Power Control for Wireless Data

Transcription

1 Power Control for Wreless Data Davd Goodman Narayan Mandayam Electrcal Engneerng WINLAB Polytechnc Unversty Rutgers Unversty 6 Metrotech Center 73 Brett Road Brooklyn, NY, 11201, USA Pscataway, NJ dgoodman@poly.edu narayan@wnlab.rutgers.edu Abstract Wth cellular phones mass-market consumer tems, the next fronter s moble multmeda communcatons. Ths stuaton rases the queston of how to do power control for nformaton sources other than voce. To explore ths ssue, we use the concepts and mathematcs of mcroeconomcs and game theory. In ths context, the Qualty of Servce of a telephone call s referred to as the "utlty" and the dstrbuted power control problem for a CDMA telephone s a "noncooperatve game". The power control algorthm corresponds to a strategy that has a locally optmum operatng pont referred to as a "Nash equlbrum." The telephone power control algorthm s also "Pareto effcent," n the termnology of game theory. When we apply the same approach to power control n wreless data transmssons, we fnd that the correspondng strategy, whle locally optmum, s not Pareto effcent. Relatve to the telephone algorthm, there are other algorthms that produce hgher utlty for at least one termnal, wthout decreasng the utlty for any other termnal. Ths paper presents one such algorthm. The algorthm ncludes a prce functon, proportonal to transmtter power. When termnals adjust ther power levels to maxmze the net utlty (utlty - prce), they arrve at lower power levels and hgher utlty than they acheve when they ndvdually strve to maxmze utlty. I. BACKGROUND AND MOTIVATION The technology and busness of cellular communcatons systems have made spectacular progress snce the frst systems were ntroduced ffteen years ago. Wth new moble satelltes comng on lne, busness arrangements, technology and spectrum allocatons make t possble for people to make and receve telephone calls anytme, anywhere. The cellular telephone success story prompts the wreless communcatons communty to turn ts attenton to other nformaton servces, many of them n the category of "wreless data" communcatons. To brng hgh-speed data servces to a moble populaton, several "thrd generaton" transmsson technques have been devsed. These technques are characterzed by user bt rates on the order

2 of hundreds or thousands of kb/s, one or two orders of magntude hgher than the bt rates of dgtal cellular systems. One lesson of cellular telephone network operaton s that effectve rado resource management s essental to promote the qualty and effcency of a system. One component of rado resource management s power control, the subject of ths paper. An mpressve set of research results publshed snce 1990 documents theoretcal nsghts and practcal technques for assgnng power levels to termnals and base statons n voce communcatons systems [1-4]. The prncpal purpose of power control s to provde each sgnal wth adequate qualty wthout causng unnecessary nterference to other sgnals. Another goal s to mnmze the battery dran n portable termnals. An optmum power control algorthm for wreless telephones maxmzes the number of conversatons that can smultaneously acheve a certan qualty of servce (QoS) objectve. There are several ways to formulate the QoS objectve quanttatvely. Two promnent examples refer to a QoS target. In one example, the target s the mnmum acceptable sgnal-to-nterference rato and n the other example the target s the maxmum acceptable probablty of error. In turnng our attenton to data transmsson, we have dscovered that ths approach does not lead to optmum results. Ths s because the QoS objectve for data sgnals dffers from the QoS objectve for telephones. To formulate the power control problem for data, we have adopted the vocabulary and mathematcs of mcroeconomcs n whch the QoS objectve s referred to as a utlty functon. The utlty functon for data sgnals s dfferent from the telephone utlty functon. Our research ndcates that when all data termnals ndvdually adjust ther powers to maxmze ther utlty, the transmtter powers converge to levels that are too hgh. To obtan better results, we ntroduce a prcng functon that recognzes explctly the fact that the sgnal transmtted by each termnal nterferes wth the sgnals transmtted by other termnals. The nterference caused by each termnal s proportonal to the power the termnal transmts. Ths leads us to establsh a prce (measured n the same unts as the utlty functon) to be calculated by termnals n decdng how much power to transmt. Termnals adjust ther powers to maxmze the dfference between utlty and prce. In dong so, they all acheve hgher utltes than when they am for maxmum utlty wthout consderng the prce.

3 II. UTILITY FUNCTIONS FOR VOICE AND DATA A utlty functon s a measure of the satsfacton experenced by a person usng a product or servce. In the wreless communcatons lterature the term Qualty of Servce (QoS) s closely related to utlty. Two QoS objectves are low delay and low probablty of error. In telephone systems low delay s essental and transmsson errors are tolerable up to a pont. By contrast, data sgnals can accept some delay but have very low tolerance to errors. In establshng a mnmum sgnal-to-nterference rato for telephone sgnals, engneers mplctly represent utlty as a functon of sgnal-to-nterference rato n the form of Fgure 1. We consder systems to be unacceptable (utlty = 0) when the sgnal-to-nterference rato ( γ ) s below a target level, γ 0. When γ >= γ 0, we assume that the utlty s constant. Our power control algorthms mplctly assume that there s no beneft to havng a sgnal-to-nterference rato above the target level. In cellular telephone systems, the target, γ 0 s system dependent. For example analog systems am for γ 0 = 18 db. In GSM dgtal systems the target can be as low as 7 db, and n CDMA t s on the order of 6 db [5]. In each case γ 0 s selected to provde acceptable subjectve speech qualty at a telephone recever utlty sgnal-to-nterference rato Fgure 1. Qualty of Servce metrc for wreless telephones represented as a utlty functon.

4 In a data system, the sgnal-to-nterference rato, γ, s mportant because t drectly nfluences the probablty of transmsson errors. When a system contans forward error correcton (FEC) codng, we consder a transmsson error to be an error that appears at the output of the FEC decoder. Because data systems are ntolerant of errors, they employ powerful error detectng schemes. When t detects a transmsson error, a system retransmts the affected data. If all transmsson errors are detected, a hgh γ ncreases the system throughput (rate of recepton of correct data), and decreases the delay relatve to a system wth a low γ. When γ s very low, vrtually all transmssons result n errors and the utlty s near 0. When γ s very hgh, the probablty of a transmsson error approaches 0, and utlty rses asymptotcally to a constant value. In addton to the speed of data transfer, a factor n the utlty of all data systems, power consumpton s an mportant factor n moble computng. The satsfacton experenced by someone usng a portable devce depends on how often the person has to replace or recharge the batteres n the devce. Battery lfe s nversely proportonal to the power dran on the batteres. Thus, we see that utlty depends on both γ and transmtted power. Of course, these quanttes are strongly nterdependent. Wth everythng else unchanged, γ s drectly proportonal to transmtted power. In a cellular system, however, many transmssons nterfere wth one another and an ncrease n the power of one transmtter reduces the sgnal-to-nterference rato of many other sgnals. To formalze these statements, we consder a cellular system n whch there are N mutually nterferng sgnals. For sgnal, = 1,2,..., N, there are two varables that nfluence utlty: the sgnal-to-nterference rato γ and the transmtted power p. Because each γ depends on p, 1, p2, K p N, the utlty of each sgnal s a functon of all of the N transmtter powers. A. The Data Utlty Functon The wreless data system transmts packets contanng L nformaton bts. Wth channel codng, the total sze of each packet s M>L bts. The transmsson rate s R b/s. At the recever of termnal, the sgnal-to-nterference rato s γ and the probablty of correct recepton s ( ) q γ, where the functon q( ) depends on the detals of the data transmsson ncludng modulaton, codng, nterleavng, rado propagaton, and recever structure. The number of transmssons

5 necessary to receve a packet correctly s a random varable, K. If all transmssons are statstcally ndependent, K s a geometrc random varable wth probablty mass functon: P K ( k) = q( γ )[ 1 q( γ )] = 0 k 1 k = 1,2,3, K otherwse. (1) The expected value of K s E[ K ] 1/ q( γ ) =. The duraton of each transmsson s M/R seconds and the total transmsson tme requred for correct recepton s the random varable KM/R seconds. Wth the transmtted power p watts, the energy expended s the random varable, [ ] p KM / R joules wth expected value E[ K ] p M / R = p M / R q( γ ). The beneft s smply the nformaton content of the sgnal, L bts. Therefore, our utlty measure s ( γ ) b/j E[beneft ] LRq = (2) E [energy cost] Mp The utlty can be nterpreted as the number of nformaton bts receved per Joule of energy expended. Zorz and Rao use an objectve that combnes throughput and power dsspaton n a smlar manner n a study of retransmsson schemes for packet data systems [6]. As a startng pont for dervng a power control algorthm, Equaton (2) has some advantages and dsadvantages. On the plus sde are ts physcal nterpretaton (bts per Joule) and ts mathematcal smplcty. Its dsadvantages derve from the smplfyng assumpton that all packet transmsson errors can be detected at the recever. Data transmsson systems contan powerful error detectng codes that make ths assumpton true, "for all practcal purposes". However, t causes problems mathematcally because the probablty of a packet arrvng correctly s not zero wth zero power transmtted. In a bnary transmsson system wth M bts per packet and p = 0, a recever smply guesses the values of the M bts that were transmtted. The probablty of correct guesses for all M bts s M 2. Therefore wth p = 0, the numerator of Equaton (2) s postve and the functon s nfnte. Ths suggests that the best approach to power control s to turn off all transmtters and wat, for the recever to produce a correct guess. Ths strategy has two flaws. One s that the watng tme for a correct packet could be months,

6 and the other s that there wll be other guesses (gnored n our analyss) that are ncorrect but undetectable by the error detectng code. To retan the advantages of Equaton (2) and elmnate the degenerate soluton, p = 0, from the optmzaton process, we modfy the utlty functon by replacng q( γ ) wth another functon f ( γ ) wth the propertes ( ) = 1 and f ( )/ p = 0, for p = 0. algorthm that maxmzes the followng utlty functon: f γ Thus we seek a power control U ( γ ) b/j LRf = (3) Mp 1 frame success rate and effcency functon fsr eff sgnal-to-nterference rato Fgure 2. Relatonshp of frame success rate to the effcency functon f ( γ ): non-coherent FSK modem, 80 bts per packet. In the numercal examples of ths paper, we have assumed a system wth no error correctng code and γ constant over the duraton of each packet. In these examples, M q ( γ ) = (1 BER ) (4)

7 where BER s the bnary error rate of transmtter-recever par. To work wth a well-behaved utlty functon, we ntroduce the followng effcency functon M f ( γ ) = (1 2BER ) (5) n our defnton of utlty. Ths functon has the desrable propertes stated above at the lmtng ponts γ = 0 and γ =, and ts shape follows that of q( ) at ntermedate ponts. For example, Fgure 2 shows f ( γ ) and q( γ ) for M = 80 and BER.5exp ( γ / 2) =, the bnary error rate of 0 a non-coherent frequency shft keyng modem. The smlar shapes of the two curves leads us to expect that a set of transmtter powers that maxmzes U n Equaton (3) wll be close to the powers that maxmze the utlty measure n Equaton (2). Note that the above formulaton of the utlty functon s general enough that other modulaton schemes can be reflected by approprately choosng the BER expresson. B. Power Control For Maxmum Utlty Our am s to derve a dstrbuted power control algorthm that maxmzes the utlty derved by all of the users of the data system. In a dstrbuted algorthm, each transmtter-recever par adjusts ts transmtter power p n an attempt to maxmze ts utlty U. For each, the maxmum utlty occurs at a power level for whch the partal dervatve of s zero: U p = 0 U wth respect to p (6) We observe n Equaton (3) that n order to dfferentate Equaton (6) wth respect to need to know the dervatve of p, we p. A general formula for sgnal-tonterference rato s γ wth respect to

8 In Equaton (7), ph ph γ = = (7) I + N N 2 p h + σ k = 1 k k k h k s the path gan from termnal to the base staton of termnal k, I s the 2 nterference receved at the base staton of termnal, and σ s the nose n the recever of the 2 sgnal transmtted by termnal. I and σ are ndependent of p. Therefore γ p = h I + N γ = p. (8) Referrng to Equatons (3) and (8), we can express the dervatve of utlty wth respect to power as U p LR = Mp df ( γ ) γ f ( γ ), (9) dγ 2 Therefore, wth p > 0, the necessary condton for termnal to maxmze ts utlty s df ( γ ) γ f ( γ ) = 0. (10) dγ Ths states that to operate at maxmum utlty a base staton recever has to have a sgnal-tonterference rato, γ *, that satsfes Equaton (10). C. Propertes Of The Maxmum-Utlty Soluton The sgnal-to-nterference rato, γ *, that maxmzes the utlty of user, s a property only of the effcency functon f( ), defned n Equaton (5). If all of the nterferng termnals use the same type of modem and the same packet length, M, they operate wth the same effcency functon. Therefore, the sgnal-to-nterference rato γ *, for maxmum effcency, s the same for all termnals. Ths s an mportant observaton because earler work on speech communcatons derves an algorthm [2-4] that allows all termnals to operate at a common sgnal-to-nterference rato. Ths algorthm drects each termnal to determne the nterference perodcally and adjust ts power to acheve ts target sgnal-to-nterference rato. After each adjustment, the other termnals adjust ther powers n the same way. Provded the number of termnals s not too

9 hgh 1, all power levels wll converge to values that produce the target sgnal-to-nterference rato at all recevers. In speech communcatons, the target s determned by consderatons of subjectve speech qualty. Our mathematcal analyss tells us that n data communcatons the modem and the packet length dctate the target. In speech, the dstrbuted power control system, leads to a globally optmum soluton. There s no set of powers that produces a better result than the set that results from the algorthm descrbed n the prevous paragraph. Ths s not the case n a data system. In a data system, we can show that f all termnals operate wth the power levels that satsfy Equaton (10), they can all ncrease ther utltes by smultaneously reducng ther power by a small (nfntesmal) amount. Ths result s formally proved n [7] and s also llustrated wth an example n Secton V. Ths mples that the dstrbuted power control algorthm for data sgnals s locally optmum but not globally optmum. As a consequence, we must extend our study to fnd power control schemes that do a better job than the sgnal-to-nterference rato balancng technque mpled by Equaton (10). To do so, we ntroduce concepts of mcroeconomcs that do not play a role n tradtonal communcatons systems engneerng, games and prces. III. GAME THEORY FORMULATION OF POWER CONTROL In the context of game theory, we say that n adjustng ts transmtter power, each termnal pursues a strategy that ams to maxmze the utlty obtaned by the termnal. In dong so, the acton of one termnal nfluences the utltes of other termnals and causes them to adjust ther powers. The dstrbuted power control algorthms we have descrbed are referred to as noncooperatve games because each termnal pursues a strategy based on locally avalable nformaton. By contrast, a centralzed power control algorthm uses nformaton about the state of all termnals to determne all the power levels. A centralzed algorthm corresponds to a cooperatve game. In game theory termnology, the convergence of the dstrbuted power control algorthm to a set of powers that maxmze the utlty of each termnal corresponds to the exstence of a Nash equlbrum for the non-cooperatve game. However, the algorthm s not Pareto effcent. Note that n optmzaton problems regardng rado resource management, 1 The lterature on power control algorthms for voce systems states a feasblty condton, whch depends on the number of termnals and ther locatons relatve to base statons. If ths condton s not satsfed t s mpossble to meet the sgnal-tonterference rato requrements for all termnals smultaneously.

10 globally optmal usually refers to a sngle unque operatng pont. However, Pareto effcency usually may refer to several ponts (whch form the Pareto fronter) some of whch may produce hgher utltes than others. From a practcal pont of vew, fndng solutons that offer Pareto mprovements may sometmes be suffcent rather than searchng for Pareto effcent ponts. Because we know that the strategy of maxmzng utlty leads everyone to transmt at a power that s too hgh, we seek a means to encourage termnals to transmt at lower power. To derve such a technque, we examne the effect of each termnal's power adjustment on the utlty of all other termnals. We defne the effect on termnal j of a power adjustment at termnal as the cost coeffcent, C j U = p j p ( j) b/j (11) Each cost coeffcent s postve because any ncrease n the power of one termnal reduces the sgnal-to-nterference rato of every other termnal, and hence decreases the utlty. The total cost, mposed on all termnals by termnal transmttng at a power level p s: N C = b/j (12) C j = j j 1 In the systems we have studed, we have dscovered that at equlbrum, the cost mposed by each termnal s a monotonc ncreasng functon of the dstance 2 of the termnal from ts base staton. Examnng termnals wth ncreasng dstances from ther base statons, we fnd: (a) ncreasng power necessary to acheve the equlbrum sgnal-to-nterference rato, (b) lower equlbrum utlty, and (c) hgher cost mposed on the other termnals. Thus f we ndex the N termnals n the system n order of ncreasng dstance from the servng base staton, where the dstance of termnal s d meters, we have at equlbrum (for d 1 < d 2 < Kd N ): 2 The dependence of varous quanttes on dstance s a property of the rado propagaton condtons of a system. The monotonc dependence of power to dstance relates to a smple propagaton model. Mathematcally, the powers, utltes and costs depend on the path gans, hj between transmtters and recevers.

11 U p C > U < p < C > K > U < K < p < K < C N N N and (13) In these nequaltes, the astersks denote equlbrum values of power, utlty, and cost. To fnd an mproved power control algorthm, we take these observatons nto account by mposng a prce on each transmsson. The prce s a tax, measured n the unts of utlty, bts per Joule, whch reduces the utlty. The nequaltes n Equaton (13) suggest that the prce should be monotonc ncreasng wth power. Moreover, by combnng Equatons (11) and (12) wth the defnton of utlty n Equaton (3), we fnd that under all condtons, not just at equlbrum, the cost mposed by termnal j on the other termnals s proportonal to p j : LR C j = t j p j b/j. (14) M Although t would be ntutvely pleasng to penalze each termnal by the value of C j n Equaton (14), ths s not feasble n a dstrbuted power control system. The value of t j depends on the current transmtter powers of all termnals n the system, and on all the path gans, h j. Therefore to determne the other termnals. t j, each termnal would need detaled nformaton about condtons at all To derve a dstrbuted algorthm that takes the costs nto account, we have adopted a prce functon that s proportonal to the power transmtted at each termnal, where the proportonalty constant s the same for all termnals: LR V j = tp j b/j. (15) M Then, we adopt a power control algorthm n whch each termnal maxmzes ts net utlty U ' = U V b/j (16)

12 IV. THE NET UTILITY FUNCTION At frst glance t appears that our task n dervng a power control algorthm s not very dfferent from the task we started wth. We began by dervng an algorthm n whch each termnal adjusts ts power to maxmze the utlty functon n Equaton (3). Now we ask for an algorthm n whch the functon to be maxmzed s the net utlty n Equaton (14), whch s smply the dfference between Equaton (3) and a term proportonal to power. However, ths prce term changes the nature of the algorthm consderably. For one thng, U', the functon to be maxmzed, can have negatve values. More mportantly, when each termnal seeks to maxmze ts own net utlty, t does not am for the same equlbrum sgnal-to-nterference rato as all the other termnals. That s because when we dfferentate the net utlty functon for each termnal, the condton correspondng to Equaton (10) contans a term that depends explctly on the power of each termnal. df ( γ ) γ f ( γ ) tp 2 = 0 (17) dγ In contrast to Equaton (10) the value of γ that satsfes ths equaton s dfferent for each termnal. It depends on all the path gans of termnal. h k n Equaton 7 and on 2 σ, the nose n the recever Ths property of the data power control algorthm takes us away from a sgnal-to-nterferencerato balancng algorthm correspondng to optmum power control for voce sgnals. In addton, we have to fnd a numercal value for the proportonalty constant t. Ths too s a departure from our orgnal stuaton n whch the functon that we maxmze depends only on observable propertes of the communcatons system: L, R, M, p, the modulaton technque (whch determnes the functon f( )), and the operatng envronment (whch determnes h j ). To fnd a good value for t, we have resorted to experments n whch we calculate transmtter powers for

13 specfc system models and then examne the effects of adoptng a range of values for t, the prce coeffcent. The followng Secton descrbes these experments. V. NUMERICAL EXAMPLES To shed lght on the salent propertes of the power control algorthms derved for wreless data transmsson, we have consdered a smple model based on a generc sngle-cell CDMA system wth no codng for forward error correcton and a fxed packet sze. Ths analyss has provded us wth nsghts nto the dfferences between power control for data sgnals and voce sgnals. Armed wth ths basc understandng, we have expanded the analyss to consder forward error correcton, varable transmsson rates, and varable packet szes. The smple system examned n ths paper has the followng desgn parameters: Number of nformaton bts per packet: L=64 Total number of bts per packet: M= 80 (wth no forward error correcton, the dfference M-L=16 s the number of bts n the cyclc redundancy check error-detectng code) Chp rate: 10 6 chps/s Bt rate: 10 4 b/s Modulaton technque: non-coherent frequency-shft keyng wth bnary error rate 0.5e -0.5γ. (Ths assumes that each sgnal encounters a non-fadng channel n whch the nterference appears as whte Gaussan nose.) Recever nose power spectral densty: 5 x W/Hz, whch produces a nose power of 2 15 σ = 5 x 10 W n a recever wth 1 MHz bandwdth. For ths system, the effcency functon s and the utlty functon s f ( γ ) [( exp( 0.5γ ))] 80 = (18) 1 U 4 80 [( 1 exp( 0.5γ ))] /80 p b/j = 64 x 10 (19) For ths effcency functon, the equlbrum sgnal-to-nose rato, found by solvng Equaton (10) s γ = 12.4 = 10.9 db. Ths s the target sgnal-to-nterference rato that all termnals am for

14 when each one seeks to maxmze ts utlty. For ths CDMA system, the feasblty condton for ths target s gven by the followng bound on the number of termnals [2]: N<= 1+(W/R)/γ* = 9.05 termnals (20) If the number of termnals transmttng to the base staton s less than or equal to 9, all termnals can operate wth γ = γ*. Moreover, when all lnks operate wth γ = γ*, all of the sgnals arrve at the base staton wth the same power: p receve = γ σ 2 ( W / R) ( N 1) γ Watts (21) The remanng quanttes that determne the propertes of ths system are the number of termnals, N, and the N path gans 3, h, 1, h 2, K h N. In the calculatons reported here, we use a smple propagaton model n whch all of the path gans are determnstc functons, wth propagaton exponent 3.6, of the dstance between a termnal and the base staton where 3.6 / d h = const (22) d meters s the dstance between termnal and the base staton. In our calculatons, the proportonalty constant n Equaton (22) s 7.75 x We chose ths value to establsh a transmt power of 10 W for a termnal operatng at 1000 meters from the base staton n a system wth N=9 termnals, all operatng wth γ = γ* = Fgure 3 shows the transmtter power as a functon of termnal-to-base staton dstance for ths system. Reflectng Equaton (22), the transmtter power n each curve vares as d 3.6. To demonstrate that the power control algorthm operatng wth a target of γ* s not globally optmum, consder a system wth N=9 termnals, all operatng wth γ = γ. Let all of the termnals reduce ther power levels by a factor of 10. By workng wth Equaton (21), we fnd that they arrve at the same sgnal-to-nterference rato, Wth γ = 11.7, the effcency 3 In general we have the notaton hk for the path gan of termnal to base staton k. In our smple example, there s only one base staton. Therefore, we smplfy the notaton to a use sngle subscrpt so that h s the path gan from termnal to the system base staton.

15 decreases from f (12.4) = 0.85 to f (11.6) = 0.80, a factor of However ths negatve effect on utlty s far outweghed by the postve effect of a 10:1 power reducton. Whle the new power control algorthm, based on a target of γ = 11.7 s more effcent (n the Pareto sense) than the algorthm wth a target of γ*, t s not an equlbrum pont of a non-cooperatve game. However, when all termnals operate wth γ = 11.7, any termnal can unlaterally mprove ts utlty by rasng ts power. For example, an ncrease n power by one termnal by a factor of 1.1, wll ncrease the sgnal-to-nterference rato transmtter power (W) dstance between termnal and base staton (km) Fgure 3. Transmtter power n a system wth N=9 termnals all operatng wth sgnal-to-nterference-rato γ = γ* = 12.4 of that termnal to 11.7 x 1.1 = 12.9 and ncrease the effcency to f (12.9)=0.88. Ths beneft to the utlty (0.88/0.80 =1.11) slghtly outweghs the negatve mpact of a 10% ncrease n power. However, ths acton by one termnal wll cause the utlty of the other termnals to decrease, whch n turn wll stmulate the other termnals to ncrease ther power levels. The chan reacton wll brng all termnals to the equlbrum sgnal-to-nterference rato of γ* = Ths stuaton motvates us to ntroduce the prce functon to create a non-cooperatve game that causes termnals to transmt at reduced powers relatve to those n Fgure 3. In ths game each

16 termnal unlaterally maxmzes ts net utlty n Equaton (16). To fnd the power transmtted by each termnal, we solve the N smultaneous equatons correspondng to Equaton (17) wth =1,2,...,N. To do so, we start wth ntal values of the N transmtter powers and fnd a numercal soluton of Equaton (17) wth =1 and p j held at the ntal values for the other values of j. We do the same thng n turn for = 2,3,,N and repeat the process untl the N power levels converge to ther equlbrum values. The results dffer from the results of the noncooperatve game that maxmzes U n that the equlbrum sgnal-to-nterference ratos are not equal. Termnals nearer the base staton have hgher values of γ at equlbrum than termnals further away. Wth unequal sgnal-to-nterference ratos, the receved powers are unequal and the power transmtted by each termnal depends not only on the dstance of that termnal from the base staton, but also on the dstances of all other termnals from the base staton prce=0 transmtter power prce=50p dstance between termnal and base staton (km) Fgure 4. Transmtter power n a system wth N= 9 termnals: comparson of equlbrum powers wth and wthout a prcng functon. These propertes of the game wth a prce functon are documented n Fgures 4 and 5. The numercal results apply to nne termnals transmttng data from dstances lsted n Table 2 n

17 whch d s proportonal to. For ths example, the prce parameter n Equaton (15) s chosen to be t = 50. Fgures 4 and 5, whch reproduce the results for the game of maxmzng utlty wthout a prce functon, demonstrate that ncorporatng the prce functon equlbrum reduces all of the equlbrum powers. The equlbrum sgnal-to-nterference ratos are also lower, but the combned effect on utlty s postve for all termnals, as ndcated n Fgure 5. Termnal Dst (km) Path gan x Utlty (b/j) pr= x Utlty (b/j) pr=50p10 5 x Net utlty (b/j) 10 5 x Table 2: Smulaton data prce=50p utlty (b/j) 10 5 prce= dstance between termnal and base staton (km) Fgure 5. Utlty n a system wth N= 9 termnals: comparson of equlbrum utlty wth and wthout a prcng functon.

18 VI. DISCUSSION OF RESULTS The numercal experments demonstrate that when each termnal operates ndependently to maxmze ts utlty, the set of transmtter powers converges to a locally optmum result, n whch all termnals obtan the same sgnal-to-nterference rato, γ*=12.4, the soluton to Equaton (10). However, we also fnd that ths result s not globally optmum. By reducng ther powers by the same factor, all termnals acheve hgher utlty. To work wthn the context of a non-cooperatve game (termnals operatng ndependently to acheve best performance), we have ntroduced a prcng functon that causes each termnal to maxmze ts net utlty, defned as the dfference between utlty and prce. In contrast to the orgnal algorthm wth zero prce, the algorthm wth a postve prcng functon converges to an equlbrum pont wth unequal sgnal-to-nterference ratos at dfferent termnals. All termnals operate wth lower power, lower sgnal-to-nterference rato, lower effcency, and hgher utlty than they do when the prce s zero. Because utlty s the rato of effcency to power, ths mples that the beneft acheved by ntroducng prcng s entrely due to reduced power. Whle all termnals acheve hgher utlty when they maxmze net utlty, rather than the utlty tself, the benefts are hghest for termnals near the base staton. Usng an algorthm wth a postve prce functon, termnals closer to the base staton operate wth hgher sgnal-tonterference ratos than termnals further away. Ths property of the power control scheme conforms to the propertes of advanced practcal wreless systems n whch Qualty of Servce s locaton-dependent. Ths dependency s ntroduced n rate adaptaton schemes, such as those ncorporated n EDGE (Enhanced Data Rates for GSM Evoluton) [11] and W-CDMA

19 (wdeband code dvson multple access) [12], and n ncremental redundancy technques for respondng to transmsson errors [13]. One drawback of power control based on prcng s that we do not have a convenent algorthm for mplementng t n practce. By followng the defnton of the algorthm, each termnal has to solve Equaton (17) perodcally and then adjust ts power accordngly. The new power s a complcated functon of the present sgnal-to-nterference rato. By contrast, the adjustments requred to converge to the soluton to Equaton (10) (correspondng to Equaton (17) wth t=0) are smple. The new power of termnal s smply the old power multpled γ*/γ, the rato of the target sgnal-to-nterference rato to the present sgnal-to-nterference rato. Most of the work reported here appears n the Master of Scence dssertaton of Vral Shah [7,8]. The dssertaton ntroduces the utlty functon used n ths paper and proves formally many of the statements n ths paper. Extensons of the work here to nclude the effects of error-correctng codng can be found n [9]. Jont transmtter power and transmsson rate control based on utlty maxmzaton as well as the effect of packet sze can be found n [10]. Investgaton of Pareto effcent prcng polces for transmt power control can be found n [14]. Whle all of the above work pertans to crcut-swtched wreless data communcatons, extensons are currently underway at WINLAB to ntroduce such a mcroeconomcs framework to packet-data wreless communcaton scenaros. Another related effort at WINLAB ncludes the study of dynamc utlty maxmzaton algorthms that take nto account moblty, channel varatons and resdual battery lfe.

20 ACKNOWLEDGEMENTS We would lke to acknowledge the followng students at WINLAB who have been nvolved n varous aspects of studyng mcroeconomc theores for rado resource management n wreless data networks: Vral Shah, Dave Famolar, Nan Feng, Cem Saraydar, Zhuyu Le and Henry Wang. REFERENCES [1] S.Grandh, R.Vjayan, D.Goodman, and J.Zander, Centralzed Power Control n Cellular Rado Systems'', IEEE Trans. on Vehcular Technology, vol. 42, no. 4, [2] R.D. Yates, A Framework for Uplnk Power Control n Cellular Rado Systems'', IEEE Journal on Selected Areas n Communcatons, vol. 13, no. 7, pp , September [3] J. Zander, Performance of Optmum Transmtter Power Control n Cellular Rado Systems'', IEEE Trans. on Vehcular Technology, vol. 41, no. 1, pp , February [4] J. Zander, Dstrbuted Co-Channel Interference Control n Cellular Rado Systems'', IEEE Trans. on Vehcular Technology, vol. 41, pp , [5] D.J. Goodman, Wreless Personal Communcaton Systems, Readng, Mass.: Addson- Wesley, [6] M. Zorz and R. Rao "Error Control and Energy Consumpton n Communcatons for Nomadc Computng" IEEE Transactons on Computers, Vol. 46, No. 3, March [7] Vral Shah, Power Control for Wreless Data Servces based on Utlty and Prcng'', M.S. Thess, Rutgers Unversty, March [8] V. Shah, N.B. Mandayam, and D.J. Goodman, Power Control for Wreless Data based on Utlty and Prcng'', n Proceedngs of Personal Indoor Moble Rado Communcatons Conference, PIMRC'98, September 1998, pp , Cambrdge, MA. [9] D. Famolar, N. B. Mandayam, D. J. Goodman, V. Shah, A New Framework for Power Control n Wreless Data Networks: Games, Utlty and Prcng'', n Wreless Multmeda Network Technologes, Kluwer Academc Publshers, Edtors: Ganesh, Pahlavan and Zvonar, pp , [10] N. Feng, N. B. Mandayam, D. J. Goodman, Jont Power and Rate Optmzaton for Wreless Data Servces Based on Utlty Functons'', n Proceedngs of Conference on Informaton Scences and Systems, CISS'99, pp , The Johns Hopkns Unversty, Baltmore, MD, March 1999.

21 [11] T.Ojanpera and R.Prasad, An Overvew of Thrd-Generaton Wreless Personal Communcatons: A European Perspectve'', IEEE Personal Communcatons, Vol. 5, no.6, pp , December [12] F.Adach, M.Sawahash, and H.Suda, Wdeband DS-CDMA for Next-Generaton Moble Communcaton Systems'' IEEE Communcatons Magazne, vol. 36, no. 9, pp , September [13] R. van Nobelen, N. Seshadr, J. Whtehead, and S. Tmr, "An Adaptve Rado Lnk Protocol wth Enhanced Data Rates for GSM Evoluton", IEEE Personal Communcatons, vol. 44, pp , Oct [14] C. Saraydar, N. B. Mandayam, D. J. Goodman, "Pareto Effcency of Prcng based Power Control n Wreless Data Networks" n Proceedngs of IEEE Wreless Communcatons and Networkng Conference (WCNC'99), New Orleans, Lousana, Sept , 1999.