Joit Schedulig ad Dyamic Power Spectrum Optimizatio for Wireless Multicell Networks Wei Yu Dept. of Electrical ad Computer Eg. Uiversity of Toroto Toroto, Otario, Caada M5S 3G4 Email: weiyu@comm.utoroto.ca Taesoo Kwo ad Chagyog Shi New Radio Access Group, Commuicatio Lab Samsug Electroics Gyeoggi-Do, South Korea, 446-72 Emails: {taesoo.kwo, c.y.shi}@samsug.com Abstract This paper proposes a joit proportioally fair schedulig ad dyamic power spectrum adaptatio algorithm for wireless multicell etworks. The proposed system allows multiple base-statios i a multicell etwork to be coordiated by exchagig iterferece pricig messages amog each other. The messages summarize the effect of itercell iterferece, ad they are fuctios of trasmit power spectra, sigal-to-oise ratios, direct ad iterferig chael gais, ad the proportioal fairess variables for each user. The use of iterferece pricig allows the trasmit power spectra ad user schedule withi each base-statio to be optimized joitly, while takig ito cosideratio both the itercell iterferece ad the fairess amog the users i multiple cells. This paper proposes two power spectrum optimizatio methods, oe based o the Karush- Kuh-Tucker KKT coditio of the optimizatio problem, ad aother based o the Newto s method. The proposed methods ca achieve a throughput improvemet of 4%-55% for users at the cell edge as compared to a covetioal per-cell optimized system, while maitaiig proportioal fairess. I. INTRODUCTION The capacity of wireless cellular etworks is fudametally limited by itercell iterferece. Traditioal cellular etworks maage iterferece with specific frequecy reuse patters so that earby cell-edge users belogig to eighborig cells do ot share the same frequecy. However, as badwidth becomes icreasigly scarce ad as moder cellular etworks move icreasigly toward the imal frequecy reuse of, where all the cells share the same frequecy, the maagemet of itercell iterferece via dyamic spectrum optimizatio is expected take a cetral role i future etworks. The aim of this paper is to study adaptive schedulig, power cotrol, ad badwidth allocatio methods for iterferece mitigatio. Traditioal cellular etworks are desiged o a per-cell basis. Trasmitters i such a etwork typically operate at fixed power-spectrum desity PSD levels, while receivers typically assume the worst-case iterferece. This paper evisios a advaced wireless cellular etwork i which base-statios cooperate with each other i the joit schedulig of users ad i the joit optimizatio of trasmit power spectra over the frequecies for all users across the cells for iterferece mitigatio. Base-statio coordiatio ca be realized by the exchage of messages amog the base-statios. The mai cotributio of this paper is a set of umerical power spectrum optimizatio methods based o a specific type of iterferece pricig messages. These messages reveal the effect of iterferece amog eighborig cells. They also help maitai fairess amog all users across the cells, ad allow frequecy allocatio ad power spectrum adaptatio to be doe i a distributed fashio. A. System Model Cosider a wireless multicell etwork as show i Fig., where users withi each cell are separated from each other usig orthogoal frequecy divisio multiple access OFDMA over a fixed badwidth. The frequecy assigmets for users withi each cell are o-overlappig. Thus, users experiece itercell iterferece oly ad o itracell iterferece. The system is assumed to employ a iitial chael estimatio ad sychroizatio phase, i which the multipath fadig chael iformatio is obtaied betwee every pair of trasmitter ad receiver i the etwork, icludig both uplik ad dowlik direct chaels withi each cell as well as the iterferig chaels betwee the base-statios ad the remote termials i eighborig cells. For example, dowlik chael estimatio may be performed usig a scheme i which sigature sequeces are sychroously trasmitted by all base-statios. The mobile users may the estimate all the dowlik chaels at the same time by matchig to the differet sequeces. I time-divisio duplex TDD systems, the uplik chael may be iferred from the dowlik chael. The joit schedulig ad power spectrum adaptatio problem ca be formulated as that of decidig for each user, which frequecy toes they should trasmit i, ad at what trasmit power level. Equivaletly, the base-statios ca be thought of makig two decisios at each frequecy toe: Schedulig: Which user should be served? Power spectrum allocatio: What is the appropriate power spectral desity level at this frequecy toe? To obtai a etwork-wide global optimal solutio, these two questios must be aswered joitly ad across the basestatios. Further, the same optimizatio problem must be solved for uplik ad dowlik separately, ad repeatedly as chaels vary over time. The service objective of a wireless cellular etwork is to provide the highest overall throughput to all users while
8 6 4 2 2 4 6 Base Statio Mobile User 8 8 6 4 2 2 4 6 8 Fig.. A multicell etwork with 9 cells ad with users distributed close to the cell edge. maitaiig fairess. Fairess is importat i a iterferecelimited eviromet as oe user s gai ofte comes at the expese of other users. This paper assumes a proportioally fair objective amog all users i all cells. Specifically, the optimizatio objective is to solve T lk log R lk lk where R lk is the log term average rate of the kth user i the lth cell, ad T lk s are a set of weights correspodig to the quality-of-service target for the kth user i the lth cell. The logarithmic fuctio is a example of a utility fuctio, which provides a balace betwee the achievable rates amog users. B. Related Work Covetioal cellular etworks are desiged as sigle-cell systems multicell coordiatio is limited to the use of soft hadoff. I a sigle-cell desig, schedulig ad power spectrum adaptatio ca be cosidered separately. For example, i proportioally fair schedulig as origially proposed i [], trasmit power is assumed fixed. The schedulig algorithm the aims to fid the best user to trasmit at each timeslot to take advatage of multiuser diversity. Power spectrum adaptatio is doe i additio ad idepedetly of schedulig. Power spectrum adaptatio becomes much more challegig i a multicell settig. I this cotext, a lot of work [2], [3], [4], [5], [6] has bee doe i the digital subscriber lies DSL settig, where each cell has oly oe user. I this case, schedulig is ot a issue; the mai challege is to fid computatioally efficiet methods for optimizig power allocatio across the frequecies. Particularly relevat to this paper are the works [5], [6] which use the idea of iterferece pricig for spectrum balacig. Iterferece pricig is a key cocept which is also used i the desig of power spectrum adaptatio algorithms proposed i this paper. The cocept of iterferece pricig first appears i the literature for wireless ad-hoc etworks with multiple trasmitterreceiver pairs sharig the same physical medium [7], [8], [9], [], [], where agai the focus is o power spectrum adaptatio ad ot user schedulig. I the cellular etwork cotext, [2], [3] deal with a sigle-cell problem ad propose a solutio to the joit schedulig ad power allocatio problem via a iteger relaxatio method. For the multicell etworks, [4] cosiders a idealized etwork ad proposes a multicell schedulig algorithm. The proposed system is related to [5] where power spectrum adaptatio methods i the DSL domai e.g. [3], [4], [5] are applied to the wireless settig. The mai cotributio of [5] is that it proposes a method of iterative optimizatio of user schedulig ad power adaptatio for imizig the weighted rate sum over all users i all cells. However, the weights are fixed i [5]. This paper proposes automatic ad cotiuous adjustmet of weights based o the proportioal fairess criterio. I additio, this paper proposes ovel power spectrum adaptatio methods with faster covergece. The proposed system is also closely related to [6], which cosiders a proportioally fair joit scheduler ad itercell power allocatio scheme. The mai differece of this paper as compared to [6] is that [6] takes a gradiet approach for power allocatio, while the system proposed i this paper takes ito accout the secod derivative iformatio i additio, ad therefore has faster covergece. II. JOINT SCHEDULING AND POWER SPECTRUM OPTIMIZATION A. Problem Formulatio Cosider a multicell eviromet with L cells ad K users per cell employig a OFDMA scheme with N toes over a fixed badwidth. Both the base-statio ad the remote users are equipped with a sigle atea oly. For simplicity, we assume chael reciprocity ad let h lmk deote the chael respose betwee the lth base-statio ad the kth remote user i mth cell i th toe for both uplik ad dowlik. The system allows oly a sigle user to trasmit or to receive at ay give toe i a cell. The uplik ad dowlik user schedules are determied by the assigmet fuctios f U l, ad f D l,, which assig a possibly differet user k to the lth cell i the th toe for the uplik ad dowlik, respectively. The uplik ad dowlik trasmit PSDs are deoted as PU,l ad P, respectively. The uplik ad dowlik are separated via TDD. For the dowlik, the proportioally fair joit schedulig ad trasmit power-spectrum adaptatio problem is that of choosig the schedulig fuctio k = f D l, ad the trasmit power spectrum P over the frequecies ad across all the cells to imize, i.e. T k log Rk l,k s.t. R k = P log + h llk 2 Γσ 2 + D lk j l P D,j h jlk 2 P P l P SD l, 2
where the summatio i the expressio for R k is over all frequecy toes assiged to the kth user i the lth cell, i.e. D lk where D lk = { k = f D l, }. Here, R k is the istataeous dowlik rate ad R k is the time averaged rate for the kth user i lth cell. Further, Pk deotes the time averaged dowlik total trasmit power summed across the frequecies, which eeds to satisfy a imum power costrait P ; Γ is the SNR gap correspodig to the choice of modulatio ad codig schemes; σ 2 is the backgroud oise; SD is the trasmit PSD costrait at the base-statios. We ote that the uplik problem ca be stated similarly. The optimizatio problem 2 is a mixed discrete user schedulig ad cotiuous power allocatio optimizatio problem. This paper provides a local optimum solutio for 2 usig a approach as i [5] that iterates betwee the schedulig ad power allocatio steps. I the schedulig step, power allocatio is assumed to be fixed. I the power allocatio step, schedulig is assumed to be fixed. B. Proportioally Fair Schedulig Proportioally fair schedulig [] is widely used for wireless systems. The mai idea is to use a schedulig policy which is a fuctio of the curret average rates of the users. For sigle-cell systems, a proportioally fair scheduler selects the user k =arg R k 3 R k i each time epoch, where R k is the user s requested rate if it were scheduled, ad R k is a expoetially weighted average rate for the kth user. This schedulig policy imizes log utility, because Rk ca be thought of as the margial icrease i the utility fuctio log R k for the kth user. This same idea ca be applied to the multiuser settig, as commoly doe i the etwork utility imizatio literature e.g. [2], [6]. I the multiuser case, the istataeous rates for all users i all cells form a capacity regio tradeoffs betwee the idividual rates withi the regio are possible. The idea of proportioal fair schedulig ca also be applied to the multicell settig i the dowlik. The key observatio that allows this to be doe is that i the dowlik, the iterferece produced by each base-statio is a fuctio of the trasmit power spectral desity oly ad is idepedet of the user assigmet at the base-statio. Thus, if the power allocatio P is fixed, user schedulig ca be doe idepedetly o a per-cell basis without affectig the iterferece level elsewhere i the etwork. For a give cell l, i order to imize the margial icrease i the proportioal fairess utility k T k log R k,the user schedulig algorithm should imize the weighted rate sum with weights set as the derivative of the utility fuctio: Tk R k. 4 R k k Sice R k is the sum of bit rates across the frequecy toes, the above imizatio decouples i a toe-by-toe basis. Now, because oly oe user ca be active i a give toe, the user schedulig algorithm should assig user k i each toe as f D l, = arg { k } T k P log + h llk 2 R k Γσ 2 + j l P D,j. 5 h jlk 2 I other words, the scheduler simply chooses k i each toe such that the weighted istataeous rate is imized. The Tk weights are computed as with R k R k = α R k + αr k 6 where < α < is a forgettig factor, ad the istataeous rate R k is computed from the fixed power spectrum allocatio as i 2. Note that the observatio that the iterferece level i the etwork is idepedet of the user schedule is true for the dowlik oly, ad is ot true for the uplik. Thus, applyig proportioally fair schedulig to the uplik requires coordiatio across the cells. To avoid excessive coordiatio, this paper proposes a heuristic approach that uses the same schedulig policy for both uplik ad dowlik, i.e. f U l, =f D l,. 7 This clearly is a suboptimal choice, but it ca be roughly justified for TDD systems where uplik ad dowlik chaels are reciprocals of each other usig a fact kow as uplikdowlik duality, i.e. uder the same sum power costrait, the rate regios of the uplik ad dowlik are the same. Although the sum power costrait does ot exactly correspod to the problem setup here, system-level simulatio idicates that this approach is reasoable. C. Power Spectrum Adaptatio The power spectrum adaptatio step assumes a fixed user schedule ad fids the optimal trasmit power spectrum i both uplik ad dowlik. The objective is agai proportioal fairess, thus the imizatio of the margial icrease i l,k T k log R k is equivalet to Tk R k 8 R l,k k for the dowlik, ad likewise for the uplik. The above step trasforms the problem ito a weighted rate sum imizatio problem with weights set as w k = T k R k 9 These weights automatically adapt to the chael coditio ad the user schedulig; they provide a atural way of balacig the competig rate requiremets from differet users. The trasformatio of the log-utility imizatio to a weighted rate-sum imizatio is crucial, as it allows the optimizatio problem to be decoupled o a toe-by-toe basis usig a dual optimizatio approach [2], [3]. More specifically,
for the dowlik, by dualizig with respect to the total power costrait, the weighted rate-sum imizatio is log l,k w k : k=f Dl, P + h llk 2 Γσ 2 + j l P D,j λ P h jlk 2 l s.t. P SD l,. which decouples ito N idepedet optimizatio problems, oe per each toe =,,N: P w k log + h llk 2 Γσ 2 + l j l P D,j h jlk 2 λ P s.t. P S D l. where k = f D l,. This per-toe problem has L variables, istead of the NL variables as the case for. Thus, it is much more maageable. Note that the purpose of λ is to esure that the power costrait is satisfied. A appropriate λ D,L ca be foud usig a subgradiet approach. For the remaiig of this sectio, we focus o the umerical solutio to for fixed λ. KKT Method: The objective i is a well kow ocovex fuctio for which fidig the global optimum solutio is believed to be difficult. This paper focuses o iterative approaches to achieve at least a local optimum solutio. Our first idea is to look at its Karush-Kuh-Tucker KKT coditio, i.e. take the derivative of the objective fuctio with respect to P ad set it to zero: w k h llk 2 P h llk 2 +Γσ 2 + j l P D,j h jlk 2 = t D,jl + λ, j l 2 with k = f D l,, forl =,,L,where t D,jl = w Γ h ljk 2 SINR D,j 2 D,jk PD,j h jjk 2 +SINR, 3 D,j ad SINR PD,j D,j = h jjk 2 Γσ 2 + i j P D,i h ijk 2, 4 with k = f D j,. The KKT coditio 2 is essetially a water-fillig coditio if the terms t D,jl are held fixed. I this case, 2 gives the followig power update equatio, which we call the KKT method : see also [5] P,ew = [ w k j l t D,jl + λ Γσ2 + j l P ] D,j S h jlk 2 D h llk 2 5 where k = f D l,, ad the otatio [x] b a deotes x upper bouded above by b ad lower bouded below by a. The secod term i the right-had side of 5 is the effective combied dowlik oise ad iterferece i the th toe of the lth base-statio, which ca be measured at the remote termial locally. Thus, to compute 5, the base-statio oly has to kow t D,jl. I this paper, we propose to pass t D,jl as messages from eighborig base-statios. I this case, P,ew ca be effectively computed i a iterative process. Note that the computatio of t D,jl requires ot oly the proportioal fairess weights, the trasmit power ad the SINR, but also the ratio of the direct ad the iterferig chael gais, which has to be estimated i the iitializatio phase. The terms t D,jl have a pricig iterpretatio, which comes from the fact that t D,jl is the derivative of the jth basestatio s data rate with respect to the lth base-statio s power, weighted by the proportioal fairess variable, i.e. w D,jk R D,jk / P,wherek = f D j,. It summarizes the iterferig effect of P o the jth eighbor. For practical implemetatio, the update accordig to 5 may be too aggressive, ad it may lead to o-covergece. We propose a damped iteratio where the ext iteratio of P is set as follows i a db scale: log P [κ +]= γ log P,ew+ γ log P[κ] 6 where the idex κ deotes the iteratio umber, ad <γ<. I practice, γ =.5 is foud to work well. The implemetatio of the algorithm depeds critically o the availability of the pricig messages t D,jl. As messages are ofte updated asychroously, computig the sum may icur delay. Observe that sice oly the sum of t D,jl eters the computatio, it is sesible to approximate the sum by j l {t D,jl }. To compesate for the fact that imum is strictly less tha the sum, we propose to adjust the imum by a costat factor c, i which case the update becomes P,ew [ = w k c j l {t D,jl } + λ Γσ2 + j l P ] D,j S h jlk 2 D h llk 2 7 which we call the -price KKT method. I practice, c =2 is foud to work well. To summarize, to solve, we propose to start with the curret power allocatios {PD,,P D,2, P D,L }, ad update the power accordig to 6 ad 5 or 7. The process iterates util covergece or util a imum umber of iteratios is reached. 2 Newto s Method: We ow propose a secod method for solvig which has a faster covergece speed tha the KKT method. The idea is to do a distributed Newto s search directly o the objective fuctio of. Let rk + =log P h llk 2 Γσ 2 + j l P D,j, 8 h jlk 2
where k = f D l,. The, r k P rd,jk P ad 2 rk P = 2 = P + = Γ h ljk 2 P D,j h jjk 2 2 + P SINR SINR, 9 SINR D,j 2 +SINR, 2 D,j 2 2 2 rd,jk P = Γ h 2 2 SINR ljk D,j 3 2 + SINR D,j 2 PD,j h jjk 2 2 + SINR 22 D,j 2 where agai k = f D l,, k = f D j,, adj l. The idea is to improve the objective fuctio gpd,,,p D,L = w k rk λ P 23 l by icremetig the trasmit power PD,,,P D,L i a Newto s directio. The Newto s directio is [ΔP D,,, ΔP D,L ]= 2 g g. 24 I practice, ivertig the Hessia matrix 2 g is computatioally expesive. Oe trick [7] is to igore the off-diagoal terms of the Hessia, ad to ivert the diagoal terms oly, i.e. ΔP = g l 2. 25 g ll However, the above method works oly if the objective fuctio g is cocave, i which case 2 g ll is egative, ad ΔP always icreases i the directio of the gradiet g l. As the objective fuctio of is ot cocave, the ΔP above does ot ecessarily give a icremet directio see e.g. [8]. For the proposed system, we modify the search directio as follows: ΔP = g l 2 g ll. 26 This heuristics works very well i practice. Now, the lth elemet of the gradiet vector is: + g l = w k P SINR t D,jl λ. 27 j l Likewise, the lth diagoal term of the Hessia matrix is: 2 g ll = w 2 k 2 + + P SINR j l w D,jk Γ h ljk 2 2 PD,j h jjk 2 SINR D,j 3 2 + SINR D,j + SINR 28 D,j 2 Substitutig 27-28 ito 26 gives the Newto s method. Note that i order to implemet the above Newto s method i a distributed fashio, the base-statios eed to pass ot oly the pricig messages t D,jl i 27, but also the additioal terms Base-to-base 2.8km.4km Distace UL DL UL DL Fixed PSD 25 29 37 42 KKT method 85 8 228 227 KKT -price 79 77 25 27 Newto s method 87 83 229 228 Newto -price 79 78 26 29 Improvemet 43% 38% 58% 54% TABLE I SUM RATE IN MBPS OVER 9 CELLS WITH 4 USERS PER CELL AT THE CELL EDGE WITH PROPORTIONAL FAIRNESS. i 28. To facilitate distributed implemetatio with messages t D,jl oly, we propose a further modificatio of the Hessia that icludes the first term of 28 oly. Fially, as i KKT method, we replace j l t D,jl by c j l{t D,jl }.These modificatios lead to the followig update equatio: wk ΔP = P + SINR c j l {t D,jl } λ w k + P 2 SINR 29 which gives the -price Newto method. Agai, c =2is foud to work well. To summarize, i Newto s method, each base-statio iteratively updates its power allocatio accordig to 2 ] S D Pκ += [ Pκ+ΔP. 3 where ΔP is computed either by 26, 27 ad 28, or by 29 util covergece. D. Overall Iterative Algorithm The power spectrum adaptatio ad the user schedulig steps are iterated util covergece. Each step is odecreasig i the optimizatio objective. Thus, the iteratios coverge. III. SIMULATION The performace of the proposed algorithm is evaluated o a wireless cellular etwork with 9 cells ad 4 users per cell over MHz badwidth. The cell radius is chose to be 2.8km correspodig to a typical WiMax or LTE settig ad.4km correspodig to a overlapped deploymet. Frequecy selective chaels with both log-ormal shadowig ad Rayleigh fadig are simulated. Both the base-statios ad the mobile users have a imum PSD costrait of 27dBm/Hz. For simplicity, o imum power costrait is imposed. Perfect chael estimatio is assumed. For evaluatio purposes, the chaels are assumed fixed throughout. Dyamic power spectrum optimizatio is expected to brig the largest beefit to cell-edge users. To illustrate the beefit for cell-edge users, the mobile users i this simulatio are distributed o purpose close to the cell edge at distaces betwee.8r to.9r, wherer is the cell radius.
3 3 Dowlik sum rate per cell Mbps 25 2 5 5 Dowlik sum rate per cell Mbps 25 2 5 5 5 5 iteratios 5 5 iteratios Fig. 2. Covergece of dowlik sum rates i each of the 9 cells usig KKT method both with -price simplificatio. Fig. 3. Covergece of dowlik sum rates i each of the 9 cells usig Netwo s method both with -price simplificatio. Table I shows the achieved sum rates across all 9 cells with 4 cell-edge users per cell with ad without the dyamic power spectrum adaptatio. Without dyamic power spectrum adaptatio, each trasmitter simply trasmits at the imum fixed PSD level to imize its ow trasmissio rate. Table I shows that all four dyamic power spectrum optimizatio algorithms proposed i this paper produce sigificat rate gais for both uplik ad dowlik as compared to the fixed PSD case. The rate improvemet is i the 4%-55% rage o average. Note that these figures pertai specifically to celledge users. Whe users are uiformly placed i the cell, the improvemet i the average rate would be about 5%-2%. Table I also shows that the performace gai is more proouced whe base-statios are closer to each other, as expected. I additio, the -price simplificatio produces ear-optimum performace, losig about 5% i sum rate. Figs. 2-3 illustrate the covergece behaviors of the price KKT ad Newto s methods. The per-cell sum rates for each of the 9 cells are plotted agaist the iteratio umber. Each iteratio step here cosists of either a power spectrum adaptatio or a user schedulig step. Up to sub-iteratios are performed per each power spectrum adaptatio step. The proportioal fairess weights are also updated at the same time. We observe that the Newto s method coverges faster, withi about 5 iteratios as compared to about 8 iteratios for the KKT method. Note that the proportioal fairess criterio esures a fair rate allocatio across all the cells. The miimum per-cell sum rate is well above Mbps. REFERENCES [] E. F. Chapoiere, P. J. Black, J. M. Holtzma, ad D. N. C. Tse, Trasmitter directed, multiple receiver system usig path diversity to equitably imize throughput, U.S. Patet 6,449,49, filed July 999. [2] R. Cedrillo, W. Yu, M. Mooe, J. Verlide, ad T. Bostoe, Optimal multiuser spectrum balacig for digital subscriber lies, IEEE Tras. Commu., vol. 54, o. 5, pp. 922 933, May 26. [3] W. Yu ad R. Lui, Dual methods for ocovex spectrum optimizatio of multicarrier systems, IEEE Tras. Commu., vol. 54, o. 6, pp. 3 322, Jue 26. [4] J. Papadriopoulos ad J. S. Evas, Low-complexity distributed algorithms for spectrum balacig i multi-user DSL etwoks, i IEEE Iter. Cof. Commu. ICC, Istabul, Turkey, 26. [5] W. Yu, Multiuser water-fillig i the presece of crosstalk, i Iform. Theory ad Appl. Workshop, Sa Diego, U.S.A., 27. [6] P. Tsiaflakis, M. Diehl, ad M. Mooe, Distributed spectrum maagemet algorithms for multi-user DSL etworks, IEEE Tras. Sigal Processig, vol. 56, o., pp. 4825 4843, Oct. 28. [7] J. Huag, R. A. Berry, ad M. L. Hoig, Distributed iterferece compesatio for wireless etworks, IEEE J. Select. Areas Commu., vol. 24, o. 5, May 26. [8] J. Yua, Optimizatio Techiques for Wireless Networks, Ph.D. thesis, Uiversity of Toroto, 27. [9] J. Yua ad W. Yu, Distributed cross-layer optimizatio of wireless sesor etworks: A game theoretic approach, i Global Telecommuicatios Cof. GLOBECOM, Sa Fracisco, U.S.A., 26. [] C. Shi, R. A. Berry, ad M. L. Hoig, Distributed iterferece pricig for OFDM wireless etworks with o-separable utilities, i Coferece Ifo. Sciece Sys. CISS, Mar. 28, pp. 755 76. [] F. Wag, M. Kruz, ad S. Cui, Price-based spectrum maagemet i cogitive radio etworks, IEEE J. Sel. Top. Sigal Processig, vol., o. 2, pp. 74 87, Feb. 28. [2] J. Huag, V. G. Subramaia, R. Agrawal, ad R. Berry, Dowlik schedulig ad resource allocatio for OFDM systems, i Coferece Ifo. Sciece Sys. CISS, Mar. 26, pp. 27 279. [3] J. Huag, V. G. Subramaia, R. Agrawal, ad R. Berry, Joit schedulig ad resource allocatio i uplik OFDM systems for broadbad wireless access etworks, IEEE J. Sel. Top. Sigal Processig, vol. 27, o. 2, pp. 226 234, Feb. 29. [4] S. G. Kiai ad D. Gesbert, Optimal ad distributed schedulig for multicell capacity imizatio, IEEE Tras. Wireless Commu., vol. 7, o., pp. 288 297, Ja. 28. [5] L. Veturio, N. Prasad, ad X. Wag, Coordiated schedulig ad power allocatio i dowlik multicell OFDMA etworks, IEEE Tras. Veh. Techol., vol. 6, o. 58, pp. 2835 2848, July 29. [6] A. L. Stolyar ad H. Viswaatha, Self-orgaizig dyamic fractioal frequecy reuse for best-effort traffic through distributed iter-cell coordiatio, i INFOCOM, Apr. 29. [7] D. Bertsekas, E. Gafi, ad R. Gallager, Secod derivative algorithms for miimum delay distributed routig i etworks, IEEE Tras. Commu., vol. 32, o. 8, pp. 9 99, Aug. 984. [8] P. E. Gill ad W. Murray, Newto-type methods for ucostraied ad liearly costraied optimizatio, Math. Prog., vol. 7, pp. 3 35, 974.