 max r Re where rate r and rate regon Re are gven n 1. III. A REDUCED PRIMAL-DUAL ALGORITHM We rewrte problem 5 n varables x and p as follows: Problem 1 Weghted Rate Maxmzaton: p e max w x log 1 +, 6 x,p X x + βp e M N subect to the per subchannel assgnment constrants n 2 and the per user power constrants n 3. Set X s gven n 4. Although the obectve functon n 6 s concave, ts dervatve s not well defned at the orgn x =, p =. Ths motvates us to look at the followng ɛ-relaxed verson of Problem 1: Problem 2 ɛ-relaxed Weghted Rate Maxmzaton: max x,p X M w x + ɛ log N 1 + p e, 7 x + βp e + ɛ where constants ɛ take small postve value for all and. The constrant set remans the same as n Problem 1.

3 By such relaxaton, the obectve functon n 7 now has dervatve defned everywhere n the constrant set X. Thanks to the contnuty of the obectve functon, the optmal value to Problem 2 can be arbtrarly close to that of Problem 1 f ɛ = [ɛ,, ] s chosen to be small enough. The constrant set of Problem 2 s convex, and the obectve functon n 7 s contnuous and non-strctly concave. 4 As such, Problem 2 has multple optmal solutons, and there s no dualty gap between t and ts dual problem. The exstence of dervatves allows us to wrte down a prmal-dual algorthm to pursue the optmal soluton to Problem 2. The Lagrangan for Problem 2 s as follows, p e Lλ, µ, x, p := w x + ɛ log 1 + x, + βp e + ɛ + λ P p + µ 1 x. 8 By strong dualty theorem, the optmal prmal and dual solutons must satsfy KKT condtons,.e., for all and, µ, x 1, µ x 1 =, 9 λ, p P, λ p P =, 1 x, p, 11 x f x, p µ, 12 p g x, p λ, 13 where f and g are gradents of the obectve functon n 7 wth respect to x and p, respectvely, and are gven by p e f x, p = w log 1 + x + βp e + ɛ w x + ɛ p e x + βp e + ɛ [x + β + 1p e + ɛ ], and g x, p = w e x + ɛ 2 x + βp e + ɛ [x + β + 1p e + ɛ ]. The last two KKT condtons n 12 and 13 become equaltes f x > and p >, respectvely. It can be verfed that the optmal solutons of Problem 2, satsfyng above KKT condtons, are exactly the saddle ponts of the Lagrangan functon n 8. Snce the prmal problem has at least one soluton, the saddle pont exsts. For notaton smplcty, we defne a + = maxa, and { a, b >, a + b = maxa,, otherwse. To pursue saddle ponts of the Lagrangan functon, we start by revewng the standard prmal-dual algorthm ntroduced n [13]. Then we derve a new reduced prmal-dual algorthm whch converges faster than the standard one. 4 Ths can be verfed by showng that the Hessan of the obectve functon n Problem 2 s negatve-semdefnte but not negatve-defnte [19]. For each and, we consder the followng standard prmal-dual algorthm: Algorthm SPD: Standard Prmal-Dual Algorthm ẋ = k x + f x, p µ, 14 x + g x, p λ ṗ = k p µ = k µ, 15 p + x 1, 16 µ + λ = k λ p P, 17 λ where k x, kp, kµ and k λ are constants representng update stepszes. Here the dervatves on the left hand sdes of 14 to 17 are defned wth respect to tme. We call a pont x, p, µ, λ an equlbrum of Algorthm SPD f and only f the correspondng dervatves n 14 to 17 are zero for all and. We can show that the set of equlbra of Algorthm SPD s equvalent to the set of global optmal solutons of Problem 2 [13]. In Algorthm SPD, all varables x, p, µ, and λ are dynamcally adapted, whch mght lead to slow convergence. One way to address ths s to reduce the number of dynamcally adaptng varables. To acheve ths, we wll constran the algorthm traectores onto a manfold that ncludes all optmal prmal and dual solutons. We study the followng manfold by settng 15 to zero,.e.,, = g x, p λ + p, 18 whch n turn mples { g x, p = λ, f p >, g x, p λ, f p =. 19 Based on the KKT condtons, the optmal prmal and dual solutons must le on the above manfold. After smplfcaton we get the followng expresson of the manfold: where h s denoted by p = h x, λ, ββ + 1 w e λ 2β + 1 h x, λ = x + ɛ 2ββ + 1e when β, and h x, λ = w e λ + x + ɛ λ e w e λ + w e λ when β =. Substtutng 2 nto 14 to 17, we obtan a new reduced prmal-dual algorthm as follows:

4 Algorthm RPD: Reduced Prmal-Dual Algorthm ẋ = k x + f x, p µ, 21 x + µ = k µ x 1, 22 µ + λ = k λ p P, 23 λ p = h x, λ. 24 Proposton 1: The set of equlbra of Algorthm RPD s the same as the set of global optmal solutons of Problem 2. Ths means that f Algorthm RPD converges, t reaches a global optmal soluton of the ɛ-relaxed weghted rate maxmzaton problem. Compared to Algorthm SPD n 14 to 17 n whch p s dynamcally adapted, p n the new Algorthm RPD s drectly computed from x and λ. Consequently, Algorthm RPD has fewer dynamcally adaptng varables, hence s expected to converge faster. Smlar as Algorthm SPD, Algorthm RPD can also be mplemented n a dstrbuted fashon by end users and the base staton. End user s responsble of updatng x s and p s as well as dual varables λ s locally. Durng each teraton, t sends the latest values of x s to the base staton, but not the p s or λ s. The base staton s responsble of updatng dual varables µ s for all subchannels and broadcastng to the users. In partcular, the base staton does not need to know users prorty weghts, power constrants, or the power allocaton. Both the communcaton complexty and computaton complexty per teraton are OMN. Next we wll show that traectores of Algorthm RPD converge to ts equlbra, and thus the global optmal soluton of Problem 2. IV. CONVERGENCE OF THE REDUCED PRIMAL-DUAL ALGORITHM The key challenge of the convergence proof s the nonstrct concavty of the obectve functon n 7. It has been well observed n lterature that although prmal-dual algorthms globally converge to the optmal soluton of strctly concave optmzaton problem, they may oscllate ndefntely and fal to converge when applyng to nonstrctly concave optmzaton problem [12] [14]. In ths secton, we study convergence of Algorthm RPD. We frst show that the traectores converge to an nvarant set that contans all global optmal solutons of Problem 2. Theorem 1: All traectores of Algorthm RPD converge to an nvarant set V globally and asymptotcally. Furthermore, let x, p, µ, λ be a global optmal soluton of Problem 2 and x, p, µ, λ be any pont n set V, the followng s true for all and, 1 x, p, µ, λ s contaned n V ; 2 µ s nonzero only f x = 1; 3 p = P, and λ s a postve constant; 4 Over set V, f x, p = f x, p = µ, and g x, p = g x, p = λ ; 5 p /x + ɛ = p /x + ɛ. The detaled proof can be found n Appendx A of the onlne techncal report [2]. Results 1 to 4 wll be used n later analyss. Result 5 s of ndependent nterest. It mples that although there can be multple global optmal solutons to Problem 2, the effectve SNR acheved by user on subchannel s the same n all solutons. Although all traectores of Algorthm RPD may converge to the desred equlbra n V, they may also converge to nonequlbrum ponts n V f there are any. We now study the condtons for V to contan only the desred equlbra, under whch Theorem 1 guarantees the convergence of Algorthm RPD to a global optmal soluton of Problem 2. Plug result 4 of Theorem 1 nto Algorthm RPD, and recall that M s the total number of users and N s the total number of subchannels. We fnd that V s exactly the set that contans all traectores of the followng lnear system n 25 to 27 over set {x, µ }. ẋ = K x A T 1 µ K x A T 1 µ, 25 µ = K µ A 1 x K µ 1, 26 λ = K λ A 2 Bx + ɛ K λ P =, 27 where K x s an MN MN dagonal matrx wth dagonal terms equal to k x s, Kµ s an N N dagonal matrx wth dagonal terms equal to k µ s, and Kλ s an M M dagonal matrx wth dagonal terms equal to k λ s. B s an MN MN dagonal matrx gven by B = dagb,,, where b = p x +ɛ. The matrx A 1 has a dmenson of N MN, and s gven by A 1 = [I N,, I N ], where I N s an dentty matrx wth dmenson N. The matrx A 2 has a dmenson of M MN and s gven by 1 1 N 1 1 N A 2 =., N where 1 1 N s an all one vector wth dmenson 1 by N. We observe the followng for the above lnear system: Lemma 1: For the lnear system n 25 to 27, we have 1 every order Le dervatve of A 2 Bx s constant, that s n : d n dt n A 2Bx = constant, where t denotes tme; 2 startng from a non-equlbrum pont, traectores of x and µ, followng 25 and 26 respectvely, do not converge and form lmt cycles. Proof: Sketch From 27, we have A 2 Bx = P A 2 Bɛ = constant. Result 1 can be derved by takng dervatves wth respect to tme on both sdes of the above equaton. For result 2, t

5 can be verfed that the transfer functon matrx of the lnear system s a product of postve dagonal matrx and a skew-symmetrc matrx. Hence, all egenvalues of the transfer matrx are purely magnary. Result 1 n Lemma 1 states that every order Le dervatve of A 2 Bx s constant. By lnear system theory, f the system state µ s completely observable from A 2 Bx, then constant A 2 Bx wll lead to µ equal to and µ beng constant. When µ s constant, ẋ s constant accordng to 25. Combnng wth the constrant that x and A 2 Bx s constant, we can show that ẋ s also zero f µ s zero over the set {x, µ }. In the followng theorem, we state condtons for µ to be completely observable from A 2 Bx, and summarze ts consequence on convergence of Algorthm RPD. Theorem 2: All traectores of Algorthm RPD converge globally and asymptotcally to the system equlbra f the followng condton holds: [ ] A 2 BK x A T 1 K µ A 1 K x A T 1 σi has rank N, 28 where σ denotes any egenvalue of matrx K µ A 1 K x A T 1. Proof: By lnear system theory, µ s completely observable from the constant A 2 Bx f and only f the complete observablty 28 holds [21]. Then the nvarant set V contans only the equlbra of the lnear system n 25 to 27, whch are the global optmal solutons of Problem 2. Consequently, all traectores of Algorthm RPD converge globally and asymptotcally to the global optmal solutons. For the problem we studed n ths paper, we can choose properly the update stepszes of Algorthm RPD to satsfy the condtons n 28. Corollary 1: Condtons 28 n Theorem 2 are satsfed f both of the followng are true K x = ki dagonal terms of K x take the same value k; all dagonal elements of K µ take dfferent values. The proof of Corollary 1 can be found n Appendx B n the onlne techncal report [2]. In ths secton, we have nvestgated the convergence of Algorthm RPD by combnng both La Salle prncple from nonlnear stablty theory and complete observablty from lnear system theory. The proof shows that Algorthm RPD can globally and asymptotcally converge to one of the global optmal solutons of Problem 2 when satsfyng the condtons n Corollary 1. V. SIMULATION RESULTS We show the convergence and optmalty of Algorthm RPD over a realstc OFDM uplnk smulator. Each user s subchannel gans e s are the product of two terms: 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 1µsec. The system bandwdth s 5.12MHz consstng of 512 tones, whch s further grouped Dual varables 1 2 Total power allocaton of all users Number of teratons Fg Dual varables 1 2 Total channel allocaton of all channels Number of teratons Prmal and dual varable convergence of Algorthm RPD nto 64 subchannels. 5 The symbol duraton s 1µsec wth a cyclc prefx of 1µsec. Unless otherwse specfed, we assume the followng parameter settng throughout all smulatons. The varables are ntalzed as x = 1/M, p = P /N, µ =, and λ =.1 max w e for all and. The update stepszes n Algorthm RPD are chosen as k x = 1 2, k µ = ε, and k λ = 1 2 for all and. Here, ε s for all channels are chosen to be very small values and dverse from each other, n order to meet the requrement of Corollary 1 as the suffcent condton for system convergence. Each user has a total transmsson power constrant P = 2Watts. Users channel condtons are randomly generated from the smulator, and users have equal weghts w = 1 for all. A. Algorthm Convergence We frst show the convergence of Algorthm RPD wth 4 users and 64 subchannels. Here we assume β =. Fg. 1 shows the convergence of dual varables upper two subgraphs, λ for 4 users and µ for 64 subchannels and prmal varables lower two subgraphs, p for 4 users and x for 64 subchannels. In Fg. 2, the upper subgraph shows how the dual value and prmal feasble value change wth teratons. The dual value s an upper bound of the global optmal soluton of Problem 2. The prmal feasble value s a lower bound and s calculated as follows: gven the prmal values of pt and xt at teraton t, normalze so that they are feasble and the resources are fully utlzed.e., p t = p tp / p t and x t = x t/ x t, and calculate the achevable rate accordngly. The bottom subfgure shows the relatve 5 Every 8 adacent tones are grouped nto one subchannel. Ths corresponds to the Band AMC mode of d/e and can help to reduce the feedback overhead. For dscussons on varous ways of subchannelzaton, see [1].

6 Dual value/prmal value-1 2 Dual value upper bound Prmal feasble value lower bound Rate bps/hz Fg. 2. Average number of teratons Total weghted rate Relatve errors Number of teratons Total weghted rate convergence of Algorthm RPD Algorthm SPD Algorthm RPD Number of users Fg. 3. Average number of teratons and standard devaton of Algorthm RPD and Algorthm SPD errors of two curves plotted n the upper subfgure. If we defne the stoppng crteron to be the relatve error less than 5 1 3, then Algorthm RPD converges n 364 teratons. B. Comparson wth the Standard Prmal-Dual Algorthm In Fg. 3, we compare the convergence speed of Algorthm RPD wth Algorthm SPD. The self-nose coeffcent s β =.1. We vary the number of users from 4 to 4. For a fxed user populaton sze, we randomly generated 1 sets of dfferent weghts and channel condtons. We plot both the average and the standard devaton.e., error bar of the number of teratons for both algorthms as the number of users changes. In all cases, Algorthm RPD converges wth much fewer teratons about 8% less than Algorthm SPD n average and a much smaller varance about 85% less than Algorthm SPD n average. VI. CONCLUSIONS AND FUTURE WORKS We presented a dstrbuted optmal prmal-dual resource allocaton algorthm for uplnk OFDM systems. The key features of the proposed algorthm nclude: a dstrbuted mplementaton by end users and base staton wth smple local updates, b global convergence despte the exstence of multple global optmal solutons, c reduced prmal-dual algorthm whch elmnates unnecessary varable updates and hence converges faster than the standard algorthm, and d ncorporatng self-nose observed n the practcal OFDM systems. The absolute convergence speed of the proposed algorthm, on the other hand, s stll slow for real-tme mplementaton. Desgnng fast and near optmal heurstc algorthms based on the proposed algorthm would be an nterestng future research drecton. REFERENCES [1] J. Huang, V. G. Subramanan, R. Agrawal, and R. A. Berry, Downlnk schedulng and resource allocaton for OFDM systems, IEEE Trans. on Wreless Communcatons, vol. 8, no. 1, pp , Jan. 29. [2] Z. Han, Z. J, and K. Lu, Far Multuser Channel Allocaton for OFDMA Networks Usng Nash Barganng Solutons and Coaltons, IEEE Trans. Comm., vol. 53, no. 8, pp , 25. [3] S. Pfletschnger, G. Muenz, and J. Spedel, Effcent subcarrer allocaton for multple access n OFDM systems, n 7th Internatonal OFDM Workshop, 22. [4] Y. Ma, Constraned Rate-Maxmzaton Schedulng for Uplnk OFDMA, Proc. IEEE MILCOM, pp. 1 7, Oct. 27. [5] K. Km, Y. Han, and S. Km, Jont subcarrer and power allocaton n uplnk OFDMA systems, IEEE Comms. Letters, vol. 9, no. 6, pp , 25. [6] C. Ng, and C. Sung, Low complexty subcarrer and power allocaton for utlty maxmzaton n uplnk OFDMA systems, IEEE Trans. Wreless Comm., vol. 7, no. 5 Part 1, pp , 28. [7] K. Kwon, Y. Han, and S. Km, Effcent Subcarrer and Power Allocaton Algorthm n OFDMA Uplnk System, IEICE Trans. Comm., vol. 9, no. 2, pp , 27. [8] L. Gao, and S. Cu, Effcent subcarrer, power, and rate allocaton wth farness consderaton for OFDMA uplnk, IEEE Trans. Wreless Comm., vol. 7, no. 5 Part 1, pp , 28. [9] R. Agrawal, and V. Subramanan, Optmalty of Certan Channel Aware Schedulng Polces, Proc. of Allerton Conference, Oct. 22. [1] A. L. Stolyar, On the asymptotc optmalty of the gradent schedulng algorthm for multuser throughput allocaton, Operatons Research, vol. 53, No. 1, pp , 25. [11] H. Kushner, and P. Whtng, Asymptotc propertes of proportonalfar sharng algorthms, n Proc. 4th Annual Allerton Conference on Communcaton, Control, and Computng, Oct. 22. [12] T. Voce, Stablty of Congeston Control Algorthms wth Mult-Path Routng and Lnear Stochastc Modellng of Congeston Control, PhD Thess, Unversty of Cambrdge, May, 26. [13] M. Chen, and J. Huang, Optmal Resource Allocaton for OFDM Uplnk Communcaton: A Prmal-Dual Approach, CISS conference, 28. [14] M. Chen, M. Ponec, S. Sengupta, J. L, and P. A. Chou, Utlty Maxmzaton n Peer-to-peer Systems, ACM SIGMETRICS, 28. [15] H. Jn, R. Laroa, and T. Rchardson, Superposton by poston, IEEE Informaton Theory Workshop, March 26. [16] J. Lee, H. Lou, and D. Toumpakars, Analyss of Phase Nose Effects on Tme-Drecton Dfferental OFDM Recevers, Proc. of IEEE GlobeCom, 25. [17] J. Huang, V. Subramanan, R. Berry, and R. Agrawal, Jont Schedulng and Resource Allocaton n Uplnk OFDM Systems for Broadband Wreless Access Networks, IEEE Journal on Selected Areas n Communcatons, vol. 27, no. 2, pp , Feb. 29. [18] W. Yu, R. Lu, and R. Cendrllon, Dual optmzaton methods for multuser orthogonal frequency dvson multplex systems, Proc. of IEEE GlobeCom, vol. 1, 24, pp [19] D. Bertsekas, Nonlnear Programmng, 2nd ed., Massachusetts: Athena Scentfc, [2] X. Zhang, L. Chen, J. Huang, M. Chen, and Y. Zhao, Optmal Prmal- Dual Algorthm for Uplnk OFDM Resource Allocaton, techncal report, whuang/publcaton/pd_ TechReport.pdf. [21] F. Caller and C. Desoer, Lnear System Theory, Sprnger-Verlag, 1991.