Uplink Power Control in QoS-aware Multi- Service CDMA Wireless Networks

Transcription

1 654 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 Uplnk Power Control n QoS-aware Mult- Servce CDMA Wreless Networks Ern Elen Tsropoulou Tmotheos Kastrnoganns and Symeon Papavasslou Member IEEE Insttute o Communcaton and Computer Systems (ICCS School o Electrcal and Computer Engneerng Natonal Techncal Unversty o Athens Athens 1578 Greece {eetsrop tmothe}@netmode.ntua.gr; papavass@mal.ntua.gr Abstract In ths paper the problem o QoS-drven power control n the uplnk o CDMA wreless networks supportng multple servces s consdered. Due to the need or supportng smultaneously varous servces wth dverse QoS requrements each user s assocated wth a nested utlty uncton whch approprately represents hs degree o satsacton n relaton to the expected tradeo between hs QoS-aware actual uplnk throughput perormance and the correspondng power consumpton. Based on ths ramework the problem s ormulated as a non-convex noncooperatve Mult-Servce Uplnk Power Control (MSUPC game where users am selshly at maxmzng ther utltybased perormance under the mposed physcal lmtatons. We rst prove the exstence and unqueness o a Nash equlbrum pont o the MSUPC game and then a decentralzed teratve algorthm to obtan MSUPC game s equlbrum pont s ntroduced. Fnally through modelng and smulaton the operaton and eatures o the proposed ramework and the decentralzed MSUPC algorthm are evaluated wth respect to the ulllment o assorted servces QoS requrements whle numercal results and relevant dscussons that demonstrate the tradeos among users channel condtons correspondng power consumpton and utlty-based perormance are presented. Index Terms CDMA networks power control QoS resource allocaton multmeda servces. I. INTRODUCTION Wth the growng demand or hgh data rate and support o multple servces wth varous qualty o servce (QoS requrements the schedulng polcy plays a key role n the ecent resource allocaton process n uture wreless networks. The nner characterstcs o wreless communcatons n code dvson multple access (CDMA cellular systems n terms o network s scarce rado resources moble nodes physcal lmtatons and users tme-varyng channel condtons have motvated the adopton o power control n both the uplnk and downlnk communcaton or procent resource allocaton and ntererence management. Recently game theoretc approaches or studyng power control n the uplnk o CDMA networks have attracted sgncant attenton [1] [] as an alternatve to tradtonal power control approaches that have been based on decsons taken n a centralzed [3]-[7] or sem- Manuscrpt receved February 6 9; accepted May centralzed [8] way at the base staton. Power control s modeled as a non-cooperatve game where users act as ndvdual players amng at maxmzng selshly ther degree o satsacton expressed by approprately dened utlty unctons. In prncple utltes assgned to users are unctons o the consumed power and the acheved sgnal to nose rato whle each user determnes hs own transmsson power level so as to maxmze hs utlty. Intutvely the choce o the utlty unctons has a great mpact on the nature o the game and how the users choose ther respectve actons. In [9] and [1] towards maxmzng system s spectral ecency users utltes are expressed as logarthmc concave unctons o ther sgnal-to-ntererence-plus-nose rato (SINR. Ths type o utlty unctons s proportonal to the Shannon capacty or each user thus treatng all ntererence as whte Gaussan nose [11]. In [1] the utlty uncton s assumed to be proportonal to the user s throughput and a prcng uncton based on the normalzed receved power o the user s proposed. S-modular power control games are studed n [13]. In [14] the authors use a utlty uncton that measures the number o relable bts that are transmtted (.e. goodput per joule o energy consumed. Ths s urther extended n [15] by ntroducng prcng to mprove the ecency o Nash equlbrum. Jont energyecent power control and recever desgn has been studed n [16] [17] and a game theoretc approach to energy-ecent power allocaton n multcarrer systems s ntroduced n [18]. Jont network-centrc and usercentrc power control s dscussed n [19] whle the stablty o dstrbuted power control or CDMA systems s addressed n []. In ths paper we study the problem o channel-aware power control n the uplnk o CDMA networks supportng multple servces (.e. both non-real-tme and real-tme wth varous and oten dverse QoS prerequstes va game theory. Followng recent research eorts and approaches we also adopt a utlty-based ramework. However one o the undamental derences n our work les n the denton and treatment o users utlty unctons n order to ulll smultaneously multple servces QoS perormance requrements n an envronment where the servce perormances are nterrelated. Speccally nstead o smply expressng the tradeo between the number o a user s relably 9 ACADEMY PUBLISHER do:1.434/jcm

2 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER transmtted bts and correspondng consumed power n our study a user s utlty that relects hs servce QoSaware perormance ecacy as a uncton o hs acheved goodput per joule o consumed energy s consdered. To acltate our goal general nested utltes o users acheved goodput are adopted wth respect to ther servce type whch may not be concave thereore leadng us to the ormulaton o a non-convex noncooperatve Mult-Servce Uplnk Power Control (MSUPC game. Ths generalzaton requres a sgncantly derent treatment than the works presented n [16] and []. By explotng the ntroduced utlty-based ramework we denty specc desgnng attrbutes that users actual throughput utltes must encompass so as to procently express ther servce QoS-aware perormance expectatons. Furthermore n our work we place specal emphass on multmeda servces QoS prerequstes ulllment n a tme-varyng envronment. It s noted that n [1] a utlty-based ramework or relectng real-rme users degree o satsacton n accordance to ther achevable goodput perormance has also been consdered however the correspondng users energy consumpton has not been taken nto account and utlty attrbutes are not studed. In [] a game-theoretc ramework has also been adopted or studyng the power control problem n the uplnk o CDMA networks supportng delay senstve real-tme servces. Nevertheless n that work and n [3] only lnear utltes o users goodput are consdered whle real-tme servces QoS requrements are expressed as statstcal delay constrants; a consderaton whch n many cases s not sucent snce as shown n [4] [5] even the delay constrants o a real-tme user are satsed the degradaton o hs servce qualty may not be avoded due to the possbly bad channel condtons and varatons. In ths work strct short-term throughput requrements are adopted or real-tme multmeda servces expressed va ther actual throughput utltes n order to smultaneously satsy both ther short-term delay and throughput prerequstes. It s noted that our approach provdes a methodology or settng and analyzng generc MSUPC games n CDMA wreless networks. We denty necessary and sucent condtons or the exstence and unqueness o a Nash equlbrum pont n ths non-convex multobjectve MSUPC optmzaton problem. Moreover a low-complexty teratve algorthm or reachng MSUPC game s Nash equlbrum s proposed and ts convergence s studed. The dstrbuted nature o MSUPC algorthm avors a user-centrc QoS-aware uplnk archtecture n sync wth uture networkng vson that regards as ts oundaton element a sel-optmzed wreless node. Furthermore va modelng and smulaton we extensvely study the propertes o system s equlbrum and as a result strong correlatons and tradeos among real-tme servces strct QoS requrements ulllment non-real-tme servces greedy throughput expectatons system s modulaton and codng schemes and users power and rate lmtatons are revealed. The rest o the paper s organzed as ollows. In secton II the system model and some background normaton are provded whle n secton III users QoS propertes are studed and mapped to approprate utlty unctons. The proposed uplnk power control noncooperatve game s ormulated n secton IV and ts soluton s presented n secton V. Secton VI contans a decentralzed low complexty teratve algorthm or attanng the proposed game s Nash equlbrum and ts convergence s proven. Fnally secton VII contans some numercal results and relevant dscussons wth respect to the perormance o the proposed approach whle secton VIII concludes the paper. II. SYSTEM MODEL &BACKGROUND INFORMATION We examne the uplnk o a sngle cell tme-slotted CDMA wreless network wth N(t contnuously backlogged users at tme slot t where S(t denotes ther correspondng set. A tme slot s a xed tme nterval and could consst o one or several packets. Users tmevaryng channels are aected by ast adng shadow adng and long-tme scale varatons and thus can be modelled as a statonary tme-varyng stochastc process. Let us denote by G (t the correspondng path gan o each user S at tme slot t. The requred normaton s assumed to be avalable to all users and the base staton at the begnnng o each tme slot t. In the ollowng assumng xed users channel condtons wthn the duraton o each tme slot we omt the notaton o the specc slot t n the notatons and dentons we ntroduce. At the begnnng o each tme slot t users Mult-Servce Uplnk Power Control (MSUPC algorthm s responsble to make decsons on ther transmsson power and resultng rate n a decentralzed manner. Note that a node s transmsson power and rate are also xed wthn the duraton o a tme slot. Let us denote by P and R the uplnk transmsson power and transmsson rate o user S n the slot under consderaton whch are both upper bounded due to moble node s physcal lmtaton (.e. P P and R R. Moreover the receved bt energy to ntererence densty rato at the base staton E I or each user S s gven by: b o R P P W GP W GP ( N R GP 1 j j GP R I I P j where denotes the orthogonalty actor (=1 W s the system s spreadng bandwdth P denotes the users power allocaton vector excludng user and I contans the background nose and ntercell ntererence. Thus I( P actually denotes the network ntererence and (1 9 ACADEMY PUBLISHER

3 656 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 background nose at the base staton when recevng data rom user and s gven by: N ( j 1 j j ( I P G P GP I In our system we consder two basc types o users namely non-real-tme (NRT users requestng delaytolerant hgh-throughput servces and real-tme (RT users wth strct short-term QoS constrants. Throughout the rest o the paper we denote by N NRT (N RT the number o non-real-tme users (real-tme users and by S NRT (S RT the correspondng set. Due to the need or supportng multple servces wth varous QoS requrements each user s assocated wth a proper utlty uncton U whch represents hs degree o satsacton n relaton to the expected tradeo between hs QoS-aware utlty-based actual uplnk throughput perormance and the correspondng energy consumpton per tme slot t. Thereore t can be expressed as: F T( R P P T( R ( P P U( R P P (3 P P F where R R ( denotes user s actual uplnk transmsson rate (.e. goodput at the under consderaton F tme slot R denotes the user s xed desgned transmsson rate (.e. R F R and s hs ecency uncton. Fnally user s actual throughput utlty T s employed to relect wthn hs power control algorthm hs degree o satsacton n terms o servce perormance expectatons and QoS requrements ullment or a gven acheved goodput as dened n detal n the ollowng secton. The ecency uncton represents the probablty o a successul packet transmsson or user and s an ncreasng uncton o hs bt energy to ntererence rato at any tme slot. A user s uncton or the probablty o a successul packet transmsson at xed data rate depends on the transmsson scheme (modulaton and codng beng used and can be represented by a sgmodal-lke uncton o hs power allocaton or varous modulaton schemes [8] [16]. Thereore a user s ecency uncton has the ollowng propertes: 1 s an ncreasng uncton o. s a contnuous twce derentable sgmodal uncton wth respect to. 3 ( = to ensure that T = when P =. 4 ( = 1. III. SATISFYING MULTIPLE SERVICES QOS REQUIREMENTS Towards desgnng a procent power control algorthm that ecently supports multple servces n ths secton we ormally dene users actual throughput utltes as well as overall utltes per type o servce. In general n our study a user s actual throughput utlty Fgure 1. Actual throughput utltes as a uncton o user s acheved goodput. T( R P P T( R s a nested (due to the exstence o non-convex uncton o hs acheved goodput R properly selected to relect wthn hs overall utlty U and respectve power control polcy hs servce perormance requrements and QoS prerequstes under the mposed physcal lmtatons. We urther assume that T has the ollowng propertes. Assumptons: a T s an ncreasng uncton o R. b T s twce contnuously derentable. c T (= to ensure that U = when R =. d T s bounded above (.e. T( R 1. e T s a sgmodal-lke 1 a strctly concave or a strctly convex uncton o R wthn ts correspondng denton set [ R ] where R denotes user s maxmum achevable goodput. Note that typcally most utlty unctons that have been used n wrelne [6] or wreless [8] [1] networks can be represented by the latter three types o unctons. Specc examples o the actual throughput utlty unctons or both real-tme and non-real-tme users are ncluded n secton VII. In the rest o ths secton we ormally dene the desgnng attrbutes that a user s actual throughput utlty should possess n order to procently express hs servce QoS-aware perormance expectatons. A. Non-Real-Tme Data Servces Concernng NRT data servces T can be ether a strctly convex or a strctly concave uncton o user s acheved goodput R n order to convey hs greedy hgh throughout perormance expectatons. On the other hand due to physcal lmtatons the maxmum goodput a NRT user can acheve ( R s nherently constraned by hs transmtter s lmtatons (.e. R R S thereore NRT 1 A uncton (x s a sgmodal uncton t has a unque nlecton pont x Inl and d ( x x and d ( x x. x x Inl x x Inl 9 ACADEMY PUBLISHER

4 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER R R when ( = 1 whch accordngly determnes hs maxmum easble degree o satsacton. Thereore to ncorporate a NRT user s rate lmtatons n hs actual throughput utlty R R ( snce ( 1 and thus T ( R = 1 (Fg.1. Wthout loss o generalty we consder: Denton 1. NRT Users Actual Throughput Utlty T( R T( R ( SNRT where R [ R ] and s a strctly convex or strctly concave uncton o R wth T ( R = T ( R =1. B. Real-Tme Multmeda Servces Wth respect to RT servces ether regardng delayadaptve (e.g. audo and vdeo or rate-adaptve real-tme applcatons [6] T s consdered as a sgmodal uncton o user s acheved goodput R to relect hs QoS perormance requrements that prmarly consst o shortterm throughput [4] and delay constrants. These can be expressed va a properly selected sgmodal actual throughput utlty shaped n accordance to the RT user s constant target actual rate per tme slot and an approprate margn actor [7]. The target actual uplnk rate R T ndcates the deal value o a user s per tme slot (to ulll hs delay constrants actual transmsson rate at whch hs servce throughput QoS requrements are ullled n order hs short-term throughput requrement to be acheved as well whle the Margn Factor (MF determnes acceptable bounds o hs actual acheved throughput devatons wth respect to the target actual rate. Speccally a RT user s margn actor determnes the mnmum acceptable levels o hs expected actual throughput (.e. RMn RT MF. Moreover we argue that when the user s acheved actual transmsson rate remans wthn the range o [ R Mn R T hs actual throughput utlty must be a slowly ncreasng uncton o hs actual data rate. On the other hand when R RMn then hs actual throughput utlty should be a rapdly ncreasng uncton o R ndcatng hs need to occupy addtonal system resources. Thereore R Mn shall determne the unque nlecton pont o user s S RT actual throughput utlty. The prevous desgn opton gves to a user s power control mechansm operatng over ast adng channels envronment the enhanced lexblty o decreasng hs actual uplnk throughput up to a certan level ( R Mn requred due to potentally bad nstantaneous channel condtons wthout however excludng the user rom transmttng data at that correspondng tme slot. The later would occur a step uncton o a RT user s actual transmsson rate was used to relect hs correspondng degree o satsacton. Furthermore we argue that an addtonal ncrement o a RT user s actual data rate rom hs target value must not correspond to an analogous ncrease o hs actual throughput degree o satsacton and thus the latter should tend asymptotcally to ts maxmum value as R R (.e. T ( R = 1. The prevous argument s based on the observaton that snce a RT servce s QoS prerequstes are ullled when the requred target bt rate per tme slot s acheved an addtonal mprovement o user s actual throughput perormance wll not urther mprove hs degree o satsacton. Moreover by restrctng up to a specc level ( R the acheved actual uplnk data rates o RT users that due to ther temporarly good transmsson envronment could receve more recourses than requred we get the advantage o reallocatng the excess system resources to the unavored RT users (e.g. wth temporally bad transmsson envronment or NRT users n order to ncrease ther perormance satsacton. For practcal purposes as s the case o NRT servces due to users hardware lmtatons we consder that a RT user s actual throughput utlty has reached a value close to the maxmum when R RT MF SRT. Thereore when SRT we set R ( RT MF ( n order T( RT MF 1 snce ( 1 (Fg.1. Wthout loss o generalty we declare: Denton. RT Users Actual Throughput Utlty T( R T(( RT MF ( SRT where R [ R MF] and s a sgmodal uncton T or wth unque nlecton pont R as ollows: T( R R R : R Mn RT MF ( R R R Mn wth T ( R = T ( R MF =1. T Fnally we should underlne that by properly selectng a RT user s modulaton and codng scheme (.e. ecency uncton hs short-term throughput and thus hs delay prerequstes can be assured as shown n [7]. In accordance to (3 the overall utlty uncton o each user S can be expressed as ollows: T R U ( P P ( P P T RT MF T( R ( (( ( S P S where R [ R ] SNRT and R [ RT MF] SRT snce U ( P or P NRT RT (4 (5 ( 1. Intutvely we consder that s a non-negatve bounded above uncton or normalzaton purposes only and thus lm U ( P P snce the user does not transmt data then hs satsacton must be. 9 ACADEMY PUBLISHER

5 658 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 IV. THE NON-COOPERATIVE MULTI-SERVICE UPLINK POWER CONTROL GAME As the system evolves at the begnnng o each tme slot a user s uplnk power control algorthm s responsble or determnng an approprate uplnk transmsson power level towards maxmzng hs overall degree o satsacton whch s relected by the correspondng values o hs utlty. In ths secton the man goals o the proposed MSUPC algorthm are analyzed and ormally dened as a generc optmzaton problem n a game theoretc ramework. Snce each user n the network ams at the maxmzaton o the expectaton o hs utlty U the correspondng goal o user s MSUPC algorthm can be dened as the maxmzaton o the objectve uncton: max E[ U ( P P ] st.. P P (6 P Assumng ndependent and dentcally dstrbuted (..d. communcatons channels the maxmzaton o the average utlty n expresson (6 s obtaned by maxmzng the utlty at every tme slot t. Moreover due to the users selsh operaton n terms o pursung optmal power values towards ther ndvdual utlty maxmzaton the overall network uplnk power control problem at each tme slot can be ormulated as a noncooperatve mult-servce uplnk power control game (MSUPC game. Let G [ S{ A}{ U}] denote the proposed non-convex non-cooperatve game where S s the set o all users and N A [ P ] s the strategy set o the th user. Furthermore or xed transmsson powers o the rest o the users the resultng per tme slot non-cooperatve game can be expressed as the ollowng maxmzaton problem: max U max U ( P P P P st.. P P or =1 N (7 In other words each player-user n game G pcks a transmsson power rom hs strategy set A and receves a payo U n accordance to hs best response polcy B ( P max U ( P P. P A We adopt Nash equlbrum approach towards seekng the soluton o the non-convex non-cooperatve MSUPC game whch s most wdely used or game theoretc problems. A Nash equlbrum pont s a set o power vectors such that no user has the ncentve to change hs power level snce ts utlty cannot be urther mproved by makng any ndvdual changes on ts value gven the powers o other users. Thereore: Denton 3. The power vector P P... 1 PN s a Nash equlbrum o the MSUPC game or every S U ( P P U ( P P or all P A. V. TOWARDS A UNIQUE NASH EQUILIBRIUM FOR THE MSUPC GAME In ths secton we study the exstence and unqueness o a Nash equlbrum pont or the proposed MSUPC game. The ollowng results reveal that users actual throughput utltes must comply wth specc attrbutes regardng ther orm at the boundares o ther correspondng denton sets n order to assert overall utltes soundness and thus or a unque Nash equlbrum o game MSUPC to exst. The absence o the Nash equlbrum means that the system s nherently unstable. A. Propertes o Users Actual Throughput Utltes Followng the prevous dscussons and analyss we ntally study the propertes o both RT and NRT users actual throughput and overall utlty unctons over ther correspondng denton domans. The ollowng two lemmas determne the attrbutes o user s nested T actual throughput utlty as a uncton o the acheved SINR. Speccally or RT servces n accordance to denton the ollowng lemma holds. Lemma 1. For RT users (.e. SRT gven: a an ecency uncton ( whch s a sgmodal uncton o and has a unque nlecton pont and b an actual throughput utlty uncton T( R whch s a sgmodal uncton o R and has a unque nlecton pont R RMn RT MF where R R ( ( RT MF ( then uncton T( T(( RT MF ( s also a sgmodal uncton o ( wth a unque nlecton pont such that: R 1 R 1 when R R : (8 R 1 R otherwse 1 R R s the mappng o uncton s where T( R nlecton pont n axs. Proo: See Appendx A. For NRT servces wth respect to denton 1 the ollowng lemma holds. Lemma. For NRT users (.e. SNRT gven: a an ecency uncton ( whch s a sgmodal uncton o and has a unque nlecton pont and b an actual throughput utlty uncton T( R whch s a strctly concave (strctly convex uncton o R where R R ( R ( then the correspondng actual throughput utlty uncton T( T( R ( s 9 ACADEMY PUBLISHER

6 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER as ollows: ( A lm T a sgmodal uncton o ( T ( lm ( (9 wth unque nlecton pont or whch: ( (1 B Inl Inl T ( lm T ( ( lm. (11 a strctly concave (strctly convex uncton o (. Proo: See Appendx B. B. Propertes o Overall Users Utltes In accordance to lemma when T( R s a strctly concave (strctly convex uncton o R and equalty (11 holds then a NRT user s actual throughput utlty s also a strctly concave (strctly convex uncton o (. The ollowng lemma demonstrates that n such cases the correspondng overall utlty contradcts wth the undamental assumptons wth respect to the denton o U n secton III (.e. lm U ( P P and U (P bounded above or P and thereore these cases are not easble. Lemma 3. When a NRT user s actual throughput utlty T( R s a strctly concave (strctly convex uncton o R ( lm T ( and T lm ( holds then or the correspondng U( P P S NRT as a uncton o P ether lm U ( P P or U( P or P (U (P s not bounded above or P whch n any case contradcts wth the basc assumptons n the denton o U. Proo: See Appendx C. Wth respect to lemma 3 we argue that a NRT user s actual throughput utlty must comply only wth the propertes n relaton (9. The ntuton behnd condton (9 s that the propertes o a user s utlty n the doman o power (.e. the basc resource that a user spends when transmttng data should also hold when hs utlty s transormed n the eld o whch relects the correspondng resource expended when these data are receved by the base staton. The latter occurs only when condtons (9 hold. Such an argument s ounded on the ntererence senstve nature o the uplnk n a CDMA system where the ndvdual amount o resources (.e. transmsson power that a user spends s not the only actor that determnes hs degree o satsacton due to others smultaneous transmssons that aect hs acheved SINR and thus hs actual servce perormance. Moreover when a user s actual throughput T utlty attrbutes ollow (11 the MSUPC game does not have a unque Nash equlbrum as t s extensvely analysed later. Based on lemma 1 and lemma and takng nto consderaton the logcal lmtatons n the denton o a user s overall utlty uncton U (.e. bearng n mnd n the rest o the paper that user s actual throughput utltes are n lne wth equaltes (9 the ollowng lemma characterzes the overall utlty maxmzaton o a sngle user when others transmsson powers are xed. Lemma 4. User s S utlty uncton U( P P s a quas-concave uncton o hs own power P. Moreover consderng that other users transmsson powers are xed U( P P or P [ P ] has a unque global maxmzaton pont: ( RT MF I( P mn{ P } S WG P R I( P mn{ P } S WG RT NRT (1 where s the unque postve soluton o the equaton T( T(. Proo: See Appendx D. Lemma 4 reveals that users goal to maxmze ther utlty-based perormance and thus ther degree o satsacton n terms o QoS-aware servce perormance and correspondng energy consumpton can be translated to a constant attempt o meetng specc SINRs (at the base staton whch eventually leads to an SINRbalanced network. Moreover accordng to (1 a user s maxmum transmsson power s not sucent or reachng the targeted due to hs potentally bad channel condtons then the best polcy B ( P s to transmt wth maxmum power. C. Exstence and Unqueness o a Nash Equlbrum The ollowng proposton asserts the exstence and the unqueness o a Nash equlbrum pont o the proposed uplnk power control game and hence determnes nodes transmsson power vector at equlbrum. Proposton 1. The Nash equlbrum o the noncooperatve game (7 s gven by P P... 1 PN where P s the unque global maxmzaton pont o the overall user s utlty uncton gven by: ( RT MF I( P P WG P mn{ } SRT and by: R I( P WG P mn{ P } SNRT. 9 ACADEMY PUBLISHER

7 66 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 Furthermore results rom the unque postve soluton o equaton ( T( T(. Proo: In accordance to lemma 4 the power level that corresponds to the maxmzaton o user s S utltybased perormance gven other users power levels equals to the power level that maxmzes the utlty n (5 and thus s gven by (1 where s the unque postve soluton o T( T(. Up to here t has been argued that at Nash equlbrum pont such a pont exsts each user s transmsson power s pursung a targeted SINR value whch depends not only on the modulaton codng and packet sze beng used (expressed through hs approprate ecency uncton but also s aected by RT or NRT users QoS requrements (relected by ther actual throughput utlty. Moreover ollowng [14] and [15] the exstence o a Nash equlbrum o the game n (7 can be shown va the quas-concavty o each node s utlty uncton n hs own power. In lemma 4 we have already shown that U( P P s a quas-concave uncton n P [ P ] and hence Nash equlbrum always exsts. Fnally or an S-shaped actual throughput utlty uncton T (gven that T obeys (9 T( T( has a unque soluton whch s the unque global maxmzer o user s utlty uncton. Due to the unqueness o and the one-to-one relatonshp between uplnk transmsson power and correspondng SINR the above Nash equlbrum s unque. Proposton 1 ndcates that all users adopt a best response polcy (7 governed by (1 game s MSUPC unque Nash equlbrum [3] wll be always reached. Such a strategy s ormally descrbed by the proposed algorthm n the ollowng secton. Concludng ths secton s analyss we provde a necessary and sucent condton or the exstence o a unque Nash equlbrum n MSUPC game. In accordance to proposton 1 the exstence and unqueness o game s MSUPC Nash equlbrum s ounded on the quas-concaveness property o users utltes U as a uncton o P and thereore the S-shape property o ther correspondng actual throughput utltes T as unctons o the acheved SINR (as lemma 4 reveals. Moreover a generc actual throughput utlty o a user as dened n secton III s always an S-shaped sgmodal uncton o and only relaton (9 holds ollowng lemmas 1 and. Thereore the satsacton o condtons n (9 not only asserts the proper denton o a user s utlty U as argued n lemma 3 but also reassures system s stablty va guaranteeng the exstence o a unque Nash equlbrum or the MSUPC game (7. Proposton. MSUPC non-cooperatve game always has a unque Nash equlbrum every user s actual throughout utlty T as dened n secton III possesses the ollowng addtonal attrbutes: a T ( lm T b lm ( T s strctly concave n R T s strctly convex n R. Proo: Followng proposton 1 and lemmas 3 and 4 the proposton holds. Note that when T s a sgmodal uncton n R no addtonal attrbutes are requred or the exstence and unqueness o the MSUPC game s Nash equlbrum. VI. MSUPC ALGORITHM &CONVERGENCE In ths secton based on our prevous analyss an teratve and decentralzed uplnk power control algorthm or reachng the Nash equlbrum or the MSUPC game G at every tme slot t s presented and ts convergence s dscussed. MSUPC Algorthm (1 At the begnnng o tme slot t user S transmts wth a randomly selected easble power (.e. ( P P power. Set k= and hence P ( S. ( Gven the uplnk transmsson powers o other users whch s mplctly reported by the base staton when broadcasts ts overall ntererence ( k ( ( k I P the user computes ( k ( ( k I P and renes hs power.e. ( k 1 computes P n accordance to (1. ( k1 ( k 5 (3 I the powers converge (.e. P P 1 then stop. (4 Otherwse set k=k+1 go to step. MSUPC algorthm can be characterzed as a snglevalued low complexty best response algorthm or every user startng rom a randomly selected easble power o hs non-empty orthogonal strategc space A (.e P ( S sucently large n order [3]. Under the assumpton that P s can be acheved by all users the convergence o MSUPC algorthm s always reached n accordance to (proposton 1 n [16]. In the general case however the convergence o MSUPC algorthm can be drawn based on the supermodularty attrbute o the under consderaton game (ollowng propostons 1 and 3 and theorem 1 n [8]. A ormal denton o a supermodular game or the case o sngle dmensonal user strategy sets s stated n the ollowng [15]. Denton 4. A game G [ S{ A}{ U}] where S s the set N o users/players and A [ P ] s the strategy set o the th user s supermodular or each user U( P P has nondecreasng derences (NDD n ( P P [15]. As explaned n [8] the sucent condtons or a supermodular game are: 9 ACADEMY PUBLISHER

8 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER D1. The strategy A s a nonempty and compact sublattc. D. The utlty uncton U s contnuous n all players strateges s supermodular n player s own strategy and has ncreasng derences between any component o player s strategy and any component o any other player s strategy. Regardng MSUPC game snce each user s strategc space A s a closed sngle-dmenson non-empty space property D1 holds. Moreover by denton U s contnuous n all users strateges or all S. Each user s U utlty uncton s (trvally supermodular n hs own one-dmensonal strategy space. The remanng ncreasng derence condton o U or any component o player s strategy and any component o any other U( P P player s strategy (.e. js holds PPj accordng to Theorem 1 n [8] snce U s a quasconcave uncton o hs own power P and contnuous nversely proportonal uncton o others power P (thus satsyng the same propertes as the utlty used n [8] (.e. proposton ( equaton (9. In accordance to proposton 3 n [8] n a supermodular game G: I the users best responses are sngle-valued and each user uses the myopc best response updates startng rom the smallest (largest elements o hs strategy space then the strateges monotoncally converge to the smallest (largest Nash equlbrum I the Nash equlbrum s unque the myopc best response updates globally converge to that Nash equlbrum rom any ntal strateges. VII. NUMERICAL RESULTS AND DISCUSSIONS In ths secton we provde some numercal results llustratng the operaton and eatures o the proposed ramework and the MSUPC algorthm. Intally or demonstraton manly purposes we assume xed predened users channel condtons n order to denty and comprehend the proposed game s G equlbrum propertes or varous users channel states and servces QoS characterstcs. Then assumng tme-varyng Raylegh adng channels system s long-term propertes under the proposed MSUPC algorthm are llustrated and the tradeos among user s average channel condtons correspondng power consumpton and utlty-based perormance are revealed. Fnally the correlatons between real-tme multmeda and data users servce perormance are studed n terms o acheved uplnk actual throughput power consumpton and utlty-based perormance when both types o servces share the same avalable resources under tme-varyng channels. Throughout our study we consder the uplnk o a sngle cell tme-slotted CDMA system supportng N=1 TABLE I SIMULATION PARAMETER VALUES Parameter ( S Value P ( P (Watt (Watt W 1 6 (Hz I n 4 TABLE II QOS PREREQUISITES OF CLASS 1(NRT USERS Users Class Characterstc Class 1 (data users c.1 R M 5.4(Mbps 5.8 (7.6 db TABLE III QOS PREREQUISITES OF CLASS (RT USERS Users Class Characterstc RT Class (multmeda users 18 (Kbps 1 (Kbps MF M T 8 A T (9.5 db M 5 contnuously backlogged users. Each smulaton lasts 1. tme slots. Unless otherwse explctly ndcated n we model users path gans as: G K d where d s the dstance o user rom the base staton n s the dstance loss exponent and K s a log-normal dstrbuted random varable wth mean and varance = 8(dB whch represents the shadowng eect [8]. Henceorth we consder the ollowng values o system parameters as shown n Table I. Two classes o users are consdered namely class 1 and class. Class 1 represents data users and class multmeda users who requre vdeo conerence servces. The two classes are derentated n accordance to ther servces characterstcs as they are relected n the orm o ther correspondng actual throughput utlty uncton and ther parameters. In the ollowng multmeda users QoS prerequstes are expressed through a sgmodal actual throughput utlty uncton ( R (.e. ( (1 A M T T R e and data users through a strctly concave uncton (.e. T( R log( cr 1. Moreover one common modulaton and codng scheme s consdered or both classes o users whch s expressed through the sgmodal ecency uncton ( (1 M e 1... N. The prevous characterstcs are summarzed n Tables II III. 9 ACADEMY PUBLISHER

9 66 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 Transmsson Power (mwatts Iteratons User 1 User User 3 User 4 User 5 User 6 User 7 User 8 User 9 User 1 Fgure. Users transmsson powers convergence at a gven tme slot Actual Transmsson Rate (Kbps Utlty Functons Iteratons Fgure 3. Users actual throughput at a gven tme slot Iteratons Fgure 4. Users utltes at a gven tme slot A. Game s Nash Equlbrum Propertes User 1 User User 3 User 4 User 5 User 6 User 7 User 8 User 9 User 1 User 1 User User 3 User 4 User 5 User 6 User 7 User 8 User 9 User 1 In the ollowng we study MSUPC algorthm s convergence at game s Nash equlbrum or a specc tme slot. Fxed channels condtons are assumed durng a tme slot and we set G 1 d or each user where d =d (m or =..1 and d 1 =3 (m. In ths way we emulate a scenaro where users channel condtons are worse as ther ID value (.e. =1 1 ncreases. For demonstraton purpose only users 1 to 7 are assumed to belong to class 1 whle users 8 to 1 belong to class. Fg. Fg.3 and Fg.4 llustrate users transmsson powers actual uplnk transmsson rates and utlty values respectvely as a uncton o the teratons requred or users MSUPC algorthm to converge at game s G equlbrum pont P. The correspondng results reveal that or users n the same class ther powers at equlbrum are nversely proportonal to ther nstantaneous channel condtons (Fg.. Moreover only ve teratons are requred or reachng equlbrum whch s an mportant attrbute o our approach due to slots small length (.e msec approxmately n HDR systems [5] n channel aware wreless systems. Furthermore even users utltes converge at values proportonal to ther channels states (Fg.4 ther actual acheved uplnk throughput s such that ther QoS expectatons are met even or the most demandng users n terms o low nstantaneous channel gans G and QoS prerequstes (.e. RT users = acheved goodput s 18 Kbps. The latter observaton reveals MSUPC algorthms lexblty n procently managng the drawbacks emergng rom users near ar eect even when varous servces are smultaneously supported. Moreover the act that or both classes o users ther actual uplnk data rates at equlbrum are n lne wth ther servces expectatons not only reveals the proper unctonalty o ther actual throughput utltes (T n ntroducng the servces QoS demands n the MSUPC algorthm but also demonstrates the utltes ablty to dstngush servces wth dverse QoS prerequstes. B. Users Power Consumpton & Overall Utlty-based / Actual Throughput Perormance Trade-o We rst study the trade-o among a user s utltybased perormance QoS requrements ulllment and correspondng power consumpton as a uncton o hs average channel state and servce type as the system evolves under the MSUPC algorthm. Thereore assumng Raylegh users tme-varyng channel condtons we let the system evolve or 1. tmeslots. Fg.5 (Fg.8 Fg.6 (Fg.9 and Fg.7 (Fg.1 llustrate users average power consumpton actual throughput rate and utlty based perormance respectvely as a uncton o users ID when dstrbutng them around the base staton wth step dstance d =d (d =d (m or =..1 and d 1 =3 (m. As n the prevous scenaro users 1 to 7 are assumed to belong to class 1 whle users 8 to 1 belong to class. In the case o class users the results reveal that as ther mean channel condtons become worse (.e. ncreasng number o RT-class users ID and n order to satsy ther QoS requrements (Fg.6 (.e. actual achevable uplnk transmsson rate o 18 Kbps per tme slot wth correspondng margn actor 1 Kbps ther energy consumpton ncreases (Fg.5 whch eventually leads to ther overall utlty-based perormance decrement (Fg.7. On the other hand as non-real-tme users mean channel condtons become worse (.e. ncreasng number o NRT-class 1 users ID both ther overall utlty and actual throughput perormance decreases whle ther power consumpton ncreases. More speccally the results show that or both NRT and RT users ther QoS expectatons are ullled n terms o hgh throughput perormance and xed transmsson rates respectvely. Ths reveals the proper unctonalty o ther actual throughput utltes (T n ntroducng the servces QoS demands n the MSUPC mechansm and thus demonstrates ther utltes ablty to dstngush and support varous servces QoS perormance metrcs. Speccally regardng data users 9 ACADEMY PUBLISHER

10 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER Users' Average Power Consumpton (Watts Users' ID Fgure 5. Users average power consumpton (step dstance 5m Users' Average Power Consumpton (Watts Users' ID Fgure 8. Users average power consumpton (step dstance 1m 6 6 Users' Average Actual Rate(Kbps Users' Average Actual Rate(Kbps Users' ID Users' ID Fgure 6. Users average actual throughput rate (step dstance 5m Fgure 9. Users average actual throughput rate (step dstance 1m Users Average Utlty Based Perormance(x1^ Users' ID Fgure 7. Users average utlty based perormance (step dstance 5m (.e. =1 7 ther optmal servce-perormance energy-consumpton tradeo (.e. utlty maxmzaton s acheved or hgh data rates even or the most dstant rom the base staton users. Such a behavor s drven by ther actual throughput utlty types that led them to get hgh transmsson powers n system s non-cooperatve game due to the correspondng hgh goodput perormance gans they attan. On the other hand data servces utlty maxmzaton s steerng them to lower energy consumpton whle ulllng ther xed expected data rates snce an addtonal ncrement on ther power and thus ther acheved goodput would not correspond to an analogous ncrement o ther actual throughput as well as overall utlty as explaned beore. Such a behavor eventually reduces overall uplnk ntererence towards mprovng both types o users perormance. Fnally a parallel and close examnaton o the two sets o gures Fg.5-Fg.7 and Fg.8-Fg.1 allows us to Users' Average Utlty Based Perormance(x1^ Users' ID Fgure 1. Users average utlty based perormance (step dstance 1m better denty the correlatons n the behavor and perormance o real-tme multmeda and non-real-tme data users when both types o servces coexst under a system wth tme-varyng avalable rado resources. We conclude that as channel s condtons become worse (Fg.8-Fg.1 the acheved actual transmsson rate o NRT users decreases (even or those close to the base staton whle the correspondng rate o RT users remans greater than ther mnmum actual data rate even those users channel condtons are worse. Such a behavor s drven by users actual throughput utlty types that avor RT users to acheve the targeted uplnk actual transmsson rate when the approprate modulaton and codng scheme s selected (as shown n proposton n [7]. Moreover n such a case data users stll mantan hgh acheved goodput but at the cost o ncreased unarness among them n terms o power consumpton and acheved utlty. 9 ACADEMY PUBLISHER

11 664 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 VIII. CONCLUDING REMARKS In ths paper we studed the ssue o ecent power control n the uplnk o a CDMA wreless network whle satsyng multple servces QoS prerequstes. To procently relect multple servces varous and oten dverse QoS prerequstes a generc utlty-based ramework was ntroduced and analyzed. The correspondng non-convex utlty-based multservce optmzaton problem was ormulated and solved through a game theoretc ramework. Towards ths drecton rst the exstence and unqueness o a Nash equlbrum o the proposed game was studed and then a decentralzed low-complexty algorthm or reachng the unque Nash equlbrum pont o the proposed game has been presented. Furthermore the ecacy o the proposed ramework n supportng both real-tme multmeda servces and non-real-tme data servces QoS demands whle achevng ther utlty-based perormance maxmzaton has been demonstrated and evaluated va analyss and smulaton. The ntroduced utlty-based ramework and proposed approach provdes us wth an enhanced lexble ramework whch can be used to unormly treat and theoretcally analyze derent access technologes (e.g. CDMA and WLAN types o resources as well as multple user servces dverse expectatons. Thereore the study and analyss o heterogeneous wreless networks jont resource allocaton and QoS provsonng mechansms under a common utlty ramework towards realzng ther seamless ntegraton becomes o hgh research and practcal mportance and s part o our current work. APPENDIX A-PROOF OF LEMMA 1 In accordance to the denton o a sgmodal uncton the ollowng propertes hold: ( Inl T( R R ( T( R R ( R R R T( R lm ( R ( lm ( T( R Inl ( R R R R ( R T( R lm ( lm Regardng the rst dervatve o T ( we have: (13 (14 (15 (16 (17 (18 T ( T( R ( T( R ( ( R ( ( R ( T( R ( R R accordng to (13 and (14. Consderng the second dervatve o T ( we have: T( R ( ( T ( T ( T ( R ( ( R ( R R ( ( ( ( T R T R R ( R R (19 Intally consderng the bounds n the denton doman o T ( [ we can easly see that: T ( lm R T R T R R R ( R R R and thus ( ( ( ( lm lm lm lm T ( lm R ( ( ( ( T R lm T R lm R lm lm R R ( R R R R snce lm R R T( R and lm R R ( R T R ( R. Furthermore wth respect to the correspondng relevance (poston o the nlecton ponts o the ntal 1 utltes (.e. and R R on the axs o ( we must study the propertes o the second dervatve o T ( as a uncton o n two derent cases. 1 Case I.1. When R R. We have the ollowng sub-cases: 1 I.1.A. For R R n accordance to (13 (14 (15 (16 t can be seen that (. 1 I.1.B. For T R R n accordance to (13 (14 (15 (16 t can be seen that T (. 1 I.1.C. For R R n accordance to prevous sub-cases (I.1.A and (I.1.B we know that T( has at least one nterrupton pont wth the 1 axs o zero when [ R R ]. In order to show that T ( s a sgmodal uncton we need to prove 9 ACADEMY PUBLISHER

12 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER that there s only one nterrupton pont wth the zero axs denoted as. Through the examnaton o (19 when 1 [ R R ] snce ( Inl the second part o the sum n (19 s always postve. Thereore only the sgn o T ( R ( R determnes and hence can alter the sgn o T(. However snce T ( R ( R alters ts sgn once we can conclude that T( has a unque cuttng pont wth the axs o zero and thereore ollowng the prevous 1 analyss we conclude that when R R T ( s a sgmodal uncton o wth a unque nlecton pont and hence 1 [ R R ]. 1 Case I.. When Inl RInl R. Followng the same thread o analyss we can easly 1 conclude that when R R T ( s a sgmodal uncton o wth a unque nlecton 1 [ R R ]. pont and hence APPENDIX B-PROOF OF LEMMA In the ollowng a common proo s provded or both the cases where T( R s a strctly concave or strctly convex uncton o R. Whenever needed we use parenthess to provde the approprate expressons or the case o strctly convex uncton whle the correspondng expressons or the case o strctly concave uncton appear wthout parenthess. In accordance to the denton o a strctly concave (strctly convex uncton the ollowng propertes hold: T( R ( R T R ( R ( T( R ( (1 ( R Moreover or the sgmodal ecency uncton ( propertes (13 (14 and (18 stll hold. Regardng the rst dervatve o T ( as shown n lemma 1 we have: T T R ( ( ( R R accordng to ( and (13. Consderng the second dervatve o T ( and ollowng lemma 1 we have: T( T( R ( ( T( R R R ( R R ( At ths pont we should study the propertes o the second dervatve o T ( as a uncton o [. Thereore n the ollowng due to the exstence o Inl we examne the values o the sgn o the second dervatve o T ( and ts correspondng propertes at the lmts o ts denton set (.e. lower (upper bound o n the ollowng complementary subsets n the axs o.e. [ ] and [. Case II.1. When (. We can easly see that T( accordng to (15 ( (1. T( Case II.. When. We consder two sub-cases (II..A and II..B wth respect to values o T ( where [ n the lower (upper bound o ts denton doman: T( T( II..A: When lm lm. In such case there always exsts an nterval wthn [ ] ( [ where T( T(. Consequently we can argue wth respect to the outcome o case II.1 that T ( has at least one nterrupton pont wth the axs o zero n the nterval [ Inl ] ( [. Moreover the unqueness o such a pont can be proven as ollows. Studyng ( when the second part o the sum n ( s always postve (negatve ollowng (15. Thereore only the sgn o T ( R ( R contrbutes to the change o the sgn o T(. However snce T ( R ( R alters ts sgn only once we conclude that T ( has a unque ntersecton pont wth the axs o. Thereore wth respect to Case II.1 and subcase II..A we can conclude the ollowng: T( T( Concluson 1: When lm lm T ( s a sgmodal uncton o [ wth a unque nlecton pont and hence [ ] [. Inl Inl T( T( II..B: When lm lm. In such case T( has no nterrupton pont wth the the axs o n the nterval [ ] ( [. To prove the above argument we must observe that rom Case II.1 and the act that T( T( lm lm T ( ether 9 ACADEMY PUBLISHER

13 666 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 never equals to zero n the nterval [ ] ( [ Inl or equals to zero (.e. nterrupts wth the axs o more than one tmes. The latter s not vald due to the act that T ( can alter ts sgn only once when [ because ts sgn s determned by T ( R ( R whch alters t sgn only once. Thereore wth respect to Case II.1 and subcase II..B we can conclude the ollowng: T( T( Concluson : When lm lm T ( s a strctly concave (strctly convex uncton o [. Conclusons 1 and complete the proo. APPENDIX C-PROOF OF LEMMA 3 In ths proo or smplcty n the presentaton let us denote the overall utlty uncton o NRT users as: T( P U( P s.t. P P P Moreover let us underlne that due to the one to one relatonshp between and P the propertes o T ( as a uncton o revealed n lemmas 1 and also hold or T (P as a uncton o P. The rst dervatve o U ( P can be expressed as: T( P T( P P T ( P U( P P P b( P P P P P T( P where b( P T( P P. Functons T (P and P T( P have been assumed as contnuous unctons o P P thereore b( P s also a contnuous uncton o P. For any two values v {1 } o user s power P denoted as Pv [ the ollowng propertes hold or a strctly concave (strctly convex T (P uncton: T( P T( P T( P ( P P P P [ P P P1 T( P T( P T( P1 ( P P1 P1 P [ P P P1 due to the strct concavty (strct convexty o T( P. Lettng P and P1 Pv or P we conclude that: and thus T( P T( T( P ( P v v P P Pv T( P T( T( Pv ( Pv P P Pv ( T( P T( P T Pv Pv T( Pv Pv P P P P v P Pv thereore b( Pv b( Pv U ( P b( P P P U( P b( P P P (3 Followng the prevous analyss a when T( P s a strctly concave uncton o P we can observe that U( P s a decreasng uncton o P P [ P n accordance to (3. In such case lm U ( P that would mply that U ( P P when P [ P U( P when P [ P or lm U ( P P n order whch contradcts wth the basc assumptons n the denton o U (secton III. b when T( P s a strctly convex uncton o P we can observe that U( P s an ncreasng uncton o P n accordance to (3. Thereore not bounded above or P whch also contradcts wth the basc assumptons n the denton o U (secton III. APPENDIX D-PROOF OF LEMMA 4 Va lemmas 1 and ollowng lemma 3 t s easly shown that T( S s a sgmodal uncton o ( regardng both RT multmeda and NRT data users. Moreover due to the strct lnear relatonshp between and P n accordance to (1 t ollows mmedately that a user s actual throughput utlty s also a sgmodal uncton o ts own power. Thereore the ollowng propertes o T( T( P P as a uncton o P when P hold: 1. Its doman s the non-negatve part o the real-lne that s the nterval [.. Its range s the nterval [1 accordng to the propertes o T( S (assumptons c and d n secton III. 3. It s an ncreasng uncton o P S ollowng lemmas 1 and respectng lemma It s strctly convex over the nterval [ P and strctly concave over the nterval [ P P] where rom (1 as well as lemmas 1 and t s straghtorward that ( RT MF I ( P PInl SRT and GW Inl R I ( P PInl SNRT. In ths paper t s GW assumed that P P S. The valdty o the 9 ACADEMY PUBLISHER

14 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER prevous property ollows drectly by takng nto account the outcome o lemma 1 and whle respectng lemma It has a contnuous dervatve snce unctons T( R and ( are assumed to have contnuous dervatves too. Snce the prevous propertes hold or T( P P S the rato T( P P P S s a quas-concave uncton o P [ P ] [9]. Thereore a user s actual throughput utlty U( P P S s a quas-concave uncton o ts own power. Moreover the rst order optmzaton condton o U( P P can be expressed as ollows: T ( T ( P T( ( P P P P ( P U P P Thereore U( P P S s maxmzed when: T( T( (4 due to the act that accordng to (1 and the strct P P lnear relatonshp between and P. Moreover due to the sgmodal type o uncton T( S (4 has a unque soluton as proven n [9]. Thereore the unque value P ˆ that maxmzes U( P P when P can be represented accordng to (1 as ollows: ( R ( T MF I P SRT ˆ GW P R ( I P SNRT GW Fnally the maxmum allowable value o P s less than P ˆ U( P P T( P P P S s the hghest value o U( P P snce the utlty uncton U s ncreasng over the nterval [ P ˆ ]. Thereore the smallest o the values o P ˆ P s the global maxmzer o U( P P S when P [ P ].e. ( R ( T MF I P mn P S GW P R ( I P mn P SNRT GW whch concludes the proo. RT ACKNOWLEDGMENT Ths work has been partally supported by EC EFIPSANS project (INFSO-ICT REFERENCES [1] D. J. Goodman and N. B. Mandayam Power control or wreless data IEEE Pers. Commun. vol. 7 pp Apr.. [] H. J and C.-Y. Huang Non-cooperatve uplnk power control n cellular rado systems Wreless Netw. vol. 4 pp Apr [3] S. Lu V. Bharghavan and R. Srkant Far schedulng n wreless packet networks IEEE/ACM Trans. on Netw. vol. 7 no. 4 pp [4] M. Andrews L. Qan and A. Stolyar "Optmal utlty based mult-user throughput allocaton subject to throughput constrants" n proc. o IEEE INFOCOM '5 vol. 4 pp Mar. 5. [5] F. Berggren and R. Jantt "Asymptotcally ar transmsson schedulng over adng channels" IEEE Trans on Wreless Commun. vol. 3 no.1 pp Jan. 4. [6] Y. Lu S. Gruhl and E. Knghtly "WCFQ: An opportunstc wreless scheduler wth statstcal arness bounds" IEEE Trans. on Wreless Commun. vol. no. 5 p.p Sept. 3. [7] X. Lu E. K. P. Chong and N. B. Shro A ramework or opportunstc schedulng n wreless networks Computer Networks vol. 41 pp Mar. 3. [8] J.-W. Lee R.R. Mazumdar and N. B. Shro "Opportunstc power schedulng or dynamc mult-server wreless systems" IEEE Trans. on Wreless Commun. vol. 5 no.6 pp Jun. 6. [9] T. Alpcan T. Basar R. Srkant and E. Altman CDMA uplnk power control as a noncooperatve game Wreless Networks vol. 8 pp Nov.. [1] S. Guntur and F. Pagann Game theoretc approach to power control n cellular CDMA n Proc. o 58th IEEE Vehcular Technology Conerence (VTC pp Orlando FL Oct. 3. [11] K.-K. Leung and C. W. Sung "An opportunstc power control algorthm or cellular network" IEEE/ACM Trans. on Netw. vol.14 no.3 pp Jun. 6. [1] C. W. Sung and W. S. Wong A noncooperatve power control game or multrate CDMA data networks IEEE Trans. on Wreless Commun. vol. pp Jan. 3. [13] E. Altman and Z. Altman S-modular games and power control n wreless networks IEEE Trans. on Automatc Control vol. 48 pp May 3. [14] D. J. Goodman and N. B. Mandayam Power control or wreless data IEEE Personal Commun. vol. 7 pp Apr.. [15] C. U. Saraydar N. B. Mandayam and D. J. Goodman Ecent power control va prcng n wreless data networks IEEE Trans. on Commun. vol. 5 pp Feb.. [16] F. Meshkat H. V. Poor S. C. Schwartz and N. B. Mandayam An energy-ecent approach to power control and recever desgn n wreless data networks IEEE Trans. on Commun. vol. 5 pp Nov. 5. [17] S. Ulukus and R.D. Yates Adaptve power control and MMSE ntererence suppresson ACM Wreless Networks vol. 4 no. 6 pp Nov [18] F. Meshkat M. Chang H. V. Poor and S. C. Schwartz A game-theoretc approach to energy-ecent power control n multcarrer CDMA systems IEEE Journal on Selected Areas n Communcatons (JSAC vol. 4 pp Jun ACADEMY PUBLISHER

15 668 JOURNAL OF COMMUNICATIONS VOL. 4 NO. 9 OCTOBER 9 [19] N. Feng S.-C. Mau and N. B. Mandayam Prcng and power control or jont network-centrc and user-centrc rado resource management IEEE Trans. on Commun. vol. 5 pp Sept. 4. [] C. W.Sung K. W. Shum and K.-K. Leung "Stablty o dstrbuted power and sgnature sequence control or CDMA systems-a game-theoretc ramework" IEEE Transactons on Inormaton Theory vol.5 no.4 pp Apr. 6. [1] X. Duan Z. Nu and J. Zheng A dynamc utlty-based rado resource management scheme or moble multmeda DS-CDMA systems n Proc. IEEE Global Telecom. Con. (GLOBECOM. Tape Tawan Nov.. [] F. Meshkat H. V. Poor S. C. Schwartz and R. Balan Energy-ecent resource allocaton n wreless networks wth Qualty-o-Servce constrants n preprnt Prnceton Unversty 5. [3] H. P. Shang and M. van der Schaar "Feedback-drven nteractve learnng n dynamc wreless resource management or delay senstve users" IEEE Trans. Veh. Tech. to appear. [4] T. Kastrnoganns and S. Papavasslou "Utlty based Short-term throughput drven schedulng approach or ecent resource ellocaton n CDMA wreless networks" n Wreless Personal Comm. Sprnger (Publshed onlne: 13 Nov. 8. [5] T. Kastrnoganns and S. Papavasslou Probablstc short-term delay and throughput requrements o multmeda servces n hgh throughput wreless networks n proc. o IEEE Sarno Symp. on Advances n Wred and Wreless Comm. Apr. 7. [6] S. Shenker Fundamental desgn ssues or the uture Internet IEEE JSAC vol.13 pp Sept [7] T. Kastrnoganns E. E. Tsropoulou and S. Papavasslou Utlty-based uplnk power control n CDMA wreless networks wth real-tme servces n Adhoc Moble and Wreless Networks (LNCS Sprgner vol p.p Sept. 8. [8] C. Long Q. Zhang B. L H Yang and X Guan "Noncooperatve power control or wreless ad hoc networks wth repeated games" IEEE JSAC vol.5 no.6 pp Aug. 7. [9] V. Rodrguez An analytcal oundaton or resource management n wreless communcaton n Proc. o GLOBECOM 3 vol. no. pp Dec. 3. [3] R.D. Yates A ramework or uplnk power control n cellular rado systems IEEE JSAC vol.13 pp Sept Ern Elen Tsropoulou receved the dploma n Electrcal and Computer Engneerng (summa cum laude rom the Natonal Techncal Unversty o Athens (NTUA Greece n 8. She s currently a PhD canddate n the ECE Department o NTUA and a research assstant n the Network Management and Optmal Desgn Laboratory at NTUA. Ms. Tsropoulou s student member o IEEE and member o Techncal Chamber o Greece. Her man research nterests nclude wreless and heterogeneous networkng ocusng on power and resource allocaton schemes. Tmotheos Kastrnoganns receved the dploma n electrcal and computer engneerng rom the Natonal Techncal Unversty o Athens (NTUA Greece n 5. He s currently a PhD canddate n the ECE Department o NTUA and a research assstant n the Network Management and Optmal Desgn Laboratory at NTUA. Hs man research nterests le n the area o wreless networks wth emphass on resource allocaton schedulng perormance analyss and multmeda servces. Symeon Papavasslou (S 9 M 96 s currently an Assocate Proessor wth the Faculty o Electrcal and Computer Engneerng Department Natonal Techncal Unversty o Athens. From 1995 to 1999 Dr. Papavasslou was a senor techncal sta member at AT&T Laboratores n Mddletown New Jersey and n August 1999 he joned the Electrcal and Computer Engneerng Department at the New Jersey Insttute o Technology (NJIT where he was an Assocate Proessor untl 4. Dr. Papavasslou was awarded the Best Paper Award n INFOCOM'94 the AT&T Dvson Recognton and Achevement Award n 1997 and the Natonal Scence Foundaton (NSF Career Award n 3. Dr. Papavasslou has an establshed record o publcatons n hs eld o expertse wth more than one hundred and ty techncal journal and conerence publshed papers. Hs man research nterests le n the area o computer and communcaton networks wth emphass on wreless communcatons hghspeed networks ad-hoc and sensor networks anomaly detecton and perormance analyss and evaluaton o stochastc systems. 9 ACADEMY PUBLISHER