Joint Resource Allocation and Base-Station. Assignment for the Downlink in CDMA Networks

Transcription

1 Jont Resource Allocaton and Base-Staton 1 Assgnment for the Downlnk n CDMA Networks 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 the jont resource allocaton and base-staton assgnment problem for the downlnk n CDMA networks wth heterogeneous data servces. We frst study a power and rate control problem that attempts to maxmze the expected throughput of the system. Ths problem s nherently dffcult because t s n fact a nonconvex optmzaton problem. To solve ths problem, we propose a dstrbuted algorthm based on dynamc prcng. Ths algorthm provdes a power and rate allocaton that s Pareto-optmal and asymptotcally optmal. We study the propertes of the proposed power and rate allocaton and the global optmal power and rate allocaton and show that they have smlar propertes. We also characterze the optmalty condtons of pure TDMA and CDMA type of transmssons, whch depend not only on the fxed system parameters but also on the channel state and the maxmum data rate of each moble. Fnally, usng the outcome of the proposed power and rate allocaton algorthm, we propose a prcng based base-staton assgnment algorthm resultng n jont resource allocaton and base-staton assgnment. In ths algorthm, a base-staton s assgned to each moble takng nto account the congeston level of the base-staton as well as the transmsson envronment of the moble. We show that t provdes hgher performance than the SIR based base-staton assgnment algorthm n whch only the transmsson envronment of the moble s consdered. I. INTRODUCTION The ncreasng demand for hgh data rate servces n wreless networks and scarcty of rado resources necesstate effcent usage of rado resources. Some of the man dffcultes n usng rado resources effcently are the tme and locaton dependent channel characterstcs and the demand to accommodate mobles Ths research has been supported n part by NSF grants ANI , ANI , and ANI

2 2 wth dverse servce requrements. Hence, for the effcent utlzaton of rado resources, t s necessary to adopt a resource allocaton scheme accordng to the channel status and the servce requrements of mobles. Moreover, f the jont allocaton of two or more resources s consdered, the performance gan could be sgnfcantly ncreased. Recently, there have been a number of papers that have studed jont resource allocaton problems [1], [2], [3], [4], [5], [6], [7]. Oh and Wasserman [1] consder a jont power and spreadng gan allocaton problem for the uplnk n Code Dvson Multple Access (CDMA) networks. They consder a sngle cell system and formulate an optmzaton problem for a sngle class of mobles. Ther objectve s to maxmze the total system throughput of mobles wth a constrant on the maxmum transmsson power of each moble. In ths work, they do not mpose any constrant on the spreadng gan (.e., data rate). They show that for the optmal soluton, mobles are selected for transmsson accordng to the channel state and f a moble s selected, t transmts at ts maxmum transmsson power. They generalze ther algorthm to mult-cellular networks n [2]. Bedekar et al. [3] and Berggren et al. [4] consder jont power and rate allocaton for the downlnk of the CDMA system wth a constrant on the total transmsson power for the base-staton. However, ther models do not consder that each moble could have a maxmum data rate constrant. They show that, for the downlnk of the system wthout the maxmum data rate constrant, selectng only one moble at a tme and allocatng the total transmsson power to that moble (.e., the pure Tme Dvson Multple Access (TDMA) type of strategy) s an optmal strategy. Jont power allocaton and base-staton assgnment problems for the uplnk have been consdered by Hanly [5] and Yates and Huang [6]. In these papers, the mnmum transmsson power satsfyng the Sgnal to Interference Rato (SIR) threshold of each moble s obtaned by jont power allocaton and base-staton assgnment. At each teraton of the algorthm, each moble s assgned to the base-staton that provdes the maxmum SIR. Saraydar et al. [7] also consder a jont power allocaton and base-staton assgnment problem for the uplnk case. They model the problem as an N-person non-cooperatve power allocaton game. Each moble selects the optmal power level and the base-staton that maxmzes ts net utlty, (.e., utlty mnus cost) wthout consderng other mobles. The performance of ther algorthm depends on the choce of the prce. However, they do not provde a strategy of how to determne the optmal prce. In ths paper, we study a jont resource allocaton and base-staton assgnment problem for CDMA networks. We focus on the downlnk problem n ths paper, snce the downlnk s typcally consdered a bottleneck due to the asymmetrc bandwdth demand between the downlnk and the uplnk for data servces [8],

3 3 [9], [10], [11]. We frst consder a jont power and rate allocaton problem by focusng on one cell of the system. Power s the most mportant resource n CDMA networks. By ncreasng the transmsson power level of a specfc moble, the performance of the moble can be mproved. But ths ncrease n transmsson power also deterorates the performance of the other mobles, snce t produces more nterference to other mobles. Hence, the power allocaton among mobles s an mportant resource allocaton problem n CDMA networks. Further, there s a trade off between the data rate and E b /I 0 (bt energy to nterference densty rato) that s related to the transmsson success probablty. At a gven transmsson power level, f the data rate s ncreased, E b /I 0 s decreased and vce versa. Therefore, by jontly optmzng the allocaton of power and data rate, the throughput of the system can be mproved sgnfcantly. In ths paper, the goal of the jont power and rate allocaton s to maxmze the total expected system throughput. We allow for constrants on the total transmsson power at the base-staton and the maxmum data rate for each moble. It turns out that ths problem s a non-convex programmng problem. In general, obtanng a global optmal soluton of a non-convex programmng problem s not easy and requres a complex algorthm. Hence, n ths paper, we propose a smple dstrbuted power and rate allocaton algorthm that provdes a Pareto-optmal power and rate allocaton as well as an asymptotc optmal power and rate allocaton. We also study the propertes of power and rate allocaton n the downlnk of the system n the presence of the constrant on the maxmum data rate and heterogeneous mobles. As mentoned above, t was proven that, for the downlnk n the dealzed system n whch there s no constrant on the maxmum data rate for each moble, the optmal strategy s to transmt only one moble at a tme. In addton, t can be easly shown that f all mobles are homogeneous, transmttng a moble n the best transmsson envronment s an optmal strategy. However, n practce, due to ether the physcal lmt of the hardware or the applcaton lmt, each moble cannot receve more than some maxmum data rate and there exsts a constrant on the maxmum data rate. Further, t s possble that each moble has a dfferent modulaton scheme and a dfferent codng level that depend on the channel state and the applcaton [12]. In ths case, each moble has a dfferent functon f, snce t depends on the modulaton and the codng schemes. Because of these two generalzatons (that are practcally mportant), a strategy that transmts only one moble n the best transmsson envronment may not be optmal. In ths paper, we characterze the optmalty condtons for pure CDMA type of transmsson n whch the base-staton transmts to all mobles smultaneously and pure TDMA type of transmsson n whch the base-staton transmts to only one moble at a tme, when there exsts a constrant on the maxmum data rate for each moble. We also study the propertes of our

4 4 moble selecton strategy, when there are heterogeneous mobles n the system. We next consder a mult-cellular system. Here we propose a prcng based base-staton assgnment algorthm. In the tradtonal SIR based base-staton assgnment, a moble s assgned the base-staton that can transmt to t at the hghest SIR. Hence, n ths scheme, only the transmsson envronment (the channel state and the nterference level) between a base-staton and a moble s consdered n determnng whch basestaton should communcate to a moble. However, when a cell s congested, by reassgnng some of the mobles n the cell to other cells that are less congested, we can expect to mprove the system performance by balancng the load of cells. Therefore, when a base-staton s assgned to a moble, by takng nto account the congeston level of the cell as well as the transmsson envronment of the moble, we can sgnfcantly mprove the performance of the system. However, obtanng optmal assgnment of base-statons may need complcated algorthm to be mplemented n practce, snce t may requre global coordnatons among basestatons. Hence, we propose a heurstc and smple algorthm n ths paper. To measure the congeston level n the cells and the transmsson envronment of mobles, we wll use the outcomes of the proposed power and rate allocaton algorthm n ths paper. Our base-staton assgnment algorthm has two dstngushng features compared wth base-staton assgnment algorthms proposed n [5], [6], [7]. Frst, t s smple and needs less sgnalng costs than the other algorthms. In our algorthm, each moble needs only base-statonspecfc nformaton that can be broadcasted from the base-staton, whle n other algorthms, each moble needs moble-specfc nformaton that requres for the base-staton to mantan sgnalng channels wth not only mobles n ts own cell but also mobles n other cells. Second, our algorthm s based on the downlnk performance, whle others are based on the uplnk performance. As mentoned before, n wreless networks, the downlnk s the bottleneck lnk. Hence, base-staton assgnment based on downlnk performance may be a more approprate approach. In many problems n wreless networks, the algorthm for the uplnk can be used for the uplnk wth smple modfcatons. However, as shown n [13], n the jont power allocaton and base-staton assgnment problems, the uplnk problem and the downlnk problem have dfferent propertes. Ths mples that the algorthm for the uplnk cannot be easly modfed for the use n the downlnk, whch necesstates the study of the downlnk problem ndependently. The rest of the paper s organzed as follows. In Secton II, we descrbe the system model. We present the jont power and rate allocaton algorthm n Secton III and study the propertes of power and rate allocaton n Secton IV. In Secton V, the base-staton assgnment algorthm s presented. We provde numercal results usng computer smulaton n Secton VI and conclude n Secton VII.

5 5 II. SYSTEM MODEL We consder the downlnk n a CDMA network consstng of B base-statons (cells) and M mobles. The system s assumed to be tme-slotted. At each tme slot (ths could be a generc nterval of tme), the power and data rate allocaton algorthm and the base-staton assgnment algorthm are executed. A tme slot n our system s an arbtrary nterval of tme and could consst of one packet or several packets. Each base-staton has ts maxmum power lmt P T and each moble communcates wth one base-staton. For a gven moble, R max s the maxmum data rate at whch t can receve and f s the probablty of packet transmsson success, whch s a functon of γ = E b /I 0 (bt energy to nterference densty rato) for moble and depends on the modulaton and codng level. We can express γ for moble that communcates wth base-staton b as: where γ (R, P (b)) = W G (b)p R θg (b) P m θg (b)p + I (b) m M(b) = W P R θ P m θp + I (b) m M(b) G (b) = W P R θ P m θp + A (b), m M(b) W θ : Chp rate. : Orthogonalty factor. P : Allocated power for moble. R : Data rate for moble. P (b) : Power allocaton vector for mobles that communcate wth base-staton b. G (b) : Path gan from base-staton b to moble. I (b) : Background nose and ntercell nterference to moble that communcates wth base-staton b. M(b) : Set of mobles that communcate wth base-staton b. A (b) : I (b)/g (b). We assume that f has the followng propertes. Assumptons: (a) f s an ncreasng functon of γ. (b) f s twce contnuously dfferentable.

6 6 (c) f (0) = 0. (d) 2 f (γ ) ( W γ 2 R + θγ )+2θ f (γ ) γ =0has at most one soluton for γ > 0. (e) If 2 f (γ ) ( W γ 2 R + θγ )+2θ f (γ ) γ =0has one soluton at γ o > 0, 2 f (γ ) ( W γ 2 R + θγ )+2θ f (γ ) γ γ <γ o and 2 f (γ ) ( W γ 2 R + θγ )+2θ f (γ ) γ < 0 for γ >γ o. > 0 for Remark 1: By the assumptons above, f M(b) P s constant, 2 f (γ ) P 2 In addton, f t has an nflecton pont P o, then 2 f (γ ) P 2 > 0 for P <P o and 2 f (γ ) P 2 has at most one nflecton pont. < 0 for P >P o. We wll show that M(b) P = P T later. Therefore, f can be one of three types: a sgmodal-lke functon of ts own power allocaton 1, a concave functon of ts own power allocaton, or a convex functon of ts own power allocaton. In most cases, the probablty of packet transmsson success can be characterzed by one of these three types of functons. In Fg. 1, we provde some examples for Bnary Phase-Shft Keyng (BPSK), Dfferental Phase-Shft Keyng (DPSK), and Frequency-Shft Keyng (FSK) modulaton schemes. In ths fgure, we assume a packet conssts of 800 bts wthout channel codng and set P T =10, θ =1, W R =16, and A (b) = III. POWER AND RATE ALLOCATION In ths secton, we study the power and data rate allocaton problem focusng on one cell of the system. For notatonal convenence, we wll omt the parameter for the base-staton n varables. We assume that mobles from 1 to M communcate wth ther target base-staton. Frst, we defne U, the utlty functon of 1 A sgmodal-lke functon means a functon f (x) that has one nflecton pont, x o and d2 f (x) dx 2.e. f (x) s a convex functon for x<x o and a concave functon for x>x o. > 0 for x<x o and d2 f (x) dx 2 < 0 for x>x o, f(p) BPSK DPSK FSK P Fg. 1. Probabltes of packet transmsson success for BPSK, DPSK, and FSK modulaton schemes.

7 7 moble as U (R, P ) = R f (γ (R, P )), =1, 2,,M, (1) whch s the expected throughput of moble. Then, the optmzaton problem for power and rate allocaton consdered n ths paper s gven by (A) M max U (R, P ) =1 M subject to P P T, =1 0 P P T, =1, 2,,M, 0 R R max, =1, 2,,M. Therefore, our goal s to maxmze the total expected system throughput (the sum of the expected throughput of all mobles) wth constrants on the total transmsson power at the base-staton and the maxmum data rate for each moble. To solve problem (A), we frst calculate the optmal data rate gven a power allocaton and, next, obtan the power allocaton. It can easly be shown that, to maxmze the total system throughput, the base-staton must transmt at ts maxmum power lmt, P T. Ths mples that M =1 P = P T and γ (R, P ) = W P R M θ P m θp + A m=1 = W R P θp T θp + A = γ (R,P ), =1, 2,,M. Note that γ (R,P ) does not depend on the power allocaton for other mobles and we can rewrte the utlty functon for moble defned n Equaton (1) as U (R,P ) = R f (γ (R,P )), (2) whch does not depend on the power allocaton for other mobles. Therefore, each moble can determne R (P ), the optmal data rate for a gven power allocaton P wthout consderng the other mobles. The optmal data rate for moble for a gven power allocaton s obtaned by the followng proposton.

8 8 Proposton 1: For the power allocaton P, R (P ), the optmal rate for moble, s gven by R (P ) = where γ = arg max γ 1 { 1 γ f (γ)}. Proof: See Appendx A. WP γ (θp T θp +A ), f P Rmax R max, otherwse, γ (θp T +A ) W +θr max γ, From Proposton 1 and Equaton (2), the utlty functon of moble at the optmal data rate allocaton for a gven power allocaton, P, U (R (P ),P ) s expressed as U R (P ) = U (R (P ),P ) = W γ P θp T θp +A f (γ ), f P Rmax R max f (γ (R max,p )), otherwse, γ (θp T +A ) W +θr max γ, (3) and problem (A) s equvalent to the followng optmzaton problem. 2 (B) max M =1 subject to U R (P ) M P P T, =1 0 P P T, =1, 2,,M. In problem (B), the problem s reduced to a power allocaton problem from the orgnal power and rate allocaton problem. After obtanng the approprate power allocaton by solvng problem (B), the optmal transmsson data rate can be obtaned va Proposton 1. U R Remark 2: In Equaton (3), U R (P ) s a convex functon for P Rmax (P ) follows the shape of f for Rmax γ (θp T +A ) W +θr max γ γ (θp T +A ) W +θr max γ and the shape of <P P T. Therefore, U R (P ) s a convex functon, or a sgmodal-lke functon of P, snce by assumpton, f s a convex functon, a concave functon, or a sgmodal-lke functon of P. Snce U R could be a sgmodal-lke functon, whch s the more nterestng case, n general, t s not easy to obtan an optmal soluton for problem (B). However, t turns out that the structure of problem (B) s smlar to that of the problem studed n [14]. In [14], a power allocaton algorthm that provdes a power allocaton that s Pareto-optmal as well as a good approxmaton of the global optmal power allocaton 2 A moble wth the fxed data rate R f can be easly accommodated n ths problem, f we assume that R (P )=R f for 0 P P T.

9 9 for sgmodal-lke, convex, and concave utlty functons. Further, t s proved that the power allocaton s asymptotcally optmal n [15]. We can hence explot the power allocaton strategy n [14] to obtan a Pareto-optmal and asymptotcally optmal power allocaton and obtan the optmal data rate for the power allocaton usng Proposton 1. In the followng, we brefly descrbe the power allocaton algorthm. We refer readers to [14], [15] for more detals. The power allocaton algorthm s a dynamc prcng based algorthm. The base-staton broadcasts λ, the prce per unt power to all mobles that communcate wth t. Based on λ, each moble requests a power level P (λ) that maxmzes ts net utlty,.e., P (λ) = arg max 0 P P T {U R (P ) λp }. (4) Based on the amount of power requested by all mobles, the base-staton updates λ to maxmze the total system utlty. The algorthm conssts of two stages, the moble selecton stage and the power allocaton stage. At the moble selecton stage, mobles are selected by the moble selecton algorthm: Moble Selecton Algorthm (a) If P 1 (λ max 1 )=P T, moble 1 s selected. (b) If k 1 j=1 P j (λ max k 1 ) <P T and k 1 j=1 P j (λ max k ) P T, mobles 1, 2,,k 1 are selected. (c) If k 1 j=1 P j (λ max k ) <P T and k j=1 P j (λ max k ) >P T, mobles 1, 2,,k 1 are selected. (d) If M j=1 P j (λ max M ) P T, mobles 1, 2,,M are selected. where λ max s the maxmum wllngness to pay per unt power of moble and s defned as and we assume that λ max 1 >λ max λ max = arg mn { max {U R (P ) λp } =0}, 0 λ 0 P P T 2 > >λ max M power of moble, snce P (λ) > 0 for λ λ max. We call λ max selecton stage, mobles are selected n a decreasng order of λ max constrant. Each moble can calculate λ max where P o λ max = as: the maxmum wllngness to pay per unt and P (λ) =0for λ>λ max. Therefore, at the moble U R (P ) P P =P, f 0 <P o <P T and P exsts, U R (P T ) P T, otherwse, as many as possble satsfyng the power s an nflecton pont of U R (P ) and P s a soluton of the followng equaton. U R (P ) P UR (P ) P = 0, P o P P T. (5)

10 10 After selectng mobles, the problem s reduced to a convex programmng problem. Thus, at the power allocaton stage, power s allocated optmally to the selected mobles usng a smple algorthm. In [14], the power allocaton algorthm was mplemented usng a bsecton algorthm. It was shown that f a condton n ether (a), (b), or (d) of the moble selecton algorthm s satsfed, the power allocaton s a global optmal power allocaton. But, f the condton n (c) of the moble selecton algorthm s satsfed, the power allocaton mght be dfferent from the global optmal power allocaton. A. Moble selectvty IV. STUDY OF THE PROPOSED POWER AND RATE ALLOCATION ALGORITHM We study the propertes of the moble selectvty n ths subsecton. Recall that A s defned by I /G, where I s background nose and ntercell nterference to moble and G s path gan from the base-staton to moble, f s the probablty of packet transmsson success for moble, and R max s the maxmum data rate for moble. Thus, A represents the degree of goodness of the transmsson envronment from the base-staton to moble. Smaller A mples that a better transmsson envronment. The next proposton shows that f moble s more effcent than moble j, then λ max λ max j and moble has a greater chance to be selected by the algorthm than moble j. Ths s the case because n the moble selecton algorthm, mobles are selected n a decreasng order of λ max. As ponted out n the prevous secton, the power allocaton s a global optmal power allocaton for the selected mobles. Therefore, wth ths effcent moble selecton combned wth the globally optmal power allocaton for the selected mobles, our algorthm can acheve hgh effcency of the system. We frst defne the effcency of mobles. Defnton 1: Moble s sad to be more effcent than moble j f U R (P ) U R j j (P ) for 0 P P T. Proposton 2: If moble s more effcent than moble j, then λ max Proof: See Appendx B. λ max j. Corollary 1: Suppose f (γ) =f j (γ) and A = A j. Then, f R max Proof: See Appendx C. >Rj max, λ max λ max j. Corollary 1 mples that f other condtons are same, the moble wth a hgher maxmum data rate has a hgher prorty to be selected than the moble wth a lower maxmum data rate. Corollares 2 and 3 can be proved n a smlar way to Corollary 1.

11 Corollary 2: Suppose R max = R max j and A = A j. Then, f f (γ) f j (γ) for all γ, λ max λ max j. In Corollary 2, f (γ) f j (γ) mples that moble needs a lower value of E b /I 0 to acheve the same transmsson success probablty as moble j,.e., moble has a more effcent transmsson scheme than moble j. Hence, Corollary 2 mples that f other condtons are same, the moble wth the more effcent transmsson scheme has a hgher prorty to be selected than the moble wth the less effcent transmsson scheme. 11 Corollary 3: Suppose f (γ) =f j (γ) and R max = Rj max. Then, f A <A j, λ max λ max j. In Corollary 3, A <A j mples that moble s n a better transmsson envronment than moble j. Hence, f other condtons are same, a moble n a better transmsson envronment has a hgher prorty to be selected than a moble n a worse transmsson envronment. Ths also mples that f all mobles are homogeneous, then mobles are selected n an ncreasng order of A,.e., mobles are selected accordng to ther transmsson envronment. It can be easly shown that the results n ths subsecton can be appled to the global optmal power and rate allocaton. Therefore, n the system wth heterogeneous mobles, the moble selectvty depends not only on the transmsson envronment of mobles but also on the maxmum data rate and the transmsson scheme of mobles. Ths mples that, n general stuatons n whch several condtons are mxed, t s not easy to determne whch mobles must be selected for the global optmal soluton. However, our moble selecton that selects mobles n a decreasng order of λ max provdes a smple and unfed strategy of moble selecton, whle provdng a good approxmaton of the global optmal soluton. B. Propertes of the proposed power and rate allocaton In ths subsecton, we study propertes of the proposed power and rate allocaton. The next proposton gves us the optmal transmsson data rate of the selected mobles. Proposton 3: Suppose that moble s selected by the moble selecton algorthm, then R, the transmsson data rate for moble s gven by Proof: See Appendx D. R = P T W γ A <R max, f A > P T W R max, otherwse. R max γ,

12 12 Proposton 3 mples that each moble can compute ts transmsson data rate when t s selected by the proposed power and rate allocaton algorthm before the power allocaton procedure, snce R does not depend on the ndvdual power level P. Ths s true even though the optmal data rate of a moble s a functon of the level of the power allocaton, as shown n Proposton 1. From the proof of Proposton 3, f R <R max, U R (P ) s a convex functon and moble s allocated the total transmsson power P T at prce λ = λ max. Ths mples that f a moble s selected for transmsson, t s always allocated the data rate ether R = R max that s a constant or R <R max that requres the total transmsson power P T. Thus, we can thnk that R does not depend on the ndvdual power level P. Usng ths property, moble can calculate ts transmsson data rate R after measurng ts transmsson envronment A and t can act lke a moble wth a fxed data rate R durng the power allocaton procedure. We also have the next corollary that ndcates that the jont power and data rate allocaton for a moble usng the proposed algorthm s one of boundary ponts of the ndvdual constrant set of the moble. Corollary 4: If moble s selected by the moble selecton algorthm and R selected and the total power s allocated to t. <R max, only moble s Corollary 4 mples that at most one moble can be allocated a data rate that s smaller than ts maxmum data rate. Further, f there exsts such a moble, other mobles cannot be selected. Therefore, f we assume that mobles are selected sequentally (n a decreasng order of the maxmum wllngness to pay) n the proposed scheme, a new moble can be selected, only when both the mobles that are already selected and the new moble can be allocated ther maxmum data rates. A smlar result can be shown for the global optmal power and rate allocaton. Proposton 4: If P =(P 1,,P M) s a global power allocaton, then at most one moble acheves utlty value n the convex regon of the utlty functon among the selected mobles for transmsson. Proof: See Appendx E. Proposton 4 mples that at the global optmal power and rate allocaton, at most one moble can be allocated a data rate that s smaller than ts maxmum data rate (smlar to the case n our scheme), snce at the concave regon of the utlty functon, a moble s allocated ts maxmum data rate. If some moble, that s allocated a smaller data rate than ts maxmum data rate, s selected for transmsson along wth other mobles, we can thnk that ths moble utlzes power most neffcently among the selected mobles. Ths has to be true, otherwse, by takng some power from the other mobles and allocatng t to the moble, we can ncrease ts

13 13 data rate up to the maxmum data rate and mprove the overall system performance. Hence, f we assume that mobles are selected for transmsson sequentally (n a decreasng order of the effcency of power utlzaton) at the global optmal power and rate allocaton, a new moble can be selected, only when mobles that are already selected can be allocated ther maxmum data rate as n our scheme. But n the global optmal scheme, a new moble can be allocated a data rate that s smaller than ts maxmum data rate, whch s not allowed n our scheme. Therefore, we can nfer that f we have a stuaton when a moble s n the convex regon at the global optmal power and rate allocaton, the proposed algorthm does not allocate power to the moble but dstrbutes power to be allocated to the moble to the other selected mobles optmally. As ponted out before, n general, the moble n the convex regon at the global optmal power and rate allocaton utlzes power less effcently than the other selected mobles and the utlty value n the convex regon s relatvely small. But fndng the moble n the convex regon at the global optmal power and rate allocaton s not easy. Hence, by gnorng the moble that s relatvely neffcent, we can obtan a good approxmaton of the global power and rate allocaton wth the smple proposed algorthm. C. Study of the optmalty condtons for TDMA and CDMA type of transmssons In ths subsecton, we study the optmalty condtons of pure TDMA type of transmsson that transmts to only one moble at a tme and pure CDMA type of transmsson that transmts to all mobles smultaneously. The next proposton gves us the optmal condtons for the TDMA type of transmsson and the CDMA type of transmsson. Proposton 5: (a) If P 1 (λ max 2 ) P T, where λ max 1 >λ max 2 > >λ max M, then selectng only moble 1 and allocatng the total transmsson power to t s an optmal strategy. (b) If M =1 P (λ max M ) P T, where λ max 1 >λ max 2 > >λ max M, then selectng all mobles and transmttng to all mobles smultaneously s an optmal strategy. Proof: See Appendx F. Note that the optmalty condtons n Proposton 5 depend not only on system parameters that are constant, but also on the transmsson envronment of each moble (tme varyng and locaton dependent) and the maxmum data rate of each moble (also tme varyng from the pont of vew of the system). Moreover, selectng a subset of mobles at a tme and transmttng only to them can be an optmal strategy dependng

14 14 on the system status. Ths mples that a statc strategy for transmsson could be neffcent n some cases and to obtan a hgh effcency of the system, we need a strategy that can be adapted to dynamc characterstcs of the system, such as the scheme proposed n ths paper. From Proposton 5 (a), we can also nfer that f R max > P T W A γ for all mobles (.e., f mobles have a hgh enough maxmum data rate or the transmsson envronment of mobles s bad because of hgh nterference or hgh channel loss), then pure TDMA type of transmsson s optmal. In ths case, R <R max for all mobles by Proposton 3 and U R (P ) s a convex functon for each moble that has P (λ max ) = P T. Therefore, by Proposton 5 (a), selectng only one moble and allocatng the total transmsson power to t s an optmal strategy. Ths also mples that f there s no constrant on the maxmum data rate for mobles, then TDMA type of transmsson s always optmal regardless of transmsson envronment of mobles as shown n [3], [4]. V. BASE-STATION ASSIGNMENT In the prevous sectons, we consdered a power and rate allocaton problem by focusng on one cell of the system. In ths secton, we consder a mult-cellular system and t brngs us two addtonal problems that were not consdered n the prevous sectons. One s the total transmsson power allocaton problem for each base-staton and the other s the base-staton assgnment problem for each moble. To maxmze the mprovement of the system performance, the power allocaton for base-statons must be done consderng the status of each cell and and the channel state of each moble. But, n practce, ths would requre a very complex power allocaton algorthm, snce t needs the cooperaton among base-statons (at least among adjacent base-statons). In addton, each base-staton may requre nformaton not only from mobles n ts cell, but also from mobles n other cells, resultng n a sgnfcant amount of sgnalng cost. For nstance, Oh et al. extend ther power control and spreadng gan allocaton algorthm for a sngle cell system [1] to a mult-cellular system [2]. To perform the algorthm, each base-staton needs to know the detaled nformaton of the status of the mobles n ts own cell and adjacent cells. However, even wth ths sgnfcant amount of nformaton, the algorthm s not guaranteed to converge to the optmal allocaton and requres much longer convergence tme than the algorthm for the sngle cell system. Therefore, n ths paper, we adopt a strategy that each base-staton tres to maxmze ts total system utlty wthout consderng the status of other cells. Usng ths strategy, each base-staton performs the power allocaton algorthm n the prevous secton by ndependently transmttng at ts maxmum power

15 15 level, whch makes the algorthm smple. Ths s typcally called a non-cooperatve stuaton and results n a Nash equlbrum operatng pont, whch mght be neffcent,.e., there mght exst another total power allocaton wth whch each moble can obtan hgher utlty than the Nash equlbrum power allocaton. If the load of each cell s unbalanced, the neffcency mght be large. For example, f one cell s heavly loaded and an adjacent cell s lghtly loaded, by decreasng the total transmsson power of the base-staton n the lghtly loaded cell, nterference for mobles n the heavly loaded cell can be decreased. Ths results n a decrease of the utlty of the lghtly loaded cell and an ncrease n the utlty of the heavly loaded cell. In such a case, generally, the ncrease can be larger than the decrease n utlty and hence the total system utlty could be hgher than the total system utlty at the Nash equlbrum. To cope wth ths stuaton, we propose a base-staton assgnment algorthm. By reassgnng some mobles n the heavly loaded cells to lghtly loaded cells, we can expect to mprove the total system utlty by balancng the load among the base-statons. We next descrbe the proposed base-staton assgnment algorthm. Each base-staton performs the power and data rate allocaton algorthm n the prevous secton ndependently. For the base-staton assgnment, we use the results of the power and data rate allocaton algorthm, whch s a prcng based algorthm. Thus, we call our base-staton assgnment the prcng based base-staton assgnment. We frst defne some varables. S c (b): Set of mobles that are not selected for transmsson by base-staton b among mobles that communcate wth base-staton b. λ max (b): Maxmum wllngness to pay per unt power of moble for base-staton b. λ (b): Equlbrum prce per unt power for base-staton b. max S w(b) = c (b){λ max (b)}, f S c (b) φ, 0, otherwse. Also, recall that A (b) ndcates the transmsson envronment between base staton b and moble. Smaller A (b) ndcates a better transmsson envronment. Assume that moble s currently communcatng to base-staton b. If moble s selected for transmsson by base-staton b, t contnues to communcate to base-staton b n the next tme slot. But, f moble s not selected for transmsson by base-staton b, at the next tme slot, t selects and connects to a base-staton d that satsfes d = arg max k g() {λmax (k) λ (k)}, (6)

16 16 where g() ={k B λ max (k) w(k)} (7) and B s the set of base-statons. Note that B may not be the set of all base-statons. It could be a subset of all base-statons. For example, B could be the set of adjacent base-statons of base-staton b or the set of base-statons from whch moble receves plot sgnal wth enough strength. If w(b) >λ max (b), moble cannot be selected by base-staton b and n Equaton (7), we exclude base-staton b B from g() whose w(b) s larger than λ (b). Therefore, g() can be thought as the set of canddate base-statons by whch moble can be selected. By Equaton (6), the moble selects a base-staton that has the maxmum value of the maxmum wllngness to pay for the base-staton mnus the equlbrum prce of the base-staton among the canddate base-statons. We can explan the ntuton of the proposed algorthm usng two nterpretatons. Frst, from the pont of vew of prcng, we can thnk that the moble selects the base-staton whch gves t the hghest proft per unt power. Snce λ max (b) can be nterpreted as the maxmum value for unt power of moble at base-staton b and λ (b) can be nterpreted as the current prce for unt power at base-staton b, λ max (b) λ (b) can be nterpreted as the proft per unt power that moble can obtan f t s selected by base-staton b. Another nterpretaton s that the moble selects the base-staton wth whch t has the relatvely best transmsson envronment consderng the congeston level of the base-staton. The next proposton tells us that λ max (b) s an ndcator of the transmsson envronment between base-staton b and moble. Proposton 6: If λ max (n) <λ max (m), then A (n) >A (m). Proof: See Appendx G. By the defnton of A (b), λ max (n) <λ max (m) mples that the transmsson envronment between basestaton m and moble s better than the transmsson envronment between base-staton n and moble. Thus, λ max (b) can be nterpreted as the goodness of the transmsson envronment of moble when t s connected to base-staton b. Also, we can nterpret λ (b) as beng the congeston level at base-staton b from the followng propostons. Proposton 7: Suppose moble s selected by base-staton n and moble j s selected by base-staton m. Further suppose that f = f j, R max P j (λ (m)). Proof: See Appendx H. = Rj max, and A (n) =A j (m). Then, f λ (n) <λ j(m), P (λ (n))

17 17 What Proposton 7 tells us s that f two mobles are n the same condton and they are selected by dfferent base-statons, a moble n the base-staton wth a lower equlbrum prce s allocated more power than a moble n the base-staton wth a hgher equlbrum prce. In other words, the base-staton wth a hgher equlbrum prce has more demand for power than the base-staton wth a lower equlbrum prce. Thus, we can nterpret λ (b) as beng the congeston level of base-staton b. Hence, λ max (b) λ (b) n Equaton (6) can be nterpreted as the relatve goodness of the transmsson envronment between moble and base-staton b, takng nto account the congeston level of the cell. From ths nterpretaton, Equaton (6) mples that moble s reassgned to base-staton d whch has the relatvely best transmsson envronment consderng the congeston level of the cell among the canddate base-statons from whch t can be selected. To perform the base-staton assgnment algorthm n Equatons (6) and (7), moble must calculate λ max (b) and know λ (b) and w(b) for b B. Moble can calculate λ max (b), f t knows A (b) and P T and A (b) can be measured at moble. Hence, the nformaton that moble needs to perform the algorthm are P T, λ (b) and w(b) for b B and these parameters must be transmtted from base-staton b. But ths s base-staton-specfc nformaton not moble-specfc nformaton and ths can be broadcasted by each base-staton. In contrast, for the algorthms n [5], [6], [7], each moble needs to know some moble-specfc nformaton that must be transmtted from the base-statons that are canddates for the assgnment, requrng each moble to mantan ts own sgnalng channels wth several base-statons. Therefore, our algorthm needs less sgnalng costs than the algorthms n [5], [6], [7]. VI. NUMERICAL RESULTS In ths secton, we provde numercal results of our power and rate allocaton algorthm and base-staton assgnment algorthm by computer smulatons. We also compare the performance of our algorthms wth the performance of Qualcomm s HDR [8], [9], whch s proposed for downlnk transmsson for hgh data rate servces. In HDR, each base-staton transmts a plot sgnal. Each moble measures ts receved SIR from the plot sgnals of several base-statons and s connected to the base-staton wth the hghest SIR. At each base-staton, for downlnk transmsson, the base-staton transmts at the maxmum power, f there s at least one moble to transmt to, wthout consderng the status of other cells. The downlnk transmssons are tme multplexed and at each tme slot, each base-staton transmts to only one moble. Therefore, gnorng farness, the optmal strategy that maxmzes the throughput n HDR s to transmt to a moble that can acheve the hghest expected throughput. We adopt ths strategy to obtan the performance of HDR, snce n ths paper, we do not consder farness.

18 18 BS BS BS BS BS BS BS BS BS Fg. 2. Cellular network model. TABLE I PARAMETERS FOR THE SYSTEM. Maxmum power (P T ) 10 Chp rate (W ) Dstance loss exponent (α) 4 Varance of log-normal dstrbuton (σ 2 ) 8 Length of the sde of the cell 1000 For the smulaton, we model the cellular network wth 9 square cells as shown n Fg. 2. We assume that the base-staton s located at the center of each cell. We model the path gan from a base-staton to a moble j, G,j as G,j = K,j, (8) 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 lognormally dstrbuted random varable wth mean 0 and varance σ 2 (db), whch represents shadowng [16]. The parameters for the system are summarzed n Table I. For the smulaton, we use a sgmod functon to represent f (γ). The sgmod functon s expressed as where we set c = 1+ea h e a h and d = 1 1+e a h 1 f (γ) = c { 1+e d a (γ h ) }, (9) for the normalzaton.

19 19 TABLE II THE PROBABILITY OF PACKET TRANSMISSION SUCCESS (a =3, h =3.5). γ(db) f(γ) We frst present the performance of the power and rate allocaton algorthm wthout consderng basestaton assgnment. We focus on the cell at the center of the system and assume that the base-statons n other cells transmt at the maxmum power lmt. In Tables III - V, we assume that there are two classes of mobles and set the orthogonalty factor, θ =1. In Tables III - IV, we have 10 mobles that are located ndependently accordng to a unform dstrbuton and mobles n each class are generated wth probablty 0.5. In Table III, each class s assumed to have the same f (γ) =f(γ) but dfferent R max.forf(γ), we set a =3and b =3.5 and the values of t for several γ s are provded n Table II. We provde smulaton results by settng R2 max = 6250 (N2 mn =16) and varyng R1 max (N1 mn ). Frst, the selecton rato of mobles of each class s provded. The results n Table III ndcate that the class wth the hgher R max has a hgher selecton rato of mobles than the class wth the lower R max. Ths mples that the proposed power and rate allocaton gves a hgher prorty to be selected to mobles wth the hgher maxmum data rate, as proved n Corollary 1. We also provde the system utltes obtaned by the proposed power and rate allocaton and HDR, and the rato of ther utltes. When R1 max = 25000, these two algorthms gve almost the same utlty. Ths s because f there exst mobles wth hgh maxmum data rate, the optmal strategy s to select one moble and TABLE III COMPARISON OF PERFORMANCES OF OUR POWER AND RATE ALLOCATION AND HDR. R max (N mn 1 ) (64) (32) (16) (8) (4) Selecton rato of class Selecton rato of class Utlty (Our algorthm) Utlty (HDR) Utlty (Our algorthm)/utlty (HDR)

20 20 TABLE IV COMPARISON OF PERFORMANCES OF OUR POWER AND RATE ALLOCATION AND HDR. h Selecton rato of class Selecton rato of class Utlty (Our algorthm) Utlty (HDR) Utlty (Our algorthm)/utlty (HDR) allocate the total transmsson power to that moble. However, as R1 max decreases, the utlty acheved by our power and rate allocaton becomes much hgher than that of HDR. The reason for ths s that f R1 max s not hgh, the optmal strategy s to select several mobles and transmt to them smultaneously. From the results, we know that HDR s optmzed only for hgh data rate servces and t could be very neffcent, f there exst heterogeneous servces, whle our algorthm can be used n any cases wth hgh effcency. In Table IV, numercal results are provded by varyng h 1. We set R1 max = R2 max = 6250 (N1 mn = N2 mn = 16), a 1 = a 2 =3, and h 2 =3.5. As presented n ths table, the class wth the lower value of h has a hgher selecton rato than the class wth the hgher value of h. By Equaton (9), the moble wth the lower h needs the lower SIR to acheve the same probablty of packet transmsson success than the moble wth the hgher h. Hence, the former has the more effcent transmsson scheme and a hgher prorty to be selected than the latter, as proved n Corollary 2. We also compare the performance of the proposed power and rate allocaton and HDR. From the results, we know that as h 1 decreases the dfference between the two algorthms gets larger. Hence, the results ndcates that as mobles get more effcent, the base-staton can select more mobles and transmt them smultaneously to mprove the system throughput. In table V, we assume that each class has the same maxmum data rate, R max = 6250, and the same functon for the probablty of packet transmsson success, f = f. We set a =3and b =3.5. But we dvde the cell nto two regons: an nner regon that s a square at the center of the cell whose length of the sde s a half of the length of the sde of the cell and an outer regon that s the remanng regon of the cell. We say that mobles n the nner regon belong to class 1 and the mobles n the outer regon belong to class 2. Thus, a moble n class 1 s closer to the base-staton than a moble n class 2 and, n general, the former s n a better transmsson envronment than the latter. As we studed n Corollary 3, the comparson of selecton

21 21 TABLE V COMPARISON OF PERFORMANCES OF OUR POWER AND RATE ALLOCATION AND HDR. Rato of class Selecton rato of class Selecton rato of class Utlty (Our algorthm) Utlty (HDR) Utlty (Our algorthm)/utlty (HDR) TABLE VI COMPARISON OF PERFORMANCES OF OUR POWER AND RATE ALLOCATION AND HDR. θ Utlty (Our algorthm) Utlty (HDR) Utlty (Our algorthm)/utlty (HDR) rato ndcates that mobles n a better transmsson envronment have a hgher prorty to be selected. The comparson of the performance of our scheme and that of HDR shows that as the number of mobles n a better transmsson envronment ncreases, our scheme more outperforms a scheme lke HDR. Ths s because mobles n a better transmsson envronment need less power to acheve the desred performance and, thus, as the number of mobles n a better transmsson envronment ncreases, the base-staton can transmt to more mobles smultaneously. In table VI, we assume that a sngle class of mobles are n the cell. We set R max = 25000, a = 3, and b =3.5. We provde the result varyng θ, the orthogonalty factor of the system. As we have a smaller orthogonalty factor, mobles have less ntracell nterference and the base-staton can transmt to more mobles smultaneously. Ths s ndcated by the result that our scheme more outperforms HDR, as the orthogonalty factor gets smaller. We now present the performance of the proposed base-staton assgnment algorthm. Recall that n the proposed base-staton assgnment algorthm, power and rate are allocated usng our power and rate allocaton n the prevous secton. We call ths Algorthm A. We also compare t wth two other algorthms. One s called

22 22 Algorthm B, n whch the power and rate are allocated usng the proposed power and rate allocaton, but each moble s assgned to the base-staton wth the hghest SIR, as n HDR. In ths case, moble s assgned to base-staton d such that d = arg mn b B {A (b)}. (10) The comparson of these two algorthms shows us the performance gan of the prcng based base-staton assgnment algorthm over the SIR based base-staton assgnment algorthm. The other algorthm that we compare wth s the one n HDR, n whch each moble s assgned to the base-staton usng Equaton (10), and each base-staton transmts wth full power to only one moble that can receve the hghest data rate. We assume that there are two classes of mobles and set a 1 = a 2 =3, h 1 = h 2 =3.5, R1 max = (N1 mn =8), and R2 max = (N2 mn = 64). Mobles n class 1 are generated wth probablty 0.2 and mobles n class 2 wth probablty 0.8. For the smulaton, we model hot spot stuaton as follows. Frst, M u mobles are located n each cell wth unform dstrbuton. We call these mobles balanced mobles. Thus, there exst a total of 9 M u balanced mobles n the system. Balanced mobles are generated n the system one by one n each cell n a sequental order, whle tryng to preserve a balanced load for each cell. Next, M h mobles are located one by one n the center cell wth unform dstrbuton. We call these mobles hot spot mobles. Therefore, a total of 9 M u + M h mobles are n the system wth M u + M h mobles n the center cell and M u mobles n each of other cells. In addton, we assume that σ 2 =0n Equaton (8), whch mples that at the SIR based base-staton assgnment, each moble s assgned the closest base-staton from t. In Fgs. 3 5, we compare the system utlty acheved by each algorthm by varyng M u. In the way we generate mobles, f the number of mobles s less than or equal to 9, each base-staton has at most one moble, and, thus, all three algorthms provde the same results. For balanced mobles (.e., up to 9 M u mobles) Algorthm A and Algorthm B acheve almost the same utlty. But, durng the hot spot perod, Algorthm A outperforms Algorthm B. Ths mples that the prcng based base-staton assgnment outperforms the SIR based base-staton assgnment at the hot spot stuaton by reassgnng mobles connected to the more congested base-staton to less congested base-statons. As shown n the sngle cell results, Algorthm B outperforms HDR, and, thus, so does Algorthm A. In Fg. 6, we provde the rato of the system utlty acheved by Algorthm A to that acheved by Algorthm B for M u =1, 2, 3, respectvely. The only dfference between Algorthm A and Algorthm B s the base-staton assgnment algorthm. Hence, the dfference n the performance of these two algorthms provdes the effect of the base-staton algorthm for

23 23 System utlty 7 x Algorthm A Algorthm B HDR System utlty 10 x Algorthm A Algorthm B HDR Number of mobles Number of mobles Fg. 3. System utlty (M u =1). Fg. 4. System utlty (M u =2). System utlty 12 x Algorthm A Algorthm B HDR Algorthm A/Algorthm B M u = 1 M u = 2 M u = Number of mobles Number of mobles Fg. 5. System utlty (M u =3). Fg. 6. The rato of utlty of Algorthm A to that of Algorthm B. the system performance. Snce, as the unbalancedness of the system ncreases (.e., as M u decreases), more mobles n the more congested cell can be reassgned to less congested cells, the performance gan of Algorthm A over Algorthm B ncreases wth the unbalancedness of the system. The results also mples that balancng the load of each cell by connectng mobles to the base-staton consderng the load of each cell and the channel state s more benefcal to ncrease the system utlty than connectng mobles to the base-staton wth the best channel state. We now compare the ncrement of the system utlty durng the hot spot perod n Fgs In these fgures, the ncrement n system utlty s the ncrement n the system utlty from the system utlty acheved when only balanced mobles exst, (.e., the system utlty acheved by all mobles mnus the system utlty when only 9 M u balanced mobles exst). The number of mobles durng the hot spot perod, M h, means that the number of mobles added at the center cell after 9 M u balanced mobles are located. Hence