AN optimization problem to maximize the up-link

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1 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 8, AUGUST Power and Rate Control wth Outage Constrants n CDMA Wreless Networks C. Fschone, M. Butuss, K. H. Johansson, and M. D Angelo Abstract A rado power control strategy to acheve maxmum throughput for the up-lnk of CDMA wreless systems wth varable spreadng factor s nvestgated. The system model ncludes slow and fast fadng, rake recever, and mult-access nterference caused by users wth heterogeneous data sources. The qualty of the communcaton s expressed n terms of outage probablty, whle the throughput s de ned as the sum of the users transmt rates. The outage probablty s accounted for by resortng to a lognormal approxmaton. A mxed nteger-real optmzaton problem P 1, where the objectve functon s the throughput under outage probablty constrants, s nvestgated. Problem P 1 s solved n two steps: rstly, we propose a mod ed problem P 2 to provde feasble solutons, and then the optmal soluton s obtaned wth an ef cent branch-and-bound search. Numercal results are presented and dscussed to assess the valdty of our approach. Index Terms CDMA, outage, dstrbuted computaton, combnatoral optmzaton, branch-and-bound search. I. INTRODUCTION AN optmzaton problem to maxmze the up-lnk throughput achevable n CDMA wreless systems by rado power allocaton s nvestgated. We express qualty of servce constrants by outage probablty for any user, whch s an mportant measure for delay-lmted transmssons [1]. The problem of power control under outage constrants dates back at least to the work presented n [2]. In [3], the authors consder a problem under outage probablty constrants n Raylegh fadng channels. Ths approach has later been further developed n [4], where the channel s modeled wth lognormal fadng, and the outage probabltysapproxmated by a Gaussan dstrbuton. Both n [3] and [4], t s shown that the power mnmzaton under outage constrants can be cast as a geometrc program. An nterestng approach to the power control wth outage constrants s nvestgated n [5], [6], where an teratve method s presented for some dstnct cases of slow and fast fadng. The outage constrants have Paper approved by D. I. Km, the Edtor for Spread Spectrum Transmsson and Access of the IEEE Communcatons Socety. Manuscrpt receved June 18, 27; revsed July 9, 28. C. Fschone and K. H. Johansson are wth the ACCESS Lnnaeus Center, Electrcal Engneerng, Royal Insttute of Technology, Stockholm, Sweden (e-mal: {carlo, kallej}@ee.kth.se). M. Butuss was wth the Unversty of Padova, Italy, when ths work was done (e-mal: bokassa@de.unpd.t). M. D Angelo s wth the Unversty of L Aqula and PARADES GEIE, Italy (e-mal: mdangelo@parades.rm.cnr.t). Ths work was done n the framework of the HYCON Network of Excellence, contract number FP6-IST The work s also partally funded by the Swedsh Foundaton for Strategc Research and Swedsh Research Councl. M. D Angelo acknowledges the support of Fondazone Ferdnando Flauro, Italy. Part of ths work was presented at IEEE ICC 27. Dgtal Object Ident er 1.119/TCOMM been relaxed wth an upper bound provded by the Jensen s nequalty. In ths paper we nvestgate a mxed nteger-real optmzaton problem whch s dfferent from those n the exstng contrbutons from the lterature. The constrants on the outage are expressed by a lognormal approxmaton studed n [7], whch allows us to solve the problem even for low outage probabltes. Our contrbuton s orgnal, for example, compared to the nterestng works [3] [5], because they do not nclude a detaled model of the Mult Access Interference (MAI) and the source data, whle we adopt a general model of the SINR, whch s representatve of mxed slow and fast fadng. We propose a new method to solve the rate maxmzaton problem, whch s based on branch-and-bound search. II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION Consder a system scenaro where K moble users transmt to a base staton. Each user = 1,...,K s assocated wth a traf c sourcetype(voce,vdeo,data,etc.),alevel of transmsson power P, and a chp tme T c. Let h be the channel gan for user. We adopt the Lee and Yeh multplcatve model [8, pag. 91]: h = l z Ω,wherel s the path loss, z s the power of the fast fadng, and Ω s the power of the shadowng. A Raylegh dstrbuton for the fast fadng s assumed. For any j, z and z j are statstcally ndependent. Let Ω =exp(ξ ),whereξ s a Gaussan random varable havng zero average and standard devaton n lnear unts σ ξ.letthebnaryrandomvarableν be the actvty status (on/off) of the source. The probablty mass functon s such that Pr[ν =1]=α and Pr[ν =]=1 α,whereα /9$25. c 29 IEEE s the actvty factor of source. Leth =[h 1,...,h K ] T and ν = [ν 1,...,ν K ] T.Independencesassumedbetweenany par of random varables of the vectors h and ν. Themodelof the physcal layer for the up-lnk of a sngle-cell asynchronous bnary phase shft keyng DS/CDMA system s summarzed by the followng expresson of the SINR [9]: SINR (h, ν) = N 2T P h + K P j, (1) j=1 G j h j ν j Observe that ths SINR model takes nto account also a rake recever wth Maxmal Rato Combnng [1]. The rate of user, =1,...,K,smodeledasR = R n, where R = 1/T s the basc rate wth basc bt tme T. n s an nteger denotng the assgned rate, whch s a power of two due to the spreadng codes [8]. The spreadng factors are expressed by G = G /n,whereg = T /T c corresponds to the basc rate R.Weassumethattherado power of a user s expressed as P = p n, where p s

2 2226 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 8, AUGUST 29 the power at the basc rate R (.e., when n = 1). Ths power model s motvated by the fact that users requrng larger rates need larger transmt powers, as shown n [9]. Ths model has been adopted also n [11] (and references theren), where t has been observed that t has the advantage of keepng the bt energy to nose rato constant wth respect to varatons of n.letp =[p 1,...,p K ] T, p = [p 1,p 2,...,p 1,p +1...p K ] T, n = [n 1,n 2,...,n K ] T and n =[n 1,n 2,...,n 1,n +1,...n K ] T. The objectve functon of the rate maxmzaton problem s the sum of the users rates, and the constrants are bounds on the maxmum value for the outage probablty, transmsson powers and rates: P : max p,n 1 T n s.t. Pr [SINR (h, ν) <γ ] O, =1,...,K p T (n E [h]) P T, p p p max, =1,...,K 1 n G, n N, =1,...,K where 1 T =[ ] R K,andl T =[l 1 l 2...l K ]. The decson varables are the powers p and rates n. O s the maxmum outage probablty that s allowed for user wth respect to the threshold γ.theconstrantp T (n E [h]) = K =1 n p l exp (σξ 2 /2) P T s due to the fact that the antenna of the base staton can receve only a maxmum amount of power wthout dstortng the sgnal [9]. For obvous physcal reasons, powers cannot be smaller than a gven value p,norlargerthanamaxmumpowerp max.therate constrant follows from the fact that G = G /n 1. Fnally, the set of rates, N, contans only power of two elements. Remark 1: We assume that an admsson control polcy s used (see e.g. [12]), so that all K users can be accommodated n the system whle transmttng at least wth the mnmum power and rate whle satsfyng all constrants of P. Consequently, Problem P s feasble. In the next secton, we propose a method to solve Problem P ef cently. A. Outage Probablty III. PROBLEM SOLUTION Knowledge of the outage probablty s requred to express the rst constrant n Problem P. Snceaclosedformexpresson for such a probablty s unknown, we adoptalognormal approxmaton of the SINR pdf, as proposed n [7]. Usng such an approxmaton, the dervaton of the outage probablty can be easly accomplshed, and t can be shown that the constrant s expressed as p γ I (n, p )=γ exp( q ζ(2, 1)σ 2 X (n, p ) + µ X (n, p)+lnp ψ(1)), (2) where q = Q 1 (1 O ), Q(x) = 1/ 2π /2 x e t2 dt s the complementary standard Gaussan dstrbuton, and µ X (n, p) = 2lnM (1) (n, p) 1 (2) 2 ln M (n, p), σx 2 (n, p )=lnm (2) (n, p) 2lnM (1) (n, p), and M (1) (n, p) = E ln [z /SINR ] and M (2) (n, p) = E ln [ ] z 2/SINR2 (see [7] for the dervaton of prevous statstcal expectatons). Expresson (2) allows us to resort to the theory of standard nterference functon [13] [14]. It s possble to show that the followng result hold: Let I(n, p) = [I 1 (n 1, p 1 ),...,I (n, p ),...,I K (n K, p K )] T. For any gven rate vector n, I(n, p) s a standard nterference functon wth respect to p, namely: 1) I(n, p) ; 2) f p p then I(n, p) I(n, p); 3) f c>1, thenci(n, p) I(n, cp). Furthermore, for any gven power vector p, fn ñ then I(n, p) I(ñ, p). Canddate solutons for Problem P can be found by explotng the propertes of the nterference functon and a mod ed optmzaton problem. A branch-and-bound search reduces ef cently the set of possble solutons, so that the optmal one can be quckly acheved. We approach these ssues n the followng. B. Mod ed problem Let us denote wth P 1 Problem P, where the outage constrant has been approxmated wth (2): P 1 : max p,n 1 T n s.t. p γ I (n, p ), =1,...,K p T (n E [h]) P T, p p p max, =1,...,K 1 n G, n N =1,...,K Problem P 1 s df cult to solve because the nteger rates and the constrant on the total power prevent us to apply teratve algorthms as n [13] and [15]. However, snce the rate vector n belongs to a dscrete set, combnatoral optmzaton technques can be used for solvng Problem P 1.Wthths goal n mnd, t can be proved that (n, p) s the soluton of Problem P 1 only f the rst set of constrants of P 1 holds wth equalty: Theorem 1: Let (n, p) be a soluton of Problem P 1,then: p = γ I (n, p ) =1,...,K. (3) Prevous theorem allows us to rewrte the outage constrants of Problem P 1 at the equalty. From ths, we propose an approach to solve Problem P 1 n two steps. Frstly, we nd feasble solutons by a mod ed problem P 2.Secondly,we solve P 1 lookng at the set of feasble solutons by branchand-bound search. Feasble rate vectors for Problem P 1 can be found by the solutons of the followng problem: P 2 : mnp (n E [h]) p s.t. p = γ I (n, p ), =1,...,K Note that n ths problem the decson varable s only p, whle n, wthn N,for =1,...,K,sgven.Consderavector p, gvenn, thatsolvesp 2.Thsparofvectorsn, p provdes us wth a feasble rate vector for P 1 f the constrants p T (n

3 FISCHIONE et al.: POWER AND RATE CONTROL WITH OUTAGE CONSTRAINTS IN CDMA WIRELESS NETWORKS 2227 E [h]) P T and p p p max,for =1,...,K,are sats ed. Therefore P 2 s useful, snce we can explore the feasblty of a rate vector n for P 1 just by solvng Problem P 2. Problem P 2 can be solved ef cently by recallng the propertes of the nterference functon. Indeed, these propertes allow us to use contracton mappngs, whch gve a low computatonal cost algorthm that solves Problem P 2 by sequences of asynchronous teratons (see [15, Pag. 431]): p (t) =I (n, p (t 1))γ =1,...,K. (4) The convergence of ths algorthm s fast. Monte Carlo smulatons show that t takes from 5 to 1 teratons. One could use (4) to solve Problem P 2 for each possble rate vector n N.Thus,theratevectorn that solves the Problem P 1 could be found usng an exhaustve search among all rate vectors n N.However,suchatechnquehaslarge computatonal costs, because the cardnalty of N may be very large. In the next subsecton, we wll present some useful propertes of the rate vectors, whch, together wth a branchand-bound search, allow us to nd the optmal soluton wth reduced computatonal cost. C. Branch-and-bound search It s possble to reduce the set N usng local knowledge about spec c valuesofn. Inpartcular,weconstructaset of a smaller sze over whch performng a search for the optmal soluton. We propose a technque for the reducton of N usng two crtera. The rst one uses the cuttng-planes dea [15], whereas the second one s based on consderatons about Problem P 1. The crtera can be summarzed n the followng propostons. Proposton 1: Let n be feasble for P 1.Aparn, p s an optmal soluton of P 1 only f 1 T n 1 T n. Gven a feasble rate vector n, from Proposton 1 follows that the search for the optmal soluton can be restrcted to the set N Σ N,where N Σ ( n) =N\ { n such that 1 T n 1 T n }. (5) Proposton 2: Let n be unfeasble for P 1.Ifn n, then n s unfeasble for P 1. Prevous proposton allows us to say that f n s unfeasble, then the set N can be reduced by de nng a set of unfeasble solutons N I N accordng to the followng rule: N I ( n) =N\{n such that n n }. (6) In practce, Propostons 1 and 2 are useful to reduce the set of feasble rates by usng a neural network archtecture, where computaton and nformaton dstrbuton s optmzed. Specfcally, a branch-and-bond algorthm can be mplemented [16], whch allows us to nd ef cently the exact soluton of Problem P 1,sncewereducethesetofcanddateratevectors to a smaller one. IV. NUMERICAL RESULTS In ths Secton we apply our method to a wreless system of thrd generaton. All the system parameters are taken from the 3GPP spec catons [17]: the chp tme s T c = s, and the maxmum spreadng factor s G =256.We rate sum TABLE I SCENARIOS CONSIDERED IN THE NUMERICAL RESULTS. Scenaro σ ξ α A.5,.5,.5,.5.4,.4,.4,.4 B.5,.5,.5,.5.2,.2,.2,.2 C.5,.5,.5,.5.7,.7,.7,.7 D.4,.4,.4,.4.4,.4,.4,.4 E.6,.6,.6,.6.4,.4,.4,.4 M random n [.4,.6] random n [.2,.7] A B C D E outage probablty constrant Fg. 1. Optmal soluton of Problem P 1 as obtaned wth the branch-andbound algorthm for the cases A, B, C, D and E. assume a thermal nose N = 17 db/hz, and we set P T = ,suchthatP T T c /N =16dB. We assume ahomogeneouspropagatonenvronmentforthepathlossl, whch s set to 9dB for each transmtter. As requred for the valdty of the multplcatve model of the fadng, µ z =1[8]. The system s consdered n outage wth γ =3.1. We consder the cases of 4, 8 and 12 users, for sx representatve scenaros, denoted wth A, B, C, D, E, and M. In partcular, scenaros from A to E are referred to the case of 4 users, wth unform standard devaton of the shadowng (σ ξ )andthesourceactvty(α ). The sxth scenaro consders the case of 4, 8 and 12 users wth mxed values of σ ξ and α.intab.i,thevaluesadoptedforσ ξ and α are reported for each scenaro. A. Rate Maxmzaton In Fg. 1 the optmal soluton of Problem P 1 as obtaned wth our approach s plotted for the scenaros A to E. Each dot of the curves s referred to a value of the outage probablty constrant O (whch s reported on the x axs, and t s assumed to be the same for each user). The maxmum throughput acheved s 25, asobtanedforthescenarob when O =.1. Thecostfunctonncreasesashghervalues of the outage constrants are allowed. Observe that better performance s obtaned when users have low actvty factors (α =.2 n the scenaro B), whle the mnmum rates are acheved when users have hgh actvty (α =.7 n the scenaro E). When users have low transmsson actvty, mult

4 2228 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 57, NO. 8, AUGUST 29 rate sum users 8 users 12 users TABLE II COMPUTATIONAL COMPLEXITY, I.E., NUMBER OF TIMES THAT PROBLEM P 2 IS SOLVED. Scenaro mn max average A B C D E M M M outage probablty constrant Fg. 2. Optmal soluton of Problem P 1 as obtaned wth the branch-andbound algorthm for the scenaro M for 4, 8, and12 users. Zero rate means that the problem s not feasble. outage probablty approxmaton smulaton B. Outage Approxmaton To check the accuracy of the outage approxmaton adopted n Secton III-A, we computed the outage probablty correspondng to the rates and powers obtaned wth the soluton of Problem P 1 for several cases of the outage requrement. In partcular, SINR samples were generated by consderng a large set of determnatons of the random varables ξ and z, for =1,...,K,andthentheactualoutageprobabltywas computed numercally wth respect to the threshold γ used n Problem P 1.InFg.3weconsderthecaseof8 users, when they transmt wth the rates and powers as obtaned by the branch-and-bound algorthm for the scenaro M. The outage probablty approxmaton s close to the actual probablty. It can be shown that the same behavor s found for all scenaros outage probablty constrant Fg. 3. Outage probablty obtaned by MonteCarlosmulatons(contnuous dotted curve). The outage constrant O on the x-axs s used n the scenaro Mfor8 users, where the branch-and-bound algorthm provdes us wth the optmal rates and powers. These are used to generates samples of SINR, for whch the outage probablty s plotted on the y-axs. access nterference s reduced, thus enablng hgher transmt rates. Also, observe that from the sequence D, A and E as the standard devatons of the shadowng ncrease, the total rate decreases. The reason s because larger uctuatons of the shadowng ncrease the mult access nterference. In Fg. 2 the optmal soluton of Problem P 1 as obtaned wth our approach s plotted for the scenaro M. Thecases wth 4 users feature the hghest transmt rates, whle the cases wth 12 users shows the lowest rates. Ths s obvously due to the fact that as the number of users ncreases, the mult access nterference s larger. In the case wth 12 users, Problem P 1 s not feasble for outage probablty constrants less than.6, and for outage constrants less than.3 for 8 users. Ths can be seen n the gures where the sum of the rates s zero. The reason of nfeasblty s that low outage requrements cannot be guaranteed when there are too many users n the system, namely the number of users exceeds the system capacty [9]. C. Complexty Analyss In ths secton, we present an analyss of the computatonal complexty of the branch-and-bound algorthm. The convergence tme of the branch-and-bound algorthm s gven by the number of tmes that Problem P 2 must be solved. Proposton 1 and Proposton 2 allow us to reduce the number of canddate solutons wth neglgble computatonal complexty. Therefore, the computaton cost of the algorthm s gven by the computaton of the soluton of Problem P 2. Consequently, we de ne as computatonal complexty the number of tmes for whch Problem P 2 s solved by the branch-and-bound algorthm. The computatonal complexty s evaluated startng the branch-and-bound algorthm wth the lowest rate vector. Ths gves the worst case for the computatonal complexty, because the actve node set s the largest, Proposton 1 and Proposton 2 delete less possble solutons, and hence Problem P 2 must be solved more tmes. In the followng, we dscuss the computatonal complexty as obtaned by smulatons. In Tab. II, the computatonal complexty s reported n terms of mnmum, maxmum, and average number for the scenaros AtoM.These gures have been computed averagng over the nterval of outage probablty requrements. Consder the scenaros A to M wth 4 users. An exhaustve search over all possble rates would have requred to solve Problem P 2 for 9 4 =6561tmes, snce there are 9 possble rates per each user. We obtaned that the average number of tmes for whch P 2 s solved vares from 382 (C) to 976 (B). Observe that the larger

5 FISCHIONE et al.: POWER AND RATE CONTROL WITH OUTAGE CONSTRAINTS IN CDMA WIRELESS NETWORKS 2229 computatonal complexty of B s n agreement wth the conclusons of the prevous Subsecton, where t was shown that the scenaro B acheves larger sum of the rates. Spec cally, startng the branch-and-bound algorthm wth the lowest rates, several possble solutons have to be explored before httng the borders of feasble solutons and ndng the optmal one. By the same argument, the scenaro C s the less complex. The average computatonal complexty for the scenaro M wth 8 users s 673, andwth12 users s 7131, whletwouldhave been needed to solve Problem P 2 anumberoftmesgven by and ,respectvely,for an exhaustve search over all the possble rates. We conclude that the good computatonal complexty of the branch-andbound algorthm s remarkable for all the scenaros we have consdered. V. CONCLUSIONS AND FUTURE WORK In ths paper, we proposed an approach to allocate the users powers to maxmze the throughput of a CDMA wreless system. A general model of the physcal layer was consdered, wth Raylegh-lognormal fadng and source actvty, and qualty of servce was expressed by the outage probablty. The problem was cast as a mxed nteger-real optmzaton program, where we expressed the constrants by a moment matchng approxmaton of the outage probablty. The optmal soluton of the problem was derved n two steps: rst a mod ed program was nvestgated to provde feasble solutons, and then an algorthm, based on branch-and-bound search, was used. Numercal results n scenaros of practcal nterest con rmed the valdty of our approach, and showed that the optmal soluton of the problem s acheved wth low computatonal complexty. REFERENCES [1] G. Care, G. Tarcco, and E. Bgler, Optmal power control for mnmum outage rate n wreless communcatons, n Proc. IEEE ICC 98, Atlanta,GA,June1998. [2] T. Hekknen, A model for stochastc power control under log-normal dstrbuton, CWIT 21, 21. [3] S. Kandukur and S. Boyd, Optmal power control n nterferencelmted fadng wreless channels wth outage-probablty spec catons, IEEE Trans. Wreless Commun., vol.1,pp46 55,Jan.22. [4] K.-L. Hsung, S.-J. Km, and S. Boyd, Power control n lognormal fadng wreless channel wth uptme probablty spec catons va robust geometrc programmng, ACC, 25. [5] J. Papandropoulos, J. Evans, and S. Dey, Optmal power control for Raylegh-faded multuser systems wth outage constrants, IEEE Trans. Wreless Commun., vol.4,no.6,nov.25. [6], Outage-based power control for generalzed multuser fadng channels, IEEE Trans. Commun., vol.54,no.4,apr.26. [7] C. Fschone and M. D Angelo, An approxmaton of the outage probablty n Raylegh-lognormal fadng scenaros, Unversty of Calforna at Berkeley, schon/papers/outage approx report.pdf, Tech. Rep., Mar. 28. [8] G. L. Stüber, Prncples of Moble Communcaton. Kluwer Academc Publshers, [9] F. Santucc, G. Durastante, F. Grazos, and C. Fschone, Power allocaton and control n multmeda CDMA wreless systems, Kluwer Telecommun. Systems,vol.23,pp.69 94,May June23. [1] A. Shah and A. M. Hamovch, Performance analyss of maxmal rato combnng and comparson wth optmum combnng for moble rado communcatons wth cochannel nterference, IEEE Trans. Veh. Technol.,vol.49,no.4,pp ,July2. [11] E. Hossan, D. I. Km, and V. K. Bhargava, Analyss of TCP performance under jont rate and power adaptaton n cellular WCDMA networks, IEEE Trans. Wreless Commun., vol.3,no.3,may 24. [12] J. S. Evans and D. Evertt, Effectve bandwdth-based admsson control for multservce CDMA cellular networks, IEEE Trans. Veh. Technol., vol. 48, pp , Jan [13] J. Herdtner and E. K. P. Chong, Analyss of a class of dstrbuted asynchronous power control algorthm for cellular wreles system, IEEE J. Select. Areas Commun., vol.18,no.3,pp ,Mar.2. [14] R. D. Yates, A framework for uplnk power contorl n cellular rado system, IEEE J. Select. Areas Commun., vol.13,sept [15] D. P Bertsekas and J. N. Tstskls, Parallel and Dstrbuted Computaton: Numercal Methods. Athena Scent c, [16] C. Fschone and M. Butuss, Power and rate control outage based n CDMA wreless networks under MAI and heterogeneous traf c sources, n IEEE Internatonal Conference on Communcatons, June 27. [17] 3GPP TS V6.1., 3rd Generaton Partnershp Project; Techncal Spec caton Group Rado Access Network; Physcal layer procedures (FDD).