Utility-Based Power Control in Cellular Wireless Systems

Transcription

1 Utlty-Based Power Control n Cellular Wreless Systems Mngbo Xao, Ness B. Shroff, Edwn K. P. Chong School of Electrcal and Computer Engneerng Purdue Unversty West Lafayette, IN 797, U.S.A. fmngbo, shroff, echongg@ecn.purdue.edu Abstract Dstrbuted power control algorthms for systems wth hard SIR constrants may dverge when nfeasblty arses. In ths paper, we present a power control framework called utlty-based power control (UBPC) by reformulatng the problem usng a softened SIR requrement (utlty) and addng a penalty on power consumpton (cost). Under ths framework, the goal s to maxmze the net utlty, defned as utlty mnus cost. Although UBPC s stll non-cooperatve and dstrbuted n nature, some degree of cooperaton emerges: a user wll automatcally decrease ts target SIR (and may even turn off transmsson) when t senses that traffc congeston s buldng up. Ths framework enables us to mprove system convergence and to satsfy heterogeneous servce requrements (such as delay and bt error rate) for ntegrated networks wth both voce users and data users. Farness, adaptveness, and a hgh degree of flexblty can be acheved by properly tunng parameters n UBPC. Keywords Sgnal-to-nterference rato (SIR), wreless, cellular system, power control, utlty functon, dstrbuted algorthm. I. INTRODUCTION The effcent management of rado resource s essental for wreless networks, whch are characterzed by scarce rado spectrum, an unrelable propagaton channel (wth shadowng, multpath fadng, etc.), and user moblty. One mportant component of rado resource management s power control. Power control has been extensvely studed n recent years, especally for CDMA systems. It has manly been used to reduce co-channel nterference and to guarantee the sgnal to nterference rato (SIR) of ongong connectons, resultng n a hgher utlzaton and/or better qualty of servce (QoS). From the vewpont of practcal applcatons, dstrbuted power control schemes are of specal nterest and mportance. One of the most well-known dstrbuted algorthms was orgnally proposed n [], and has been further studed n several papers ncludng [], [3]. Ths algorthm s dstrbuted and autonomous because t reles only on local nformaton. It s also standard (n the sense defned n []), and asynchronously convergent (wth geometrc rate) to the Pareto optmal power assgnment, when the system s feasble [3], []. However, f there s no feasble power assgnment, ths algorthm can dverge. For nfeasble systems, the dstrbuted power control algorthm dverges because of the hard SIR requrements that cannot be acheved n such systems no matter how hgh the transmtted power s. In practce, although achevng satsfactory QoS s mportant for users, they may not be wllng to acheve t at arbtrarly hgh power levels, because power s tself a valuable commodty. Thus, user satsfacton wll depend on both QoS and power consumpton. Ths observaton motvates a reformulaton of the whole problem usng concepts from mcroeconomcs and game theory [5]. Our problem naturally fts n ths context: the QoS objectve can be referred to as a utlty functon, whch represents the degree of user satsfacton to the servce qualty; the cost s n terms of power consumpton; and the dstrbuted power control essentally becomes a non-cooperatve N-person game, n whch each user tres to maxmze ts net utlty (.e., utlty mnus cost). The result s a utlty-based dstrbuted power control algorthm. Ths algorthm keeps major merts of the exstng algorthm n [], but t s exempt from the dvergence problem. It s worth comparng our algorthm and those presented n [5] and [6]. Reference [6] proposes a framework for uplnk power control n cellular systems, to obtan better QoS usng less power. The objectve functon s a concave functon of power and SIR, whch decreases n power and ncreases n SIR. In addton, a suffcent condton for convergence s mposed. The objectve functon to be used n ths paper satsfes none of these condtons. Reference [5] proposes a power control algorthm for wreless data, to provde power-effcent transmsson for data users. The utlty functon s essentally the number of effectve bts transmtted per unt of energy. Ths functon may be problematc and leads to a complcated power updatng algorthm wth low Pareto effcency. In ths paper, we wll use a natural (sgmod) utlty and a lnear cost. The resultant algorthm s flexble and smple to mplement. It wll automatcally decrease the target SIR (and may even turn off transmsson) of some user when the traffc congeston s buldng up. Nether algorthm n [5] or [6] has ths property, whch facltates system convergence and admsson control. Moreover, our algorthm has several tunable parameters n utlty and cost functons to acheve farness, adaptveness, and heterogeneous servce requrements (such as delay and bt error rate) for ntegrated networks wth both voce users and data users. The rest of the paper s organzed as follows. In Secton, we frst present the system model and explan why a hard SIR requrement may result n dvergence. In Secton 3, we reformulate the power control problem as a non-cooperatve game maxmzng the net utlty, whch leads to a utlty-based power control algorthm. We also dscuss the convergence and effects of some parameters n ths secton. Important extensons and dscussons of UBPC are made n Secton. Numercal results

2 are gven n Secton 5. Fnally, the last secton concludes the paper. II. SYSTEM MODEL AND RELATED WORK We consder a power-controlled cellular system where the transmtted powers are contnuously tunable. In the system, every user s assocated wth a base staton (called ts home base staton). To mantan a relable connecton between the user and ts home base staton, the sgnal to nterference rato (SIR) at the recever should be no less than some threshold, whch corresponds to a QoS requrement such as the bt error rate. We consder only downlnk transmssons n ths paper, because the uplnk case can be treated smlarly [7], [8]. Assume that there are n users n the system, and let P be the transmtted power level at the downlnk of user. Let G j denote the gan from the home base staton of user j to user. Then, the nterference power receved at user from the downlnk of user j s G j P j. Let be the background nose receved at user, and let be ts desred SIR threshold. Then we have SIR P Pj6= G jp j + : () Note that ths model s general enough to represent DS- CDMA systems wth matched-flter recevers [9], [] or TDMA/FDMA systems [], by gvng specfc nterpretatons to the parameters. For the system consdered n [3], [7], [], the constrant SIR s enforced for each user. The objectve of a power control scheme s to fnd the mnmum power satsfyng ths constrant. To ths end, there s a well-known power control algorthm gven by: P (k + ) = SIR (k) P (k), for = ; : : :; n, () where P (k) corresponds to the power level for user at the kth teraton. Ths algorthm and ts stablty have been studed extensvely by Foschn and Mljanc [], Mtra [3], and Bambos, Chen, and Potte [], [7]. The algorthm s dstrbuted and autonomous because t reles only on locally avalable nformaton. It allows each user to have dfferent target SIR values. It has also been shown n [3] to be asynchronously convergent (wth geometrc rate) to the Pareto optmal power assgnment, when the system s feasble (.e., when there exsts a power assgnment such that SIR for all ). However, f the system becomes nfeasble, ths algorthm dverges. Dvergence occurs when the system s nfeasble because the SIR requrement s hard, and has to be satsfed at any cost. Intutvely, what user does through () s to adjust ts transmtted power P such that ts SIR just acheves the threshold n the next step. In an nfeasble system, every user blndly adjusts ts power wthout realzng that t s mpossble to satsfy these SIR requrements smultaneously, and transmtted powers buld up hgher and hgher durng ths procedure. It may appear reasonable to attrbute ths blndness to the fact that the algorthm s dstrbuted and only has local nformaton avalable. Actually, ths s not really the case, because a user can recognze that nfeasblty would be lkely from the extremely hgh nterference receved. However, each user stll contnues to ncrease ts power because ts goal s to acheve a SIR value no lower than the threshold. To avod ntroducng nfeasble users nto the above system, two dstrbuted admsson control schemes were proposed n []. The basc dea s that before the new user enters the system and begns to tune ts power n full gear accordng to (), t tentatvely transmts at a fxed level, then decdes (n a dstrbuted way) f the system wll become nfeasble after ts jonng. Only f the user wll not cause nfeasblty, s t admtted. Although no nfeasble user can slp n now, there are stll some problems that persst. In a wreless cellular system, users can be n constant movement, and an ntally feasble user may later become nfeasble. Also, due to moblty, the Pareto optmal power assgnment acheved by () may no longer be optmal and may even be nfeasble a moment later, whch may result n a hgh outage probablty []. We solve these problems by softenng the hard SIR requrements. III. UTILITY-BASED DISTRIBUTED POWER CONTROL Although achevng satsfactory qualty of servce (QoS) s mportant for users, they may not be wllng to acheve t at arbtrarly hgh power levels, because power s tself a valuable commodty. Ths observaton motvates a reformulaton of the whole problem usng concepts from mcroeconomcs and game theory [5], [6]. In ths secton, we wll use such a reformulaton to present a mechansm for power control where the desre for ncreased SIR s weghed aganst the assocated cost. We wll pont out the dfferences between our work and that n [5], [6] and we wll also dscuss some mportant mplcatons of these dfferences. A. Problem Formulaton and Basc Algorthm Instead of enforcng the constrant SIR as n the hard constrant case, we use a utlty functon U to represent the degree of satsfacton of user to the servce qualty, and ntroduce a cost functon C to measure the cost ncurred. The goal s to maxmze the net utlty N U defned as N U = U? C by adjustng the transmtted power P. Snce each user n the system wll try to maxmze ts own net utlty, regardless of what happens to the other users, ths problem s a typcal non-cooperatve N-person game [3]. Generally, the QoS depends on SIR, so we let the utlty U be a functon of SIR satsfyng: U () = ; U () = ; and that U (SIR ) ncreases n SIR. We wll use the utlty gven n Fgure (called Sgmod utlty) n ths paper, because t s smlar n shape to the capture probablty []. However, t should be noted that our scheme s also applcable to many other utlty functons. We choose C, the cost for user, as a functon of power because, as mentoned before, power by tself s a valuable commodty. Cuttng power consumpton not only prolongs the lfe-

3 U (SIR ) SIR Fg.. Sgmod utlty (and cost) versus SIR for user. tme of the battery, and allevates health concerns about electromagnetc emsson, but also decreases the nterference to other users. The specfc cost functon should reflect the expenses of power consumpton to the user. There are at least two requrements for the cost functon: C () =, and that C (P ) ncreases n power P. In ths paper, we wll use a lnear cost functon,.e., C (P ) = P ; (3) where s the prce coeffcent. Although s a constant ndependent of P, t can be a functon of envronment factors such as user locaton and receved nterference. In fact, ths knd of adaptve prce settng can be very helpful n achevng farness and robustness. As s the case for the utlty functon, other forms of cost functons would also work for our scheme (although not as smply). The net utlty of user s N U (SIR ; P ) = U (SIR )? C (P ): The power control problem for user s formulated as max N U : () P If a postve power P s a local optmum for problem (), we U (SIR ) = ; (5) where = P j6= G jp j + s the total receved nterference of user. Because the rght hand sde of (5) s known or locally measurable, we can fnd the soluton to be dsir = f? ( ); (6) where f (SIR ) = U (SIR ) n the concave part of U, where a local maxmum s possble. Then the correspondng power assgnment s I ^P = SIR d = f? ( ): (7) Clearly, ^P s a functon of I, for gven prce coeffcent. If we put n the tme ndex, then equaton (7) becomes ^P (k + ) = d SIR (k) (k) (k) = d SIR (k) SIR (k) P (k): (8) Note that P = s on the constrant boundary of (), so the optmal power at step k, P (k), s ether ^P (k) or, whchever results n a larger net utlty. Ths gves us a dstrbuted power control algorthm for optmzng net utlty wth no ntercellular communcaton overhead. We call ths algorthm Utlty-Based Power Control (UBPC), and next llustrate t graphcally. Also shown n Fgure s the cost functon. Because I C (P ) = P = SIR, the slope of the cost lne n I Fgure s. By changng the slope, we have a dfferent poston of C relatve to utlty U. Note that by defnton, so the cost lne has a postve lower bound as shown by lne, the slope of whch s K =. If the cost lne (e.g., lne ) les between lne and lne, there wll be some postve net utltycorrespondng to ^P,.e., P = ^P. If the cost lne reaches lne, the maxmum net utlty s, whch s acheved at power levels ^P and. If the cost lne (e.g., lne 3) s beyond lne, the best choce s to keep P =, because all other powers wll result n negatve net utlty. We wll call the SIR assocated wth cost lne the turnoff SIR of user, and denote t by SIR, whch s the lowest SIR value user wll acheve when transmttng. To be consstent wth the hard requrements SIR, we assume that SIR = from now on, because under ths settng user always acheves SIR hgher than when transmttng. Correspondngly, we denote the slope of lne by K. In summary, the teratve procedure of the utlty-based power control for user s as follows: Algorthm UBPC. Measure receved nterference (k), update path gan (k) and prce coeffcent (k), then calculate d SIR (k) usng (6). If dsir (k) <, turn off transmsson (.e., P (k + ) = ), and go to step 3. Otherwse, go to step.. Calculate ^P (k + ) usng (8), and tune the power level to ^P (k + ). 3. Let k k +, and go to step. The nformaton requred by UBPC from the utlty of user s the curve of the target SIR, d SIR va =, whch s shown n Fgure 3. Hence, n mplementaton, a user only has to provde the curve nstead of the utlty. Ths smplfes mplementaton, and also enhances flexblty to meet heterogeneous QoS requrements, because the curve of target SIR can have other forms, e.g., star form.

4 .35 5 U (SIR ) Target SIR..5 α / 5 SIR 5 Fg.. Dervatve functon of Sgmod utlty versus SIR for user α / Fg. 3. Target SIR versus transmsson envronment of user. Our utlty s a functon of only SIR, and we also ntroduce a cost term of power n the objectve functon N U. Ths arrangement not only acheves separaton (between utlty and cost) and smplfes dervaton, but also benefts the system performance and facltates network resource management. Before elaboratng these ponts, we wll frst show that UBPC overcomes the dvergence problem of the power control algorthm n () whle keepng all of ts major merts. B. Convergence Analyss of UBPC UBPC s non-cooperatve and dstrbuted. However, by encompassng a penalty on power as a cost term, we do ntroduce some cooperaton among users. To llustrate ths clearly, we show the dervatve functon U (SIR ) of the Sgmod utlty functon n Fgure. The horzontal lnes n ths fgure correspond to the cost lnes n Fgure, respectvely. Lnes,, and each has two ntersectons wth U (SIR ), but only the ntersecton on the rght sde qualfes for ^P because U s concave n ths part. In fact, the left sde ntersectons are the mnmum ponts. It s very mportant to note that, as llustrated n Fgure 3, the target SIR, SIR d I wll decrease as ncreases, and remans after reachng the turnoff SIR. Contrastng (8) wth (), we see that wth UBPC the target SIR value wll decrease automatcally for a worse transmsson envronment (.e., (k) (k) for larger ), whle wth the power control algorthm gven by (), t remans the same regardless of the envronment. When the transmsson envronment becomes very hostle, there wll be no gan (postve net utlty) n transmttng, and the transmsson wll be totally shut off by UBPC. Ths just mples that UBPC wll never result n powers blowng up. However, because the system cannot accommodate an nfnte number of users, we can also defne feasblty of the power-controlled system under UBPC. A system s sad to be feasble under UBPC f there exsts a power assgnment P = [P ; P ; : : :; P n ] T, such that for all user, SIR d SIR N U > : We call such a power assgnment P feasble under UBPC. As dscussed earler, unlke n the case of (), UBPC does not blow up even when the system s nfeasble. We next study how a feasble system performs under UBPC. In [], Yates proposes a framework for power control called standard power control. Every power control algorthm under ths framework has several desrable propertes, ncludng convergence n both synchronous and asynchronous cases for feasble systems. A power control algorthm P(k + ) = A(P(k)) s sad to be standard f A satsfes the followng propertes for all P : Postvty: A(P) >. Monotoncty: If P P, then A(P ) A(P). Scalablty: For all >, A(P) > A(P). Ths framework has been proposed for systems wth a hard constrant P A(P), where A(P) s called the nterference functon. A typcal example of a standard power control s (). We next show that UBPC s standard under very mld condtons. Theorem : UBPC s standard f f? (x) x s an ncreasng functon n [K ; K ] for all. Proof: From (7), the nterference functon of UBPC s A(P) = [A (P); A (P); : : : ; A n (P)] T, where P = [P ; P ; : : : ; P n ] T, and A (P) = (9) f? ( ); () where = P j6= G jp j +. The postvty property s mpled by the nonzero background recever nose. If P P, then I, where I = P j6= G jp j +. Because the functon f? (x) x s an ncreasng functon n

5 [K ; K ], we get A(P ) A(P) for P P. (As a remnder, the quanttes K and K are the slopes of lne and lne n Fgure, respectvely.) For all >, we have A(P) P j6= = G jp j + f? G P j6= < G jp j + f? < = A(P); P j6= G jp j + f? P j6= ( G jp j + ) G P j6= ( G jp j + ) G P j6= ( G jp j + ) where the frst nequalty holds because f (x) s a decreasng functon n [K ; K ]. Thus, UBPC s standard f f? (x) x s an ncreasng functon n [K ; K ]. For a fxed prce coeffcent, the functon f? (x) x s ncreasng f and only f the shaded area (under lne and to the left of ts rght sde ntersecton) n Fgure ncreases n the heght of lne, ( ). Ths s a very mld condton, because a user n practce usually has a large a or a large (see Secton IV-A). Hence, UBPC s standard for almost all practcal stuatons of nterest. Lke other standard power control algorthms, UBPC appled to a feasble system also has the followng propertes that can be proved n exactly the same way as n []. Propertes of UBPC. There s a unque fxed pont P.. If the ntal power assgnment P s feasble, then UBPC generates a monotone decreasng sequence of feasble power assgnments that converges to P. Ths mples that the fxed pont P s Pareto optmal (componentwse mnmum) n the set of all feasble power assgnments. 3. If the ntal power assgnment s the all-zero vector, then UBPC generates a monotone ncreasng sequence of power assgnments that converges to P.. UBPC converges from any ntal power assgnment to the unque fxed pont P n both synchronous and asynchronous cases. The standard power control framework was orgnally proposed for systems wth a hard constrant (e.g., SIR ), whch drectly leads to the feasblty condton and the nterference functon. However, systems under UBPC do not have a hard requrement, and the feasblty condton n (9) for such systems s mpled by the power control nstead. In ths sense, our work here essentally generalzes the standard power control framework n []. Although the feasblty condton n (9) for systems under UBPC s somewhat subtle, we can nfer a physcal meanng for ths feasblty defnton from the followng theorem. Theorem : A system s feasble under UBPC f and only f no user s turned off when startng from the all-zero power assgnment. Proof: If the system s not feasble under UBPC, then for any power assgnment P(k) we wll have ether N U or SIR (k) < SIR d (k) for all. The former condton means that user has to be turned off. If ths happens, we fnsh the proof of ths part. So we only have to consder the case where N U > n all steps,.e., SIR (k) < SIR d (k). By (8), ths condton means that UBPC generates an ncreasng sequence of power assgnments. If the ncreasng sequence s bounded from above, then the sequence wll converge [5] to a power assgnment P, at whch SIR = SIR d for all. However, ths contradcts the assumpton that the system s nfeasble. Thus, the ncreasng of the power (and of the cost) could be unlmted. Then there always exsts a user j that has N U j at some step of teraton, because U j. Ths just means that user j s turned off by UBPC at ths step. The other drecton of the theorem follows from Property 3 of UBPC. In fact, we need not start from the all-zero assgnment at all. If the ntal power assgnment s arbtrary, UBPC appled to a feasble system may turn off some user temporarly, but eventually all users wll be turned on. Theorem mples that an nfeasble system always has some user turned off, but ths does not exclude the possblty that a user that s swtched off retres transmsson, whch may result n power oscllaton. Fortunately, we can avod such a stuaton by forbddng mmedate retres or by settng a hgh prce coeffcent (see Secton IV-B). Then, unlke the standard power controls wth hard constrants such as (), UBPC can acheve convergence even when nfeasblty arses. Moreover, a system convergng under UBPC always reaches the Nash equlbrum [3], whch s straghtforward to check from the defnton of Nash equlbrum. IV. DISCUSSIONS AND EXTENSIONS OF UBPC In ths secton, we wll llustrate that several desrable propertes and a hgh degree of flexblty can be acheved by properly tunng the parameters n UBPC. We also brefly dscuss some rado resource management technques that can be jontly consdered wth UBPC. A. Effects of Parameters n Utlty Functon The utlty functon represents the user' s degree of satsfacton wth the servce qualty, so t should be able to reflect the requrement of servce. There are two tunable parameters n the Sgmod utlty U : the steepness a (the maxmum tangent slope of U ) and the center. Note that the turnoff SIR s unquely determned by a and, and s a predefned QoS requrement for user. Hence, we often use a and as the two parameters to characterze the utlty functon U. We llustrate the utlty functons and the correspondng dervatve functons for two dfferent values of a n Fgures and 5, respectvely. Applyng UBPC to two users wth utlty functons shown n Fgure, we fnd that for the user wth a larger value of a, the I dsir decreases more slowly as ncreases. Thus, a user havng a utlty wth large value of a s rgd, and s hard to

6 U (SIR ) U (SIR ) a =, β = a =, β = 5 5 SIR Fg.. Sgmod utlty versus SIR wth dfferent a. a =, β = a =, β = 5 5 SIR Fg. 5. Dervatve functon of Sgmod utlty versus SIR wth dfferent a. be turned off (especally when s small). If user has a step utlty and no cost (.e., = ), then our UBPC just reduces to the power control algorthm (), because the target d SIR s fxed to. Ths type of rgd user can never be turned off by UBPC. If we fx a and change, the utlty functons wll have the same shape, but wth dfferent turnoff SIR' s. Clearly, a user wth hgher n utlty wll get hgher SIR when n servce, but s more lkely to be turned off by UBPC. Based on the effect of the two parameters on our power control, we make the followng mportant observaton: UBPC s sutable for ntegrated wreless systems wth both voce users and data users, mportant n the thrd-generaton wreless networks. To llustrate ths pont clearly, we start wth the wellknown dfference between voce and data n ther servce requrements. For a voce user, the essental objectve s low delay, and transmsson errors are tolerable up to a relatvely hgh pont. Thus, the voce user does not want to be easly turned off, but ts target SIR can be relatvely low. Ths means that for a voce user, the cost should be low, and the utlty functon should be steep and wth low turnoff SIR,.e., s small, a s large and s small. Note that a must be large enough, but need not be nfnte (n contrast to what s used n [5]). On the other hand, a data user can accept some delay but has very low tolerance to errors, whch can be satsfed by a utlty functon wth a smaller a and larger. A user wth such a utlty can acheve a hgh SIR when transmttng, but t must allow for beng turned off and for delayng ts transmsson untl later. Thus, we can characterze the servce requrements of both voce users and data users by choosng dfferent utlty parameters. In ths way, these two dfferent types of users have a unfed and comparable measure for sharng rado resource. It should also be ponted out that under the framework of UBPC, voce users can get preemptve prorty over data users due to ther rgd behavor, whch s obvously desrable n ths case. Moreover, some factors such as voce actvty factor and data burstness can be automatcally exploted through the receved nterference. We beleve UBPC also works for hybrd systems wth both crcutswtched and packet-swtched users. So far, we have assumed that the prce coeffcent s fxed, and we have not consdered ts effect. However, the prce need not be fxed. As wll be descrbed n the followng sectons, adaptve prce settng mechansms can be helpful n solvng problems such as farness, robustness, and admsson control. We should also pont out that prce settng can be userdependent. For example, an automoble-mounted phone has a much larger supply of power than a handheld one, so ts user may choose a lower prce coeffcent. B. Adaptveness and Farness Consderatons B. Adaptve Prce Settng It has been shown that SIR d I decreases as the quantty ncreases; and when ths quantty exceeds some crtcal pont K (the slope of lne n Fgure ), there wll be no postve net utlty, and user should stop transmsson. Clearly, the prce coeffcent has the same effect as I does on the power control. In other words, transmsson s dscouraged by a large value of as well as by hostle transmsson envronment (n terms of ). Ths s ntutvely true who s wllng to pay more for the same servce? However, as to be explaned soon, a hgher prce coeffcent also means a more robust system, whch can facltate network management n congested cases. From the network pont of vew, the network can use the prce coeffcent as an effectve way to manage resource and to maxmze revenue. When the transmsson envronment s desrable, we should set a low prce, allowng the user to enjoy good qualty of servce (hgh SIR or hgh transmsson rate). On the other hand, when congeston bulds up, we should set a hgh prce, to mprove system robustness. In fact, f the prce coeffcent s not hgh enough, under heavy traffc stuatons, a user may re-

7 peat the procedure of beng turned off and retryng alternatvely, resultng n oscllaton. Thus, t s desrable to have a prce coeffcent that s adaptve to the transmsson envronment. A good measure for the transmsson envronment experenced by user s I, so we should set the prce coeffcent as an ncreasng functon of ths quantty. For example, a smple adaptve settng can be = ; () where s a constant (may be provded by the base staton). Then, we have a dfferent power control algorthm, ^P = SIR d = f? I! : () Our smulatons show that UBPC wth the lnear adaptve settng works well under a large range of traffc loads. B. Near-far Farness The basc UBPC scheme exhbts a knd of unfarness, the so-called near-far unfarness. A user far from the base staton (called far user ) normally has a small path gan, so t s more lkely to get lower SIR or to be turned off. Ths s benefcal to the network, because the total throughput can be mproved n ths way. In fact, t has been shown by Oh and Wasserman [6] that the optmal scheme to maxmze throughput s that near users transmt at hghest power whle far users totally turn off. Of course, these arrangements are extremely unfar to users close to cell boundares, who may never get a chance. In moble cellular systems, t may be especally undesrable to turn off far users, because handoff happens around cell boundary, and t s generally agreed that handoff users should be granted hgh prorty rather than turned off, to avod the undesrable stuaton of call droppng [7]. Moreover, the reason to use power control n the frst place s to overcome the near-far problem n CDMA systems. Thus, we must acheve near-far farness even at some cost of throughput. To ths end, we should set a lower prce to the further user. One smple and natural scheme s where s a constant. Then we have = ; (3) = = ; ().e., we perform power control based only on the receved nterference (see Fgure 3), and have no prejudce aganst far users. At the same tme, we also acheve handoff prortzaton. If ths weghtng s stll not enough, we can further decrease the prce for handoff users. Other farness problems (e.g., deadlne farness) can be solved smlarly. Snce farness s often acheved at the cost of throughput, what prce to use should depend on the tradeoff between farness and throughput. B.3 Combned Prce Settng To acheve both adaptveness and farness, we only have to combne schemes () and (3), to get the followng prce settng: = = ; (5) where s a constant ndependent of and, and can be provded by the base staton. In (5), the prce coeffcent adopted by a user s proportonal to ts receved nterference. Now the new power control algorthm becomes ^P = d SIR = f? It stll depends on, but more on. C. Integrated Resource Management ( I ): (6) It has been shown that power control can greatly mprove system capacty [8]. On the other hand, resource management of power-controlled systems often becomes more challengng because of the dynamcally varable capacty and lmted (locally) avalable nformaton [8], []. In the followng, we wll llustrate that UBPC can be readly ntegrated wth other resource management technques, and facltate ther mplementatons. The task of admsson control (AC) s to decde whether or not to grant access to a newly arrvng user. A good admsson control scheme should admt as many users as possble whle mantanng the qualty of ongong users. It s nterestng to note that UBPC automatcally has some AC functon. Rejecton happens when a new user fnds that ts cost lne s beyond (hgher than) lne n Fgure. If there are enough resources to accommodate a new user, t s accepted to receve servce better than ts mnmum requrement. As a new user gets admtted, the ongong users may yeld a lttle to make space for t. It s also possble for an nfeasble new user to get servce by turnng an exstng user off. Such a stuaton allows a user of hgh prorty (e.g., voce user) to preempt a user of best-effort servce. The user beng turned off frst s usually the bottleneck user, whch has the worst transmsson envronment. As mentoned before, the transmsson envronment can be changed by tunng the prce coeffcent. Hence, to provde more protecton to ongong users or handoff users, we only have to lower ther prces or set hgher ntal prces to new users. In short, power-controlled systems under UBPC are hghly autonomous, and the behavor of a user depends on ts utlty, prce coeffcent, and nteractons wth other users n the system. Lke the dstrbuted power control scheme (), UBPC can also be solved jontly wth dynamc base staton and channel assgnment [9], [9]. The man dfference s that we frst choose the base staton and channel resultng n the best transmsson envronment (n terms of = ). These dynamc assgnments, along wth the adaptveness of UBPC, promse good performance n the face of moblty. Consderng the hostle envronment encountered n wreless systems, t s desrable for users to have adjustable transmsson

8 rates []. Wth rate control, systems under UBPC can take advantage of hgh SIR n desrable transmsson envronments, and translate t nto hgh transmsson rate to ncrease throughput. V. SIMULATION RESULTS In ths secton, we smulate the evoluton of power and SIR for dfferent algorthms. The purpose of the smulatons s to show that UBPC overcomes the dvergence problem, and s flexble to acheve farness and adaptveness usng dfferent parameters and dfferent prce coeffcent settngs. Capacty mprovement of power-controlled systems has been demonstrated n [8], so we wll not cover t n ths space-lmted paper. For smplcty, we consder a one-channel lnear cellular system consstng of cells. Base statons use omndrectonal antennas and are located at the center. The dstrbuton of users s llustrated n Fgure 6. In ths fgure, each vertcal lne represents a user, and the heght of a lne llustrates the SIR threshold of the correspondng user. Each user s numbered accordng to arrval order. Base statons are marked by X' s on x-axs, and each user s assgned to the closest base staton. The path gan G j s modeled as G j = A j =d j, where d j s the dstance between user and home base staton of user j, and the attenuaton factor A j models the power varaton due to shadowng. We assume all A j to be ndependent and dentcally log-normally dstrbuted random varables wth db expectaton and 8 db log-varance as n [], [9]. The path gan matrx G of our smulated system s O(?5 ) :37 O(? ) O(?3 ) O(? ) O(?5 ) 53:67 O(?5 ) O(?3 ) O(?5 ) O(?5 ) :9 O(?5 ) 66:6 O(? ) O(? ) O(? ) O(? ) O(?3 ) O(? ) 8556:8 O(?5 ) O(? ) O(?3 ) O(?5 ) O(? ) O(?5 ) 38:3 7:8 O(?3 ) O(?5 ) O(? ) O(?5 ) 5:3 7: ; where O(?x ) means a quantty on the order of?x. From ths matrx, we know that users, 5, and 6 are far users, and the latter two are closely coupled. It s easy to check that the Perron-Frobenous egenvalue of the system s :8 >,.e., the system s nfeasble [], []. Note that the remanng system after removng user 5 or 6 s feasble. We frst apply the power control algorthm () wthout AC, and the results are shown n Fgure 7. Each lne n ths fgure shows the evoluton of power or SIR of a user. The leftmost lne corresponds to user, the rghtmost lne corresponds to user 6, and so on (smlarly for the other smulaton plots n ths secton). We observe from the fgure that algorthm () works very well untl an nfeasble user s admtted, after whch the power control algorthm becomes unstable. Ths s shown by the power blow-up and SIR oscllatons of users 5 and 6 n the fgure. To mplement UBPC, each user should specfy a utlty functon and a prce coeffcent. We assume that the sx users n the system use Sgmod utlty, wth turnoff SIR' s equal to the correspondng SIR thresholds, and steepness beng.99,.35,.93,.,.66, and., respectvely. Clearly, user 6 s more rgd than user 5. Fgure 8 plots the evoluton of power and SIR for UBPC, wth prce coeffcent for all users. It works smlarly to () when Power γ SIR users base statons x x x x x x x x x x x x x x x x x x x x 5 5 Fg. 6. Dstrbuton of users n the system. 6 8 Iteratons 5 user user 5 5 user user 3 user user Iteratons user 5 user 6 Fg. 7. Evoluton of power and SIR usng power control (). the system s feasble. However, as the system reaches the pont of nfeasblty, the admsson of user 6 forces ongong user 5, whch s less rgd, to turn off. The turnoff of user 5 allows user 6 to get satsfactory servce at lower power. Now we ncrease the the prce coeffcent to for all users. The results are shown n Fgure 9, where users and 5 are turned off (at teratons and ). As ponted out before, users and 5 are far users, whch are admssble only when prce s low. Hence, hgh prce tends to make conservatve admsson, leadng to hgh system robustness. However, a hgh prce wll also dscourage users to transmt at hgh SIR even when the traffc load s low. To solve ths problem, we use adaptve prce coeffcent gven by (), where = 5; for all users. As shown n Fgure, user 5 s turned off, and all other users acheve hgher SIR. Thus, ths prce coeffcent acheves transmsson envronment adaptveness and robustness at the same tme.

9 6 x 3 6 x 3 Power Power 5 5 Iteratons Iteratons SIR 5 SIR Iteratons Iteratons Fg. 8. Evoluton of power and SIR, usng UBPC wth prce =. Fg.. Evoluton of power and SIR, usng UBPC wth adaptve prce. Power SIR 8 x Iteratons Iteratons Fg. 9. Evoluton of power and SIR, usng UBPC wth prce =. All the above schemes demonstrate near-far unfarness. To gve far users more far shares of resource, we use UBPC wth combned prce coeffcent gven by (5), where =. Fgure shows the results. Under ths prce settng, nether user nor user 5 s turned off. Instead, user 6 s rejected for mantanng the feasblty of the system of exstng users, even though user 6 s more rgd. All these seem to ndcate that the combned prce settng s superor to all others above as to robustness, farness, and actve lnk protecton. Fnally, n Fgure we compare the SIR's acheved by dfferent schemes. There are sx groups of bars n the fgure, where the th group corresponds to user. In each group, the frst bar represents the SIR threshold, the mnmum SIR requred; the remanng bars correspond to the acheved SIR under power control al- Power SIR 5 5 Iteratons Iteratons Fg.. Evoluton of power and SIR, usng UBPC wth combned prce. gorthm (), UBPC wth fxed prce, wth fxed prce, wth adaptve prce, and wth combned prce, respectvely. The empty slots n some groups are due to turned-off users. Ths fgure llustrates that for nfeasble systems, algorthm () leads to SIR' s lower than ther correspondng thresholds, whle UBPC guarantees QoS to most users by turnng off some bottleneck users. We also verfy that a low prce encourages transmttng, and adaptve prce settngs generally perform better. VI. CONCLUSIONS In ths paper, we frst demonstrate that when nfeasblty arses, the well-known dstrbuted power control algorthm () dverges, due to the hard constrant of SIR requrement. Usng ths algorthm, every user tres to acheve ts requred SIR value, no matter how hgh the power consumpton, whch g-

10 SIR SIR threshold F M Algorthm UBPC, α= UBPC, α= UBPC, adaptve α UBPC, combned α User number Fg.. Comparson of acheved SIR under dfferent power control schemes. nores the basc fact that power s tself a valuable commodty. By softenng the SIR requrement usng utlty functons, and by ntroducng a cost functon for power, we propose a power control framework called utlty-based power control (UBPC). Although UBPC s stll non-cooperatve and dstrbuted n nature, some degree of cooperaton emerges: a user wll automatcally decrease ts target SIR (and may even turn off transmsson) when t senses that traffc congeston s buldng up. It s ths cooperaton that prevents the system from blowng up when nfeasblty arses. At the same tme, under very mld condtons UBPC s standard (n the sense defned n []), whch mples asynchronous convergence of the algorthm when appled to a feasble system. Whle UBPC s of best-effort flavor, the degree of yeldablty s tunable. Several tunable parameters n UBPC enable ths scheme extremely flexble to satsfy dfferent servce requrements (such as farness, delay, and bt error rate) n ntegrated networks wth both voce and data users. It also allows ntegraton wth some network resource management technques. Sgnfcant mprovements over the exstng algorthm are demonstrated by analyss and smulaton. Whle UBPC provdes a promsng framework for dstrbuted power control of cellular wreless systems, t s stll not welldeveloped. How to translate dfferent QoS requrements nto utlty and cost functons that lead to a solvable power control problem, how to acheve system optmalty, and how to relate a cost term to practcal prcng schemes are all topcs requrng further research. cellular rado systems, n Proc. th Wnlab Workshop Thrd Generaton Wreless Informaton Network, Rutgers Unversty, Oct. 993, pp [] R. D. Yates, A framework for uplnk power control n cellular rado systems, IEEE Journal on Selected Areas n Communcatons, vol. 3, pp. 3 37, Sept [5] D. Goodman and N. Mandayam, Power control for wreless data, IEEE Personal Communcatons, vol. 7, pp. 8 5, Apr.. [6] H. J and C.-Y. Huang, Non-cooperatve uplnk power control n cellular rado systems, Wreless Networks, vol., pp. 33, Apr [7] N. Bambos, S. C. Chen, and G. J. Potte, Rado lnk admsson algorthm for wreless networks wth power control and actve lnk qualty protecton, n Proceedngs of IEEE INFOCOM, Boston, MA, Apr. 995, pp. 97. [8] J. Zander, Performance of optmum transmtter power control n cellular rado systems, IEEE Transactons on Vehcular Technology, vol., pp. 57 6, Feb. 99. [9] C.-Y. Huang, Rado resource management n power controlled CDMA systems, Ph.D. thess, Rutgers Unversty, Pscataway, NJ, 996. [] D. Km, Rate-regulated power control for supportng flexble transmsson n future CDMA moble networks, IEEE Journal on Selected Areas n Communcatons, vol. 7, pp , May 999. [] M. Xao, N. B. Shroff, and E. K. P. Chong, Dstrbuted connecton admsson control for power-controlled cellular wreless systems, n Thrtyseventh annual Allerton conference on Communcaton, control, and computng, Montcello, IL, Sept. 999, pp [] M. Andersn and Z. Rosberg, Tme varant power control n cellular networks, n Proc. PIMRC ' 96, Tape, Chna, Apr. 996, pp [3] D. Fudenberg and J. Trole, Game Theory, MIT Press, Cambrdge, MA, 99. [] F. Bogonovo, L. Fratta, P. Mlano, M. Zorz, and A. Acampora, Capture dvson packetaccess: a new cellular access archtecture for future PCNs, IEEE Communcatons Magazne, vol. 3, pp. 5 6, Sept [5] H. L. Royden, Real Analyss (3rd Ed.), Macmllan, New York, 988. [6] S.-J. Oh, T. Lennon Olsen, and K. M. Wasserman, Dstrbuted power control and spreadng gan allocaton n CDMA data networks, n Proceedngs of IEEE INFOCOM, Tel Avv, Israel, Mar.. [7] E. C. Posner and R. Guern, Traffc polces n cellular rado that mnmze blockng of handoff calls, n Proc. th Teletraffc Conf. (ITC-), Kyoto, Japan, Sept [8] N. Bambos, Toward power-senstve network archtectures n wreless communcatons: concepts, ssues, and desgn aspects, IEEE Personal Communcatons, vol. 5, pp. 5 59, June 998. [9] A. Lozano and D. C. Cox, Integrated dynamc channel assgnment and power control n tdma moble wreless communcaton systems, IEEE Journal on Selected Areas n Communcatons, vol. 7, pp. 3, Nov REFERENCES [] G. J. Foschn and Z. Mljanc, A smple dstrbuted autonomous power control algorthm and ts convergence, IEEE Transactons on Vehcular Technology, vol., pp. 6 66, Nov [] S. C. Chen, N. Bambos, and G. J. Potte, Admsson control schemes for wreless communcaton networks wth adjustable transmtter powers, n Proceedngs of IEEE INFOCOM, Toronto, Canada, June 99, pp. 8. [3] D. Mtra, An asynchronous dstrbuted algorthm for power control n