Downlink Throughput Maximization: TDMA versus CDMA

Transcription

1 00 Conference on Informaton Scences and Systems, The Johns Hopkns Unversty, March, 00 Downlnk Throughput Maxmzaton: TDMA versus CDMA Changyoon Oh and Ayln Yener Department of Electrcal Engneerng The Pennsylvana State Unversty Unversty Park, PA 0 e-mal: changyoon@psu.edu yener@ee.psu.edu Abstract We consder the downlnk throughput maxmzaton problem for nterference lmted multuser systems. Our goal s to characterze the optmum base staton transmsson polcy (TDMA or CDMA), and fnd the correspondng power allocaton. We model the nterference by the ad of the orthogonalty factor and determne the transmsson strategy for a range of the values of the orthogonalty factor, and users channel gans, so as to maxmze the system throughput, subject to a total power constrant. Although the resultng optmzaton problem may turn out to be non-convex dependng on the value of the orthogonalty factor, we show that valuable observatons regardng the optmum soluton can be obtaned based on the propertes of the problem. In partcular, we propose an exact and a near-exact algorthm to determne whether TDMA s the optmum strategy, or smultaneous users (CDMA) should be transmtted to. I. Introducton Wreless servces are becomng more data centrc and often requre hgher data rate for downlnk communcatons [, ]. Data servces typcally requre hgh relablty but are delay tolerant. The delay tolerance can be exploted by effectvely schedulng transmssons to users n favorable channel condtons to ncrease the overall system throughput []. In ths paper, we consder the power allocaton problem for the downlnk of delay tolerant CDMA systems, by takng the channels of the users nto account. For the CDMA downlnk, typcally, orthogonal codes are used to communcate to each user []. However, due to multpath fadng, orthogonalty between codes s not preserved at the moble sde. In such cases, a parameter defned as the orthogonalty factor s often used to desgnate the level of multpath fadng, representng the degree of nterference from other users []. We focus on ths scenaro, where the orthogonalty between the codes s not mantaned. Specfcally, our am s to fnd the optmum transmsson polcy,.e., whch user(s) wll be transmtted to and the power allocaton n each tme slot so as to maxmze the total system utlty, subject to a total power constrant at the base staton. The total system utlty s defned smply as the sum of the ndvdual utltes. The ndvdual utlty we consder n ths paper s the transmsson rate to a user, whch s an ncreasng functon of the user s transmt power. Utlty based power control for data servces has prevously been consdered n [, ], for the uplnk, where the qualty of servce (QoS) of each user, e.g. sgnal to nterference rato (SIR), s defned as the utlty. In references [, 9], one-at-a-tme transmsson for downlnk throughput maxmzaton s consdered for delay tolerant servce, and the optmalty of ths soluton s shown. All avalable power from the base staton s transmtted for the sake of ncreased throughput, and no compensaton for the ntracell nterference s needed [, 9]. To overcome the possble unfarness that may result from the one-at-a-tme transmsson strategy, reference [0] mposes a mnmum servce rate requrement for each user, and consders the throughput maxmzaton problem for downlnk multrate CDMA system where the multrate s acheved by ether multcode or orthogonal varable spreadng factor (OVSF). The optmalty of greedy power allocaton s shown n ths scenaro [0]. Ths polcy s ntutvely pleasng when the user rate and power have a lnear relatonshp such as n a CDMA system wth varable processng gan or multcodes. Reference [0] also provdes numercal results whch show the effect of orthogonalty factor to the system performance. Agrawal et al. n [] consder the optmzaton of the sum of the ndvdual weghted throughput values n the downlnk of a multrate CDMA system wth orthogonal codes. It s assumed that the orthogonalty s preserved at the recever sde. The resultng problem s a convex program, whch for the specal case of the equal weght scenaro,.e., throughput maxmzaton, produces one-at-a-tme transmsson as the optmum polcy []. In contrast wth [], we consder the downlnk of an nterference lmted CDMA system and determne the optmum transmsson strategy. The key observaton s that the optmum polcy s a functon of the orthogonalty factor. Specfcally, gven the channel realzaton of the users, we determne whether the TDMA-mode,.e., the base staton transmts to one user at a tme, or the CDMA-mode,.e., smultaneous transmssons, should be chosen. If the CDMA-mode s optmum, that s, f multple nterferng users should be served by the base staton smultaneously, then the correspondng power allocaton s also found. The paper s organzed as follows. Secton II descrbes the system model as well as the problem formulaton and the performance metrc. Optmum power allocaton s consdered n secton III. Secton IV provdes numercal examples that support our analyss, and Secton V summarzes the results and concludes the paper. II. System Model and Problem Formulaton We consder a sngle cell CDMA system wth K users. The base staton has a maxmum power budget, P T. Ths power budget s assgned to each user such that p PT, where p s the allocated power for user. Fgure depcts the model of the CDMA downlnk. An orthogonal code s assgned to each user and s used to modulate the sgnal transmtted to that user. However, due to multpath fadng, orthogonalty between the codes (users) s lost at the recever sde. The

2 User User User K g g g K p p p K Base Staton Orthogonal codes code code code K where we assume that g > g >... > g K, such that user wth the lower ndex has a hgher channel gan. We note that the actual outcome of the above optmzaton problem s power vector. The optmum transmsson strategy s mplctly ncluded n the optmum power vector n that the users that belong to the subset of users that the base staton transmts to, end up wth non-zero power, and the rest wth zero power. If the TDMA-mode turns out to be optmum, the power allocaton s such that transmsson power to only one user s postve. If the CDMA-mode turns out to be optmum, the power allocaton s such that transmtted power values to multple users are postve. We frst note that any power vector p wth p < P T can not be the optmum power vector. Defne a power vector p wth p = βp (β > ), =,, K such that βp = P T. It s easy to see that p ncreases all ndvdual utltes and hence the system utlty as compared to p. Thus, () can be rewrtten as: Fg. : Downlnk System model degree of nonorthogonalty s descrbed by the orthogonalty factor, (0 ) []. Under the assumpton of ndependent dentcal channels for all users, t s reasonable to work wth the same average orthogonalty factor for all users. We assume that each user employs a matched flter recever. The sgnal to nterference rato at the recever of user s expressed as p g SIR = j pjg + I () where g denotes channel gan of user. I denotes the sum of the thermal nose and the ntercell nterference. We defne the utlty of user as follows: p g U (p) = log( + k ) () pjg + I j where log denotes the natural logarthm. We note that the ndvdual utlty s a functon of the power vector p = [p, p,..., p K]. The utlty s an achevable rate for user, specfcally by treatng the nterference as nose [9]. It s assumed that adaptve modulaton s employed to enable the rate determned for each user []. The factor k captures the product of the adaptve modulaton ndex and the fxed processng gan. Ths resembles the structure of HSDPA where multrate s acheved by multcode and adaptve modulaton. We should also note that, n a practcal system employng M- ary modulaton, the transmsson rate (utlty) s a dscrete quantty. For smplcty of analyss, and smlar to prevous work [, ], we wll assume contnuous rate values. We wll examne the effect of ths assumpton n the numercal results. The problem we consder s to determne the optmum allocaton of the total transmt power from the base staton to the users so as to maxmze the overall system utlty,.e., sum of ndvdual utltes. Formally, the optmzaton problem s formulated as: log( + k p g max p s.t. j p jg +I ) () p PT, p 0, =,, K () max p s.t. log( + k p ) () (p T p )+ g I p = PT, p 0, =,, K () Observe that the utlty of user s a functon of the power for user only,.e., U (p) = U (p ). Also note that, although the utlty s concave n SIR, t need not be concave n power. Therefore, the optmzaton problem s n general not a convex program. Fgure emphaszes ths pont by plottng the ndvdual utltes for users wth dfferent channel gans. III. Optmum Transmt Strategy and Power Allocaton We examne the throughput maxmzaton problem n terms of the ndvdual utlty and the system utlty, whch we term the user centrc approach, and the system-wde approach, respectvely. We frst consder the user centrc approach, and we observe the behavor of the ndvdual utlty n power by utlzng the dervatve of the ndvdual utlty functon. Our observatons motvate us to consder the system-wde approach, n whch we nvestgate the system utlty n terms of the best user s power. The system-wde approach s useful to characterze the system utlty when the utltes of all users are not concave n power. Fgure depcts such a case. III.A The User Centrc Approach Consder the utlty functon of user and ts frst and second dervatves n terms of p, the power of user. The frst dervatve of the utlty functon s postve,.e., U (p ) s an ncreasng functon of p. We are nterested n whether the utlty U (p ) s an ncreasng concave or convex because ths plays a crucal role n characterzng the behavor of the utlty n power. Whether U (p ) s concave ( U (p ) < 0 ) or convex p ( U (p ) p > 0) depends on the sgn(a ) of the numerator of U (p ). A p s gven by A = ( k)((p T p ) + I g ) + ((p T p ) + I g + kp ). By lettng I t = (p T p ) + I g, we have A = ( k)i t + (I t + kp ). Thus, we need to have p I t > k ()

3 . λ mn 0 9. λ max p s wthn concave regon p n p s wthn convex regon λ max gan λ mn Utlty. gan Utlty gan. λ max λ max gan gan gan 0. gan gan Transmt power[watts] Transmt power[watts] Fg. : gan > gan > gan > gan. An example where ndvdual utlty functons for all users are concave. Fg. : gan > gan > gan > gan. An example where ndvdual utlty functons for some users are not concave. for U (p ) to be an ncreasng convex functon of p and p I t < k for U (p ) to be an ncreasng concave functon of p. Clearly, by observng U (p ) p p =P T, we can dentfy the behavor of functon U (p ). When U (p ) p p =P T < 0, U (p ) s an ncreasng concave functon n 0 p P T as n Fgure. When p =P T > 0, U (p ) s an ncreasng concave functon U (p ) p n 0 p p n, whle t s ncreasng convex functon n p n p P T, where p n denotes the nflecton pont n U (p ) as n U (p ) shown n Fgure. Fgure shows the ndvdual utltes of dfferent channel gans. In ths example, the ndvdual utltes of all users are concave n power. It s clear that when the ndvdual utlty s concave n power, among multple users, rather than allocatng the entre power P T to one user, sharng the power P T wll yeld a hgher total utlty. In ths case, the optmzaton problem n () s a convex program. However, ths s no longer the case when the ndvdual utltes of some users are not concave, as depcted n Fgure. In ths case, we need to have a closer look at the components that contrbute to the system utlty. The followng termnology s used extensvely n the sequel. U (p ) s convex f U (p ) > 0 n 0 p p P T U (p ) s concave/convex f U (p ) p p =0< 0 and U (p ) p p =P T > 0 U (p ) s concave f U (p ) p p =P T < 0 p s sad to be wthn the concave regon of U (p ), f U (p ) p p =p < 0 p s sad to be wthn the convex regon of U (p ), f U (p ) p p =p > 0 () Fgure depcts two possble power regons of user (concave/convex user). Followng from the above, we defne λ mn = U (p ) p p =p whenu(p)s concave/convex, (9) λ mn = λ max = and, and U (p ) p p =p = 0 ; U (p ) p p =pt whenu(p)s concave (0) U (p ) p p =0 () p (λ) = arg p ( U (p ) p p =p (λ) = λ) whenλ < λ max() = 0 whenλ > λ max() where λ mn mples the mnmum dervatve value of utlty of user, whle λ max denotes the maxmum dervatve value of utlty of user. Fgures and depct λ mn and λ max. Gven λ, power of user p (λ) s obtaned by optmzng the ndvdual utlty U (p ) over p. The optmum polcy s a functon of the orthogonalty factor as well as users channel gans n general. Below we state the propostons that lead to characterzng the optmum polces. The proofs of all the propostons n ths paper can be found n []. Our frst observaton states that when the orthogonalty factor s larger than a certan threshold, TDMA-mode s always optmum, regardless of users channel gans: Proposton III.: If k, TDMA-mode wth p = PT yelds the optmum system utlty. Whle, t s clear that the TDMA-mode s optmum when k, and no further power allocaton s necessary, when k, we need to characterze the optmum polcy whch conssts of the transmsson strategy,.e., TDMA-mode or CDMA-mode, and the correspondng power allocaton. Frst, we observe the followng. Proposton III.: The optmum power allocaton p = (p, p,..., p K) s such that p > p >... > p K when p > 0.

4 When the ndvdual utltes of multple users are concave/convex as n Fgure, we have the followng observaton. Proposton III.: Suppose p s the optmum power allocaton. Then, at most one user s power (the user wth the best channel gan) s wthn the convex regon of the utlty functon. From the precedng observatons, we conclude that the optmum power allocaton p = [p, p,..., p K] s one of two cases n terms of the best user,.e., p can be ether wthn the concave regon or wthn the convex regon of U (p ). Ths observaton motvates our approach to focus on the the system utlty n terms of the best user. III.B System-Wde Approach We defne the system utlty n terms of the best user,.e., p : J(p ) = U (p ) + Z(p ) () Z(p ) = max[ j= Uj(pj)] j= p j=p T p. () We note that J(p ) expends all avalable power, P T, p for user and P T p for the rest of the users. Our am s to fnd the maxmzer of J(p ) over 0 p P T. Proposton guarantees that U j(p j) (j ) s always concave at optmum power allocaton. Hence, Z(p ) can be maxmzed wth the power constrant pj(λ) = PT p where λ s j= the optmum Lagrange multpler. We note that no power s assgned to the user when λ max < λ. Dependng on user s channel condton, U (p ) can be ether concave or concave/convex. When U (p ) s concave, the resultng optmzaton problem s convex and the optmum power allocaton s such that pj(λ) = PT. If U(p) s concave/convex j= as n Fgure, the optmum power allocaton s one of three possbltes. Observaton III.: If U (p ) s concave/convex as n Fgure, the optmum power allocaton s one of three local optmum solutons: () p s wthn the concave regon wth 0 p < p (λ mn), () p s wthn the convex regon wth p (λ mn) < p < P T, or () p = P T. Transmsson strategy for the frst two cases s CDMA, whle the thrd case (p = P T) s TDMA. Gven the fact that when U (p ) s concave/convex, the resultng optmzaton problem s non-convex, the three possble cases descrbed above need to be consdered n detal to maxmze J(p ). Consder p(λ),.e., the sum of the powers transmtted to all users, gven λ. Observng whether p(λ) < PT, or p(λ) > PT provdes us wth the nformaton whether J(p ) s ncreasng or decreasng at p = p (λ): Proposton III.: If p(λ) < PT for a gven λ, J(p) s an ncreasng functon at p = p (λ). Proposton III.: If p(λ) > PT for a gven λ, J(p) s a decreasng functon at p = p (λ). From the prevous two propostons, by observng K p(λ), we can always determne whether J(p) s an ncreasng functon or decreasng functon at p = p (λ). For nstance, f p(λ mn) > P T, J(p ) s a decreasng functon at p = p (λ mn) and the optmum power allocaton s one of three possbltes as descrbed by observaton III.. If p(λ mn) < P T, ths mples p(λ) < PT for λ > λ mn where p (λ) s wthn the concave regon, snce p (λ mn) > p (λ) when U (p ) s ncreasng concave. In ths case, J(p ) s an ncreasng functon when p s wthn the concave regon,.e., 0 p p (λ mn). Hence, no p value wthn the concave regon can be optmum. The optmum power allocaton dctates that p be ether p (λ mn) < p < P T, or P T. Fndng the local optmum soluton wthn the convex regon of U (p ) requres ntensve computatonal complexty. If we fnd a way to dentfy p = P T,.e., the optmalty of TDMA, we can reduce the complexty of dentfyng the optmum polcy. Consder the case of p(λ) < PT for λ where p(λ) s wthn the convex regon of U (p ). In ths case, J(p ) s an ncreasng functon for 0 p < P T. Observaton III.: If p(λ) < PT for λ where p(λ) s wthn the convex regon of U (p ), then p = P T s optmum power allocaton. Observaton III. guarantees the optmalty of TDMA, but stll requres consderable computatonal complexty. Notce that, as descrbed above, when p(λ mn) < P T, the optmum power allocaton does not fall wthn the concave regon of U (p ). By adoptng TDMA as the optmum polcy whenever p(λ mn) < P T, we can sgnfcantly reduce computatonal complexty n fndng local optmum power allocaton n the convex regon of U (p ). In the next secton, we wll term ths short-cut, the near-exact algorthm. The numercal results n Secton IV valdate the accuracy of near-exact algorthm. III.C Algorthms for Transmt Strategy and Power Allocaton Followng the precedng dscusson n Secton III, we conclude that the optmalty of TDMA s determned by checkng p(λ) < PT for all λ where p(λ) s wthn the convex regon of U (p ). Instead of ths potentally computatonally ntensve search, we propose to smply check K p(λ mn) < P T. Clearly, the performance gap between the optmum soluton and the near-optmum soluton utlzng the nearexact algorthm of determnng TDMA optmalty results from the case when the optmum power allocaton s such that p (λ mn) < p < P T. Ths s because the near-exact algorthm decdes that TDMA-mode,.e., p = P T s optmum only f p(λ mn) < P T, regardless of max p J(p ) for p(λ mn) < p P T. In our numercal results, we have observed that such cases,.e., the optmum power allocaton s such that p (λ mn) < p < P T, are rare and that n most cases, the optmum power allocaton s ether 0 < p < p (λ mn) or p = P T. The power allocaton step s done accordng to the mode that s desgnated to be the optmum strategy. When TDMA-mode s optmum, no further power allocaton s necessary. When CDMA-mode s optmum, the optmum power allocaton needs to be found. We should note that observaton III. s a suffcent condton for TDMA optmalty. Hence, even though j= pj(λ mn) > P T, TDMA can be optmum. However, as the orthogonalty factor ncreases, ths condton accounts for almost all of the cases where TDMA s optmum, as demonstrated by numercal results n Secton IV. The followng summarzes the proposed algorthm to maxmze the system utlty, whch we term the system utlty maxmzer, A-SUM.

5 9 9 =0. =0. =0. =0. =0. =0. System Utlty System Utlty (Dscrete) number of users n the system number of users n the system Fg. : System Utlty versus Number of users n the system. Fg. : System Utlty versus Number of users n the system. A-SUM: STEP. If k, then TDMA mode s optmum, STOP. STEP. Fnd λ max for =,, K, and λ mn. STEP. (convex regon) If j= pj(λ mn) < P T, then, TDMA s optmum, STOP. STEP. (concave regon) Fnd the power allocaton n the concave regon J (p (λ)) = max(j(p(λ))), 0 p (λ) p (λ mn) va A-PACR. STEP. Choose max(j (p (λ)), J(P T)). The near-exactness of the algorthm arses from the fact that n STEP, we smply check the suffcent condton for optmalty of TDMA, nstead of solvng for max p J(p ), p (λ mn) < p < P T. The optmum power allocaton (STEP ) entals selectng the users to whch the base staton transmts wth postve power. In other words, f zero power s assgned to one user, that mples that user s not scheduled for transmsson. The algorthm to determne the power allocaton n the concave regon (A-PACR) s summarzed below. It s assumed n A- PACR that j= pj(λ mn) > P T. If ths s not the case, there s no soluton n the concave regon, and A-PACR smply determnes TDMA,.e., p = P T. A-PACR: STEP. Fnd ˆk = arg k max k j= pj(λk max), subject to k j= pj(λk max) P T, k =,, K. STEP. Fnd p j(λ) ( j ˆk) such that ˆk j= pj(λ) = P T where λ mn < λ < λˆk max. The optmum number of users to whch the base staton transmts s selected n STEP. Then, optmum power values are allocated n STEP. Note that a numercal method, e.g., bsecton [] can be used to determne the value of λ. IV. Numercal Results We provde smulaton results for a range of the orthogonalty factor values. Maxmum total power at the base staton s P T = 0 watts. Total nose and ntercell nterference s I = 0 watts. Factor k n the utlty functon s k =.. Thus, when 0., the ndvdual utltes of all users are convex n power, and the TDMA-mode s optmum, regardless of users channel gans. Channel gan from a base staton to moble s modeled as g = r where d d denotes the dstance between moble and the base staton, whch s unformly dstrbuted between 00m and 000m. r s the realzaton of the lognormal fadng coeffcent wth varance db. We have frst examned the accuracy of the near-exact algorthm determnng TDMA optmalty for 0,000 channel gan realzatons. The rato of the near exact algorthm whch correctly determnes the TDMA optmalty n percentage are 9.%, 9.%, 99.9% for = 0., 0., 0., respectvely. As ncreases, as expected, the accuracy ncreases,.e., the case when the optmum power allocaton s such that p (λ mn) p < P T dsappears. For example, when =0., the near-exact algorthm almost always correctly determnes the optmalty of TDMA. Fgure shows the system utlty, gven and the number of users K n the system. The system utlty s averaged over 0,000 channel gan realzatons. The optmum polcy selects the number of users and the power levels for the scheduled users. In general, not all users n the system are smultaneously scheduled. When = 0.,.e., a low orthogonalty factor, smultaneous transmssons do not result n much nterference between users, and the optmum polcy chooses CDMA-mode. However, f the best user has a much hgher channel gan as compared to the rest, the optmum polcy may end up to allocatng all power to that best user. As ncreases, smultaneous transmssons result n hgher nterference, and TDMA-mode becomes the preferred mode so as to avod the nterference. For example, for > 0., TDMAmode s optmum n most cases. For value of > 0., for example, when = 0.,.e., all ndvdual utltes are convex, and TDMA-mode s always optmum. Fgure shows that the

6 system utlty, when = 0., s lower than the system utlty value for smaller values that facltate smultaneous transmssons. The gap between the system utlty for ( < 0.) and the system utlty for = 0. can be nterpreted as the gan of optmum polcy that results n hybrd CDMA/TDMA over the one that results n TDMA, as a result of the dfference n the orthogonalty factor values. Fgure shows the system utlty where the resultng ndvdual utlty s dscretzed to the nteger value. Wth ths, we attempt to nvestgate the performance of the proposed algorthm n a practcal settng where only dscrete rates are avalable. We observe that dscretzaton does not lead to a sgnfcant loss n system utlty especally for large values. Ths s because for large, TDMA results most often, and the quantzaton loss n utlty s due to one user only. Fgure shows the system utlty gan of CDMA-mode over TDMA-mode. When the orthogonalty factor s low, smultaneous transmssons has a gan over the one-at-a-tme transmsson. The gan ncreases as the number of users n the system ncreases up to a certan lmt. As the orthogonalty factor ncreases, however, we observe that the gan dsappears, as eventually TDMA-mode becomes optmum. V. Concluson In ths paper, we consdered the optmum polcy whch conssts of the user(s) the base staton selects to transmt to, and the power allocaton for total utlty maxmzaton n the downlnk. The ndvdual utlty we consdered s the rate to the user. Dependng on the channel condtons,.e., the orthogonalty factor and the users channel gans, the optmum polcy chooses ether TDMA mode or CDMA mode and the correspondng power allocaton. The hgher the orthogonalty factor, the larger the nterference caused by smultaneous transmssons. Ths causes the TDMA-mode beng optmal when the orthogonalty factor s hgh. In contrast, smultaneous transmssons yeld a hgher overall utlty when the orthogonalty factor s low. We observed that dependng on the channel condtons, the overall system utlty may or may not be a concave functon and care must be gven to characterzng the soluton. We took the approach of examnng the system utlty n terms of the best user, and observed propertes that helped us dentfy the optmal polcy. We presented numercal results to support our analytcal fndngs, n partcular to the beneft of allocatng transmsson to smultaneous users n certan scenaros, and evaluate the frequency of TDMA mode optmalty for dfferent system parameters. References [] M. Andrews, K. Kumaran, K. Ramanan, A.L. Stolyar, R. Vjayakumar and P. Whtng. Provdng Qualty of Servce over Shared Wreless Lnk. IEEE Communcatons Magazne, vol. 9, pages 0-, 00. [] P. Bender, P. Black, M. Grob, R. Padovan, N. Sndhushayana and A. Vterb. CDMA/HDR: a bandwdth-effcent hgh-speed wreless data servce for nomadc users. IEEE Communcatons Magazne, pages 0-, 000. [] X. Lu, E. Chong and N. Shroff. Transmsson Schedulng for Effcent Wreless Utlzaton. IEEE Infocom, 00. [] H. Holma. W-CDMA for UMTS:Rado Access for Thrd Generaton Moble Communcatons. Wley and Sons, 00. [] N.B. Mehta, L.J. Greesten, T.M. Wlls and Z. Kostc. Analyss Results for the Orthogonalty Factor n WCDMA Downlnks. IEEE Transactons on Wreless Communcatons, pages - 9, November, 00. System Utlty 0 9 CDMA(=0.) CDMA(=0.) CDMA(=0.) TDMA mode Number of users n the system Fg. : CDMA-mode gan over TDMA-mode for dfferent value. [] C.U. Saraydar, N.B. Mandayam and D.J. Goodman. Prcng and Power Control n a Multcell Wreless Data Network. IEEE Journal on Selected Areas n Communcatons, pages -9, October 00. [] N. Feng, S.C. Mau and N.B. Mandayam. Prcng and Power Control for Jont Network-Centrc and User-Centrc Rado Resource Management. IEEE Transactons on Communcatons, pages -, September 00. [] A. Bedekar, S. Borst, K. Ramanan, P. Whtng and E. Yeh. Downlnk schedulng n CDMA data networks. IEEE Globecom 999. [9] F. Berggren, S.L. Km, R. Juntt and J. Zander. Jont Power Control and Intra-cell Schedulng of DS-CDMA Non-real tme Data. IEEE Journal on Selected Areas n Communcatons, pages 0-0, 00. [0] M.K. Karakayal, R. Yates and L.V. Razoumov. Throughput maxmzaton on the downlnk of a CDMA system. IEEE WCNC 00. [] R. Agrawal and V. Subramanan and R. A. Berry. Jont Schedulng and Resource Allocaton n CDMA Systems. Submtted, 00. Avalable at rberry/opt.pdf. [] X. Qu and K. Chawla. On the Performance of Adaptve Modulaton n Cellular Systems. IEEE Transactons on Communcatons, pages -9, 999. [] S.J. Oh and K.M. Wasserman. Optmalty of Greedy Power Control and Varable Spreadng Gan n Mult-class CDMA Moble Networks. ACM/IEEE Mobcom, 999. [] S. Ulukus and L.J. Greensten. Throughput maxmzaton n CDMA uplnks usng adaptve spreadng and power control. IEEE ISSSTA, 000. [] C. Oh and A. Yener. Downlnk Throughput Maxmzaton for Interference Lmted Multuser Systems: TDMA versus CDMA. submtted to IEEE Transactons on Wreless Communcatons, February 00. [] D.P. Bertsekas. Nonlnear programmng. Athena Scentfc, 99.