Downlink Resource Allocation and Pricing for Wireless Networks

Transcription

1 Downlnk Resource Allocaton and Prcng for Wreless Networks Peter Marbach and Randall Berry Abstract Ths paper consders resource allocaton and prcng for the downlnk of a wreless network. We descrbe a model that apples to ether a tme-slotted system (e.g. Qualcomm s HDR proposal or a CDMA system; the man feature of ths model s that the channel qualty vares across the users. We study usng a prcng scheme for the allocaton of rado resources. We show that to maxmze revenue n such a system, the base staton should allocate resources n a dscrmnatory manner, where dfferent users are charged dfferent prces based n part on ther channel qualty. However optmally allocatng resources n ths way s shown to requre knowledge about each user s utlty functon. We consder a suboptmal scheme whch does not requre knowledge of the users utlty functons, and show that ths scheme s asymptotcally optmal, n the lmt of large demand. Moreover such a scheme s shown to maxmze socal welfare. We also consder a heurstc scheme for the case of small demand, whch does not requre perfect knowledge about the users utlty functons. We provde numercal results that llustrate the performance of ths heurstc. I. INTRODUCTION In ths paper we study the use of prcng for resource allocaton n a wreless network. We focus on a downlnk communcaton to a group of untethered users from a sngle wred network access pont, such as a base staton n a cellular network or a hub n a wreless LAN. The followng s restrcted to the downlnk for two reasons. Frst, t s expected that for many data applcatons, such as down-loadng web pages or multmeda, the traffc load on the downlnk wll be much larger than over the uplnk, thus effcently utlzng ths lnk s of more mportance. The second reason s that the resource allocaton for the uplnk nvolves several fundamentally dfferent ssues than for the downlnk, e.g. on the downlnk all communcaton n a sngle cell orgnates at one pont so ssues of coordnaton and nterference are of lesser mportance than on the uplnk. A basc feature of a wreless network s that channel characterstcs wll vary across the user populaton. Ths varaton s due to dfferences n the proxmty of a user to the transmtter as well as mult-path fadng and shadowng effects. In ths settng, the transmtter can explot knowledge of the channel qualty to send at hgher rates to users wth better channels, for example by usng dfferent modulaton and codng choces for dfferent users. Channel knowledge can also be used for allocatng rado resources such as tme-slots, bandwdth or transmsson P. Marbach s wth the Dept. of Computer Scence, Unversty of Toronto, 10 Kng s College Rd., Toronto, Ontaro, M5S 1A4, Canada. E-mal: marbach@cs.toronto.edu R. Berry s wth the Dept. of Electrcal and Computer Engneerng, Northwestern Unversty, 2145 Sherdan Rd., Evanston, IL E-mal: rberry@ece.nwu.edu. Hs work was supported n part by the Northwestern- Motorola Center for Communcatons power, among the users. One example of such a system s the Qualcomm Hgh Data Rate (HDR scheme [1], whch s the bass for the IS-856 wreless data standard [3]. In HDR, the base staton transmts to a sngle user durng any gven tme slot; the transmsson rate a user can receve n a tme slot s determned by estmates of the channel condtons whch are fed back from the moble to the base staton. A schedulng algorthm at the base staton s used to decde whch user s transmtted to durng any gven tme slot, based n part on the channel condtons of the users. In the followng, we consder a model whch ncorporates several features of such a system, and we study the role of prcng on resource allocaton n ths model. The use prcng as a means for allocatng resources n communcaton networks has receved much attenton n recent years. For wre-lne networks, a samplng of ths lterature ncludes the work n [7], [11], [10], and [12]. A number of authors have also studed prcng n wreless networks, n partcular the role of prcng to ad n mplementng dstrbuted power control for the uplnk n a CDMA settng; examples of ths work nclude [4], [13], [6], and [5]. Regardng related work on the downlnk of wreless networks, we menton the work n [9], [17] n whch prcng for the downlnk of a CDMA network s studed; these papers address both welfare and revenue maxmzaton. The emphass n [9], [17] s on characterzng optmal prces and the resultng resource allocaton gven knowledge of the users utlty functons. We also menton work by Borst and Whtng [2] on schedulng for HDR systems; ths work shows that a form of revenue-based schedulng s optmal for maxmzng the normalzed long-term expected throughput. Algorthms for adaptvely calculatng the optmal revenue vector are also gven n [2]. Our focus here s on prcng strateges for revenue maxmzaton,.e. how should a servce provder prce resources to maxmze revenue. One key dfference between ths work and much pror work s that we do not restrct the base staton to charge the same prce per resource to all users. Indeed we show that to maxmze revenue t s often advantageous to employ dscrmnatory prcng; that s to charge users dfferent prces based on the users preferences and channel condtons. It s a well known fact n mcroeconomcs that prce dscrmnaton can allow a frm wth some degree of monopoly power to ncrease revenue. In the wreless settng we show that the optmal prce charged per user depends on both the user s demand for bandwdth as well as the channel condtons. Another dfference wth much of the lterature s that we consder a prcng framework that s recever drven,.e. we vew the recever, rather than the sender, as payng for servce. For the applcatons mentoned above, such as down-loadng web

2 pages, ths seems to be the natural approach. We note that a recever drven approach precludes a straghtforward extenson of prcng technques as n [11] and [12], n whch a prce s dentfed wth each packet at the transmtter. Also, snce we allow the recever to charge dfferent prces to varous users, ths complcates a markng approach as n [8]. Instead of the above approaches, we ntroduce a prcng scheme, where users submt bds and resources are allocated based on these bds. The paper s organzed as follows. In Secton II we descrbe our model for both a tme-slotted system and a CDMA system. Next, we look at some prelmnary approaches for resource allocaton. In Secton IV, we ntroduce a utlty framework to explctly model a user s preferences. Wth an approprate choce of varables, the problem of maxmzng socal welfare s shown to be mathematcally equvalent to the problem studed by Kelly et al. n [7]. In Secton V we ntroduce a prcng scheme for allocatng rado resources as dscussed above. In Secton VI an optmal revenue maxmzng strategy s gven for ths prcng scheme; ths strategy depends on the base staton havng perfect knowledge of the user s utltes. In Secton VII we study suboptmal strateges that do not requre complete knowledge of the user s utltes. In partcular, we gve an allocaton strategy whch requres no knowledge about the utltes, and show that ths strategy s asymptotcally optmal n the regme of large demand. Furthermore such a strategy also maxmzes socal welfare. Fnally n Secton VIII we provde some numercal results to llustrate these deas. II. MODEL In ths secton we descrbe a dealzed model for the downlnk n a sngle cell of a wreless network; we choose a smple model whch hghlghts the possble dsparty between users n such an envronment. We frst descrbe ths model n the context of a tme-slotted system. The resultng model s shown to also apply to a CDMA system smlar to that studed n [9]. Tme slotted system: Consder a system where the base staton transmts to a sngle user at any any gven tme. Assume there are M actve users, and for = 1,...,M the channel between the base staton and user s parameterzed by a state, h. Assume that ths state s known exactly at the base staton. Based on ths state, let r(h be the rate n packets per second at whch the user can receve data n a tme slot wth acceptable relablty. 1 We assume that over the tme scales of nterest, the channel state of each user stays fxed; we note ths s dfferent than the assumpton n work such as [2] and n partcular prohbts us from explotng mult-user dversty due to tme varatons n the users channels [15]. Suppose that the base staton schedules transmssons over a fxed length tme frame of T seconds. Durng a frame, the base staton allocates for each user a tme t [0,T] durng whch t transmts to that user at rate r(h, where x t = T. (1 r(h 1 In ths paper we gnore the possblty of packet losses; one way of ncorporatng such losses s by assumng that r(h s the long-term average rate at whch packets can be successfully sent. Here x s the throughput (n number of packets of user. Note we allow the transmsson tme gven to each user to be of an arbtrary length; n practcal systems the transmsson tme would lkely be constraned to be an nteger multple of some elementary tme-slot, but we gnore ths constrant here. CDMA system: Consder the downlnk n a CDMA system, where users are assgned orthogonal spreadng codes. 2 In ths case, assume that the base staton transmts to each user durng the entre tme frame of T seconds. Let P be the transmsson power allocated to user, and assume that the base staton has a constrant on ts total transmsson power so that P P. Let g represent the channel gan of user, so that g P s the receved power at user. Assume that the base staton can send to user wth a rate, r, (n packets/sec. that s proportonal 3 to the receved power,.e. r = Kg P, for some constant K. User wll then receve x = r T packets n the tme frame; thus we have that x 1,...,x M are constraned so that x TKg P. (2 Clearly by dentfyng r(h wth TKg and T wth P these models are mathematcally equvalent. We wll descrbe the followng results n the context of the model n (1, but the reader should keep n mnd that they apply equally for the second model as well, or any other other stuaton where a user s transmsson rate depends on some resource n a lnear fashon and there s a lnear constrant on ths resource. In a wre-lne network, a smlar lnk model can be developed (see e.g. [12] where the throughputs, {x }, are constraned as x CT, (3 and C s the constant transmsson rate n packets/sec of the lnk. Thus the key feature that dfferentates the above model from the wre-lne case s that n the wreless model, the packets of dfferent users can requre a dfferent porton of the lnk resources (dependng on the values of r(h, and g, respectvely, whle n the wre-lne case each user s packets need the same amount of network resources. III. MAXIMIZING THROUGHPUT/REVENUE Suppose that the base staton has an nfnte supply of data to send to each user, and consder the problem of allocatng resources among the users to maxmze the total system throughput. For the tme-slotted model, ths corresponds to choosng a 2 We gnore any constrants on the avalable number of spreadng codes n the followng. Such consderatons can be taken nto account through a code constrant as n [9]. 3 A lnear relatonshp between rate and power s reasonable for a wde-band system, provded the base staton cannot send at too hgh of a rate. At a hgh enough rates, channel capacty consderatons tell us that ths relatonshp wll no longer hold.

3 transmsson tme allocaton (t 1,...,t M to maxmze the total throughput of the system. To acheve ths, clearly the base staton should always allocate the entre tme frame to the user, or users, wth the best channel,.e. the user(s wth the maxmal transmsson rate r(h. We note that ths s smlar n sprt to the concluson that have been drawn n several other settngs, such as [16]. Although the resultng allocaton acheves the maxmal throughput, ths polcy s extreme and unfar as most of the users wll not be able to receve any data, n partcular n the case where the channel states do not change over the tme scales of nterest. Moreover t does not account for the relatve preferences (servce requrements of each user. Suppose nstead that recevers (users agree to pay the base staton (servce provder a prce u, = 1,...,M, per receved packet. In ths scenaro, the base staton maybe nterested n maxmzng ts revenue, rather than the throughput. The correspondng maxmzaton problem s gven as follows maxmze subject to: x u x 0, x r(h T, = 1,...,M. In order to maxmze the revenue, the entre tme slot should be allocated to the user, or users, wth the hghest ndex n = u r(h. Note that n can be nterpreted as the revenue per unt tme the base staton receves when t transmts packets of user. It s nterestng to note that n the case where the base staton wants to maxmze the revenue, tme slots are not necessarly allocated to the users who pay the hghest prce per packet or the user wth the best channel, but to the users who pay the largest amount per unt tme. Ths means that users wth a bad channel (small r(h are stll able to receve data when they are wllng to pay a hgher prce u per packet. In the case where one user has a hgher ndex n than all other users, ths revenue maxmzng polcy s agan extreme and unfar as the whole tme frame gets allocated to only one user, and all others are starved. In the case where the ndces of all users are equal, any feasble schedule such that M t = T generates the same revenue. Notce that the throughput optmzaton problem can be thought of as a specal case of ths revenue maxmzaton problem where all users pay the same amount u per receved packet. IV. MAXIMIZING SOCIAL WELFARE The above allocaton schemes are optmal n the sense that they maxmze ether the throughput or the revenue. However, as we noted, the resultng allocatons are not balanced,.e. most users are starved, except possbly f the ndces of all the users are equal. Also, we dd not take nto account how the prces u are generated for the revenue maxmzaton problem. In ths secton, we ntroduce utlty functons as a means of explctly characterzng the servce preferences of a user. We then explore an approach, where rather than optmzng the throughput, or revenue, the base staton maxmzes the so called socal welfare, or the total utlty over all users. We consder users wth elastc traffc [14],.e. users who perceve qualty of servce solely as a functon of the throughput. For ths case, we can characterze the servce preferences of each user through a qualty ndcator or utlty functon that depends on a sngle varable: the throughput. We wll assume that these utlty functons are ncreasng n the throughput. Examples of users wth elastc traffc are users sendng emal, transferrng fles, and browsng the Web. Consder a fxed user {1,...,M} and suppose that x s the throughput of user. Then, we assocate wth user the utlty functon U (x. We make the followng assumpton. Assumpton 1: For each user = 1,...,M, the functon U : R + R + satsfes the followng condtons: a. U s ncreasng, strctly concave and twce dfferentable, wth U (0 = 0. b. There exsts constants, K 1, K 2, such that for all,x, U (x K 1 and U (x K 2. Utlty functons for elastc traffc wth these characterstcs are commonly used n the prcng lterature, see e.g. [7]. The most restrctve assumpton here s condton b, whch states that the frst and second dervate of the utltes are unformly bounded. Ths assumpton wll be used n Secton VII. Notce that the frst dervatve of U s bounded f and only f U (0 <. Also notce that Assumpton 1 does not requre that all users have the same utlty functon. We defne the demand functon D : R + R + of user as follows. For = 1,...,M, let the functon D be defned such that D (u s the optmal soluton, x, to the maxmzaton problem max {U (x xu}, u R +. x 0 The goal of ths maxmzaton problem s to optmze the user s net beneft gven by utlty mnus cost. Note that n ths problem the user faces a trade-off between achevng a hgh utlty (by choosng a large throughput x and keepng ts cost low (by choosng a small throughput x. Under Assumpton 1, the above maxmzaton problem has a unque fnte soluton for all u, so D (u s well defned. D (u can be nterpreted as the rate (n packets per second would request when the prce for recevng one packet s equal to u. Furthermore, from Assumpton 1.b t follows that, for each, there exsts some constant u,max such that D (u = 0, u u,max. In other words, when user s charged more than u,max, t wll request zero rate. Assume now that the servce provder wants to optmze the total users utlty. Ths objectve s captured by the followng optmzaton problem. maxmze subject to: U (x x 0, x r(h T, = 1,...,M. (4

4 Let us compare ths to a sngle lnk case of the utlty maxmzaton problem studed by Kelly n [7] for a wre-lne network. In that case, the constrant n (1 s replaced by (3 resultng n the optmzaton problem maxmze subject to: U (x x CT, x 0, = 1,...,M. For each, defne Û(t = U(t r(h ; the quantty Û(t can be nterpreted as an ndcator of utlty as a functon of the amount of tme allocated to a user. Note f two users and j have dentcal utltes, U (x = U j (x j, but r(h r(h j then Û(x Ûj(x j,.e. they wll have dfferent utltes as a functon of t. Usng ths notaton, the optmzaton n (4 can be re-wrtten as maxmze subject to: Û (t t T, t 0, = 1,...,M. In ths form, ths problem can be seen to be mathematcally dentcal to the problem n (5. Thus the results n [7] can be adapted n the current settng as well. In partcular, from Theorem 1 n [7] and the above dentfcaton, the followng proposton drectly follows. Proposton 1: Let Assumpton 1 hold. Then the above maxmzaton has an unque soluton (ˆx 1,..., ˆx M. In addton, there exsts a parameter ˆλ such that where ˆx = D (u, u = = 1,...,M, ˆλ r(h. The parameter ˆλ can be nterpreted as the (optmal Lagrange multpler for the unconstraned optmzaton problem: ( M maxmze U (x ˆλ x r(h T. The frst order condtons for the above optmzaton problem are then gven by U (x ˆλ r(h = U (x u = 0, = 1,...,M. It can be shown that ˆλ ncreases when the total demand D(u = D (u, u R +, (5 (6 ncreases. The parameter ˆλ can then also be thought of as a congeston prce where each user s charged the same prce per tme slot n the optmal soluton (but dfferent prces, u, per packet; ths prce ncreases as the demand ncreases. We also pont out that makng the equvalent dentfcaton for the CDMA model, results n users beng charged an equal prce per unt of power under the optmal soluton. The above procedure for allocatng resources may not be practcal for two reasons. (a Frst, t s not clear why the base staton should be nterested n optmzng the total user utlty, rather than the throughput or the revenue. (b Even when the base staton wants to optmze the total user utlty, t may be unrealstc to assume that the base staton knows the utlty functons of the ndvdual users to carry out the above optmzaton drectly. In [7], Kelly proposes a prcng mechansm for wre-lne packet networks that allows the network to optmze socal welfare wthout requrng the network to have any knowledge about the users utlty functons. In ths approach, users bd for resources by ndcatng a wllngness to pay, and the network allocates network resources accordngly. Here, we pursue a smlar approach to allocate resources for the downlnk of a wreless network. However, rather than consderng welfare maxmzaton, we assume that the servce provder s objectve s revenue maxmzaton. V. RESOURCE ALLOCATION THROUGH PRICING In ths secton, we consder a recever drven prcng mechansm for allocatng the transmsson tmes t, = 1,...,M. As dscussed n the ntroducton, ths scheme requres users (the recevers to compete for resources through a bddng mechansm. Specfcally, users submt a prce bd and the base staton allocates transmsson tmes based n part on these bds. In each frame, users pay a prce that s equal to ther bd, ndependent of the amount of data they receve,.e. the prce s a prce per frame, as opposed to a prce per packet. For our analyss we assume that users behave n a selfsh way (.e. each user s only nterested n maxmzng ts own net beneft and users act ndependently (.e. they do not collaborate durng the bddng process. User adjust ther bds over tme, based on the allocaton they receve. The goal of the base staton s to allocate resources n such way that the resultng bds maxmze revenue. In Secton III, we consdered optmzng revenue gven fxed prces per packet. The mechansm consdered here dffers from ths n several ways. Frst, the users submt a prce per frame (of length T versus a prce per packet, and second, the base staton takes nto consderaton how the users bds change over tme. For ths framework, whch we descrbe n more detal below, we nvestgate the followng questons, (a How wll the users bd? (b Based on the users bd, how should the base staton allocate the transmsson tme n order to maxmze ts revenue?

5 A. Prcng Mechansm We consder the followng prcng mechansm. In each frame, users bd for resources by submttng prce bd w R +. The base staton then allocates resources accordng to a functon f(w = (f 1 (w,...,f M (w such that f (w T, for all w R M +, where t = f (w s the tme that the base staton allocates n each frame to transmt packets for user. Notce that we do not requre the the entre frame to be allocated. Indeed, some examples can be found where the revenue maxmzng allocaton does not allocate the entre frame. To study ths prcng scheme, we proceed n a smlar manner as n [7] and defne a user problem and a base staton problem. The user problem addresses the ssue of how user chooses the bd w (based on the last prce bd and the tme t = f (w n the last frame. For the base staton problem, the goal s to fnd an allocaton strategy f that maxmzes the revenue. B. User Problem We consder a fxed allocaton strategy f = (f 1,...,f M. Let w (k = (w (k 1,...,w(k M be the bd vector n the kth frame, and f(w (k = (f 1 (w (k,...,f M (w (k the transmsson tmes allocated to users n ths frame. The prce per packet, u (k, that user pays n the kth frame s then equal to and the transmsson rate x (k u (k w (k = f (k (wr(h, x (k s equal to = w(k. u (k In the above, f f (k (w = 0, we set u (k ndependent of the other varables. (k + 1th frame, user chooses a bd w (k+1 net beneft under the prce u (k maxmzaton problem, max w 0 {U ( = and x (k = 0, We assume that n the to maxmze ts,.e. user solves the followng w u (k w } We note that ths problem s equvalent to the user problem from the second decomposton n [7]. Gven the allocaton strategy f, we then defne an equlbrum bd vector as a vector w (f = (w1(f,...,w M (f such that for all users = 1,...,M we have where w (f = arg max w 0 u = { U ( w u w (f f (w r(h.. w }, Note that under an equlbrum vector w (f = (w1(f,...,w M (f, each user maxmzes ts own net beneft and therefore has no ncentve to devate from ts bd w (f. The followng proposton gves a useful alternatve characterzaton of an equlbrum bd vector. Proposton 2: A bd vector w s an equlbrum bd vector for an allocaton strategy f f and only f, where u = f (w = D (u r(h, w f (w r(h. = 1,...,M, The proof of ths follows drectly from the defntons above. C. Base Staton Problem Next, we consder the followng queston. What allocaton strategy should the base staton choose such that t maxmzes the revenue? We wll proceed n two steps. Frst, we assume that the base staton has perfect global knowledge (knows the utlty functon of all users, and derve an allocaton f whose equlbrum bddng vector maxmzes the base staton s revenue. In the second step, we derve an allocaton strategy ˆf for the case where the base staton has only mperfect nformaton (does not know the users utlty functons. One would expect that the revenue P(f under the allocaton strategy f s n general larger than the revenue P( ˆf under the allocaton strategy ˆf. However, as we wll show n the followng, the revenue under the allocaton ˆf s close to f when many users are actve. VI. OPTIMAL ALLOCATION STRATEGY In ths secton, we derve an optmal allocaton strategy for the case where the base staton knows each user s utlty functon. Let U be the utlty functon of user, and let D (u be the correspondng demand (as defned n Secton IV. The revenue maxmzaton problem s then gven by maxmze subject to: u D (u u 0, D (u r(h T, = 1,...M. It can be shown that an optmal soluton (u 1,...,u M exsts for ths maxmzaton problem. If for each, u D (u s strctly convex, then ths optmum wll be unque. Otherwse, there may be multple optmal solutons, n whch case, consder (u 1,...,u M to be one optmal soluton, pcked arbtrarly. From the pont of vew of revenue maxmzaton, whch optmal s chosen does not matter. Let λ = u r(h be the prce per unt tme the base staton charges user, and consder the the followng allocaton strategy. Gven the bd vector w = (w 1,...,w M, set f (w = w φ λ, = 1,...,M, (7

6 where and λ s such that φ = λ λ, = 1,...,M, 1 M w φ λ = We then have the followng result. D (u r(h. Proposton 3: Let Assumpton 1 hold. Then there exsts a unque equlbrum bddng vector w = (w1,...,w M for the strategy f. In addton, the revenue under the equlbrum bddng vector maxmzes the base staton s revenue,.e. we have u = w f (w r(h. Proof: Assume an equlbrum bddng vector w exsts for the allocaton strategy f. Usng the above defntons, the prce per packet pad by user at ths equlbrum s gven by w f (w r(h = λ λ u. 1 From Proposton 2, f w s an equlbrum bddng vector t must be that for = 1,...,M, f (w = 1 ( λ r(h D λ u. 1 Thus, from the defnton of the allocaton strategy, ( 1 λ r(h D λ u = D (u 1 r(h. Snce the demand functon of each user s strctly decreasng on [0,u,max ], t follows that there exsts a unque λ that satsfes ths equaton, and λ = λ 1. Thus f a equlbrum for f exsts, t s unque and the equlbrum allocaton maxmzes revenue. To see that an equlbrum exsts, let w = u D (u ; under the allocaton strategy f, ths bddng vector can be shown to satsfy Proposton 2. Thus t s an equlbrum bddng vector. The strategy f allows the base staton to maxmze revenue. Under the revenue maxmzng strategy, the users may be charged a dfferent prce per tme slot; ths dffers from the welfare maxmzaton case dscussed above. The dfference n prce depends on the parameters φ, = 1,...,M. An example where these parameters vary across the users s gven n Sect. VIII. To calculate these parameters requres that the base staton knows the utlty functon of each user. Also note that even f all users had the same utlty functon they could stll be charged a dfferent prce per unt tme, based on ther channel condtons. As we noted above, assumng that the base staton knows the utlty functons of each user may not be practcal. In the next secton, we propose a strategy ˆf whch does not requre the base staton to know the users utlty functon, and show that ths strategy s close to optmal for the case where many users are actve. VII. MANY USER CASE Suppose that the base staton uses the allocaton strategy ˆf gven by ˆf (w = w λ, = 1,...,M, where λ s such that w λ = T. Note ths allocaton strategy does not depend on the users utlty functons. In the followng, we analyze ths allocaton strategy and show that t has the followng propertes. In equlbrum, the above strategy, ˆf, (a maxmzes the total users utlty, and (b maxmzes the base staton s revenue (n the lmt for the case where many users are actve and demand on transmsson resources ncreases (to nfnty. The frst property follows mmedately from our dscusson n Secton IV; we summarze ths n the followng proposton. Proposton 4: Let Assumpton 1 hold. Then there exsts a unque equlbrum bddng vector ŵ = (ŵ 1,...,ŵ M for the strategy ˆf. In addton, the rate vector (ˆx 1,..., ˆx M under the allocaton ( ˆf 1 (ŵ,..., ˆf M (ŵ maxmzes the total user s utlty. Next, we address the second property above. We make the followng assumpton. Assumpton 2: For all ū 1,...,ū M, such that we have that ū D (ū = max u 0 u D (u, = 1,...,M, D (ū r(h T. Ths assumpton mples that when (u 1,...,u M s an optmal soluton to the revenue maxmzaton problem gven by Eq. (7, then we have that D (u = T. r(h Intutvely, ths states that there are enough users actve to ensure that the base staton wll fully utlze the system (allocate the whole frame duraton T under a strategy that maxmzes the revenue. In the next proposton, we show that n the case where many users are actve (and demand on transmsson resources ncreases to nfnty, then the ˆf s close to optmal. By Assumpton 1.b, there exsts a postve constants L such that for all users = 1,...,M we have U (x U (x L x x, for all x,x R +. For each λ 0, defne D T (λ = max{r(h D (λ/r(h }.

7 Ths can be nterpreted as the maxmum amount of tme any user would demand when charged λ per unt tme. From Assumpton 1, the functon D T (λ s well-defned, decreasng, and there exsts λ max such that D T (λ = 0, for all λ λ max. Proposton 5: Let Assumpton 1 and 2 hold. Then we have that P(f P( ˆf L D T (ˆλ, where ˆλ s the prce that the base staton charges each user n equlbrum per unt tme under the strategy ˆf. We provde an outlne for a proof of ths proposton n the appendx. As the number of actve users ncreases, t can be shown that ˆλ approaches λ max. Remember that the functon D T (λ s decreasng n λ and lm DT (λ = 0. λ λ max Ths mples that the the dfference P(f P( ˆf vanshes as the number of actve users ncreases. Thus Proposton 5 states that the strategy ˆf s close to optmal durng perods when the demand s hgh and the base staton acheves the hghest revenue (under the optmal strategy. In the followng, we provde an nformal dervaton for Proposton 5. Usng Lagrange multplers, we can rewrte that revenue maxmzaton problem gven by Eq. (7 as maxmze ( M u D (u λ subject to: u 0, = 1,...M. D (u r(h T The optmal prces u, = 1,...,M, are then gven by and for λ = u r(h we have u = λ r(h D (u D (u (8 λ = λ D (u D (u r(h, = 1,...,M. (9 Note from Assumpton 1.b, D (u 1 K 2 and D (u 0 as u ncreases. Ths means that when the number of actve users ncreases (to nfnty, then the term D(u D (u wll decrease to zero. Ths mples that when the number of actve users s large, we have (approxmately that λ = λ j = λ, = 1,...,M, and the strategy ˆf becomes (essentally optmal. A. A Heurstc Allocaton Strategy We showed n Proposton 5 that n the case where many users are actve, the revenue maxmzaton and welfare maxmzaton problem are equvalent (n the lmt as the the number of user ncreases to nfnty. However, n the case where a small number of users are actve, the strategy ˆf may perform sgnfcantly worse than f. For ths case, the above nformal argument can be used to derve a heurstc. In Eq. (9, the term D (u /D (u s the recprocal of the demand elastcty of user. Eq. (9 llustrates that under the optmal polcy f the base staton charges users wth nelastc demand a hgher prce to maxmze ts revenue. Assume the base staton does not have the exact knowledge of the utlty functon of ndvdual users, but some estmate ˆα about the demand elastcty of user. The base staton could then allocate to user the transmsson tme t = f (w gven by w f (w =, = 1,...,M, λ + r(h ˆα where λ s such that w λ + r(h ˆα = T. Notce that ths strategy always fully utlzes the frame (gven that there s enough demand, whereas the optmal strategy does not necessarly allocate the full frame duraton, T. The term D (u /D (u s a functon of u ; therefore, the base staton should (dynamcally change the estmate α as the number of actve users changes over tme. The hope s that when the base staton can form a good estmates of the demand elastcty of ndvdual users, then ths strategy should perform close to optmal. VIII. CASE STUDY We llustrate the above results usng a case study. We assume that the user demands are pecewse-lnear functons gven by D (u = We then have that D (u/d (u = { C a u, 0 u C a 0, otherwse. { C a u, 0 u C a, 0, otherwse. Lettng λ be the Lagrange multpler n (8 and settng u = λ r(h, the optmal prce λ of (9, as a functon of λ, s gven by λ = { λ 2 + Cr(h 2a, 0 λ Cr(h a λ, otherwse. In the case where the optmal strategy f allocates the whole frame duraton T, the heurstc that we ntroduced n Secton V s optmal when the base staton can exactly estmate parameters C and a for each user. However, ths means that the base staton has exact knowledge of the users demand functons, or at least perfect knowledge of the rato C 2a. In practce, the base staton mght not know the exact rato C 2a, = 1,...,M, but only be able to obtan an estmate ˆα.

8 Fg. 1. Revenue under the optmal strategy f as a functon of of the number of actve users of each group gven by N(k = 10(k 1+100, k = 1,..., Fg. 3. Revenue dfference P(f P( ˆf under the heurstc f and the strategy ˆf as a functon of of the number of actve users of each type gven by N(k = 10(k , k = 1,..., 60. The estmate ˆR s equal to 1.3R (bottom curve and 0.8R (top curve Fg. 2. Revenue dfference P(f P( ˆf under f and ˆf as a functon of of the number of actve users of each type gven by N(k = 10(k , k = 1,..., 60. We compare, va a numercal study, the performance under the optmal strategy f, the strategy ˆf, and the heurstc strategy f, for the above type of demand functon. We set D (u = 0.01(1 u, = 1,...,M. We assume that there are M = 3N actve users whch we can classfy nto three dfferent groups correspondng to dfferent channel states; each group wth a total of N users. For users of group 1 we set r(h = 1, for users of group 2 we set r(h = 0.5, and for users of group 3 we set r(h = 0.3. We vary the number of actve users of each group accordng to N(k = 10(k , k = 1,...,60. The frame length s equal to T = 1. Fgure 1 shows the revenue under the optmal strategy as a functon of the number of actve users. As expected, the revenue ncreases as the number of actve users of each group ncreases. Fgure 2 shows compares the revenue under the optmal strategy f and the strategy ˆf. The strategy f always performs better than ˆf. The dfference between the two strateges ntally ncrease as more users are added. However, as the number of actve users gets large, the dfference decreases to zero, as predcted by Proposton 5. Fgure 3 llustrates how estmaton errors of the rato R = C 2a affect the performance of the heurstc strategy. In our case study, ths rato s gven by R = R = 0.01 = 0.5, = 1,...,M Let ˆR be the estmate by of the rato R. We consder two scenaros where we assume that the estmate ˆR s equal to 1.3R, and 0.8R, respectvely. Fgure 3 that the heurstc outperforms strategy ˆf when the number of actve users s small. When the number of actve users s large, then the performances of the two strateges are dentcal and equal to the optmal performance. However for ˆR = 1.3, the heurstc can be worse when the number of actve users s moderately large. A. Conclusons We have presented a prcng scheme for the downlnk n a wreless network where dfferent users can receve data at dfferent rates. In ths scheme, users bd n each frame by submttng a prce bd and the base staton allocates resources to maxmze ts revenue. We propose a revenue maxmzng strategy f for the base staton. Ths scheme however requres that the base staton know the utlty functon of each user. We also propose a suboptmal strategy ˆf whch does not requre knowledge of the user s utlty functons, and show that ths scheme s asymptotcally optmal, n the lmt of many users and large demand. Moreover, such a scheme s shown to maxmze socal welfare. We also consder a heurstc strategy f, whch does not requre perfect knowledge of the users utlty functons. Usng a case study, we show that the revenue under ˆf s close to f for large demand. The case study also llustrates the the heurstc f performs better than ˆf for small demand, gven that the base staton can accurately estmate the elastctes of the ndvdual users.

9 REFERENCES [1] P. Bender, et al., CDMA HDR:A Bandwdth-Effcent Hgh Speed Wreless Data Servce for Nomadc Users, IEEE Comm. Mag., July, 2000, pp [2] S. Borst and P. Whtng, Dynamc Rate Control Algorthms for HDR Throughput Optmzaton, IEEE Infocom 2001, Anchorage, Alaska, Aprl, [3] TIA/EIA IS-856 CDMA 2000: Hgh rate packet data ar nterface specfcaton, Nov., [4] D. Famolar, N. Mandayam, and D. Goodman. A new Framework for power control n wreless data networks: games, utlty and prcng, Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, Sept., [5] T. Hekknen, Optmal Qualty of Servce and prcng n the wreless Internet, ITC semnar 2000, Lllehammer, [6] H. J, An Economc Model for Uplnk Power Control n Cellular Rado Networks, Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, Sept., [7] F. P. Kelly, Chargng and Rate Control for Elastc Traffc, European Trans on Telecommun 8, pp , [8] F. P. Kelly, A. K. Maulloo, and D. K. H. Tan, Rate Control for Communcaton Networks: Shadow Prces, Proportonal Farness and Stablty, Journal of Operaton Research Socety 49(1998, [9] P. Lu, M. Hong, and S. Jordan, Forward-Lnk CDMA Resource Allocaton Based on Prcng, IEEE Wreless Communcatons and Networkng Conference, Chcago, IL, Sept., [10] S. Low and P. Varaya, A new approach to servce provsonng n ATM networks, IEEE/ACM Trans. on Networkng, vol. 1, no. 5, pp , [11] J. K. MacKe Mason and H. R. Varan, Prcng Congestble Network Resources, IEEE Journal of Selected Areas n Communcaton. vol. 13, no. 7, pp , Sept [12] P. Marbach, Prcng Prorty Classes n Dfferentated Servces Networks, Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, Sept., [13] C. Saraydar, N. Mandayam, and D. Goodman, Pareto effcency of prcng based power control n wreless data networks, Wreless Communcaton and Networkng Conference, [14] S. Shenker, Fundamental desgn ssues for the future Internet, IEEE J. Selected Areas Comm., vol. 13, pp , [15] D. Tse, Multuser Dversty n Wreless Networks, Talk at 2001 IEEE Commun. Theory Workshop [16] D. Tse, Optmal Power Allocaton over Parallel Gaussan Broadcast Channels, Internatonal Symposum on Informaton Theory, Ulm, Germany, June, [17] C. Zhou and M. Hong and S. Jordan, Two-Cell Power Allocaton for Wreless Data Based on Prcng, Allerton Conference on Communcaton, Control, and Computng, Montcello, IL, Sept., APPENDIX Let f be the allocaton strategy from Secton VI that maxmzes the base staton s revenue. Let w ndcate the equlbrum bd vector for f and let { w λ f = (w f f (w > 0, (10 u,max r(h f f (w = 0. Thus for all actve users, λ s the equlbrum prce user pays per unt tme under f, and for all nactve users, λ s the mnmum prce per unt tme such that the demand of that user s zero. Thus, usng Proposton 2 and Assumpton 2, t follows that ( 1 λ r(h D = T. (11 r(h Assume that under f, for user m, we have, λ m λ, for all = 1,...,M, and for user n we have that f n(w > 0 and λ n λ, for all such that f (w > 0. Note that λ n > 0. Under the strategy ˆf and any gven bd vector, notce that all users are charged the same prce per unt tme; let ˆλ ndcate ths prce for the equlbrum bd vector under ˆf. Agan usng Proposton 2, t follows that 1 r(h D Usng (11 and (12, we then have Thus, ( ˆλ = T. (12 r(h λ m ˆλ λ n. P(f P( ˆf T λ m ˆλ T λ m λ n. (13 Thus f λ m = λ n the proof s done. Therefore, n the followng we assume that λ n λ m. By the above defntons, we have that f n(w > 0 and thus f n(w < T. Let t be a gven small constant such that 0 < t < mn{f n(w,t f m(w }. Furthermore, let the constant w m s chosen so that fm(w + t = 1 ( r(h m D w m m (fm(w + tr(h m and let the constant w n s chosen so that fn(w t = 1 ( r(h n D n w n (f n(w tr(h n. Consder a new allocaton strategy f, defned as follows: f (w f n,m, and w w, f f (w = m(w + t f = m and w m w m, fn(w t f = n and w n w n. 0 otherwse. The strategy f can be seen to have the equlbrum bd vector w (f = (w1,..., w m,.., w n,...,wm, ths results n the allocaton: f (w f m,n, f (w (f = fm(w + t f = m, fn(w t f = n. Let λ, 1,...,M be defned as n (10 for the equlbrum under f,.e. { w (f f λ = (w (f f f (w (f > 0, u,max r(h f f (w (f = 0. We then have that λ f m,n, λ = λ m m f = m, λ n + n f = n,

10 for some m > 0, n > 0. Let P(f ndcate the revenue under strategy f. Then we have ( P(f P(f = t λ m λ n m fm(w + n f n(w t( m + n. Usng Assumpton 1.b, there exst a Lpschtz constant L > 0 such that m Lr(h m 2 t, and smlarly n Lr(h n 2 t. Thus ( P(f P(f t λ m λ n Lr(h m 2 fm(w + o( t. From ths t follows that strategy f would acheve a hgher revenue than strategy f for some t when λ m λ n > Lr(h m 2 f m(w. As f s an optmal strategy, t must be that λ m λ n Lr(h m 2 fm(w = Lr(h m D m ( λ m r(h m L D T (λ m L D T (ˆλ. Here we have used that (see Proposton 2 f m(w = 1 r(h D m and the defnton of D T (λ. ( λ m r(h m