Wreless Networks 8, 659 670, 2002 2002 Kluwer Academc Publshers. Manufactured n The Netherlands. CDMA Uplnk Power Control as a Noncooperatve Game TANSU APCAN, TAMER BAŞAR and R. SRIKANT Coordnated Scence aboratory, Unversty of Illnos, 1308 West Man Street, Urbana, I 61801, USA EITAN ATMAN INRIA, B.P. 93, 06902 Sopha Antpols cedex, France, and CESIMO, Facultad de Ingenera, Unversdad de os Andes, Merda, Venezuela Abstract. We present a game-theoretc treatment of dstrbuted power control n CDMA wreless systems. We make use of the conceptual framework of noncooperatve game theory to obtan a dstrbuted and market-based control mechansm. Thus, we address not only the power control problem, but also prcng and allocaton of a sngle resource among several users. A cost functon s ntroduced as the dfference between the prcng and utlty functons, and the exstence of a unque Nash equlbrum s establshed. In addton, two update algorthms, namely, parallel update and random update, are shown to be globally stable under specfc condtons. Convergence propertes and robustness of each algorthm are also studed through extensve smulatons. Keywords: power control, CDMA, game theory, prcng, resource allocaton 1. Introducton In wreless communcaton systems, moble users respond to the tme-varyng nature of the channel, descrbed usng shortterm and long-term fadng phenomena [9], by regulatng ther transmtter powers. Specfcally, n a code dvson multple access (CDMA) system, where sgnals of other users can be modeled as nterferng nose sgnals, the maor goal of ths regulaton s to acheve a certan sgnal-to-nterference (SIR) rato regardless of channel condtons whle mnmzng the nterference due to user transmt power level. Hence, there are two maor reasons for a user to exercse power control: the frst one s the lmt on the battery energy avalable to the moble, and the second reason s the ncrease n capacty, whch can be acheved by mnmzng the nterference. Power control n CDMA systems are n ether open-loop or closed-loop form. In open-loop power control, the moble regulates ts transmtted power nversely proportonal to the receved power. In closed-loop power control, on the other hand, commands are transmtted to the moble over the downlnk to ncrease or decrease ts uplnk power [12, p. 182]. A specfc proposal to mplement dstrbuted power control made by Yates [13] reles on each user updatng ts power based on the total receved power at the base staton. It has been shown n [13] that the resultng dstrbuted power control algorthm converges under a wde varety of nterference models. Another dstrbuted power control scheme has been ntroduced n [7], whch s adaptve and uses local measurements of the mean and the varance of the nterference. The authors have shown that ths algorthm s convergent provded that a certan condton s satsfed. Game theory provdes a natural framework for developng prcng mechansms of drect relevance to the power control Correspondng author. problem n wreless networks. In such networks, the users behave noncooperatvely,.e., each user attempts to mnmze ts own cost functon (or maxmze ts utlty functon) n response to the actons of the other users. Ths makes the use of noncooperatve game theory [1] for uplnk power control most approprate, wth the relevant soluton concept beng the noncooperatve Nash equlbrum. In ths approach, a noncooperatve network game s defned where each user attempts to mnmze a specfc cost functon by adustng hs transmsson power, wth the remanng users power levels fxed. The man advantage of ths approach s that t not only leads to dstrbuted control as n [13], but also naturally suggests prcng schemes, as we wll see n ths paper. Possble utlty functons n a game theoretcal framework, and ther propertes for both voce and data sources have been nvestgated n detal n [4], whch formulates a class of utlty functons that also account for forward error control, and shows the exstence of a Nash equlbrum and the unqueness of an optmal response. One nterestng feature of ths framework s that t provdes utlty functons for wreless data transmsson, where power control drectly affects the capacty of mobles data transmsson rates. Falomar et al. [4] also propose a lnear prcng scheme n order to acheve a Pareto mprovement n the utltes of mobles. In an earler study [10], Nash equlbra acheved under a prcng scheme have been characterzed by usng supermodularty. It has been shown that a noncooperatve power control game wth a prcng scheme s superor to one wthout prcng. One defcency of ths game setup, however, s that t does not guarantee socal optmalty for the equlbrum ponts. near and exponental utlty functons based on carrer (sgnal)-to-nterference rato are also proposed n [5]. The exstence of a Nash equlbrum s shown under some assumptons on the utlty functons, and an algorthm for solvng the noncooperatve power control game s suggested.
660 APCAN ET A. In ths paper, we propose a power control game smlar to the ones n [4,5]. In the model we adopt, however, we use a cost functon defned as the dfference between a lnear prcng scheme proportonal to transmtted power, and a logarthmc, strctly concave utlty functon based on SIR of the moble. Furthermore, the utlty functon s made user-specfc by multplyng t wth a utlty parameter reflectng the ndvdual user preferences. We then rgorously prove the exstence and unqueness of a Nash equlbrum. We also nvestgate possble boundary equlbrum solutons, and hence derve a quanttatve crteron for admsson control. As n [13], one way of extendng the model s to nclude certan SIR constrants. As an alternatve, we suggest a prcng strategy to meet the gven constrants, and analyze the relaton between prce, SIR, and user preferences as reflected by the utlty parameter. Thus, we address not only the power control problem, but also prcng and allocaton of a sngle resource among several users. Furthermore, we study dfferent prcng strateges, and obtan a dstrbuted and market-based power control mechansm. Fnally, under a suffcent condton we prove the convergence of two algorthms, parallel update (PUA) and random update (RUA), to the unque Nash equlbrum. In order to llustrate the convergence, stablty and robustness of the update algorthms, we use extensve smulatons usng MATAB. Moreover, we study the effect of the varous parameters of the model, especally dfferent prcng schemes. In order for the smulatons to capture realstc scenaros, we ntroduce feedback delay and modelng dsturbances, where the latter s caused by varatons n the number and locaton of users n the network. The next secton descrbes the model adopted and the cost functon. In secton 3, we prove the exstence and unqueness of the Nash equlbrum. We present update algorthms for mobles n secton 4, whereas secton 5 ntroduces dfferent prcng strateges at the base staton. The smulaton results are gven n secton 6, whch s followed by the concludng remarks of secton 7. 2. The model and cost functon We descrbe here the smple model adopted n ths paper for a sngle cell CDMA system wth up to M users. The number of users s lmted under an admsson control scheme that ensures the mnmum necessary SIR for each user n the cell. For the th user, we defne the cost functon J as the dfference between the utlty functon of the user and ts prcng functon, J = P U. The utlty functon, U, s chosen as a logarthmc functon of the th user s SIR, whch we denote by γ. Ths utlty functon can be nterpreted as beng proportonal to the Shannon capacty [9,11] for user, fwe make the smplfyng assumpton that the nose plus the nterference of all other users consttute an ndependent Gaussan nose. Ths means that ths part of the utlty s smply lnear n the throughput that can be acheved (or approached) by user usng an approprate codng, as a functon of ts transmsson power. Ths logarthmc functon s further weghted by a user-specfc utlty parameter, u > 0, to capture the user s level of desre for SIR. The prcng functon defnes the nstantaneous prce a user pays for usng a specfc amount of power that causes nterference n the system. It s a lnear functon of p,the power level of the user. Accordngly, the cost functon of the th user s defned as J (p,p ) = λ p u ln(1 + γ ), p 0, (2.1) where p denotes the vector of power levels of all users except the th one, and γ s the SIR functon for user,gvenby h p γ = h p + σ 2. (2.2) Here, = W/R s the spreadng gan of the CDMA system, where W s the chp rate and R s the total rate; we assume throughout that >1. The parameter h,0<h < 1, s the channel gan from user to the base staton n the cell, and σ 2 > 0 s the nterference. For notatonal convenence, let us denote the th user s power level receved at the base staton as y := h p, ntroduce the quantty y := y,and further defne a user specfc parameter (a ) for the th user as a := u h σ 2 λ. (2.3) 3. Exstence and unqueness of Nash equlbrum The th user s optmzaton problem s to mnmze ts cost, gven the sum of powers of other users as receved at the base staton, y, and nose. The nonnegatvty of the power vector (p 0 ) s an nherent physcal constrant of the model. Takng the dervatve of the cost functon (2.1) wth respect to p, we obtan the frst-order necessary condton: J (p,p ) u h = λ p h 0. (3.1) p + h p + σ 2 In the case of a postve nner soluton, (3.1) holds wth equalty. It s easy to see that the second dervatve s also postve, and hence the nner soluton, f t exsts, s the unque pont mnmzng the cost functon. The boundary soluton, p = 0, s the other possble optmal pont for the constraned optmzaton problem. If the user s cost functon, J (p,p ), attans ts mnmum for a power value less than zero, p,mn < 0, the optmal soluton wll be the boundary pont. Solvng equaton (3.1) and nvokng the postvty constrant p 0, we obtan the reacton functon,,oftheth user: p = (y,a ) [ 1 a 1 ] = h y, f y a, 0, else. (3.2) The reacton functon s the optmal response of the user to the varyng parameters n the model. It depends not only on the user-specfc parameters, lke u, λ,andh, but also on
CDMA UPINK POWER CONTRO AS A NONCOOPERATIVE GAME 661 Theorem 3.1. In the power game ust defned (wth M users), let the ndexng be done such that a <a >, wth the orderng pcked arbtrarly f a = a.etm M be the largest nteger M for whch the followng condton s satsfed: Fgure 1. A smplfed block dagram of the system. the network parameter,, and total power level receved at the base staton, M =1 y. Actually, (3.2) shows dependence only on y, but addng ( p /) to both sdes, and dvdng both sdes by (1 1/), one can express the response of the th user as a functon of the quantty M =1 y. The base staton provdes the user wth total receved power level usng the downlnk. If the frequency of user updates s suffcently hgh, ths can be done ncrementally n order to decrease the overhead to the system. A smplfed block dagram of the system s shown n fgure 1. Smlar to the transmsson control protocol (TCP) n the Internet [8], there s an nherent feedback mechansm here, bult nto the reacton functon of the user. In ths model, the total receved power at the base staton provdes the user wth nformaton about the demand n the network, whch s comparable to congeston n case of the TCP. However, one maor dfference s that here the reacton functon tself takes the place of the wndow based algorthms n the TCP. In order for the th moble to be actve, or p > 0the followng condtons from (3.2) have to hold: a > 0and y a. An ntutve nterpretaton for these condtons s the followng: f the prce, λ, s set too hgh for a moble, the moble prefers not to transmt at all, dependng on ts channel gan and utlty parameter, and the spreadng gan and nterference level. For any equlbrum soluton, the set of fxed pont equatons can be wrtten n matrx form by explotng the lnearty of (3.2). In case of a boundary soluton, the rows and columns correspondng to users wth zero equlbrum power are deleted, and the equaton below nvolves only the users wth postve powers. Hence, we have (assumng here that all M users have postve power levels at equlbrum): 1 h 1 h 2 1. h 2 h 1 h 3 h 1 h 1 h M h 2 h M. h 3 h 2 h M h 1 h M h 2.... h M 1 1 h M c 1. = c 2. p 1 p 2 p M c M Ap = c, (3.3) where the varable c s defned as c = a /h. Note that c > 0fa s postve. a M > 1 ( + M 1) M =1 a. (3.4) Then, the power game admts a unque Nash equlbrum (NE), whch has the property that users M + 1,...,M have zero power levels, p = 0, M + 1. The equlbrum power levels of the frst M users are obtaned unquely from (3.3) wth M replaced by M, and are gven by p = 1 { h 1 [ a 1 + M 1 M a ]}, M := { 1, 2,...,M }. (3.5) If there s no M for whch (3.4) s satsfed, then the NE soluton s agan unque, but assgns zero power level to all M users. Proof. We frst state and prove the followng lemma, whch wll be useful n the proof of the theorem. emma 3.2. If condton (3.4) s satsfed for M = M, t s also satsfed for all M such that 1 M < M. Proof. Suppose that condton (3.4) holds for M = M; then we argue that t also holds for M = M 1. Substtutng n (3.4) M for M, we rewrte t as ( + M 2 ) M 2 a M a M 1 > =1 Due to the ndexng of users, we have a M 1 a M. Substtutng a M 1 for a M above, we obtan ( + M 3 ) M 2 a M 1 > =1 Hence, (3.4) s satsfed for M = M 1. The proof then follows by nducton on M. Returnng to the proof of the theorem, we frst show that the matrx A n equaton (3.3) s full rank and hence nvertble, and thereby the soluton to (3.3), p, s unque. Then we show that the soluton s strctly postve (.e., p > 0 M) f, and only f, condton (3.4) s satsfed for M = M. Fnally, we relax condton and allow for boundary solutons, and conclude the proof by provng the unqueness of the boundary soluton. In order for the matrx A n(3.3)tobefullrankand hence nvertble, there should not exst a nonzero vector a. a.
662 APCAN ET A. q = (q 1 q 2...q M ) T 0suchthatAq = 0. Ths equaton can be wrtten as h q + h q = 0 ( 1)h q + M h q = 0. (3.6) =1 Summng up ths set of equatons over all users ( = 1,...,M), we arrve at ( M ) ( 1 + M) h q = 0. =1 The term 1 + M above s nonzero, and hence, the sum M=1 h q has to be zero. Snce the channel gans are strctly postve, h > 0, and>1, t follows from (3.6) that q = 0. Accordngly, the matrx A s full rank and hence nvertble, whch leads to a unque soluton for equaton (3.3). Smple manpulatons lead to (3.5), wth M = M, for ths unque soluton. If the NE exsts and s strctly postve, then (3.3) has to have a unque postve soluton, whch we already know s gven by (3.5). Hence, (3.5) has to be postve, whch s precsely condton (3.4) n vew of also the ndexng of the users. On the other hand, f (3.4) holds for M = M, then we obtan from (3.5) that the equlbrum power level of each user s strctly postve. The exstence and unqueness of the NE follows from (3.3). We thus conclude that condton (3.4) wth M = M s both necessary and suffcent for the exstence of a unque postve Nash equlbrum. To complete the proof for the case M = M, possble boundary solutons need to be nvestgated to conclude the unqueness of the nner Nash equlbrum. We have to show that there cannot be another NE, wth a subset M of M users transmttng wth postve power, and the remanng M M users havng zero power level. In ths case, the reactve power level of the th moble, M, sgvenby(3.5) wth M = M. For any th moble, / M, n order for the zero power level to be part of a Nash equlbrum, condton y a (3.7) should fal accordng to the reacton functon (3.2) of the moble. Summng up the equlbrum power levels as receved by the base staton of M users wth postve power levels (from (3.5) wth M = M) results n 1 1 y = a. (3.8) + M 1 M M Substtutng n (3.7) the expresson (3.8) for y yelds 1 a a. (3.9) + M 1 M On the other hand, from lemma 3.2, and (3.4), we have foranyth usern the ndexedset {1,..., M + 1} the followng: 1 a + M 1 M =1 a. (3.10) Also, from the ndexng of the users t follows that M =1 a M a. Usng ths n (3.10), we see that nequalty (3.9) s satsfed for any th user, {1,..., M + 1}, regardless of the choce of the subset M. We note that there exsts at least one user belongng to the set {1,..., M + 1}, but not the subset M. Thus, the power of that moble must be postve, and hence the boundary soluton cannot be a Nash equlbrum. As ths argument s vald for any subset M, all boundary solutons fal smlarly for beng an equlbrum, ncludng the trval soluton, the orgn. We thus conclude that the nner Nash equlbrum s unque. Ths completes the proof for the case M = M. If M <Mn condton (3.4), then the equlbrum (whenever t exsts) wll clearly be a boundary pont. If condton (3.4) fals for users M + 1,...,M where users are ndexed as descrbed n theorem 3.1, then these users use zero power n the equlbrum. Hence, for any th user among M + 1,...,M, condton (3.7) should fal. It was shown above that equaton (3.8) holds wth M = M. As condton (3.4) does not hold for the th user, equaton (3.9), and hence (3.7) fals. Thus, from (3.2) power level of the th user s zero, p = 0, at the equlbrum. As ths holds for any { M + 1,...,M}, the equlbrum power levels for these users are zero. We now argue that the gven boundary soluton s unque. One possblty s the exstence of an th user, where 1 M, to have zero power. Ths cannot be a Nash equlbrum, as t follows from (3.8) and (3.9) wth M = M. Another possblty s the exstence of an th user, where M M, transmttng wth postve power level. Ths cannot be an equlbrum, ether, as t was shown above that (3.7) fals for such an th user, and p = 0 follows drectly from the reacton functon (3.2). All possble boundary solutons can be captured by varous combnatons of these two cases. Consder the case where a subset of users among 1,...,M use zero power whereas some of the users among M + 1,...,M use postve power levels. Snce for the subset of users wth postve power levels among M + 1,...,M condton (3.4) does not hold, they cannot be n equlbrum followng an argument smlar to the one above. Otherwse, as condton (3.4) holds for the subset of users wth zero power level among 1,...,M, they cannot be n equlbrum, ether. We conclude, therefore, that the boundary soluton s unque. Fnally, n the case where no M exsts satsfyng condton (3.4), all users fal to satsfy (3.4), and the only soluton s the trval one, p = 0.
CDMA UPINK POWER CONTRO AS A NONCOOPERATIVE GAME 663 4. Update schemes for mobles, and stablty In ths secton, we nvestgate the stablty of the Nash equlbrum n the gven model under two relevant asynchronous update schemes: parallel and random update. We establsh a suffcent condton whch guarantees the convergence to the unque equlbrum pont for both algorthms. 4.1. Parallel Update Algorthm (PUA) In the PUA, users optmze ther power levels at each teraton (n dscrete tme ntervals) usng the reacton functon (3.2). If the tme ntervals are chosen to be longer than twce the maxmum delay n the transmsson of power level nformaton, t s possble to model the system as a delay-free one. In a system wth delays, there are subsets of users, updatng ther power levels gven the delayed nformaton. The algorthm s gven by p (n+1) ( ( = y (n),a ) = max 0, 1 [ a 1 h ]) y (n), y (n) = h p (n), (4.1) or equvalently by y (n+1) ( = max 0,a 1 ) y (n), whose global stablty s establshed n the next theorem. Ths means that PUA converges to the unque Nash equlbrum of theorem 3.1 gven as [ p = max (0, 1h a 1 ]) h p (4.2) from any feasble ntal pont, p 0. Theorem 4.1. PUA s globally stable, and converges to the unque equlbrum soluton from any feasble startng pont f the followng condton s satsfed: M 1 < 1. (4.3) Proof. et us defne the dstance between the th user s power level receved n the base staton at any tme (n) and receved equlbrum power level as y (n) := y (n) y.we consder frst the case when y > 0 for an arbtrary th user. Then, gven the receved power levels of all users except the th one at tme n, y (n), we have the followng from (4.1) and (4.2): (n+1) < 1 y (n), f a < y(n), y = 1 y (n), else. Thus, we obtan y (n+1) 1 (n) y. (4.4) Next, we consder the case when the receved equlbrum powerlevelforanarbtraryth user s zero, y = 0. Then, from (4.1) and (4.2) t follows that (n+1) 1 y y (n), f a > y(n), 0, else. Thus, the nequalty (4.4) holds for any th user at any tme nstant n for both cases. We now show that (4.3) s a suffcent condton for the rght-hand sde of (4.4) to be a contracton mappng. et y denote the l -norm of the vector ( y 1 y 2... y M ) T,.e., Then, from (4.4), y (n+1) 1 max y = max y. (4.5) y (n) M 1 y (n). Hence, (4.4) s a contracton mappng under condton (4.3), whch leads to the stablty and global convergence of the PUA (4.1). We fnally note that usng an ntal admsson control mechansm and user droppng scheme, whch lmts the number, M, of users n the cell, ths condton can easly be satsfed for a gven. Thus, the stablty and convergence of the algorthm follows. 4.2. Random Update Algorthm (RUA) Random update scheme s a stochastc modfcaton of PUA. The users optmze ther power levels n dscrete tme ntervals and nfntely often, wth a predefned probablty 0 <π < 1. Thus, at each teraton a set of randomly pcked users among the M update ther power levels. The system wth delay s also smlar to PUA. The users make decsons based on delayed nformaton at the updates, f the round trp delay s longer than the dscrete tme nterval. The RUA algorthm s descrbed by y (n+1) = ( (n) h y,a ), wth probablty π, y (n), wth probablty 1 π, where was defned n (4.1). We already know from the proof of theorem 4.1 that f user updates, then (4.4) holds. Hence, for each = 1,...,M, E y (n+1) { y (n+1) = E user updates at tme n }π { y (n) + E user does not update at tme n }(1 π ) π E (n) y + (1 π )E (n) y. (4.6)
664 APCAN ET A. Usng agan the l -norm defned n (4.5), but wth y replaced by E y (that s, y := max E y ), and followng steps smlar to the ones of PUA, we obtan max E (n+1) y M 1 whch leads to y (n+1) y (n) max π + max(1 π ) y (n) ( ) M 1 y π + (1 π ) (n), where π<1andπ > 0 are the upper and lower lmts for the update probablty of the th user respectvely, π <π < π. Therefore, M 1 π + (1 π) <1 (4.7) s a suffcent condton for the rght-hand sde of (4.6) to be a contracton mappng, and for the stablty and convergence of RUA n norm. We also note that when all users have the same update probablty, π = π, ths condton smplfes to (M 1)/ < 1, same suffcent condton (4.3) as the one for PUA. We show next a stronger result, almost sure (a.s.) convergence of RUA, under condton (4.7). By the Markov nequalty and usng the defnton of the l -norm, we have P ( y (n) >ε ) E y (n) ε n=1 1 ε n=1 y (n), (4.8) n=1 where P( ) denotes the underlyng probablty measure. Snce E y (n) s a contractng sequence wth respect to the l -norm as shown, y (n) α y (n 1) α n y (0), where 0 <α<1. Usng ths n (4.8), t follows that P ( y (n) >ε ) α n y(0) ε n=1 n=1 = y(0) ε(1 α), and thus, follows that P ( (n) ) y K >ε ε(1 α), n=1 where K s a constant (actually, K = y (0) ). Hence, the ncreasng sequence of partal sums N n=1 P( y (n) >ε)s bounded above by K/(ε(1 α)). Thus, t converges for every ε>0. From the Borel Cantell lemma [2,3], t then follows that ( { P lm sup ω: y (n) } ) >ε = 0. n Hence, RUA converges also a.s. under condton (4.7). 4.3. Comparson of PUA and RUA One mportant feature of PUA s that t ascrbes a myopc behavor to the users, that s they optmze ther power levels based on nstantaneous costs and parameters, gnorng future mplcatons of ther actons. Ths behavor of users s realstc for the analyzed wreless network as t may not be feasble or even possble for a moble to estmate future values of total nterference n the cell or future varatons n ts own channel gan. In the case of RUA, the users are agan myopc and update ther power levels based on nstantaneous parameters. But, not all of them act at every teraton; whether a partcular user responds or not s determned probablstcally. In the lmtng case when all update probabltes, π, are equal to 1, RUA s the same as PUA. An advantage of RUA, however, s that through t one can nvestgate the convergence of theproposed scheme when there are random delays n the system. Such delays may be due to dfferences n the processng or propagaton tmes. As we wll see n the smulatons ncluded n secton 6, n a delay-free system f all the users have the same ntal power level, then RUA performs better than PUA. Ths s due to the myopc behavor of users, as well as the nherent randomzaton n the case of RUA. On the other hand, the opposte s true for a system wth delay as varatons n delay provde suffcent randomzaton, and PUA becomes more advantageous due to frequent updates. More detaled observatons on the convergence of both algorthms can be found n secton 6 for both delay-free and delayed cases. 5. Prcng strateges at the base staton In a noncooperatve network, prcng s an mportant desgn tool as t creates an ncentve for the users to adust ther strateges, n ths case power levels, n lne wth the goals of the network. In the CDMA system we are studyng here, the prce per unt power of the th user, λ, s determned by the base staton n a manner to be dscussed shortly. We ntroduce a prcng scheme where the prce charged to each user s proportonal to the receved power from the user at the base staton. Thus, the prce s proportonal to the channel gan of the th user, λ = k h. The nner Nash soluton by tself does not guarantee that the users wth nonzero power levels wll meet the mnmum SIR requrement to establsh a connecton to the base staton. Achevng the necessary SIR level s obvously crucal to the successful operaton of the system. Furthermore, one has to recognze that dfferent communcaton applcatons n wreless systems leads to dfferent types of users and SIR requrements n addton to the mnmum SIR level.
CDMA UPINK POWER CONTRO AS A NONCOOPERATIVE GAME 665 In vew of these consderatons, we wll consder n ths secton two dfferent prcng schemes: () A centralzed prcng scheme. Users are dvded nto classes, wth all users belongng to a partcular class havng the same utlty functon parameter (u ). Further, all users wthn a class have the same SIR requrement. The role of the base staton s to set prces for these dfferent classes such that, under the resultng Nash equlbrum, the SIR targets of the users are met. () Decentralzed, market-based prcng. The base staton sets a sngle prce for all users, and the users choose ther wllngness to pay parameter, u, to satsfy ther QoS requrements. As compared to the centralzed scheme, ths one s more flexble, and allows users to compete for the system resources by adustng ther ndvdual u s. 5.1. Centralzed prcng schemes and admsson control Frst, we consder the symmetrc-user case where every moble has the same SIR requrement, and for convenence we let u = 1. It s possble to fnd a smple prcng strategy by pckng the prce drectly proportonal to the channel gan, λ = kh, where the prcng factor, k, s user ndependent. The parameter k s a functon of the number of users and the desred SIR level. Notce that ths approach s equvalent to centralzed power control as the prces are adusted by the base staton n such a way that the mobles use the power levels determned by the unque Nash equlbrum as a result of ther ndvdual optmzaton. Moreover, the base staton can set the prces such that the SIR requrements of the users are satsfed. A precse result coverng ths case s now captured by the followng theorem. Theorem 5.1. Assume that the users are symmetrc n ther utltes, u = 1, they have the same mnmum SIR requrement, γ, and are charged n proporton wth ther channel gan, λ = kh. Then the maxmum number of users, M, the system can accomodate s bounded by M < + 1. (5.1) γ Moreover, the prcng parameter k under whch M M users acheve the SIR level γ s k = λ = γ (M 1) h σ 2 (γ. (5.2) + 1) Proof. Solvng for the user-ndependent y from (3.2), we have y = (/k) σ 2 + M 1. Combnng ths result and the SIR functon n (2.2), and takng the mnmum SIR, γ, as nput, we obtan (5.2) for a sngle class of users n a cell. To ensure that (5.2) s well defned, we requre the condton n (5.1). Based on (5.2), condton (5.1) satsfes (3.4). Thus, both the necessary and suffcent condtons for a unque postve Nash equlbrum are satsfed f (5.1) holds. Then, the unque soluton s strctly postve accordng to theorem 3.1, and all M M users attan the desred SIR level, γ. We note that f M>M, all users fall below the desred SIR level (γ ) due to the symmetry. In ths case, droppng some of the users from the system n order to decrease the number of users M below the threshold (M ) would lead to a vable soluton. Next, we consder the case where the network may provde multple servce levels and multple prcng schemes. For ths more general case, t s convenent to splt the mobles n a cell nto multple groups accordng to ther need for bandwdth, or n our context, ther desred SIR levels, where the users wthn each group are symmetrc. Usng a multple prcng scheme, a soluton capturng multple user groups can be obtaned straghtforwardly. 5.2. Market-based scheme It s natural to thnk of each user wthn a cell havng dfferent SIR requrements, whch can be quantfed wth the userspecfc utlty parameter u. The base staton can mplementa natural prcng strategy by formulatng the prcng parameter drectly proportonal to the channel gan, λ = kh.however, t s mpossble n ths case for the base staton to calculate the parameter k, as the user preferences are unknown to the base staton. Hence, after the base staton sets an approprate value for prce (k), each user dynamcally updates ts power level by mnmzng ts cost under parallel update (PUA) or random update (RUA) algorthms. As a result, a dstrbuted and market-based power control scheme s obtaned. Due to the nterference n the CDMA system, each user affects others. Hence, the th moble can adust ts utlty parameter, u, dynamcally accordng to ts mnmum SIR level, γ, gven the nterference at the base staton. From (3.2) and (2.2), t follows that u > λ ( γ h + 1 )( y + σ 2). The parameter u s bounded below by the total receved power at the base staton. Ths can be nterpreted as follows. If a moble s n a cell where the nterference s low, the moble can acheve the desred SIR level wth a low power, hence payng a lower prce. However, n a stuaton where many users compete for the SIR, the moble has to use more power, and pay a hgher prce to reach the same SIR level. In the latter case, the user s wllngness to spend more can be ustfed wth a hgher u based on (3.2). We note that, together wth the utlty functon, the utlty parameter u quantfes the user s desre for the SIR. The base staton can lmt aggressve requests for SIR even n the case when a user pays for ts excessve usage of power, by settng an upper-lmt, y max, for the receved power of the th user at the base staton: y y max. Hence, unresponsve users can be punshed by the base staton n order to preserve network resources. From (3.2), we can obtan an upper-bound
666 APCAN ET A. on the value of u. Furthermore, ths bound depends only on user-ndependent parameters, such as the upper lmt of the total receved power at the based staton, maxmum number of mobles, M max, and the spreadng gan,, f proportonal prcng s used: u k [ σ 2 ] + ( + M max 1)y max. When ths bound s combned wth a smple admsson control scheme, lmtng the number of mobles to M max, the base staton can provde guarantees for a mnmum SIR level γ mn : y max γ mn = (M max 1)y max + σ 2. A tradeoff s observed n the choce of the desgn parameters γ mn versus M max. If the network wants to provde guarantees for a hgh SIR level, then t has to make a sacrfce by lmtng the number of users. In addton, users may mplement a dstrbuted admsson scheme accordng to ther budget constrants and desred SIR levels. If the prce necessary to reach a SIR level exceeds the budget, B, of the user, that s kγ ( y + σ 2) B, then the user may smply choose not to transmt at all. 6. Smulaton studes The proposed power control scheme has been smulated numercally usng MATAB. Here, we frst nvestgate dfferent prcng schemes for symmetrc mobles n the fxed-utlty case. Then, we analyze the robustness of the system under varyng parameters such as nose, the number of users, and channel gans. Furthermore, the rate of convergence of both update schemes, PUA and RUA, are studed both n the delayfree and delayed cases. Fnally, we nvestgate a system consstng of users wth varous utlty parameters. All results of the smulatons are vald for both update schemes, PUA and RUA, where the only dfference between the two s the convergence rate. Smulaton parameters are chosen as follows, unless otherwse stated. The spreadng gan = 128 s chosen n accordance wth IS-95 standard [9]. Nose factor s σ 2 = 1 whereas the stoppng crteron or dstance to equlbrum s gven by ε = 10 5. The users are assumed to be located randomly n the cell where the dstance of the th user to the base staton, d, s unformly dstrbuted between d 0 = 10 and d max = 100. The channel gans of users are determned by a smple largescale path loss formula h = (d 0 /d ) 2 where the path loss exponent s chosen as 2 correspondng to open ar path loss. Under the fxed-utlty case, users have the same utlty parameter, u = 1. The ntal condton for smulatons s p = 1, an estmated value for establshng ntal communcaton between the moble and the base staton. In the smulatons, a dscrete tme scale s used. The delay-free case s characterzed by a tme span that s long enough for perfect Fgure 2. Comparson of power and SIR fnal values of the mobles for the fxed and proportonal prcng schemes. nformaton flow to users. Subsequently, delay s ntroduced to the system to make the settng more realstc. 6.1. Effect of the prcng parameters In the frst smulaton, proportonal and fxed prcng schemes are compared. For smplcty, we frst choose the users beng symmetrc under both fxed prcng, λ = λ, and proportonal prcng, λ = kh. For llustratve purposes the number of users s chosen small, M = 20. In fgure 2, the equlbrum power and the SIR values of each user can be seen under both prcng schemes. In the top graph, power values of the users wth dfferent channel gans are almost the same under fxed prcng. Hence, the users wth lower channel gans acheve lower SIR values. In contrast, all users meet a mnmum SIR level under proportonal prcng, regardless of ther channel gan. An ntutve explanaton s that under proportonal prcng the dstant users are allowed to use more power to attan the necessary SIR level. We also note that proportonal prcng s far n the sense that the users are not affected by ther dstance to the base staton. Convergence of users power levels to ther equlbrum values s demonstrated n fgure 3(a) under PUA, and fgure 3(b) under RUA wth update probablty beng 0.6. In both cases there are 10 users and = 20. The effect of prcng s nvestgated n the next smulaton for a sngle class of users by varyng the prcng parameter, k, under proportonal prcng. Equvalently, ths smulaton can be nterpreted as varyng the utlty parameter, u. Both parameters play a crucal role n the system by affectng the overall power and SIR levels. From (3.2), the effect of u on the system s nversely proportonal to k. In fgure 4 t can be observed that a gradual ncrease n k from 1 to 4,.e., an ncrease n prce, affects the system n a such a way that both power and SIR values decrease. Snce, wth an ncrease n the prce, the users decrease ther powers to the same extent leadng to lower SIR values gven a constant nose level. Equvalently,
CDMA UPINK POWER CONTRO AS A NONCOOPERATIVE GAME 667 (a) (b) Fgure 3. Convergence of users ndvdual power levels to the equlbrum values versus number of teratons under (a) PUA and (b) RUA wth π = 0.6. Fgure 4. Effect of the prcng parameter k (utlty parameter 1/u)ontheSIR and the power levels of users. a decrease n u, the users level of request for SIR, gves the same result. Furthermore, the observatons match theoretcal calculatons for the sngle class case n accordance wth (5.2). 6.2. Convergence rate and robustness of algorthms 6.2.1. Smulatons wthout delay The convergence rate of the two update schemes s of great mportance, as t has a drect effect on the robustness of the system. We have smulated PUA and RUA for dfferent numbers of symmetrc users under a sngle prcng scheme. In fgure 5, the number of teratons to the equlbrum pont s shown for dfferent probablty values of RUA and also for PUA (whch corresponds to RUA wth the update probablty equal to one). As the number of users ncrease the optmal update probablty decreases. Ths result s n accordance wth Fgure 5. Convergence rate for dfferent update probabltes and ncreasng numbers of users. = 800. the one n [6] where t s shown that n a quadratc system wthout delay, an approxmate value for the optmal update probablty s 2/3, as the number of users goes to nfnty. On the other hand, f the number of users s much smaller than the spreadng gan, M, then PUA s superor to RUA. Next, we nvestgate the robustness of the system n the delay-free case. Frst, we analyze t under ncreasng nose, σ 2. The background nose s ncreased step by step up to 100% of ts ntal value. Accordngly, the base staton allows users to ncrease ther powers by decreasng the prces by the same percentage n the fxed-utlty case. The smulaton s repeated wth N = 20 users under a proportonal prcng scheme. We observe n fgure 6(a) that the power values ncrease n response to the ncreasng nose to keep the ntal SIR constant. Smlarly, we ncrease the number of mobles n the system threefold n fgure 6(b). It has the same effect as ncreasng the nose due to the nature of CDMA. Agan by
668 APCAN ET A. (a) (b) Fgure 6. Power and SIR fnal values for ncreasng nose (a) and numbers of users (b). adustng the prces accordngly, all users keep ther SIR levels. Same results are obtaned equvalently under the marketbased prcng scheme, where users adust ther utlty parameter, u, dynamcally whle the prcng parameter determned by the base staton s kept constant. As a concluson, these observatons confrm the robustness of the proposed power control scheme. Fnally, we smulate the system n a realstc settng under a sngle prcng scheme. The number of users, N = 10 taken as ntal value, s modeled as a Markov chan. Arrval of new mobles s chosen to be Posson wth an average of 2 new users per tme nterval. Call duratons are exponentally dstrbuted wth an average of 20 tme ntervals. We observe the average percentage dfference between the theoretcal equlbrum and the current operatng pont of the system n terms of power values of users for some perod of tme. In the smulaton, PUA s chosen as the update algorthm. The ntal condton s the equlbrum pont for users. The smulated system operates wthn 1% range of the equlbrum ponts, and the results are very smlar to those of n fgure 7. Heretofore, robustness of the system was nvestgated for statc mobles. The movements of the users wthn the cell result n varyng channel gans. In the next smulaton, the locatons, hence, channel gans of users are vared randomly. The movement of th user s modelled after a random walk where a random value s added to the dstance of the user to the base staton d at each tme nstant. Hence, we obtan d (n + 1) = d (n) + x(n) where x(n) [ 1, 1] s unformly dstrbuted. Furthermore, the settng used n ths smulaton s the same as prevous one. From fgure 7, the system agan operates wthn 1% dstance to equlbrum. 6.2.2. Smulatons wth delay We ntroduce the delay factor nto the system n the followng way: users are dvded nto d equal sze groups, and each group has an ncreasng number of unts of delay. Frst, the convergence rates of the two update schemes are compared Fgure 7. Average percentage dstance to the equlbrum pont versus tme. Channel gans, h, are vared based on random movements of users. and contrasted under delay-free and delayed condtons. The update probablty of RUA s chosen as 0.66 whch corresponds to the optmal update probablty for a large number of users. In the delay-free case RUA outperforms PUA as the number of users ncreases. In the delayed case, however, PUA s always superor to RUA. Then, the smulaton nvestgatng the convergence rate of RUA for varous update probabltes s repeated n the delayed case. The result shown n fgure 8 s dfferent from the prevous one n fgure 5. Here, PUA converges faster than RUA for any number of users verfyng prevous results. Fnally, a market-based prcng scheme wth proportonal prcng at the base staton, k = 1, s nvestgated. There are two groups of users, whch are symmetrc wthn themselves. Users n each group have dfferent utlty parameters, u. The group wth hgher u s labeled as the prorty user group, whle the other one s called the regular user group. In or-
CDMA UPINK POWER CONTRO AS A NONCOOPERATIVE GAME 669 Fgure 8. Convergence rate for dfferent update probabltes and ncreasng numbers of users (wth delay). = 800. Fgure 9. SIR and power levels at the base staton (or prces) of two selected users from prorty and regular user groups versus tme. der to observe the effect of varyng number of users on the SIR levels, we let a sample user from each group make a long enough call. At the same tme, the number of users n each group and channel gans of the users are vared smlar to those n prevous robustness smulatons to create realstc dsturbances n the system. For smplcty, the values of the utlty parameters are kept constant throughout the smulaton. In fgure 9, t s observed that a prorty user always obtans a hgher SIR than a regular user. Another observaton s that prorty users use a hgher power level, and therefore pay more than regular users, as expected. The fluctuaton n the power levels s due to the varyng number of users, and varyng total demand for SIR n the system. 7. Concluson In ths paper, we have developed a mathematcal model wthn the framework of noncooperatve game theory, and have obtaned dstrbuted, asynchronous control mechansms for the uplnk power control problem n a sngle cell CDMA wreless network. Exstence of a unque Nash equlbrum has been proven, and convergence propertes of parallel and random update schemes have been nvestgated analytcally and numercally. Moreover, condtons for the stablty of the unque equlbrum pont under the update algorthms have been obtaned and analyzed accordngly. We have shown that the unque Nash equlbrum has the property that, dependng on the parameter values, only a subset of the total number of mobles are actve. Some of the users are dropped from the system as a result of the power optmzaton. By defnng a utlty functon and a utlty parameter, user requests for SIR were modeled dynamcally. Furthermore, the relatonshp between the SIR level of the users and the prcng has been nvestgated for two dfferent prcng schemes for the fxed and varyng utlty cases. It has been shown both analytcally and through smulatons that choosng an approprate prcng strategy guarantees meetng the mnmum desred SIR levels for the actve users n the fxed-utlty case. In addton, the prncples for an admsson scheme have been nvestgated under the market-based scheme. The results obtaned ndcate that the proposed framework provdes a satsfactory decentralzed and market-based soluton. The algorthms n ths model are practcally mplementable, as the only nformaton a user requres to update ts power other than own preferences and fxed parameters s the total receved power level from the base staton. Ths nformaton may be conveyed ncrementally to reduce the overhead n the case of frequent updates. Although a specfc cost structure s chosen n ths paper, most of the results may be extended to more general cost functons. Another possble extenson to ths work s to a multple cells model, where the effect of neghborng cells are taken nto account. A further topc of nterest s the development of the counterparts of the results n the case of multple base statons, whch brngs up the challengng ssue of handoff. Acknowledgements Research supported n part by grants NSF ANI 98-13710, NSF INT 98-04950, NSF CCR 00-85917, AFOSR MURI AF DC 5-36128, and ARMY OSP 35352-6086. References [1] T. Başar and G.J. Olsder, Dynamc Noncooperatve Game Theory, 2nd ed. (SIAM, Phladelpha, PA, 1999). [2] P. Bllngsley, Probablty and Measure, 2nd ed. (Wley, New York, 1986). [3] J.. Doob, Stochastc Processes (Wley, New York, 1953). [4] D. Falomar, N. Mandayam and D. Goodman, A new framework for power control n wreless data networks: Games utlty and prcng, n: Proc. Allerton Conf. on Communcaton, Control, and Computng, Illnos, USA (September 1998) pp. 546 555. [5] H. J and C. Huang, Non-cooperatve uplnk power control n cellular rado systems, Wreless Networks 4(3) (Aprl 1998) 233 240.
670 APCAN ET A. [6] R.T. Maheswaran and T. Başar, Mult-user flow control as a Nash game: Performance of varous algorthms, n: Proc. 37th IEEE Conf. on Decson and Control (December 1998) pp. 1090 1095. [7] D. Mtra and J. Morrson, A novel dstrbuted power control algorthm for classes of servce n cellular cellular CDMA networks, n: Proc. 6th WINAB Workshop on 3rd Generaton Wreless Informaton Networks, New Jersey, USA (March 1997). [8].. Peterson and B.S. Dave, Computer Networks: A System Approach, 2nd ed. (Morgan Kaufmann, San Francsco, CA, 2000). [9] T.S. Rapaport, Wreless Communcatons: Prncples and Practce (Prentce Hall, Upper Saddle Rver, NJ, 1996). [10] C.U. Saraydar, N. Mandayam and D. Goodman, Pareto effcency of prcng-based power control n wreless data networks, n: WCNC (1999). [11] C.W. Sung and W.S. Wong, Power control for multrate multmeda CDMA systems, n: Proc. of IEEE INFOCOM, Vol. 2 (1999) pp. 957 964. [12] A.J. Vterb, CDMA Prncples of Spread Spectrum Communcaton (Addson-Wesley, Readng, MA, 1995). [13] R.D. Yates, A framework for uplnk power control n cellular rado systems, IEEE Journal on Selected Areas n Communcatons 13 1341 1347 (September 1995) 1341 1347. Tansu Alpcan receved the B.S. degree n electrcal engneerng from Boğazç Unversty, Istanbul, Turkey, n 1998, and the M.S. degree n electrcal and computer engneerng from the Unversty of Illnos at Urbana-Champagn (UIUC) n 2001. Hs research nterests nclude game theory, and control and optmzaton of wrelne and wreless communcaton networks. He s the recpent of a Fulbrght scholarshp n 1999, and s a student member of IEEE. Currently, he s a Ph.D. canddate n electrcal and computer engneerng at UIUC. E-mal: alpcan@control.csl.uuc.edu Tamer Başar receved B.S.E.E. degree from Robert College, Istanbul, and M.S., M.Phl., and Ph.D. degrees n engneerng and appled scence from Yale Unversty. After stnts at Harvard Unversty and Marmara Research Insttute (Gebze, Turkey), he oned the Unversty of Illnos at Urbana- Champagn n 1981, where he s currently the Fredrc G. and Elzabeth H. Nearng Professor of Electrcal and Computer Engneerng. He has publshed extensvely n systems, control, communcatons, and dynamc games, and has current nterests n robust nonlnear and adaptve control, modelng and control of communcaton networks, control over wreless lnks, resource management and prcng n networks, rsksenstve estmaton and control, and robust dentfcaton. Dr. Başar s currently the Deputy Edtor-n-Chef of Automatca, Edtor of the Brkhäuser Seres on Systems & Control, Managng Edtor of the Annals of the Socety of Dynamc Games, and a member of edtoral and advsory boards of several nternatonal ournals. He has receved several awards and recogntons over the years, among whch are the Medal of Scence of Turkey (1993), Dstngushed Member Award of the IEEE Control Systems Socety (CSS) (1993), and the Axelby Outstandng Paper Award of the same socety (1995). He s a member of the Natonal Academy of Engneerng, Fellow of IEEE, and a past presdent of CSS. E-mal: tbasar@control.csl.uuc.edu R. Srkant receved hs B. Tech. from the Indan Insttute of Technology, Madras, Inda, n 1985, and the M.S. and Ph.D. degrees from the Unversty of Illnos n 1988 and 1991, respectvely, all n electrcal engneerng. He was a Member of Techncal Staff n AT&T Bell aboratores from 1991 to 1995. Snce August 1995, he has been wth the Unversty of Illnos where he s currently an Assocate Professor n the Department of General Engneerng and Coordnated Scence aboratory, and an Afflate n the Department of Electrcal and Computer Engneerng. He s the Char of the 2002 IEEE Computer Communcatons Workshop, and serves as an assocate edtor of Automatca and IEEE/ACM Transactons on Networkng. Hs research nterests nclude communcaton networks, queueng theory, nformaton theory and stochastc control. He receved an NSF CAREER award n 1997. E-mal: rsrkant@uuc.edu Etan Altman receved the B.Sc. degree n electrcal engneerng (1984), the B.A. degree n physcs (1984) and the Ph.D. degree n electrcal engneerng (1990), all from the Technon-Israel Insttute, Hafa. In 1990, he further receved hs B.Mus. degree n musc composton n Tel-Avv Unversty. Snce 1990, he has been wth INRIA (Natonal Research Insttute n Informatcs and Control) n Sopha-Antpols, France. He has been on sabbatcal snce 2001 n CESIMO, Facultad de Ingenera, Unversdad de os Andes, Merda, Venezuela. Hs current research nterests nclude performance evaluaton and control of telecommuncaton networks, stochastc control and dynamc games. In recent years, he has appled control theoretcal technques n several ont proects wth the French telecommuncatons company France Telecom. E-mal: altman@sopha.nra.fr