Downlink Power Allocation for Multi-class. Wireless Systems

Transcription

1 Downlnk Power Allocaton for Mult-class 1 Wreless Systems Jang-Won Lee, Rav R. Mazumdar, and Ness B. Shroff School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN 47907, USA {lee46, mazum, shroff}@ecn.purdue.edu Abstract In ths paper we consder a power allocaton problem n mult-class wreless systems. We focus on the downlnk of the system. Each moble has a utlty functon that characterzes ts degree of satsfacton for the receved servce. The objectve s to obtan a power allocaton that maxmzes the total system utlty. Typcally, natural utlty functons for each moble are non-concave. Hence, we cannot use exstng convex optmzaton technques to derve a global optmal soluton. We develop a smple (dstrbuted) algorthm to obtan a power allocaton that s asymptotcally optmal n the number of mobles. The algorthm s based on dynamc prcng and conssts of two stages. At the moble selecton stage, the base-staton selects mobles to whch power s allocated. At the power allocaton stage, the base-staton allocates power to the selected mobles. We provde numercal results that llustrate the performance of our scheme. In partcular, we show that our algorthm results n system performance that s close to the performance of a global optmal soluton n most cases. Index Terms Power allocaton, downlnk, wreless networks, and non-convex optmzaton. I. INTRODUCTION In recent years, the area of power control n wreless networks has receved sgnfcant nterest from both academc and ndustral researchers. Power control plays an mportant role n the effcent management of code dvson multple access (CDMA) networks. Snce voce has been the man servce provded by Ths research has been supported n part by NSF grants ANI , ANI , and ANI

2 2 wreless networks thus far, most research efforts have been devoted to voce systems. In voce systems, typcally all users have the same qualty of servce (QoS) requrements and t s mportant that the sgnal to nterference and nose rato (SINR) exceeds some mnmum threshold. Hence, the man purpose of power control n such systems s to elmnate the near-far effect by equalzng the SINR of each user by settng t at the mnmum SINR threshold [1], [2]. In the next generaton of wreless networks, t s expected that servces wll have sgnfcantly dfferng characterstcs from the current voce-domnated systems. Already, the demand for varous servces wth dfferent QoS requrements such as vdeo and data s ncreasng. The requred bandwdth for these servces s much hgher than that for voce servces, further compoundng the scarcty of resources n wreless systems. Therefore, to more effcently accommodate servces wth dfferent characterstcs, we need a new approach for power control n next generaton wreless networks. A dstngushng feature of many of these new servces s ther elastcty,.e., they can adjust transmsson rates (to some degree) based on the channel condtons and the congeston level of the system. Hence, by approprately explotng the elastcty of such servces, we can mantan hgh network effcency and prevent network congeston. Moreover, such servces are hghly asymmetrc, requrng more bandwdth n the downlnk than the uplnk. Ths mples that, n the next generaton of wreless networks, effcent resource allocaton for the downlnk becomes an mportant ssue [3], [4], [5]. Recently, the concept of utlty (and prcng) from economcs has been used to develop network control algorthms by explotng the elastcty of the servces. The utlty represents the degree of a user s (servce s) satsfacton when t acqures a certan amount of the resource, and the prce s the cost per unt resource that the user needs to pay. Hence, servces wth heterogeneous QoS requrements (elastcty) can be modeled wth dfferent utlty functons. The basc dea of these algorthms s to control the users behavor by prcng resources approprately to obtan the desred results, (e.g., hgh utlzaton for the overall system and farness among users). In wrelne networks, utlty and prcng based algorthms have been studed for dstrbuted flow control of best effort servces [6], [7], [8]. In these works, the utlty functon s assumed to be a concave functon of the allocated rate, whch results n a convex programmng problem. Hence, the Karush-Kuhn-Tucker (KKT) condtons or the dualty theorem can be used to obtan the optmal soluton. Utlty (and prcng) based control algorthms can also be appled to the power control problem n wreless networks. However, the man dffculty n solvng the problem s that, n general, t cannot be formulated as

3 3 a convex programmng problem, snce the utlty functon may not be concave [9], [10], [11], [12]. Thus, nether the KKT condtons nor the dualty theorem provdes a suffcent condton for the optmal soluton. In most works on utlty and prcng for power control, only Nash equlbra, whch are neffcent from the pont of overall system utlty [13], have been obtaned. In [9], [10], [11], a power control problem s formulated as a non-cooperatve N-person game n whch each moble transmts a power level that maxmzes ts (net) utlty wthout consderng the behavor of the other mobles. They show that ther algorthms converge to Nash equlbra. Further, n [10], [11], the authors show that, by ntroducng prcng, system effcency can be mproved. In these works, the basestaton nforms each moble of a fxed prce per unt power and each moble transmts at a power level that maxmzes ts net utlty (utlty mnus cost for power allocaton). They show that the system utlzaton sgnfcantly depends on the choce of prce. However, they do not provde a systematc algorthm to fnd an optmal prce. In [12], [14], a downlnk resource allocaton problem s consdered wth restrcted types of utlty functons. In [12], only voce servces are consdered and utlty functons are modeled as step functons and n [14], utlty functons are modeled as concave functons. In these works, the authors obtan the optmal prces for maxmzng the total system utlty and the total revenue. In [15], [16], capacty regons and optmal power and rate allocaton schemes are studed from an nformaton theoretc pont of vew. In ths paper, we study the downlnk power allocaton problem for mult-class wreless networks. We use a utlty based framework as n other works. However, the stuaton consdered here dffers from prevous works n many aspects. Prmarly, we consder general types of utlty functons that are sutable for multclass systems and may be non-concave. Ths generalzaton requres a sgnfcantly dfferent analyss than the works of [12], [14]. We also study the problem of maxmzng total system utlty for heterogeneous users that provdes a hgher system utlty than those consdered n [9], [10], [11]. We put an emphass on the effcency of the system. However, due to the non-convexty of the problem, obtanng a global optmal power allocaton s dffcult and, f feasble, would requre a very complex algorthm. Therefore, we develop a smple (dstrbuted) algorthm that provdes an asymptotcally (n the number of mobles) optmal power allocaton. Ths algorthm can be mplemented n ether a dstrbuted or centralzed way. If mplemented n a centralzed way, the base-staton must know certan nformaton about the mobles, such as path gan from the base-staton to the moble, the nterference level at the moble, the utlty functon of the moble, and so on. In addton the computatonal burden s mposed all on the

4 4 base-staton. If mplemented n a dstrbuted way, the base-staton need not know the detaled nformaton about the mobles, and the computatonal burden can be dstrbuted among the base-staton and mobles. Ths s sutable for the case when the base-staton does not know the utlty functon of the moble [17], [18]. However ths requres teratve communcaton between the base-staton and mobles for the algorthm to converge. In ths case, our problem can be expressed as a utlty and dynamc prcng problem. The dynamc prcng attrbute s also another dstngushng feature of ths work compared wth other works. The rest of the paper s organzed as follows. In Secton II, we descrbe the system model consdered n ths paper and formulate the basc problem. In Secton III, we present the power allocaton algorthm, whch conssts of the moble selecton stage and the power allocaton stage. In Secton IV, we study the asymptotc optmalty (n the number of mobles) and the lower bound on the performance of our power allocaton. In Secton V, we study a specal case when all mobles are homogeneous. Numercal results are provded n Secton VI. Fnally, we conclude n Secton VII. II. SYSTEM MODEL AND PROBLEM DESCRIPTION Our objectve s to determne the approprate power levels at whch the base-staton should communcate to the dfferent mobles (the downlnk power allocaton problem n a mult-class wreless network). We focus on a sngle cell consstng of a sngle base-staton and M mobles. The system s assumed to be tme-slotted. At each tme-slot, the power allocaton algorthm s executed. A tme-slot n our system s an arbtrary nterval of tme and could consst of one packet or several packets. We focus on a tme-slot assumng that the path gan, background nose, and ntercell nterference for each moble do not change durng ths tmeslot. Each moble communcates wth the base-staton. For downlnk communcaton, the base-staton has a maxmum power lmt, P T. It allocates power to each moble wthn the power lmt (.e., the sum of the power allocated to each moble cannot exceed the power lmt). Each moble, = 1, 2, M, has ts own utlty functon, U that represents the degree of moble s satsfacton of the receved QoS and s a functon of the generc sgnal qualty for moble. We frst defne γ, the generc sgnal qualty for moble as follows: γ ( P) = = N G P G θ( M m=1 P m P ) + I N P θ( M, (1) m=1 P m P ) + A where

5 5 P : Allocated power for moble. P : Power allocaton vector, (P 1,P 2,,P M ). N : Constant for moble. G : Path gan from the base-staton to moble. I : Background nose and ntercell nterference to moble. A : Goodness of the transmsson envronment of moble, whch s defned by I /G. M: Number of mobles n the cell. θ: Orthogonalty factor (0 θ 1). Note that f θ 0, γ and the utlty functon U depend not only on moble s own power allocaton but also on the power allocatons of all the other mobles. In the above equaton, f N = 1, then the sgnal qualty metrc γ represents the SINR for moble. If N s the processng gan for moble, whch s defned by W/R, where W s the chp rate for the CDMA network and R s the data rate at whch the base-staton transmts to moble, then γ represents the bt energy to nterference densty rato of moble, (E b /I 0 ), n the CDMA system. If N = W, then γ = (E b /I 0 ) R of moble n the CDMA system. In ths case, for a gven power allocaton,.e., for a gven γ ( P), R and (E b /I 0 ) have an nversely proportonal relatonshp. Hence, there exst approprate R and (E b /I 0 ) for a gven γ ( P), and they may have dfferent values for dfferent γ ( P). Hence, n ths case, each moble may receve varable data rates (.e., varable processng gans) and they can be adjusted approprately based on the power allocaton. Thus, the utlty value can depends on R and (E b /I 0 ). We further assume that U has the followng propertes. Assumptons: (a) U s an ncreasng functon of γ. (b) U s twce contnuously dfferentable. (c) U (0) = 0. (d) U s bounded above. (e) If M =1 P = P 1 T, then U (γ ( P)) s one of three types: a sgmodal-lke 2, a strctly concave, or a strctly convex functon of P, ts own power allocaton. 1 We wll show ths n Lemma 1. 2 A functon f(x) s sad to be a sgmodal-lke functon f t has one nflecton pont, x o and d2 f(x) dx 2 x > x o. > 0 for x < x o and d2 f(x) dx 2 < 0 for

6 f(p) BPSK DPSK FSK P Fg. 1. Probabltes of packet transmsson success for BPSK, DPSK, and FSK modulaton schemes. Note that typcally, most utlty functons used n wrelne or wreless networks can be represented by three types of functons n assumpton (e) [11], [19]. For nstance, we can defne the utlty functon of each moble as ts expected throughput, whch s defned as U (γ (P)) = R f (γ (P)), where R s the data rate receved at moble and f (γ (P)) s the probablty of packet transmsson success of moble. In Fg. 1, we provde the probablty of packet transmsson success for varous modulaton schemes such as Bnary Phase-Shft Keyng (BPSK), Dfferental Phase-Shft Keyng (DPSK), and Frequency-Shft Keyng (FSK) [20]. We assume that a packet conssts of 800 bts wthout channel codng and set P T = 10, θ = 1, N = 16, and A = As shown n ths fgure, the probablty of packet transmsson success s represented by a sgmodal-lke functon of ts power allocaton. Hence, n ths case, we have sgmodal-lke utlty functons. The goal of ths paper s to obtan the power allocaton for each moble that maxmzes the total system utlty (.e., the sum of the utltes of all mobles). The basc formulaton of ths problem s gven by the followng optmzaton problem: (A) max U (γ ( P)) P =1 subject to P P T, =1 0 P P T, = 1, 2,,M. In problem (A), f we defne the utlty functon of the moble as ts expected throughput, the objectve of ths problem wll be to maxmze the total expected throughput of the system. Further, f each corresponds to each sub-carrer n an Orthogonal Frequency Dvson Multplexng (OFDM) system, problem (A) can be appled to power allocaton for sub-carrers n the OFDM system.

7 7 III. POWER ALLOCATION We consder only the dstrbuted soluton,.e., a user based approach. However, the algorthm can be easly executed n a centralzed way at the base-staton, f each moble nforms the base-staton of the goodness of ts transmsson envronment, A and ts utlty functon, U. Our power allocaton algorthm conssts of two stages. In the frst stage, mobles to whch power s allocated are selected, and, then, power s allocated to the selected mobles n the second stage. Before we descrbe the detals of our power allocaton algorthm, we decompose problem (A) as a moble problem and a base-staton problem. To do ths, we need certan results outlned next. The followng lemma wll show that to maxmze the total system utlty, the base-staton must transmt at ts maxmum power lmt, P T. Lemma 1: If P = (P 1,P 2,,P M ) s a power allocaton and M =1 P < P T, then we can fnd another power allocaton P = (P 1,P 2,,P M) such that M m=1 P m = P T and M =1 U (γ ( P )) > M =1 U (γ ( P)). Proof: See Appendx A. Hence, the base-staton always transmts at the maxmum power level, P T and M =1 P = P T. So, we can rewrte γ ( P) n (1) as γ ( P) = = N P θ( M m=1 P m P ) + A N P θ(p T P ) + A = γ (P ), = 1, 2,,M. Note that γ (P ) does not depend on the power allocaton for the other mobles and so problem (A) s equvalent to the followng problem. (B) max U (γ (P )) P =1 subject to P P T, =1 0 P P T, = 1, 2,,M. Snce M =1 P = P T, from assumpton (e) on the utlty functon, U (γ (P )) s one of three types: a

8 8 sgmodal-lke, a strctly concave, or a strctly convex functon of P. We now defne P o as P o = the nflecton pont of U (γ (P )), f U (γ (P )) s sgmodal-lke 0, f U (γ (P )) s concave. P T, f U (γ (P )) s convex Note that snce we allow non-concave utlty functons, n general, (B) s a non-convex optmzaton problem. We wll develop a smple (dstrbuted) algorthm that attempts to approxmate the performance of the global optmal soluton, and show that the performance of ths algorthm asymptotcally (n the number of mobles) converges to the global optmum. To that end, we wll use the followng result. Lemma 2: Let us defne a Lagrangean functon assocated wth problem (B) as L( P,λ) = U (γ (P )) + λ(p T P ), =1 =1 S = { P 0 P P T }, and Y (λ) = { x S L( x,λ) = max{l( P,λ)}}, P S where 0 = (0, 0,, 0) and P T = (P T,P T,,P T ). Then, for any λ 0, P(λ) Y (λ) s a global optmal soluton of the followng problem. max U (γ (P )) P =1 subject to P P (λ) =1 =1 0 P P T, = 1, 2,,M, (2) where P(λ) = (P 1 (λ),p 2 (λ),,p M (λ)). Proof: Ths mmedately follows from Property 6.6 n [21]. Lemma 2 mples that f we fnd a λ above such that M =1 P (λ ) = P T (when P T s the threshold n problem (A)), the global optmal soluton of problem (A) can be obtaned. However, when we cannot fnd such a λ (ths case s descrbed later n ths secton), P(λ) s a global optmal soluton of the perturbed problem that dffers from problem (A) by P T M =1 P (λ) on the constrant. From Theorem 5.4 n [21], we can show that M =1 U (γ (P o )) M =1 U (γ (P (λ))) λ(p T M =1 P (λ)), where P o = (P o 1,P o 2,,P o M) s a global optmal soluton of problem (A). Hence, f P T M =1 P (λ), we expect that M =1 U (γ (P o ))

9 9 M=1 U (γ (P (λ))). Therefore, n ths paper, we wll attempt to mnmze ths quantty by consderng the followng problem, and later on, we wll also show that t provdes an asymptotcally (n the number of mobles) optmal power allocaton. (C) mn λ {P T subject to P (λ)} =1 P(λ) = arg max {L( P,λ)} 0 P P T P (λ) P T. =1 We frst consder the equaton, P(λ) = arg max 0 P P T {L( P,λ)}. Snce L( P,λ) s separable n P, P(λ) solves the equaton f and only f t solves the followng problem. (D ) P (λ) {0 q P T L (q,λ) = max 0 P P T L (P,λ)}, = 1, 2,,M, where L (x,λ) = U (γ (x)) λx. Note that the parameters n problem (D ) correspond only to moble. By ths property, we can decompose problem (C) as the moble problem (D ) for each moble and the followng base-staton problem. (E) mn{p T P (λ)} λ =1 subject to P (λ) P T. =1 We can nterpret the decomposed problems as follows. Based on λ, the prce per unt power, each moble tres to maxmze ts net utlty (.e., the utlty mnus the cost) by solvng problem (D ). Ths s a greedy procedure and s typcally known as a non-cooperatve property. In our formulaton, by solvng problem (E) based on the power request of each moble, the base-staton adjusts the prce λ dynamcally to reduce the performance dfference between the global optmal power allocaton and ts power allocaton by mnmzng {P T M =1 P (λ)}. Therefore, ths problem can be nterpreted as a utlty and dynamc prcng problem. Usng ths nterpretaton, we can mplement the power allocaton algorthm n a dstrbuted way. However, a soluton to problem (C) (or equvalently problems (D ) and (E)) may result n an neffcent power allocaton,.e., M =1 P < P T. Further, due to the dscontnuty and non-unqueness of P (λ) (we wll show ths later), f we mplement the dstrbuted soluton usng standard gradent descent technques, the resultant power allocatons could oscllate (.e., there would be no equlbrum soluton). Hence, we wll devse a strategy to ensure that our soluton wll n fact have an effcent power allocatons ( M =1 P = P T ) as well as have

10 10 a stable soluton. To that end, we dvde the algorthm n two stages. The frst stage s the moble selecton stage. In ths part, mobles that can be allocated postve power are selected. The second stage s the power allocaton stage. Here, only selected mobles partcpate at the power allocaton stage and power s optmally allocated to the selected mobles. A. Moble Selecton Before, we develop an algorthm for moble selecton, we frst study the propertes of P (λ) n problem (D ). We defne λ max for moble as: The parameter λ max by λ max = mn{λ 0 max 0 P P T {U (γ (P)) λp } = 0}. wll play an mportant role n moble selecton. From Appendx B, t can be calculated λ max = du (γ (P)) dp P=0, f P o = 0 du (γ (P)) dp P=P, f 0 < P o < P T and P exsts U (γ (P T )) P T, otherwse where P s a soluton of the followng equaton., (3) U (γ (P)) P du (γ (P)) dp = 0, P o P P T. Further, we defne λ mn as λ mn = max{λ 0 P (λ) = P T }. We now summarze the propertes of P (λ). Detals are provded n Appendx C. (P1) (P2) P (λ) s dscontnuous and has two values (zero and postve) at λ = λ max, f U s a convex or a sgmodal-lke functon. In ths case, the postve value s greater than or equal to P o. P (λ) s contnuous functon of λ, f U s a concave functon. (P3) (P4) (P5) P (λ) s a postve, contnuous, and decreasng functon of λ for λ mn P (λ) = 0 for λ > λ max. P (λ) = P T for λ λ mn. When the prce s λ max λ < λ max., P (λ max ) can have two values. One s zero and the other s postve. In the sequel, unless explctly mentoned, P (λ max ) wll denote the postve value. Hence, wth a slght abuse of the

11 11 notaton, we redefne P (λ) n problem (D ) as Note that there exsts a λ max P (λ) = arg max 0 P P T {U (γ (P)) λp }. such that P (λ) = 0 for λ > λ max λ max the maxmum wllngness to pay of moble. and P (λ) > 0 for λ < λ max. Hence, we call Usng these propertes of P (λ), we can characterze the optmal moble selecton for problem (C), where the optmal moble selecton s defned as follows. Defnton 1: We call a subset of mobles S an optmal moble selecton for an optmzaton problem, f there exsts a λ that makes P = (P 1,P 2,,P M) a global optmal soluton of the problem, where P P (λ ), f S =. 0, otherwse In the followng, wthout loss of generalty, we assume that λ max 1 > λ max 2 > > λ max M 3. Proposton 1: Selectng mobles 1 from K for power allocaton s an optmal moble selecton for problem (C), where Further, f or K = max{1 j M K =1 j =1 P (λ max j ) P T }. P (λ max K+1) P T, K < M (4) K = M, (5) t s an optmal moble selecton for problem (A). Proof: See Appendx D. Proposton 1 mples that the mobles are selected n a decreasng order of λ max. By usng Proposton 1, we can develop a dstrbuted algorthm for moble selecton. Moble Selecton Algorthm (MSA) () The base-staton broadcasts ts maxmum power lmt, P T, to all mobles. () Each moble reports ts λ max 3 Snce λ max to the base-staton. of each moble depends on ts channel condton, n general, each moble has dfferent λ max. If some mobles have the same λ max, they can be ordered randomly.

12 12 () Let K = 1. (v) If K = M, select mobles from 1 to K and stop. (v) The base-staton broadcasts prce, λ max K+1. (v) Each moble reports ts power request P (λ max K+1) to the base-staton. (v) If K+1 =1 P (λ max K+1) > P T, select mobles from 1 to K and stop. Otherwse, let K = K + 1 and go to (v). The MSA needs O(M) teratons for selectng mobles. B. Power Allocaton for the Selected Mobles After the base-staton selects mobles usng the MSA n the prevous subsecton, t allocates ts power to the selected mobles. In ths subsecton, we assume that mobles, = 1, 2,,K are selected and λ max 1 > λ max 2 > > λ max K. In the proof of Proposton 1 n Appendx D, we have shown that f the condton n (4) or (5) s satsfed, to solve problem (C), we have to fnd a λ such that K =1 P (λ ) = P T, and t s also a global optmal power allocaton for problem (A). Further, we have shown that otherwse,.e., f K =1 P (λ max K+1) < P T and K+1 =1 P (λ max K+1) > P T, K < M, (6) the optmal soluton of problem (C) s λ max K+1 and K =1 P (λ max K+1) < P T. Hence, n ths case, the amount of power that s allocated to the selected mobles s less than P T at the optmal soluton of problem (C). However, from Lemma 1, we can ncrease the total system utlty by allocatng resdual power to the mobles. Hence, the purpose of ths stage s to fnd a λ that satsfes K =1 P (λ ) = P T. If there exsts such a power allocaton, from Lemma 2, t s a global optmal power allocaton for the selected mobles. To that end, the base-staton problem (E) for the selected mobles can be rewrtten as (F) K mn P T P (λ) λ =1 K subject to P (λ) P T, =1 0 λ λ max K. Hence, n the power allocaton stage, the base-staton solves problem (F) and each selected moble, = 1, 2,,K solves ts problem (D ). The next proposton wll show that the soluton of problem (F) and problem (D ) s a global optmal soluton for the set of selected mobles.

13 13 Proposton 2: There exsts a power allocaton, P K (λ ) = (P 1 (λ ),P 2 (λ ),,P K (λ )), whch s a soluton of problem (F) and problem (D ). Further, t satsfes K =1 P (λ ) = P T,.e., t s a global optmal soluton of the followng optmzaton problem: (G) K max U (γ (P )) P =1 K subject to P P T, =1 0 P P T, = 1, 2,,K. Proof: See Appendx E. As we have dscussed before, Proposton 2 also mples that f the condton n (4) or (5) s satsfed, t s also a global optmal power allocaton for all mobles. But, when the condton n (6) s satsfed, t may not be a global optmal power allocaton for all mobles. However, we wll show that our power allocaton asymptotcally optmal n the number of mobles. The power allocaton algorthm can be mplemented n several ways. Frst, f we consder problem (F), we can use lne search algorthms such as a golden secton algorthm [21], snce P T K =1 P (λ) s a unmodal functon. Secondly, snce we know that P T K =1 P (λ) has a unque root, λ for 0 λ λ max K and t s an optmal soluton of problem (F), we can use root fndng algorthms such as a bsecton algorthm. Fnally, f we consder problem (G), we can use a gradent based algorthm [8] or a penalty based algorthm [6], snce problem (G) s equvalent to the followng convex programmng problem. (H) max P K U (γ (P )) =1 Snce P (λ max K subject to K =1 P P T, P (λ max K ) P P T, = 1, 2,,K. ) P (λ max ) P o, = 1, 2,,K, U (γ (P )) s a concave functon for P (λ max ) P P T, = 1, 2,,K, whch makes problem (H) a convex programmng problem. In ths subsecton, we mplement the power allocaton algorthm usng a smple bsecton algorthm. Power Allocaton Algorthm Let ǫ be a small postve constant. () Set a (1) = 0, b (1) = λ max K and n = 1. () The base-staton broadcasts the prce λ (n) = a(n) +b (n) 2 to all selected mobles. K

14 14 () Each moble reports ts power requests P (λ (n) ) to the base-staton. (v) If b (n) a (n) 2ǫ or P T = K =1 P (λ (n) ), allocate power to the selected mobles as P K (λ (n) ) = (P 1 (λ (n) ),P 2 (λ (n) ),,P K (λ (n) )) and stop. Otherwse, go to (v). (v) If P T < K =1 P (λ (n) ), set a (n+1) = λ (n) and b (n+1) = b (n). Otherwse, set a (n+1) = a (n) and b (n+1) = λ (n). (v) n = n + 1 and go to (). If the power allocaton algorthm stops at teraton n, we have λ n λ ǫ, where λ s an optmal soluton of problem (F). Hence, a smaller value of ǫ can provde a more accurate soluton. Further, we can 1 easly show that 2n λmax K 2ǫ. Hence, n = mn{n log λmax K log 2ǫ, n = 1, 2, }. log 2 IV. ASYMPTOTIC OPTIMALITY AND A LOWER BOUND ON THE PERFORMANCE In ths secton, we frst study the asymptotc optmalty n the number of mobles of our power allocaton and, then, also study the lower bound on the worst case performance. Before we show the asymptotc optmalty of our power allocaton, we frst study the upper bound on the global optmal power allocaton of problem (A). Let us defne U u (P) as U u (P) = We now consder the followng optmzaton problem. λ max P, f 0 P P (λ max ). (7) U (γ (P)), f P (λ max ) P P T (U) max U u (P ) =1 subject to P P T, =1 0 P P T, = 1, 2,,M. Snce we can easly show that for each moble, U u (P ) U (γ (P )), P, problem (U) gves us an upper bound on the total achevable system utlty,.e., =1 U (γ (P )) =1 U u (P u ), where P = (P 1,P 2,,P M) and P u = (P u 1,P u 2,,P u M) be optmal solutons of problems (A) and (U), respectvely.

15 15 We now obtan a bound on the dfference between our power allocaton and the upper bound on the global optmal power allocaton. Proposton 3: Let P p = (P p 1,P p 2,,P p M) be our power allocaton and P u = (P u 1,P u 2,,P u M) be an optmal soluton of problem (U). Then, =1 where u max = max 1 M {U (γ (P T ))}. Proof: See Appendx F. U u (P u ) =1 U (γ (P p )) u max, Proposton 3 shows that the maxmum dfference between the system utlty obtaned by our power allocaton and the upper bound on the system utlty s at most the utlty of one moble. Further, snce we assume that the utlty functon of each moble s bounded, from Proposton 3, our power allocaton can be shown to be asymptotcally optmal n the followng sense. Corollary 1: Let P p = (P p 1,P p 2,,P p M) be our power allocaton and P = (P 1,P 2,,P M) be an optmal soluton of problem (A). If M =1 U (γ (P )) as M, M=1 U (γ (P p )) M=1 1, as M. U (γ (P )) Corollary 1 mples that f there are many mobles requrng a small amount of power n the system (.e., f the orthogonalty factor of the system s small, or f each moble has a large processng gan or a good transmsson envronment), our power allocaton scheme wll yeld a soluton close to the global optmal soluton. We now study the worst case performance of our algorthm. The next proposton provdes us a lower bound on the performance of our algorthm. Proposton 4: Let P p = (P p 1,P p 2,,P p M) be our power allocaton and P = (P 1,P 2,,P M) be an optmal soluton of problem (A). Then, M=1 U (γ (P P )) M=1 U (γ (P )) u mn, u max + u mn where u mn = mn 1 M {U (γ (P T ))} and u max = max 1 M {U (γ (P T ))}. Proof: See Appendx G.

16 16 The mplcaton s that the worst case performance can be poor when there are only few mobles n the system wth utlty functons that are qute dfferent from each other and maxmum wllngness to pays that are nversely proportonal to utlty values. However, n general, utlty functons are comparable wth each other and a moble wth a hgher utlty value has a hgher maxmum wllngness to pay than a moble wth a lower utlty value. Such a stuaton s unlkely to occur, especally when network provders wll lkely requre that users pck utlty functons from a pre-defned set. V. SPECIAL CASE: SINGLE CLASS OF MOBILES In ths secton, we study, for llustraton, a specal case of our method n whch all mobles are homogeneous,.e., each moble has the same U = U and the same N = N. We present ths case because t provdes some nsght. In the homogeneous case, we can show the followng propertes. Detals are provded n Appendx H. Let P = (P 1,,P M) be a global optmal power allocaton. (S1) (S2) If A < A j, then λ max > λ max j. If A < A j, then γ (P ) γ j (P j ). (S3) If P k = 0, then P j = 0 for all j such that A j > A k. Property (S1) shows the relatonshp between A and λ max. Ths mples that n the homogeneous case, mobles are selected n an ncreasng order of A by the MSA snce mobles are selected n a decreasng order of λ max by the MSA. Ths also mples that the moble n a better transmsson envronment has a greater chance to be selected by the MSA than the moble n a worse transmsson envronment. Furthermore, from property (S2), the former acheves a hgher utlty than the latter. Property (S3) mples that, at the global optmal soluton, mobles are selected n an ascendng order of A. By propertes (S1) and (S3), the order of moble selecton n our power allocaton s the same as that of the global optmal soluton. Hence, the set of mobles selected by the MSA s a subset of the set of mobles selected by the global optmal soluton and the relatonshp between mobles n each set s as follows: A j A, for,j V,j Z and Z, where V s the set of mobles selected at the global optmal soluton and Z s the set of mobles selected at our power allocaton. Ths mples that the MSA excludes only those mobles that obtan relatvely low utlty n the global optmal moble selecton and, thus, the dfference between ther acheved performance should be small.

17 17 BS BS BS BS BS BS BS BS BS Fg. 2. Cellular network model. VI. NUMERICAL RESULTS In ths secton, we provde numercal results of our power allocaton scheme for the CDMA network. Hence, the parameters N and γ n (1) correspond to the processng gan and E b /I 0 for moble, respectvely. For smplcty, we model the cellular system wth nne square cells, as shown n Fg. 2. We assume that the base-staton s located at the center of each cell and that each base-staton has the same maxmum power lmt, P T. We focus on the cell at the center of the system assumng that the base-statons n the other cells transmt at the maxmum power level, P T. We model the path gan from a base-staton to a moble j, G,j as follows: G,j = K,j, d α,j where d,j s the dstance from the base-staton to moble j, α s a dstance loss exponent, and K,j s the log-normally dstrbuted random varable wth mean 0 and varance σ 2 (db) that represents shadowng [22]. The parameters for the system are summarzed n Table I. For the smulaton, we use a sgmod utlty TABLE I PARAMETERS FOR THE SYSTEM Maxmum power (P T ) 10 Orthogonalty factor (θ) 1 Dstance loss exponent (α) 4 Varance of log-normal dstrbuton (σ 2 ) 8 Length of the sde of the cell 1000

18 a 1 = 1 a 2 = b 1 = 3 b 2 = Utlty 0.5 Utlty γ γ Fg. 3. Sgmod functons wth dfferent a (b = 5). Fg. 4. Sgmod functons wth dfferent b (a = 3). TABLE II COMPARISON OF UTILITY FOR THE HOMOGENEOUS CASE (b = 7(dB), N = 64, M = 10, 95% CONFIDENCE) a Our ± ± ± ± ± Global ± ± ± ± ± Upper ± ± ± ± ± Our/Global Our/Upper functon. The sgmod utlty functon s expressed as 1 U(γ) = c{ d}. (8) 1 + e a(γ b) We normalze the sgmod utlty functon such that U(0) = 0 and U( ) = 1 by settng c = 1+eab e ab d = 1 1+e ab. The sgmod utlty functons wth dfferent values for a and b are provded n Fgs. 3 and 4, respectvely. For each experment, we run the smulaton program 10 4 tmes and tabulate the average values (e.g., the total system utlty and the selecton rato of mobles n each class, whch s defned as the rato of the number of selected mobles to the number of mobles n each class). At each tme epoch of the smulaton, each moble s generated at a new locaton (wth new path gan) n the cell va an ndependent unform dstrbuton. We frst provde smulaton results for the sngle class case. We compare our power allocaton, the global optmal power allocaton, and the upper bound on the global power optmal allocaton. For the global and

19 19 TABLE III COMPARISON OF UTILITY FOR THE HOMOGENEOUS CASE(a = 3, N = 64, M = 10, 95% CONFIDENCE) b(db) Our ± ± ± ± ± Global ± ± ± ± ± Upper ± ± ± ± ± Our/Global Our/Upper TABLE IV COMPARISON OF UTILITY FOR THE HOMOGENEOUS CASE (a = 3, b = 7(dB), M = 10, 95% CONFIDENCE) N Our ± ± ± ± ± Global ± ± ± ± ± Upper ± ± ± ± ± Our/Global Our/Upper optmal power allocaton, we use an exhaustve search method. However by propertes n Secton V, the search regon can be reduced sgnfcantly for the sngle class case. In Tables II IV, we provde the total system utltes for each power allocaton, varyng the values of a, b, and N. Table II ndcates that as the value of a ncreases, the total system utlty ncreases. As shown n Fg. 3, as the value of a ncreases, less power s requred to acheve the same utlty for the concave regon and more power for the convex regon. In general, n our power allocaton, mobles that are allocated postve power are n the concave regon, as shown n Lemma 3 n Appendx B. Hence, generally, f other condtons are same, a moble wth a larger value of a n ts utlty functon requres less power than a moble wth a smaller value of a to acheve the same utlty. We say that, n ths case, the former s more effcent than the latter. Hence, the results ndcate that as the mobles n the system get more effcent, the total system utlty ncreases. Smlar results are provded n Tables III and IV. If other condtons are same, a moble wth a smaller value of b requres less power than a moble wth a larger value of b to acheve the same utlty, as shown n Fg. 4, and the moble

20 20 TABLE V COMPARISON OF PERFORMANCES OF TWO CLASSES (a 1 = 2, b 1 = b 2 = 7(dB), N 1 = N 2 = 64, M = 10, 95% CONFIDENCE) a Selecton rato of class ± ± ± ± ± Selecton rato of class ± ± ± ± ± Our ± ± ± ± ± Upper ± ± ± ± ± Our/Upper TABLE VI COMPARISON OF PERFORMANCES OF TWO CLASSES (a 1 = a 2 = 1, b 1 = 9(dB), N 1 = N 2 = 64, M = 10, 95% CONFIDENCE) b Selecton rato of class ± ± ± ± ± Selecton rato of class ± ± ± ± ± Our ± ± ± ± ± Upper 7.23 ± ± ± ± ± Our/Upper wth a smaller value of b s more effcent than the moble wth a larger value of b. Therefore, as the value of b decreases, the total system utlty ncreases, as shown n Table III. Also, f other condtons are same, a moble wth a larger value of N requres less power than a moble wth a smaller value of N to acheve the same γ (and thus, the same utlty) from (1) and the moble wth a larger value of N s more effcent than the moble wth a smaller value of N. Therefore, as the value of N ncreases, the total system utlty ncreases, as shown n Table IV. We also provde the rato of the system utltes of our power allocaton to that of other allocatons n Tables II- IV. As shown n these tables, the ratos are qute close to 1 n most cases, whch mples that the system utlty acheved by our power allocaton s qute close to that acheved by the global optmal power allocaton. In Tables V VII, smulaton results for a system wth two classes of mobles are provded. Each class s generated wth probablty 0.5. For the utlty functon of mobles n class, we set a = a, b = b and

21 21 TABLE VII COMPARISON OF PERFORMANCES OF TWO CLASSES (a 1 = a 2 = 1, b 1 = b 2 = 7(dB), N 1 = 64, M = 10, 95% CONFIDENCE) N Selecton rato of class ± ± ± ± ± Selecton rato of class ± ± ± ± ± Our ± ± ± ± ± Upper ± ± ± ± ± Our/Upper N = N. In ths case, we do not provde the system utlty acheved by the global optmal power allocaton, snce t s not easy to obtan. However, as shown n the tables, the ratos of the system utlty acheved by our power allocaton to that acheved by the upper bound on the global optmal power allocaton are qute close to 1 n most cases. Ths mples that, even n mult-class cases, the system utlty acheved by our power allocaton s close to that acheved by the global optmal power allocaton. In fact, as n the sngle class cases, the rato between these two allocatons s much closer to 1 than the rato between our power allocaton and the upper bound on the global optmal power allocaton. We also provde the moble selecton rato for each class n Tables V VII. The results show that the class of mobles wth a larger value of a (a smaller value of b, or a larger value of N) has a hgher selecton rato than one wth a smaller value of a (a larger value of b, or a smaller value of N). Ths mples that n our power allocaton, the moble that s more effcent has a hgher prorty to be selected than the moble that s less effcent. Ths effcent utlzaton of power results n our power allocaton achevng hgh system utlty. On the other hand, from the results, our power allocaton n whch only the effcency of the system s consdered could be unfar to some mobles (that are less effcent). VII. CONCLUSION In ths paper, we have developed a downlnk power allocaton algorthm for mult-class wreless networks by usng a utlty based framework allowng general types of utlty functons. The algorthm can be mplemented n a dstrbuted way usng a utlty and dynamc prcng framework. We have shown that t provdes an asymptotcally (n the number of mobles) optmal power allocaton. Further, numercal results show that ts performance s close to that of the global optmal power allocaton.

22 22 Power s a fundamental resource n wreless networks and other resource allocaton problems n wreless networks, such as data rate and tme must be studed based on the power allocaton scheme. Therefore, even though n ths paper, we consder only the power allocaton problem n wreless networks, our framework can be extended to other resource allocaton problems as well [23]. In ths paper, we have focused only on the effcency of the system wthout consderng farness among the mobles, whch could result n unfar power allocaton for some mobles. However, n resource allocaton, consderng farness as well as effcency such as the opportunstc schedulng schemes n [24], [25], [26] s an mportant ssue and s a topc for future research. A. Proof of Lemma 1 If M =1 P < P T, there exsts an α > 1 such that We defne P = αp for = 1, 2,,M, then APPENDIX P < α P = P T. =1 =1 γ ( P ) = = θ( j=1 N P P j P ) + A αn P θ( αp j αp ) + A j=1 αn P > θ( αp j αp ) + αa j=1 = γ ( P), = 1, 2,,M. Therefore, U (γ ( P )) > U (γ ( P)) for all, snce U s an ncreasng functon of γ. B. Proof of (3) We frst prove the followng lemma, from whch f moble requests postve power P (λ) at prce λ, then P (λ) = P T or U (γ (P (λ))) s n the concave regon. Lemma 3: P (λ) = 0 or P o P (λ) P T.

23 Proof: If 0 < P(λ) < P T, t must satsfy the frst and the second order necessary condtons for optmalty [21],.e., du (γ (P)) dp mples that P (λ) = 0 or P o P (λ) P T. P=P(λ) = λ, d2 U (γ (P)) dp 2 P=P(λ) 0, snce P(λ) s an nteror pont. Ths From Lemma 3, f the utlty functon U, of moble, s convex, then moble wll always request a power level of 0 or P T. We now prove (3). We frst defne w (λ) as 23 w (λ) = max {U (γ (P)) λp }, P o P P T whch s a non-ncreasng functon of λ. Then, by Lemma 3, and max 0 P P T {U (γ (P)) λp } = max{0,w (λ)} λ max = mn{λ 0 w (λ) 0}. (9) We now defne q (λ) = arg max P o P P T {U (γ (P)) λp }, λ T = du (γ (P)) dp P=PT, and λ o = du (γ (P)) dp P=P o. du (γ (P)) dp Then, snce U (γ (P)) s a concave functon for P o P P T, s a decreasng functon for P o P P T. Hence, λ T λ o and P T, f λ < λ T q (λ) = q (λ), f λ T λ λ o, (10) P o, f λ > λ o where q (λ) s a unque soluton of du (γ (P)) dp = λ, P o P P T. (11) Therefore, w (λ) = U (γ (P T )) λp T, f λ < λ T U (γ (q (λ))) λq (λ), f λ T λ λ o U (γ (P o )) λp o, f λ > λ o. (12) Snce, by the assumptons on the utlty functons, du (γ (P)) dp s a contnuous functon, q (λ) s a contnuous functon and, thus, w (λ) s a contnuous functon. Further, we can easly show that w (λ) s a decreasng

24 24 functon for λ λ o, w (λ o ) 0, and w (0) > 0. Ths mples that w (λ) = 0 has a unque soluton for 0 λ λ o and, by (9), λ max We now consder the followng equaton: s ts soluton. Hence, there exsts a unque λ max for moble. U (γ (P)) du (γ (P)) P = 0, P o P P T. (13) dp Then, by (10) - (12), there exsts a soluton P of the equaton n (13) f and only f there exsts a soluton λ of w(λ) = 0, λ T λ λ o. We frst assume that there exsts a soluton P of the equaton n (13) and let λ = du (γ (P)) dp P=P. Hence, q (λ ) = P and w (λ ) = 0. Ths mples that λ max = λ = du (γ (P)) dp P=P. We now assume that there s no soluton of the equaton n (13). Ths mples that w (λ) 0, λ T λ λ o. However, snce we have shown that w (λ) = 0 has a soluton for 0 λ λ o, t has a soluton λ max < λ T. Hence, by (12), λ max = U (γ (P T )) P T. If P o = 0,.e., U (γ (P)) s a concave functon, the equaton n (13) always has a soluton at P = P o = 0. Hence, n ths case, λ max = λ o = du (γ (P)) dp P=0. If P o = P T,.e., U (γ (P)) s a convex functon, we can easly show that U (γ (P T )) < du (γ (P)) dp P=PT P T. Hence, the equaton n (13) has no soluton and, thus, λ max = U (γ (P T )) P T. C. Propertes of P (λ) In ths subsecton, we study the propertes of P (λ). Throughout ths subsecton, λ T, λ o, q (λ), and w (λ) are defned as n Appendx B and we wll use ther propertes that have been shown there. For convenence, we summarze some useful propertes as follows: (B1) λ T λ o. (B2) w (λ max ) = 0. (B3) If there s a soluton P of the equaton n (13), then q (λ max ) = P and λ T λ max λ o. (B4) (B5) (B6) If there s no soluton of the equaton n (13), then λ max < λ T. If P o = 0, then λ max = λ o and there exsts a soluton P = 0 of the equaton n (13). If P o = P T, then there s no soluton of the equaton n (13). (B7) w (λ) s a decreasng functon for λ λ o. Further, by usng the defntons of w (λ) and q (λ), we can represent P (λ) as {0}, f w (λ) < 0 P (λ) {0,q (λ)}, f w (λ) = 0. (14) {q (λ)}, f w (λ) > 0

25 25 Property 1: P (λ max ) {0}, f P o = 0 {0,P }, {0,P T }, otherwse f 0 < P o < P T and P exsts, where P s a soluton of the equaton n (13). Proof: From (B2) and (14), P (λ max ) {0,q (λ max )}. If there exsts a soluton P of the equaton n (13), then q (λ max ) = P by (B3). Otherwse, λ max < λ T by (B4) and, thus, q (λ max ) = P T by (10). Hence, f P o = 0, then, P (λ max ) {0} by (B5). If P o = P T, then P (λ max ) {0,P T } by (B6). Property 2: P (λ) {0} for λ > λ max. Proof: λ > λ max If P o = 0, then q (λ) = P o = 0 for λ > λ max by (B5) and (10). Hence, P (λ) {0} for by (14). If P o 0, then we can easly show that w (λ) s a decreasng functon. Ths mples that w(λ) < 0 for λ > λ max by (B2). Hence, P (λ) {0} for λ > λ max by (14). Property 3: P (λ) s non-ncreasng n λ. Moreover, P (λ) s a decreasng and contnuous functon of λ for λ mn λ < λ max, f λ mn λ max, where λ mn = max{λ 0 P (λ) = P T }. Proof: We wll prove ths by consderng two dfferent cases. We frst suppose that there s no soluton of the equaton n (13) or that these exsts P, a soluton of the equaton n (13) and P = P T. By Property 2, P (λ) {0} for λ > λ max {0,P T }. In ths case, we can show that λ max (B1), (B2), and (B7) and q (λ) = P T for λ < λ max and by Property 1, P (λ max ) λ T by (B4) and (3). Hence, w (λ) > 0 for λ < λ max by (10). Ths mples that P(λ) {P T } for λ < λ max by (14). Hence, P(λ) s non-ncreasng n λ. Further, by the defnton of λ mn, n ths case, λ max = λ mn. λ T We now suppose that there exsts P, a soluton of the equaton n (13) and P P T. In ths case, λ max λ o by (B3). However, snce P P T, λ T < λ max by λ o. By Property 2, P (λ) {0} for λ > λ max. By Property 1 and (B3), P (λ max ) = {0,P } and P = q (λ max ). In a smlar way to the above case, we can show that w (λ) > 0 for 0 λ < λ max. Ths mples that P (λ) {q (λ)} for 0 λ < λ max by (14). Hence, by (10), P (λ) {P T } for λ < λ T. Agan, by (10), q (λ) s a soluton of du (γ (P)) dp = λ, P o P P T

26 26 for λ T λ λ max. Snce, by the assumptons on the utlty functon, du (γ (P)) dp s a contnuous and decreasng functon for P o P P T, q (λ) s a contnuous and decreasng functon for λ T λ λ max. Hence, P (λ) s a contnuous and decreasng functon for λ T λ T λ < λ max λ < λ max. Further, by the defnton of λ T, P (λ T ) {P T }. Ths mples that λ mn P(λ) s a non-ncreasng n λ and t s a contnuous and decreasng functon for λ mn and P (λ) > q (λ max ) for λ < λ max. = λ T. Hence, D. Proof of Proposton 1 When K = max{1 j M j =1 P (λ max ) P T }, we can have the followng three cases. j We frst consder the case when K = M. Then, M =1 P (λ max M ) P T. Snce each P (λ), = 1, 2,,M s a contnuous and non-ncreasng for λ λ max M, we can fnd a λ such that M =1 P (λ ) = P T and λ λ max M. Ths mples that λ s an optmal soluton of problem (C). Hence, selectng mobles from 1 to M s an optmal moble selecton for problem (C). Further, snce M =1 P (λ ) = P T, by Lemma 2, t s an optmal moble selecton for problem (A). We now consder the case when K < M and K =1 P (λ max K+1) P T. In ths case, n a smlar way to the above case, we can fnd a λ such that M =1 P (λ ) = K =1 P (λ ) = P T and λ K+1 λ λ K (f λ = λ max K+1, P K+1 (λ ) mples zero). Hence, selectng mobles from 1 to K s an optmal moble selecton for problems (A) and (C). Fnally, we consder the case when K < M and K =1 P (λ max K+1) < P T. Then, by the defnton of K, K+1 =1 P (λ max K+1) > P T. In ths case, due to the non-ncreasng property of P (λ), λ = λ max K+1 s an optmal soluton of problem (C) and selectng mobles from 1 to K s an optmal moble selecton for problem (C). However, snce M =1 P (λ ) = K =1 P (λ ) < P T (where P K+1 (λ ) mples zero) and M =1 P (λ ) = K+1 =1 P (λ ) > P T (where P K+1 (λ ) mples postve), n ths case, there s no λ o such that M =1 P (λ o ) = P T and ths moble selecton may not be an optmal moble selecton for problem (A). E. Proof of Proposton 2 By Proposton 1, K =1 P (λ K ) P T and P (λ), = 1, 2,,K s a non-ncreasng and contnuous functon for 0 λ λ K. Hence, there always exsts λ λ max K that satsfes K =1 P(λ ) = P T. Therefore, by Lemma 2, P K (λ ) = (P 1 (λ ),P 2 (λ ),,P K (λ )) s a global optmal soluton for problem (G).

27 27 F. Proof of Proposton 3 We assume that mobles from 1 to K are selected by the MSA and λ max 1 > λ max 2 > > λ max M. We defne P,u (λ) and λ max,u and as P,u (λ) = arg max 0 P P T {U u (P) λp } λ max,u = mn{λ 0 max 0 P P T {U u (P) λp } = 0}. Then, we can easly show that P,u (λ) = P (λ) (> 0), f λ < λ max {P 0 P P (λ max )} ( P (λ)), f λ = λ max P (λ) (= 0), f λ > λ max. (15) Ths mples that max 0 P PT {U u (P) λp } > 0 for λ < λ max λ λ max. Hence, λ max,u = λ max. and max 0 P PT {U u (P) λp } = 0 for We frst suppose that the condton n (4) or (5) s satsfed. In ths case, from the proof of Proposton 1 n Appendx D, there exsts a λ such that P (λ ) = P p, = 1, 2,,M and M =1 P (λ ) = P T. Snce, P p P,u (λ ) by (15) and M =1 P p = P T, by Lemma 2, P p s a global optmal soluton of problem (U). Hence, we can take P u = P p. Further, f P p > 0, then P p P (λ max ). Ths mples that, by the defnton of U u n (7), U u (P p ) = U (γ (P p )) for = 1, 2,,M. Hence, =1 U u (P u ) = =1 U (γ (P p )). We now suppose that nether of the condton n (4) nor (5) s satsfed,.e., from (6), K =1 P (λ max K+1) < P T and K+1 =1 P (λ max K+1) > P T, K < M. In ths case, from Propostons 1 and 2, P p = P (λ ) for = 1, 2,,K and P p = 0 for = K + 1,K + 2,,M, where λ satsfes K =1 P (λ ) = P T. Ths mples that λ λ max K+1 and, thus, P p P (λ max K+1) for = 1, 2,,K. Hence, =1 U (γ (P p )) = K =1 U (γ (P p )) K =1 U (γ (P (λ max K+1))). (16) We now defne P u = P (λ max K+1), f = 1, 2,,K P T K =1 P (λ max K+1), f = K , f = K + 2,K + 3,,M

28 Then, M =1 P u = P T and P u P (λ max K+1) for = 1, 2,,M by (15). Hence, by Lemma 2, t s a global optmal soluton of problem (U). Further, snce P (λ max K+1) P (λ max ) for = 1, 2,,K, by the defnton of U u n (7), 28 Therefore, by (16) and (17), =1 U u (P u ) =1 U (γ (P p )) U u (P u ) = U (γ (P (λ max K+1))), = 1, 2,,K. (17) K =1 U u (P (λ max K+1)) + U u K+1(P T = U u K+1(P T U u K+1(P T ) K =1 = U K+1 (γ K+1 (P T )) P (λ max K+1)) K =1 P (λ max K+1)) K =1 U (γ (P (λ max K+1))) max {U (γ (P T ))}. 1 M G. Proof of Proposton 4 By Proposton 3, M=1 U (γ (P p )) M=1 U (γ (P )) M=1 U (γ (P p )) u max + M =1 U (γ (P p )). Snce by Proposton 2, our power allocaton s a global optmal power allocaton for the selected mobles, =1 U (γ (P p )) U j (γ j (P T )) u mn, where moble j s one of the selected mobles by the MSA. Ths mples that M=1 U (γ (P p )) M=1 U (γ (P )) H. Proof of Propertes n the Homogeneous Case u mn u max + u mn. 1) Proof of property (S1): Snce U (γ) = U j (γ) and N = N j, A < A j mples that U(γ (P)) > U(γ j (P)) for 0 P P T. By the defnton of λ max, U(γ (P)) λ max P 0, 0 P P T. Hence, U(γ j (P)) λ max P < 0, 0 P P T.