2454 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 7, JULY Changyoon Oh, Member, IEEE, and Aylin Yener, Member, IEEE

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1 25 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO. 7, JULY 2007 Downlnk Throughput Maxmzaton for Interference Lmted Multuser Systems: TDMA versus CDMA Changyoon Oh, Member, IEEE, and Ayln Yener, Member, IEEE Abstract We consder the downlnk throughput maxmzaton problem for nterference lmted multuser systems. Our goal s to characterze the optmum base staton transmsson strategy,.e., whether the base staton transmts to one-user TDMA or multple users CDMA. Specfcally, we am at determnng the optmum number of users to be scheduled and fndng the correspondng power allocaton. We model the nterference by the ad of the orthogonalty factor, and determne the throughput maxmzng transmsson strategy for a range of the values of the orthogonalty factor, and the channel gans, subject to a total power constrant. Although the resultng optmzaton problem may turn out to be non-convex, we show that valuable observatons regardng the structure of the optmum soluton can be obtaned by examnng the performance metrc from an ndvdual user s pont of vew. We propose an exact and a near-exact algorthm to determne whether one-user-transmsson s the optmum strategy, or more than one user should be transmtted to. Numercal results to support our analyss, as well as the modfcatons to the proposed algorthms n the presence of ndvdual power constrants are presented. Index Terms Throughput maxmzaton, schedulng, TDMA, CDMA. I. INTRODUCTION DUE to the scarcty of wreless resources, effcent resource management s crucal for hgh speed wreless communcatons. Power control s one mportant resource management technque [2] []. For voce CDMA servces, the man purpose of power control s to mantan the sgnal to nterference rato SIR to satsfy the mnmum qualty of servce for all voce users constantly. Current and future wreless servces are becomng more data centrc, and often requre hgher data rate for downlnk communcatons [5] [7]. Data servces typcally requre hgher relablty but are delay tolerant. Conventonal power control as descrbed above s not effcent for data servces n that, the transmt power would be wasted to compensate for constant nterference. Instead, Manuscrpt receved July 27, 2005; revsed December 2, 200; accepted March 1, The edtor coordnatng the revew of ths paper and approvng t for publcaton was A. Scaglone. Ths work was supported n part by NSF grant CCF CAREER , and was presented n part at the Conference on Informaton Scences and Systems, CISS 05, Baltmore, MD, March 2005 [1]. C. Oh was wth the Wreless Communcatons and Networkng Laboratory WCAN, Department of Electrcal Engneerng, Pennsylvana State Unversty, Unversty Park, PA 102 USA. He s now wth the Advanced Research Laboratory, Global Standards and Research Team, Telecommuncatons R&D Center, Samsung Electroncs, Seoul, Korea e-mal: changyoon.oh@samsung.com. A. Yener s wth Pennsylvana State Unversty, Unversty Park, PA 102 USA e-mal: yener@ee.psu.edu. Dgtal Object Identfer /TWC /07$25.00 c 2007 IEEE delay tolerance can be exploted by effcently schedulng users n favorable channel condtons to ncrease overall system throughput []. In ths paper, we consder the schedulng and 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 [5]. However, due to multpath fadng, orthogonalty between codes s not preserved at the moble sde. In such cases, orthogonalty factor s often used to represent the level of nterference from other users [9]. Ths s precsely the scenaro we focus on wth the am to fnd the optmum transmsson polcy,.e., the users the base staton wll transmt to and the correspondng transmt power levels so as to maxmze the total system utlty. The total system utlty s defned smply as the sum of the ndvdual utltes. The ndvdual utlty we consder 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 [10] [12], for the uplnk, where the qualty of servce QoS of each user, e.g. sgnal to nterference rato SIR, s defned as the utlty. Power allocaton problem for CDMA data servces s consdered n the uplnk n [13], [1] and the downlnk n [15] [1]. References [13], [1], consder the jontly uplnk power control and spreadng gan allocaton problem for the non-real tme users. The jontly power and spreadng gan control results n the optmum spreadng gan beng nversely proportonal to the user s SIR,.e., the optmum rate s proportonal to the user s SIR. When no mnmum spreadng gan constrant exsts, users are served n the order of decreasng channel gan [13]. Power allocaton problem n the downlnk dffers from that of the uplnk n that a total power constrant s requred n downlnk, whle typcally an ndvdual power constrant s mposed n the uplnk. In references [15], [1], all avalable power from the base staton s transmtted to one user for the sake of ncreased throughput, and no compensaton for the ntracell nterference s needed. To overcome the possble unfarness that may results from the one-user-transmsson strategy, reference [17] 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 [17]. Also provded n

2 OH et al.: DOWNLINK THROUGHPUT MAXIMIZATION FOR INTERFERENCE LIMITED MULTIUSER SYSTEMS: TDMA VERSUS CDMA 255 [17] are numercal results whch demonstrate that the value of the orthogonalty factor sgnfcantly affects the system performance. References [15] [17] promote a formulaton where the transmsson rate and the power have a lnear relaton. In references [1], [19], on the other hand, rate and power have a logarthmc relaton. Reference [1] consders 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 then becomes a convex program, whch for the specal case of the equal weght scenaro,.e., throughput maxmzaton, produces one-user-transmsson as the optmum polcy [1]. Reference [19] nvestgates the multuser schedulng gan,.e., gan by transmssons of a fracton of users over transmssons of all users for a gven orthogonalty factor, wth the assumpton of equal power transmsson for the scheduled users wthout the consderaton of optmum schedulng and power allocaton. In ths paper, we wll adopt a model smlar to [1], [19], but consder the downlnk of an nterference lmted CDMA system. We determne the optmum number of users to be scheduled and fnd the correspondng power allocaton. The key observaton s that the optmum polcy s a functon of the orthogonalty factor. Specfcally, gven the channel realzaton of the users and the orthogonalty factor, the TDMA-mode,.e., the base staton transmts to one-user, or the CDMAmode,.e., the base staton transmts to multple users, can be the optmum strategy. If the CDMA-mode s optmum, that s, f more than one-user should be served by the base staton, then the correspondng power allocaton s also found. A smlar approach for the effect of orthogonalty factor s consdered n uplnk n [20] wthout the consderaton of the optmum power allocaton. The paper s organzed as follows. Secton II descrbes the system model as well as the problem formulaton and the performance metrc. In Secton III, we obtan the algorthm to fnd the optmum schedulng polcy and the correspondng power allocaton. Ths s done va examnng the utlty maxmzaton problem from an ndvdual user s perspectve and reflectng our fndngs to the system-wde optmzaton problem. In Secton IV, we extend our approach to the scenaro where ndvdual power constrants are n place. Secton V provdes numercal examples that support our analyss. Secton VI summarzes the results and concludes the paper. II. SYSTEM MODEL AND PROBLEM FORMULATION 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 users such that =1 p P T, where p 0 s the allocated power for user. Fgure 1 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 degree of nonorthogonalty s descrbed by the orthogonalty factor, 0 1 [9]. Under the assumpton of ndependent dentcal channels for all users, t s User 1 User 2 User K Fg. 1. g 1 g 2 g K Downlnk system model. Base Staton Orthogonal codes p 1 p 2 p K code 1 code 2 code K reasonable to work wth the same average orthogonalty factor for all users. The sgnal to nterference rato at the recever of user s expressed as p g SIR = j p 1 jg + 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 1+k j p 2 jg + I where log denotes the natural logarthm. We note that the ndvdual utlty s a functon of the power vector p = [p 1,p 2,..., p K ]. The utlty s an achevable rate for user, specfcally by treatng the nterference as nose [15]. It s assumed that adaptve modulaton s employed to enable the rate determned for each user [21]. Followng reference [22], we denote the factor, that captures the product of the SNR gap Γ and the processng gan N, k =Γ N, whereγ s derved from the target bt error rate BER, Γ= ln5ber 1.5. We should 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 [13], [1], 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: K p g max log 1+k p j p 3 jg + I s.t. =1 K p P T, p 0, =1,,K =1 whereweassumethatg 1 > g 2 >... > g K, such that user wth the lower ndex has a hgher channel gan. We note

3 25 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO. 7, JULY λ 1 mn λ 1 max 7 p 1 s wthn concave regon p n 1 p 1 s wthn convex regon 3 λ 2 max gan1 λ 1 mn Utlty 2.5 gan2 Utlty 5 2 gan λ 3 max λ max gan3 3 2 gan2 gan3 0.5 gan 1 gan Transmt power[watts] Transmt power[watts] Fg. 2. Utlty versus transmt power. gan 1 >gan 2 >gan 3 >gan. An example where ndvdual utlty functons for all users are concave. that 3 consders only the total power constrant. Ths s the problem we wll consder n Secton III. In Secton IV, we wll consder addtonal ndvdual power constrants. The actual outcome of optmzaton problem n 3 s the 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 =1 p <P T can not be the optmum power vector. Let us defne a power vector p wth p = βp β>1, =1,,K such that K =1 βp = P T. It s easy to see that p ncreases all ndvdual utltes, and hence the system utlty, as compared to p. Thus, 3 can be rewrtten as: K p max log 1+k p P =1 T p + I g K s.t. p = P T, p 0, =1,,K. =1 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 3 emphaszes ths pont by plottng the ndvdual utltes for users wth dfferent channel gans. III. OPTIMUM TRANSMIT STRATEGY AND POWER ALLOCATION At the outset, does not appear to be easy to solve. In an effort to gan understandng towards ts optmum soluton, we frst consder the behavor of the ndvdual utlty by observng ts dervatve. Our observatons motvate us to consder the system-wde approach, n whch we nvestgate the system Fg. 3. Utlty versus transmt power. gan 1 >gan 2 >gan 3 >gan. An example where ndvdual utlty functons for some users are not concave. utlty n terms of the best user s power. The nvestgaton provded n Secton III-A and III-B leads to the algorthms n Secton III-C. A. User Centrc Approach Consder the utlty functon of user and ts frst and second dervatve n terms of p, the power of user : p U p =log 1+k P T p + I 5 g U p p p T + I g = k p T p + I g + kp p T p + I g > 0 2 U p p 2 where A and B are defned as = kp T + I g A B 7 A = kp T p + I +p T p + I + kp g g B =p T p + I g + kp 2 p T p + I g 2. 9 Clearly, U p s an ncreasng functon of p.wearenterested n whether the utlty U p s an ncreasng concave or convex n p [0,P T ].WhetherU p s concave 2 U p p 2 < 0 or convex 2 U p p 2 sgna, the numerator of 2 U p p 2 > 0 depends on the, defned n. By lettng I t = p T p + I g,wehavea = ki t +I t +kp. Thus, U p s an ncreasng convex functon of p f p I t > 1 2 k 10 and an ncreasng concave functon of p otherwse. Clearly, by examnng 2 U p p 2 p=p T, we can dentfy the behavor of functon U p.when 2 U p p 2 p=p T < 0, U p s an ncreasng concave functon n 0 p P T as depcted n Fgure 2. When 2 U p p 2 p=p T > 0, U p s an ncreasng

4 OH et al.: DOWNLINK THROUGHPUT MAXIMIZATION FOR INTERFERENCE LIMITED MULTIUSER SYSTEMS: TDMA VERSUS CDMA 257 concave functon n 0 p p n, whle t s an ncreasng convex functon n p n p P T, wth p n denotng the nflecton pont n U p. Fgure 2 shows the ndvdual utltes correspondng to users wth dfferent channel gans. In ths example, the ndvdual utltes of all users are concave n power. Observe that when each ndvdual utlty s concave n power,.e., the case shown n Fgure 2, rather than allocatng the entre power P T to one user, sharng the power P T among multple users 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 3. 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 at p f 2 U p p 2 U p s concave/convex f 2 U p p 2 p=p T > 0 > 0 for 0 p P T p=0< 0 and 2 U p p 2 U p s concave f 2 U p p 2 p=p T < 0 p s sad to be wthn the concave regon of U p,f 2 U p p 2 p=p < 0 p s sad to be wthn the convex regon of U p,f p=p > 0 2 U p p 2 Fgure 3 shows the two possble power regons for user 1 a concave/convex user. Followng from above, we defne λ 1 mn = U 1p 1 p 1 p1=p f U 1p 1 s concave/convex, and 2 U 1p 1 p 2 1 p1=p =0 U 1p 1 p 1 p1=p T f U 1 p 1 s concave 11 λ max = U p p p=0 12 and, p λ = { arg p Up p p=p λ = λ f λ<λ max 0 f λ>λ 13 max where λ 1 mn s the mnmum dervatve value of utlty of user 1, whle λ max denotes the maxmum dervatve value of utlty of user n the concave regon of U.. Fgures 2 and 3 depct λ 1 mn and λ max. Havng set the stage wth these defntons, we now move to presentng our fndngs. The optmum schedulng polcy s a functon of the orthogonalty factor as well as users channel gans. Below we state the propostons that lead to characterzng the optmum polcy. 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 3.1: If k 2, TDMA-mode wth p 1 = P T yelds the maxmum system utlty. Proof: k 2 mples U p s a convex functon for, because 2 U > 0 for all 0 p p 2 P T.WhenU p s convex for, the overall utlty s convex. In ths case, the maxmzer of the overall utlty functon s at p 1 = P T,and p =0,>1. Whle, t s clear that the TDMA-mode s optmum when k 2, and no further power allocaton s necessary, when k 2, we need to characterze the optmum polcy whch conssts of the transmsson strategy,.e., TDMA-mode or CDMA-mode, and the correspondng power allocaton. The followng propostons help n that regard. Proposton 3.2: Suppose p = p 1,p 2,..., p K s the optmum power allocaton where the n users are scheduled,.e. p > 0 for =1,,n and p =0for = n +1,,K. Then, p 1 >p 2 >... > p n. Proof: Suppose the optmum power allocaton s such that p <p j g >g j,for, j {1,,n}. Frst, we note that Up p p=p > Ujpj p j pj =p,.e., at a gven power level, the rate of change n the utlty of the user wth the better channel s larger. Ths can be shown smply by evaluatng for user and user j at p = p j = p>0, and observng that the dfference of the two terms s strctly postve. Ths n turn means that by exchangng the power between user and j, wegetu p j +U jp >U p +U jp j,.e., we can always ncrease the system utlty. Therefore, the optmum power allocaton p must be such that the better channel user gets assgned a hgher power value among the users wth nonzero power. When the ndvdual utltes of multple users are concave/convex as n Fgure 3, the followng holds. Proposton 3.3: At p, the optmum power allocaton, at most one user s power s wthn the convex regon of ts utlty functon. Ths s the user wth the best channel gan. Proof: Suppose the optmum power allocaton s such that two users power values are wthn the convex regon of ther utlty functons,.e., p =[p 1,p 2,p 3,..., p K ] where p 1 and p 2 are wthn the convex regon of U 1 p 1 and U 2 p 2. Consder an alternatve power allocaton p =[p 1 + p, p 2 p, p 3,..., p K ],.e., the power of user 1 and 2 are ncreased by p and p, respectvely, whle the rest of users power values reman the same. Snce, by assumpton, U 1 p 1 and U 2 p 2 are convex at p 1 and p 2, and as explaned n the proof U of Proposton 3.2, 1p 1 p 1 p1=p > U2p2 p 2 p2=p, t follows that U 1 p 1 + p+u 1 p 2 p >U 1 p 1+U 1 p 2. Thus, when the power p 2 s wthn the convex regon of U 2p 2,we can always ncrease the total utlty. Therefore, ths cannot be the optmum polcy, and ndeed at the optmum pont, only the best user s power p 1 can be wthn the convex regon of U 1 p 1. From the precedng two propostons, we conclude that the optmum power allocaton p = [p 1,p 2,..., p K ] has to be one of two cases n terms of the best user,.e., p 1 can be ether wthn the concave regon or wthn the convex regon of U 1 p 1 1, whle the rest of the power values cannot be n the convex regon of ther respectve utlty. Ths fact motvates our approach to focus on the the system utlty n terms of the best user. B. System-Wde Approach Let us rewrte the system utlty n terms of the power allocated to the best user,.e., p 1 : 1 Ths possblty ncludes p 1 = P T

5 25 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO. 7, JULY 2007 Jp 1 =U 1 p 1 +Zp 1 1 Zp 1 =max[ j=2 U jp j ]P Kj=2 p j=p T p We note that Jp 1 expends all avalable power, P T, p 1 for user 1 and P T p 1 for the rest of the users. Our am s to fnd the maxmzer of Jp 1 over 0 p 1 P T. Proposton 3.3 guarantees that {U j p j j 2} are all n ther concave regon at the optmum power allocaton. Hence, Zp 1 can be maxmzed wth the power constrant j=2 p jλ =P T p 1 where λ s the optmum Lagrange multpler. We note that no power should be assgned to the user when λ max <λ. Dependng on user 1 s channel condton, U 1 p 1 can be ether concave or concave/convex. When U 1 p 1 s concave, the resultng optmzaton problem s convex and the optmum power allocaton s such that j=1 p jλ =P T.IfU 1 p 1 s concave/convex as n Fgure 3, we have the followng observaton. Observaton 3.: If U 1 p 1 s concave/convex, the optmum power allocaton s one of three local optmum solutons: p 1 s wthn the concave regon wth 0 p 1 <p 1 λ 1 mn, p 1 s wthn the convex regon wth p 1 λ 1 mn <p 1 <P T, or p 1 = P T. The transmsson strategy for the frst two cases s CDMA, whle the thrd case p 1 = P T s TDMA. Gven the fact that when U 1 p 1 s concave/convex, the resultng optmzaton problem s non-convex, the three possble cases descrbed above need to be consdered n detal to maxmze Jp 1. Consder =1 p λ,.e., the sum of the powers transmtted to all users, gven λ. Observng whether =1 p λ < P T,or =1 p λ >P T provdes us wth the nformaton whether Jp 1 s ncreasng or decreasng at p 1 = p 1 λ: Proposton 3.5: If =1 p λ <P T for a gven λ, Jp 1 s an ncreasng functon at p 1 = p 1 λ. Proof: We can fnd λ such that p 1 λ+ j=2 p jλ = P T where λ < λ, because U j p j j 2 s ncreasng concave at the optmum power values as depcted by proposton 3.3. Ths mples we can ncrease the system utlty by ncreasng p 1 and decreasng p j j>2, whle mantanng j=1 p jλ =P T. Proposton 3.: If =1 p λ >P T for a gven λ, Jp 1 s a decreasng functon at p 1 = p 1 λ. Proof: Ths s the opposte stuaton to the prevous case. We can fnd λ such that p 1 λ+ j=2 p jλ =P T where λ >λ. Ths mples we can ncrease the system utlty by decreasng p 1 and ncreasng the p j j>2, whle mantanng j=1 p jλ =P T. It follows from the prevous two propostons, by comparng K =1 p λ wth P T, we can always determne whether Jp 1 s ncreasng or decreasng at p 1 = p 1 λ. In partcular, f K =1 p λ 1 mn > P T, Jp 1 s a decreasng functon at p 1 = p 1 λ 1 mn and the optmum power allocaton s one of three possbltes as descrbed by Observaton 3.. If, on the other hand, =1 p λ 1 mn <P T,thsmplesthat =1 p λ <P T for all λ>λ 1 mn where p 1λ s wthn the concave regon 2. In ths case, Jp 1 s an ncreasng functon when p 1 s wthn the concave regon,.e., 0 p 1 p 1 λ 1 mn. 2 Recall that p λ 1 mn >p λ when U p s ncreasng concave. Ths means that, no p 1 value wthn the concave regon can be optmum. We know that the optmum power allocaton dctates that p 1 be ether p 1 λ 1 mn < p 1 < P T, or p 1 = P T, and that we would have to fnd the local optmum soluton of the sum utlty functon, =1 U p, wthn the convex regon of U 1 p 1. Gven that the overall utlty functon s not necessarly unmodal, ths task n general requres exhaustve search and the computatonal complexty assocated wth ths task may be prohbtve snce locatng the optmum λ would requre a search wth fne resoluton. Thus, f we fnd a computatonally nexpensve way to dentfy whether p 1 = P T s the optmum strategy,.e., the optmalty of TDMA, we can reduce the complexty of dentfyng the optmum polcy. Next, we set out to accomplsh ths task, and make the followng observaton. Observaton 3.7: If =1 p λ <P T for λ where p 1 λ s wthn the convex regon of U 1 p 1, then p 1 = P T s optmum power allocaton. Observaton 3.7 follows from the fact that, n ths case, Jp 1 s an ncreasng functon for 0 p 1 P T. Although the condton n observaton 3.7 guarantees the optmalty of TDMA, t stll requres consderable computatonal complexty to verfy. Notce that, as descrbed above, when K =1 p λ 1 mn <P T, the optmum power allocaton does not fall wthn the concave regon of U 1 p 1. Thus, by adoptng TDMA as the optmum polcy whenever =1 p λ 1 mn < P T, we can sgnfcantly reduce computatonal complexty n fndng the local optmum power allocaton n the convex regon of U 1 p 1. In the next secton, we wll term ths shortcut, the near-exact algorthm. The numercal results n Secton V valdate the accuracy of near-exact algorthm. 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 =1 p λ <P T for all λ where p 1 λ s wthn the convex regon of U 1 p 1. Instead of ths potentally computatonally ntensve search, we propose to smply check =1 p λ 1 mn <P T, and declare that p 1 = P T and p = 0, 2 whenever ths condton s satsfed. Ths acton,.e., decdng that p 1 = P T s optmum whenever =1 p λ 1 mn <P T, regardless of max p1 Jp 1 for pλ 1 mn < p 1 P T results n sub-optmum performance whenever the optmum power allocaton s such that p 1 λ 1 mn < p 1 < P T. In our numercal results, we have observed that cases where the optmum power allocaton s such that p 1 λ 1 mn <p 1 <P T, are rare and that n most cases, the optmum power allocaton s ether 0 < p 1 < p 1 λ 1 mn or p 1 = P T. We should note that there may be the cases when even though j=1 p jλ 1 mn > P T, TDMA can be optmum. However, as the orthogonalty factor ncreases, the condton =1 p λ 1 mn <P T accounts for almost all of the cases where TDMA s optmum, as demonstrated by numercal results n Secton V. The followng descrbes the steps of the proposed algorthm to maxmze the system utlty, whch we term the system utlty maxmzer, A-SUM.

6 OH et al.: DOWNLINK THROUGHPUT MAXIMIZATION FOR INTERFERENCE LIMITED MULTIUSER SYSTEMS: TDMA VERSUS CDMA 259 A-SUM: STEP 1. If k 2, then TDMA mode s optmum, p 1 = P T, STOP. STEP 2. Fnd λ max for =1,,K,andλ 1 mn. STEP 3. STEP. convex regon If j=1 p jλ 1 mn <P T, then, declare that TDMA mode s optmum, STOP. concave regon Fnd the power allocaton J p 1 λ = maxjpλ, 0 p 1 λ p 1 λ 1 mn,usnga-pacr. STEP 5. Choose maxj p 1 λ,jp T. We re-emphasze that the reducton n the computatonal complexty of the near-exactness of the algorthm arses from the STEP 3, nstead of solvng for max p1 Jp 1, p 1 λ 1 mn < p 1 <P T. The optmum power allocaton STEP entals selectng the users to whch the base staton transmts wth postve power. If zero power s assgned to one user, that user s not scheduled for transmsson. The algorthm to determne the power allocaton n the concave regon A-PACR s gven below. It s assumed n A-PACR that j=1 p jλ 1 mn >P T. If ths s not the case, there s no soluton n the concave regon, and A-PACR smply determnes p 1 = P T. A-PACR: STEP 1. Fnd ˆk =arg k max k j=1 p jλ k max, s.t. k j=1 p jλ k max P T, k =1,,K. Fnd p j λ 1 j ˆk STEP 2. s. t. ˆk j=1 p jλ =P T and λ 1 mn <λ<λˆk max. The optmum number of users to whch the base staton transmts s selected n STEP 1. Then, optmum power values are allocated n STEP 2. Note that a numercal method, e.g., bsecton [23] should be used to determne the value of λ. IV. DOWNLINK TRANSMISSION POLICIES WITH INDIVIDUAL POWER CONSTRAINTS In ths secton, we consder the effect of havng mnmum and maxmum lmts on the powers that the base staton transmts to ndvdual users. Specfcally, we extend our approach and present modfcatons to the algorthms presented n Secton III. We assume that each scheduled user,.e., each user that the base staton s desgnated to transmt wth nonzero power, has a maxmum and/or a mnmum transmt power constrant. We emphasze ths to convey the fact that we stll may end up wth non-scheduled users. Ths s n contrast to havng a mnmum power constrant for each user, whch automatcally sets the transmsson polcy to CDMA wth all users smultaneously transmtted to by at least the mnmum power. As a result, the optmzaton problem now has addtonal constrants, p mn p p max p > 0 1 where p mn ensures all users selected for transmsson to acheve mnmum requred rate U mn, whle p max sgnfes a practcal lmt,.e., an arbtrary hgh data rate cannot be acheved, even f a user has a very hgh channel gan. The lmts would be obtaned from the mnmum and maxmum rate, U mn and U max,va U mn U p U max 17 where and U mn =log 1+k U max =log 1+k p mn P T p mn p max P T p max + I g + I g, Observe the above assumes that full power s expended at the base staton. We wll elaborate on ths ssue n the next secton. A. Observatons on the Constraned Problem Frst, we consder the case when only maxmum power constrants exst. Observaton.1: When only maxmum power constrants exst and the base staton transmts to more than one user, full power, P T should be expended for the optmum power allocaton. Proof: Suppose the optmum power allocaton expends ˆP T ˆP T = =1 p < P T, whle one user, say user 1, s at ts maxmum allowable utlty value,.e. U 1 = U max. Observe that, we can concentrate on ths case, wthout loss of generalty, because f =1 p <PT and U 1 <U max,by proportonally ncreasng transmt power of all scheduled users such that U 1 reaches U max, the utlty of each scheduled user s further ncreased. Thus, =1 p <PT and U 1 <U max cannot be the condton for optmum power allocaton. Note that, from 19, t follows that by expendng P T, U 1 = U max would be acheved wth p max <p 1. Specfcally, U max p max 1 = log 1+k P T p max 1 + I g 1 p 1 = log 1+k ˆP T p I g 1 Snce, ˆPT < P T, 20 mples that p max 1 < p 1. Ths means that wth leftover power, P T ˆP T +p 1 p max 1, by proportonally ncreasng the transmt power of all other scheduled users such that j=2 βp j = P T p max 1, the utltes of these scheduled users are ncreased, whle U 1 s kept at U max. Thus, f ˆPT = =1 p < P T and U 1 = U max, the system utlty can be further ncreased by expendng full power P T. Therefore, full power P T should be expended for the optmum power allocaton. Observaton.2: When both the maxmum and the mnmum power constrants exst, the optmum polcy can be such that the base staton transmts less than full power. To see why ths observaton s vald, consder the followng argument. For smplcty of analyss, assume each scheduled user experences nterference as f full power P T s transmtted from the base staton. Then, the utlty of each scheduled user s a functon of ts power only and the mnmum and maxmum power of user, p mn and p max are obtaned for each scheduled user from 19. Consder the case that the optmum polcy schedules the frst n users to be transmtted wth ther maxmum power correspondng to the maxmum utlty value U max, whle satsfyng n =1 pmax < P T. In ths case, the optmum

7 20 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO. 7, JULY 2007 polcy may try to schedule the next user wth leftover power P T n =1 pmax. If ths acton volates the total power constrant, n =1 pmax + p mn n+1 >P T, the transmsson polcy ends up wth expendng less than full power. However, we note that p max = 1,,n values were obtaned wth the the assumpton that the base staton transmts full power, P T. Notng the fact that less than full power transmsson s optmum, we can see that the actual nterference scheduled user experences s n fact less than P T p max + I g.in turn, the maxmum utlty U max can be acheved by user wth power value p opt <p max. Fortunately, the power vector p opt =[p opt 1,,popt n ] smply s the downlnk power vector that satsfes a gven SIR value,.e. γ = eumax 1 k = 1,,n, and can be easly determned []. For convenence, we term ths Power Control for Fxed Rate PCFR. B. Utlty Maxmzaton wth Indvdual Constrants Proposton 3.3 where no power constrant s assumed states that at most one user can be wthn the convex regon of the utlty functon. By mposng a maxmum power constrant, we may have more than one user wthn the convex regon. In ths case, the utlty wth hgher channel gan can be ncreased up to the maxmum utlty value by ncreasng the power of the user wth hgher channel gan, and decreasng the power of the user wth lower channel gan, whle the sum of the power of these two users remans the same. After the utlty wth hgher channel gan reaches the maxmum utlty value, the power of the user wth lower channel gan can be ether wthn the concave or the convex regon. Ths mples that, under the maxmum power constrant, there can be more than one user wthn the convex regon. However, at most one user expends less than maxmum power wthn the convex regon at the optmum power allocaton. Otherwse, the system utlty can be ncreased as descrbed by Proposton 3.3. The observaton below follows from the fact that by U p p p=p mn Uj pj > p j pj=p mn g j >g j, and the fact that at most one user has less than maxmum power wthn the convex regon. Observaton.3: If p mn for some users are wthn the convex regon, the optmum polcy employs greedy packng of these users n the order of decreasng channel gans,.e., allocaton of maxmum allowable power n the order of decreasng channel gans. We term such users greedy users and the resultng process convex greedy packng. If all avalable power s expended n ths process, then the resultng power allocaton s the optmum power allocaton. If there s a leftover power, but s not enough to support the next user wth the mnmum requred power, the maxmum system utlty s acheved wth less than full power and wth convex greedy packng. After greedy packng, f the leftover power s enough to support the next user wth mnmum requred power, we need to consder the optmzaton problem n terms of each concave/convex user by allowng each concave/convex user to have the maxmum power n the order of decreasng channel gans. To that end, we modfy 1 and 15 such that Jp j M = <j U p max +U j p j +Zp j 21 K Zp j =max[ U l p l ]P P pl =P T pmax p j,p l p mn l l>j 22 where the notaton Jp j M denotes the fact that we have M users wth mnmum power requrement. The frst term n 21 ncludes users whch expend ther maxmum power p max wthn the convex regon. Smlar to 15, Jp j M s optmzed over p j wth remanng power P T pmax M l> pmn l. The resultng soluton max Jp j M only consders the case when p = p max < j. Recall that that at most one user expends less than maxmum power wthn the convex regon at the optmum power allocaton. Hence, by solvng max Jp j M for all concave/convex users, we can fnd the optmum soluton. Ths means that we need to solve max max Jp j M j = k +1,, mnk,m 23 j p j where k, k denotes the number of greedy users and concave/convex users, respectvely. If all ndvdual utltes are concave, the resultng optmzaton problem s convex, and the soluton of 23 s gven n Secton III. Note that, due to the mnmum power constrant, the followng modfcatons n our prevous defnton Secton III are needed: λ max = U p p p=p mn 2 λ mn = U p p, U p p=p when 2 p 2 p=p =0 and U p s concave/convex. 25 { arg p λ = p Up p p=p λ = λ f λ<λ max p mn f λ>λ max 2 C. Algorthms wth Indvdual Power Constrants Followng the observatons n Secton IV-A, we propose modfed versons of A-SUM and A-PACR n the presence of power constrants, A-SUMPC and A-PACRPC, respectvely. Let k and k k k denote the number of greedy users and the number of concave/convex users respectvely. A-SUMPC: STEP 1. Fnd λ max =1,,K and λ mn ; fnd T 1 = U 1 p 1 = mnp max 1,P T, p 2 =0,,p K =0,setM = k +1. STEP 2. If k > 0, Apply Convex Greedy Packng. STEP 3. M-user system If k =1 pmax + M j=k +1 pmn j >P T, M = M 1, GO TO STEP 5. T M =max j maxpj Jp j M, j = k +1,, mnm,k, va A-PACRPC. STEP. If M<K, M = M +1 and GO TO STEP 3. STEP 5. max 1 M T. In the algorthm above, T denotes the maxmum system utlty of the -user system. In STEP 2, convex greedy packng gven below s appled to greedy users. Feasblty of the M-user system s checked n STEP 3, then optmum power allocaton of M-users system s obtaned va A-PACRPC. STEP 5 selects the largest among T =1,,M. When

8 OH et al.: DOWNLINK THROUGHPUT MAXIMIZATION FOR INTERFERENCE LIMITED MULTIUSER SYSTEMS: TDMA VERSUS CDMA 21 the optmum power allocaton expends less than full power P T, the actual optmum power vector s found va solvng M lnear equatons,.e. PCFR. Convex Greedy Packng: STEP 1. P re = P T. =1. STEP 2. p =mnpmax,p re, P re = P re p If p <pmn, p =0. STEP 3. If P re =0or p <pmn, STOP. STEP. If <k,then = +1, GO TO STEP 2. In STEP 2 of Convex Greedy Packng, the maxmum allowable power s assgned to greedy users n the order of decreasng channel gans. If the algorthm stops n STEP 3, and P re =0, then the resultng soluton s optmum power. If P re 0, the actual power vector s obtaned va PCFR. A-PACRPC: STEP 1. = k +1. STEP 2. convex regon 1 M p =mnpmax,p T p max j p mn j, j=1 j=+1 Fnd p j λ <j M s.t. 1 j=1 pmax j + p + M j=+1 p jλ =P T, and the correspondng system utlty TM convex. If 1 j=1 pmax j + M j= p jλ mn <P T, TM concave =0and GO TO STEP. STEP 3. concave regon Fnd p j λ j M s.t. 1 j=1 pmax j + M j= p jλ =P T and the correspondng system utlty TM concave. STEP. T M =maxtm convex,tm concave If <mnm,k, = +1, GO TO STEP 2. STEP 5. T M =max 1 j mnm,k T Mj. Note that the near exactness of the algorthm A-PACRPC arses from the fact that n STEP 2, we smply take p = mnp max,p T 1 j=1 pmax j M j=+1 pmn j, nstead of solvng max p Jp M over all p wthn the convex regon. V. NUMERICAL RESULTS We provde smulaton results for a range of the orthogonalty factor values. The total power at the base staton s P T =10watts. Total nose and ntercell nterference s I =10 11 watts. Factor k n the utlty functon s k =1.2, correspondng to Γ=0.15 and processng gan N =. Thus, when 0., the ndvdual utltes of all users are convex n power, and the TDMA-mode s optmum, regardless of the channel gans. Channel gan from the 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 100m and 1000m, and 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. Our experments over 10,000 channel gan realzatons have demonstrated that the near-exact algorthm correctly determnes TDMA optmalty 97.5%, 99.%, 99.9% of the tme for = 0.1, 0.2, 0.3, respectvely. As ncreases, as expected, the accuracy gets Percentage of TDMA mode Number of users n the system =0.1 =0.3 =0.5 Fg.. Percentage of channel realzatons n whch the TDMA mode s selected. System Utlty Fg. 5. =0.1 =0.3 = number of users n the system System Utlty versus number of users n the system. better snce the lkelhood of the case when the optmum power allocaton s such that p 1 λ 1 mn p 1 <P T gradually dsappears. To gve the dea about the savngs n complexty of the near-exact algorthm versus the optmum soluton, we were able to locate the optmum soluton by evaluatng 1000 λ values wthn the convex regon of U 1 p 1 n STEP 3 of A-SUM. On the other hand, the near-exact algorthm requres a sngle evaluaton at λ mn. Fgure shows the percentage of channel realzatons n whch the polcy our proposed algorthm fnds s TDMA. As the number of users n the system ncreases, the possblty that multple users may have hgh channel gan values ncreases. Accordngly, the percentage decreases. As expected, the frequency of TDMA beng selected ncreases as the orthogonalty factor ncreases. Fgure 5 shows the system utlty, gven and the number of users K n the system. The system utlty s averaged over 10,000 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

9 22 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL., NO. 7, JULY =0.1 =0.3 = NPC PCMIN PCMAX PC 10 System Utlty Dscrete 7 5 System Utlty number of users n the system number of users n the system Fg System Utlty versus number of users n the system. CDMA =0.1 CDMA =0.2 CDMA =0.3 TDMA mode Fg.. System utlty wth no power constrant NPC, mnmum power constrant PCMIN, maxmum power constrant PCMAX and maxmum and mnmum power constrants PC. k =2. Γ=0.15,N = 1, = NPC PCMIN PCMAX PC System Utlty 5 3 System Utlty Dscrete 2 Fg Number of users n the system CDMA-mode gan over TDMA-mode for dfferent values number of users n the system scheduled. When =0.1,.e., a low orthogonalty factor, smultaneous transmssons do not result n much nterference, and the optmum polcy chooses the CDMA-mode. However, f the best user has a much hgher channel gan as compared to the rest, the optmum polcy may stll end up to allocatng all power to that best user. As ncreases, more than one user transmssons result n hgher nterference, and TDMA-mode becomes the preferred mode so as to avod the nterference. For example, for >0.5, TDMA-mode s optmum n most cases. For any value of > 0., for example, when = 0.7, TDMA-mode s always optmum: Fgure 5 shows that when =0.7 the maxmum system utlty s lower than the maxmum system utlty values acheved for smaller values that facltate transmssons to multple users. The gap between the system utlty for <0. and the system utlty for =0.7 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. As the number of users n the system ncreases, the chance of selectng users wth hgher channel gans, multuser dversty, ncreases. Thus, the system utlty ncreases. Fg. 9. Dscrete system utltes wth no power constrant NPC, mnmum power constrant PCMIN, maxmum power constrant PCMAX and maxmum and mnmum power constrants PC. k = 2. Γ = 0.15,N = 1, =0.1. 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 7 shows the system utlty gan of CDMA-mode over TDMA-mode. When the orthogonalty factor s low, multpleuser transmssons has a gan over the one user transmsson. The gan ncreases as the number of users n the system ncreases. As the orthogonalty factor ncreases, however, we observe that the gan dsappears, as eventually TDMA-mode becomes optmum. Fgures and 9 show the system utltes and the dscretzed system utltes wth no power constrant NPC, mnmum

10 OH et al.: DOWNLINK THROUGHPUT MAXIMIZATION FOR INTERFERENCE LIMITED MULTIUSER SYSTEMS: TDMA VERSUS CDMA 23 power constrant only PCMIN, maxmum power constrant only PCMAX and mnmum and maxmum power constrant PC for =0.1 and k =2. Γ =0.15 and N =1. Note the loss n total system utlty we have due to the presence of the maxmum power constrants, because these constrants lmt the throughput values of the users wth hgh channel gans. VI. CONCLUSION In ths paper, we have consdered an nterference lmted downlnk and nvestgated the total utlty throughput maxmzng polcy whch conssts of the users the base staton selects to transmt to, and the correspondng power allocaton. 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 multpleuser transmssons, renderng TDMA-mode beng optmal. In contrast, multple-user-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 when t s not. In partcular, we took the approach of examnng the system utlty n terms of the best user, and dentfed propertes that helped us desgn an algorthmc method of fndng the optmal polcy. Snce dentfyng the optmum polcy may be prohbtvely complex, we also dentfed a near-exact algorthm wth reduced complexty, whch s observed to fnd the optmum polcy n an overwhelmng majorty of numercal examples.we also consdered the effect of mposng mnmum and maxmum ndvdual power constrants and observed that CDMA-mode becomes the choce more often when maxmum power constrants are mposed. REFERENCES [1] C. Oh and A. Yener, Downlnk throughput maxmzaton: TDMA versus CDMA, n Proc. 39th Annual Conference on Informaton Scences and Systems CISS, March [2] H. Alav and R. W. Nettleton, Downstream power control for a spread spectrum cellular moble rado system, n Proc. IEEE Globecom 192. [3] J. Zander, Performance of optmum transmtter power control n cellular rado systems, IEEE Trans. Veh. Technol., vol. 1, pp. 57 2, Feb [] R. Yates, A framework for uplnk power control n cellular rado systems, IEEE J. Sel. Areas Commun., vol. 13, pp , Sept [5] H. Holma, W-CDMA for UMTS: Rado Access for Thrd Generaton Moble Communcatons. Wley and Sons, [] M. Andrew, K. Kumaran, K. Ramanan, A. L. Stolyar, R. Vjayakumar, and P. Whtng, Provdng qualty of servce over a shared wreless lnk, IEEE Commun. Mag., vol. 39, no. 2, pp , Feb [7] 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 Commun. Mag., pages 70 77, [] F. Berggren and R. Jantt, Asymptotcally far transmsson schedulng over fadng channels, IEEE Trans. Wreless Commun., vol. 3, pp , Jan [9] N. B. Mehta, L. J. Greensten, T. M. Wlls, and Z. Kostc, Analyss results for the orthogonalty factor n WCDMA downlnks, IEEE Trans. Wreless Commun., vol. 2, no., pp , Nov [10] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, Prcng and power control n a multcell wreless data network, IEEE J. Sel. Areas Commun., vol. 50, no. 2, pp , Oct [11] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, Effcent power control va prcng n wreless data networks, IEEE Trans. Commun., vol. 50, no. 2, pp , Feb [12] N. Feng, S. C. Mau, and N. B. Mandayam, Prcng and power control for jont network-centrc and user-centrc rado resource management, IEEE Trans. Commun., vol. 52, no. 9, pp , Sept [13] S. J. Oh and K. M. Wasserman, Optmalty of greedy power control and varable spreadng gan n mult-class CDMA moble networks, n Proc. ACM/IEEE Mobcom [1] S. Ulukus and L. J. Greensten, Throughput maxmzaton n CDMA uplnks usng adaptve spreadng and power control, n Proc. IEEE ISSSTA [15] F. Berggren, S. L. Km, R. Juntt, and J. Zander, Jont power control and ntra-cell schedulng of DS-CDMA non-real tme data, IEEE J. Sel. Areas Commun., pp , [1] A. Bedekar, S. Borst, K. Ramanan, P. Whtng, and E. Yeh, Downlnk schedulng n CDMA data networks, n Proc. IEEE Globecom [17] M. K. Karakayal, R. Yates, and L. V. Razoumov, Downlnk throughput maxmzaton n CDMA wreless networks, IEEE Trans. Wreless Commun., vol. 5, no. 12, Dec [1] R. Agrawal, V. Subramanan, and R. A. Berry, Jont schedulng and resource allocaton n CDMA systems submtted 200; avalable at rberry/opt.pdf. [19] F. Berggren and R. Jantt, Multuser schedulng over raylegh fadng channels, n Proc. IEEE Global Telecommuncatons Conference [20] K. Ramanan and L. Qan, Uplnk schedulng n CDMA packet-data systems, n Proc. IEEE Infocom [21] X. Qu and K. Chawla, On the performance of adaptve modulaton n cellular systems, IEEE Trans. Commun., pp. 95, [22] A. J. Goldsmth and S. G. Chua, Varable-rate varable-power MQAM for fadng channels, IEEE Trans. Commun., vol. 5, pp , Oct [23] D. P. Bertsekas, Nonlnear programmng. Athena Scentfc, Changyoon Oh S01, M0 receved the B.S. degree n Electrcal Engneerng from Yonse Unversty, Seoul, Korea, n 1999, and the M.S. and Ph.D. degrees n Electrcal Engneerng from The Pennsylvana State Unversty, Unversty Park, PA, n 2001 and 2005, respectvely. Currently, he s wth Samsung Electroncs as a senor Engneer. Hs research nterests nclude rado resource management and physcal layer ssues. Hs current responsbltes nclude standardzaton actvtes of IEEE Ayln Yener S 90, M 00 receved the B.S. degrees n Electrcal and Electroncs Engneerng, and n Physcs, from Boğazç Unversty, Istanbul, Turkey, n 1991, and the M.S. and Ph.D. degrees n Electrcal and Computer Engneerng from Rutgers Unversty, NJ, n 199 and 2000, respectvely. Durng her Ph.D. studes, she was wth Wreless Informaton Network Laboratory WINLAB n the Department of Electrcal and Computer Engneerng at Rutgers Unversty, NJ. Between fall 2000 and fall 2001, she was wth the Electrcal Engneerng and Computer Scence Department at Lehgh Unversty, PA, where she was a P.C. Rossn assstant professor. Between , she was an assstant professor wth the Electrcal Engneerng department at The Pennsylvana State Unversty, Unversty Park, PA, where she currently serves as an assocate professor. Dr. Yener s a recpent of the NSF CAREER award n She s an assocate edtor of the IEEE Transactons on Wreless Communcatons. Dr. Yener s research nterests nclude performance enhancement of multuser systems, nformaton theory of moble ad hoc networks, wreless communcaton theory and wreless networkng n general.