Service Outage Based Power and Rate Allocation

Transcription

1 1 Service Outage Based Power and Rate Allocation Jiangong Luo, Lang Lin, Roy Yates, and Predrag Spasojević Abstract Tis paper combines te concepts of ergodic capacity and capacity versus outage for fading cannels, and explores variable rate transmissions under a service outage constraint in a block flat fading cannel model. A service outage occurs wen te transmission rate is below a given basic rate. We solve te problem of maximizing te expected rate subject to te average power constraint and te service outage probability constraint. Wen te problem is feasible, te optimum power policy is sown to be a combination of water filling and cannel inversion allocation, were te outage occurs at a set of cannel states below a certain tresold. Te service outage approac resolves te conflicting objectives of ig average rate and low outage probability. Index Terms Adaptive transmission, block-fading cannel, ergodic capacity, outage capacity, service outage I. INTRODUCTION Wireless communication cannels vary wit time due to mobility of users and canges in te environment. For a time varying cannel, dynamic allocation of resources suc as power, rate, and bandwidt can yield improved performance over fixed allocation strategies. Indeed, adaptive tecniques are employed in EDGE [1], GPRS [2], and HDR [3], and are proposed as standards for next generation cellular systems. Te system performance criterion is usually application specific; terefore, different classes of applications will benefit from specific adaptive transmission scemes. In order to differentiate real-time service from non real-time service, tree capacity measures ave been defined in te literature: ergodic capacity [4], delay limited capacity [5], and capacity versus outage probability [9 11]. A compreensive survey of tese concepts can be found in [6]. Te ergodic capacity [4] was developed for non real-time data services. It determines te maximum acievable rate averaged over all fading states. Te corresponding optimum power allocation is te well known water filling allocation [7, 8]. Te ergodic capacity may not be relevant for real-time applications in a slow fading environment, were substantial delay can occur wen averaging over all fading states. Delay limited capacity [5] and te capacity versus outage probability [9 11] were developed for constant-rate real-time applications. Te delay limited capacity specifies te igest acievable rate wit a decoding delay independent of fading correlation structures [5]. Te outage capacity in te capacity versus outage probability problem determines te ɛ-acievable rate [12] of te M-block fading cannel. Te corresponding optimum power allocation was derived in [1] for M parallel flat fading blocks (frequency diversity or space diversity), and in [11] for M consecutive flat fading blocks (time diversity). Te zero-outage capacity in [1, 11] is also referred to as te delay-limited capacity. Toug te outage capacity studies te capacity for constantrate real-time applications, te constant-rate assumption may not be essential for many real-time applications. For applications wit simultaneous voice and data transmissions, for example, as soon as a basic rate r o for te voice service as been guaranteed, any excess rate can be used to transmit data in a best effort fasion. For some video or audio applications, te source rate can be adapted according to te fading cannel conditions to provide multiple quality of service levels. Typically tese applications require a nonzero basic service rate r o to acieve a minimum acceptable service quality. Motivated by tese variable-rate real-time applications, we study variable rate transmission scemes subject to a basic rate requirement in a slow fading environment. By allowing variable rate transmissions, te variation of te fading cannel can be exploited to acieve an average rate iger tan te outage capacity. By imposing a basic rate requirement, te system can be guaranteed to operate properly. Since infinite average power is needed to acieve any nonzero rate at all times in a Rayleig fading cannel, we impose a probabilistic basic service rate requirement, tat we call a service outage constraint. Te service is said to be in an outage wen te information rate is smaller tan te basic service rate r o. Service quality is acceptable as long as te probability of te service outage is less tan ɛ, a parameter indicating te outage tolerance of te application. Unlike te information outage in te capacity versus outage probability problem [1, 11], te bits transmitted during te service outage may still be valuable in tat tey will be transmitted reliably and will contribute to te average rate. For variable-rate systems, te expected rate determines ow muc rate can be transmitted on te average and is a meaningful measure of system performance. Terefore, in tis paper, te allocation problem is to find te power and rate allocation tat maximize te expected rate subject to te service outage constraint and te average power constraint. Under te assumptions of a block flat fading AWGN cannel model and perfect cannel state information at te transmitter, we verify tat te outage sould occur at bad cannel states below a certain tresold. Te resulting optimum power allocation is sown to be a combination of cannel inversion and water filling wen te allocation problem is feasible. Te service outage approac strikes good balance between te average rate and te outage probability. Tis approac as been generalized to te case of code words spanning multiple blocks in [13]. Altoug a continuous fading distribution is assumed in tis paper, te results can be extended into te case of discrete fading distributions by employing a probabilistic power allocation, as examplified by te policies in [1]. Altoug our problem as been motivated by real time applications, it also caracterizes coverage versus capacity tradeoffs. In particular, mobility in cellular systems results in can-

2 2 nel variations due to canges in distance attenuation. An important objective of a cellular system is to provide a basic service rate over as muc of te service area as possible. In tis case, te service outage constraint caracterizes te spatial coverage requirement of te system. Te objective is ten to maximize te average rate over all geograpic locations subject to meeting te service outage constraint. Te remainder of tis paper is organized as follows. In Section II, te system model and te optimization problem are presented. In Section III, te optimum allocation policy is derived. In Section IV, a supporting teorem for te optimum allocation policy is proved. Furter discussion of te optimum solution is presented in Section V. II. THE ALLOCATION PROBLEM In tis work, we employ te block flat fading Gaussian cannel (BF-AWGN) model [9]. In te BF-AWGN cannel, a block of N symbols experiences te same cannel state, wic is constant over te wole block, but may vary from block to block. Witin eac block we ave te time-invariant Gaussian cannel y = x + n. Here x is te cannel input, y is te cannel output, n is wite Gaussian noise wit variance σ 2, and is te cannel state. Let f() denote te probability density function of te cannel state and F () denote te corresponding cumulative distribution function (CDF). Here, we consider te case were F () is a continuous function and te power allocation is a deterministic function of cannel state. We make te following assumptions: Te cannel state information is known perfectly at bot transmitter and receiver. One codeword spans one fading block and te block size N is large. Te fading process is ergodic over te time scale of te application. As pointed out in [1], it makes sense to study te BF-AWGN cannel as N, since for typical practical systems N is fairly large and outage is te dominant error event wen using an actual code. Let p() denote te transmission power at cannel state. Ten te maximum acievable rate at eac block is te capacity of Gaussian cannel wit received power p(), and is denoted as R[p()], were R[P ] = 1 2 log 2 (1 + Pσ ) 2. (1) Terefore, te resource allocation problem requires finding only te optimum power allocation p (). Under te assumption of te ergodicity of fading process, te time average rate of te system can be caracterized by te expected rate. Terefore, given te average power P av, te basic service rate r o, and te allowable service outage probability ɛ, we wis to maximize te expected code rate, as follows: R = max p() E {R[p()]} (2) subject to: E {p()} P av (2a) p() Pr{R[p()] < r o } ɛ. (2b) (2c) In te absence of te service outage constraint (2c), R would be te ergodic capacity for te fading cannel, and te well known water filling allocation [7, 8] would be te corresponding optimum power assignment. III. OPTIMUM POWER AND RATE ALLOCATION In tis section, we derive an optimum power allocation p () for problem (2). Te difficulty in deriving p () is primarily due to te probabilistic nature of te constraint (2c). Here, we sow ow an optimum power allocation can be derived based on a problem analogous to (2) wit a deterministic constraint on te assigned rate. Given a basic service rate r o and a power policy p(), te service set is defined as H s (p()) = { R[p()] r o }, and te outage set is H o (p()) = { R[p()] < r o }. Our approac will be to sow tat tere is an optimal solution to problem (2) wit a particular form of a service set. Prior to sowing tis, we solve te following subproblem in wic it is required tat te service set contains an arbitrary set H a. R (H a ) = max p() E {R[p()]} (3) subject to: E {p()} P av (3a) p() (3b) R[p()] r o H a. (3c) Let p (, H a ) denote an optimum solution to problem (3). Terefore, p (, H a ) acieves te igest average rate among all te scemes wose service set contains H a. Problem (3) does not necessarily ave a solution for a given (P av, r o, H a ). Constraint (3c) implies tat a feasible allocation p() must satisfy p() σ2 (2 2ro 1) H a. (4) Tis implies tat te minimum average power needed to meet te constraint (3c) for a given (r o, H a ) is σ 2 (2 2ro 1) P min (r o, H a ) = f() d. (5) H a Consequently, problem (3) as a solution only if P av P min (r o, H a ). Wen P av = P min (r o, H a ) te corresponding power allocation is te on-off cannel inversion policy σ 2 (2 2ro 1) p (, H a ) = H a, (6) oterwise. Wen P av > P min (r o, H a ) te corresponding power allocation is given by te following teorem. We use te notation (x) + = max(x, ). Teorem 1 Wen P av > P min (r o, H a ) te optimum solution for problem (3) is: σ 2 (2 2ro 1) H a { } 2 2ro, p (, H a ) = ( 1 σ oterwise. ) (7)

3 3 were is te solution of E {p (, H a )} = P av. Teorem 1 follows from standard variational arguments; te proof appears in te Appendix. Note tat wen P av = P min (r o, H a ) te resulting power allocation (6) can be viewed as a limiting case of expression (7) as. Te power allocation p (, H a ) is a combination of cannel inversion and water filling allocations. To obtain a ig average rate, we would like to allocate power in te form of te water filling allocation, wile to meet te service constraint (3c), we must allocate power no less tan te cannel inversion allocation witin te set H a. Te solution p (, H a ) balances tese two factors. To caracterize te solution to te optimization problem (2), we define ɛ as te solution to F ( ɛ ) = ɛ. Te tresold ɛ partitions te cannels into a set H ɛ = { ɛ } of good cannels and te complementary set H ɛ = { < ɛ } of bad cannels. In te following, we sow tat te solution of (3) wit H a = H ɛ, specifically p () = p (, H ɛ ), is an optimum solution of problem (2). In order to prove tis, we define te partial ordering and sow a number of preliminary results. Definition 1 H 1 H 2 if 1 < 2 for all 1 H 1 and 2 H 2. Teorem 2 Problem (2) as an optimal solution p () wit te outage set H o (p ()) and te service set H s (p ()) satisfying H o (p ()) H s (p ()). Teorem 2 sows tat tere exists an optimum power allocation suc tat te outage occurs wen te cannel state is worse tan a particular tresold. Proof of Teorem 2 involves a somewat complicated two-step construction and is deferred to Section IV. Using Teorem 2 and te fact tat Pr{H s (p ())} 1 ɛ by constraint (2c), it is easy to sow te following corollary. Corollary 1 Problem (2) as an optimum solution p () suc tat H ɛ H s (p ()). Now we can prove p () = p (, H ɛ ) by sowing tat R = R (H ɛ ). Wit H a = H ɛ in te outage constraint (3c), te service set of p (, H ɛ ) must contain H ɛ. Tus p (, H ɛ ) satisfies te outage constraint (2c) and is a feasible power allocation sceme for problem (2), implying R (H ɛ ) R. On te oter and, Corollary 1 implies tat problem (2) as an optimal solution p () acieving an average rate of R tat satisfies constraint (3c) wit H a = H ɛ. Tus, p () is a feasible power allocation sceme for problem (3) and R R (H ɛ ). Consequently, R = R (H ɛ ). In conclusion, an optimum solution is p () = p (, H ɛ ) and te following conclusions apply to problem (2). Problem (2) is feasible if only if (P av, r o, ɛ) satisfies P av P min (r o, H ɛ ) = σ 2 (2 2ro 1) ɛ Wen P av = P min (r o, H ɛ ) we ave σ 2 (2 2ro 1) p () = ɛ, < ɛ. f() d, (8) (9) Wen P av > P min (r o, H ɛ ), we can apply Teorem 1 wit H a = H ɛ yielding an optimum solution to problem (2) of te form p () = σ 2 (2 2ro 1) ɛ min{ ɛ, 22ro }, ( 1 σ oterwise, ) (1) were is te solution of E {p ()} = P av. As P av approaces P min (r o, H ɛ ), and te power allocation (1) will reduce to te on-off cannel inversion allocation (9). IV. OPTIMUM SERVICE SETS In tis section, we will prove Teorem 2, wic implies tat we can find an optimal solution wose service set H s (p ()) includes te good cannel states H ɛ. Our approac will be to sow tat given an arbitrary feasible power allocation sceme ˆp(), we can always construct a better sceme p () wic satisfies H o (p ()) H s (p ()). Tis implies tat tere is an optimum power allocation sceme p () wit H o (p ()) H s (p ()). Let Ĥ s denote te service set and ˆR te average rate for te policy ˆp(). Feasibility of ˆp() implies E {ˆp()} P av and Pr{Ĥs} 1 ɛ. We use a two-step construction. First, we construct p () from ˆp() by setting H a = Ĥs in problem (3), yielding te solution p () = p (, Ĥs) σ 2 (2 2ro 1) = Ĥs { } <, ( 22ro 1 σ oterwise. ) (11) were is te solution of E {p (, Ĥs)} = P av. Here in te case of P av = P min (r o, Ĥs), p () can be expressed by (11) as. Clearly, p () is feasible and acieves a iger average rate tan ˆp(). Second, we construct p () by decomposing p () into a water filling component and a residual power component. Given, we define te following functions over te wole cannel state space: ( 1 p wf () = σ2 1 +, (12) ) ( σ p 2 + (2 2ro 1) res() = p wf()). (13) Te function p wf () is a water filling allocation over te wole cannel space. Te function p res() is te nonnegative difference of cannel inversion and water filling allocations. From (11), we observe tat water filling alone meets te service condition R[p ()] r o over te set of cannel states H wf = { 2 2ro}. (14)

4 4 ^ R(p()) r p () p () service set (a) residual power residual power (b) (c) 2r 2 2r 2 water filling power water filling power Fig. 1. (a) Rate allocation R[ˆp()] for policy ˆp(), (b) te improved policy p () given by (16) wit water filling p wf () and residual power 1 ( H inv )p res (), (c) te new policy p () given by (17) wit water filling p wf () and residual power 1 ( H inv )p res (). In particular, p res() = for H wf wile residual power p res () > is needed to meet te service condition over te cannel inversion set H inv = Ĥs\H wf. (15) Tus, in terms of te indicator function 1 (x) suc tat 1 (x) = 1 wen x is true, and oterwise, p () can be rewritten in te form p () = p wf() + 1 ( H inv)p res(). (16) Here, we call 1 ( H inv )p res() te residual power allocation for p (). As sown in Figure 1, p () can be viewed as a twolayer allocation: te first layer is te water filling allocation over te wole cannel space and te second layer is te residual power allocation over H inv. Based on p (), we construct p () by preserving te first layer water filling allocation and redistributing te residual power. Intuitively, te best allocation sceme for te residual power is to allocate it to te good cannel states. Since p res() is strictly positive witin < 22ro, we will allocate te residual power over te interval [ b, 2 2ro ] were b is cosen to consume te total residual power. As sown in Figure 1, we ave were p () = p wf () + 1 ( H inv )p res (). (17) H inv = { b < 2 2ro}, (18) and b is te solution to p res ()f() d = H inv H inv p res ()f() d. (19) Note tat (16), (17), and (19) imply tat p () as te same total power as p (). Let R and R denote te average rates for p () and p (), respectively. Te following lemma gives us te properties of p (). Lemma 1 Te power sceme p () as te following properties: (a) E {p ()} = E {p ()} = P av (b) H o (p ()) H s (p ()) (c) R R (d) Pr{H s (p ())} Pr{H s (p ())}. Proof of Lemma 1 is given in Appendix. Hence, we summarize te proof: 1) Start wit arbitrary ˆp() wit average rate ˆR and service set Ĥs. 2) Set H a = Ĥs and solve (3) yielding p () wit rate R ˆR and service set H s (p ()) containing Ĥs. 3) Decompose p () into water filling p wf () and residual power components 1 ( H inv )p res (). 4) Fix te water filling component p wf () and reallocate te residual power to generate p (). Te power policy p () satisfies Pr{H s (p ())} Pr{H s (p ())} and R R. Hence, p () is a better power allocation sceme tan p () for problem (2). We can conclude tat from any feasible ˆp() we can obtain a better power allocation p () in wic H o (p ()) H s (p ()) olds. Tis implies tat problem (2) as an optimum solution p () satisfying H o (p ()) H s (p ()). V. PROPERTIES OF THE OPTIMUM POLICY In Section III, we derived te optimum allocation sceme for problem (2). In tis section, we will discuss tis optimum solution, and sow ow problem (2) in tis paper is related to te capacity versus outage probability problem. Te optimum power allocation sceme (1) includes a combination of cannel inversion and water filling. For a given probability distribution f(), te optimum solution belongs to one of te following possible types depending on te value of (P av, r o, ɛ): 1 I Wen P av = P min (r o, H ɛ ), p () includes no transmission for < ɛ and cannel inversion for ɛ. II Wen P av > P min (r o, H ɛ ) but ɛ, p () includes no transmission for < ɛ, cannel inversion for ɛ <, and water filling for 22ro 22ro. III Wen P av is sufficiently ig suc tat ɛ 2 2ro < < ɛ, p () includes no transmission for <, water filling for < ɛ, cannel inversion for ɛ < 2 2ro, and water filling for 2 2ro. 1 In te case of r o = or ɛ = 1, te solution types II and III will degenerate into solution type IV, wic is te pure water filling allocation.

5 5 p () I R[p ()] I ergodic capacity service outage approac wit r =.5 bits/symbol service outage approac wit r =.7 bits/symbol outage capacity r o 1.6 ε ε 1.4 II r o II R av (bits/symbol) ε 2 2r o ε 2 2r o.8 III III.6 r o P av (db) IV ε 2 2r o IV ε 2 2r o Fig. 3. Comparison of service outage approac wit oter capacity notions in te Rayleig fading cannel, for a fixed ɛ =.1.. ε r o ε.35.3 r =.5 bits/symbol r =.7 bits/symbol ε=.1 Fig. 2. For optimum solution types I-IV, power policies are given on te left and corresponding rate allocation are on te rigt. IV Wen P av is ig enoug for ɛ2 2ro, p () is just te water filling allocation. Tese four types of power allocation scemes as well as te corresponding rate allocations are depicted in Figure 2. For solution types I, II, and III, te optimum service set is H s (p ()) = H ɛ and te resulting outage probability is ɛ, wile for type IV solution H ɛ H s (p ()) and te resulting outage probability is less tan ɛ. Type I solution is te on-off cannel inversion allocation. In tis case, we ave just enoug average power to satisfy te service outage constraint. Wen we ave extra power beyond P min (r o, H ɛ ), we can allocate te power in a more efficient way to obtain a iger average rate and, at te same time, to meet te service outage constraint. Wen P av is sufficiently ig for te water filling allocation to satisfy te service outage constraint, ten it must also be te optimum solution for problem (2). Tus, for a given pair (r o, ɛ), te optimum power allocation sceme gradually canges from te onoff cannel inversion allocation to te water filling allocation as P av increases. Now we examine te connection of te service outage problem wit te outage capacity in [1] and te ergodic capacity in [4]. Te outage capacity C ɛ (P av ) in [1] specifies te maximum supportable rate for a given average power P av wit outage probability ɛ, wic implies tat te basic service rate in tis work must satisfy r o C ɛ (P av ). It is easy to see tat te above condition is equivalent to te feasibility condition (8), tat is, P av P min (r o, H ɛ ). Furtermore, we can see tat te resulting average rate R canges from r o (1 ɛ) to te ergodic capacity wit increasing P av for a given (r o, ɛ). In Fig 3, te expected rate acieved by te service outage approac is plotted against te ergodic capacity and te outage capacity in Rayleig fading cannel wit normalized mean for cannel gain and normalized noise variance. As we can see, for a given outage prob- Outage probability ε P av (db) Fig. 4. Outage probability acieved by te water filling allocation for different basic rate r. ability ɛ =.1, te outage capacity as nearly a 5 db loss in average power compared to te ergodic capacity for a given rate. Between te outage capacity and te ergodic capacity, a number of service outage approaces wit different basic rates exist. Te outage probability for different r acieved by te water filling allocation is also plotted against te service outage approac wit a given ɛ =.1 in Fig 4. It can be observed tat, for a range of P av, te service outage approac acieves a rate very close to te ergodic capacity, and at te same time significantly reduces te outage probability. Hence, te service outage approac strikes good balance between average rate and outage probability. APPENDIX Proof: Teorem 1 Wen P av P min (r o, H a ), problem (3) is feasible and can be, equivalently, translated into te following

6 6 problem: max E {R[p()]} (2) p() subject to E {p()} = P av (2a) p() p() σ2 (2 2ro 1) H a. (2b) (2b) Tis is a standard variational optimization problem [14]. Te objective function is concave on p() and te constraints are linear functions of p(). Ten p (, H a ) is te optimum solution iff it satisfies te te Kun-Tucker conditions [15]. Using a Lagrange multiplier g(p(),, ) = 2log(2)σ 2 >, we define [ ] R[p()] 2 log(2)σ 2 p() f(). (21) Let H denote te set wit non-boundary points as { } H = H a p (, H a ) > σ2 (2 2ro 1) { / H a p (, H a ) > }. (22) It is easy to verify tat wen H te p (, H a ) satisfies dg(p (, H a ),, ) dp (, H a ) oterwise te p (, H a ) satisfies dg(p (, H a ),, ) dp (, H a ) =, (23). (24) Terefore, te p (, H a ) is te optimum solution. Proof: Lemma 1 Power scemes p () and p () differ in te allocation of te residual power. In order to sow tat p () allocates te residual power in a better way tan p (), we define te following power efficiency function for p res() over its strictly positive space. Definition 2 Te power efficiency function η() for p res() is η() = r o R[p wf ()] p res () < 22ro. (25) Te power efficiency function indicates te rate increment corresponding to a unit power assigned from p res (). We ave te following property for η(). Proposition 1 Te power efficiency function η() is a strictly increasing function of over te interval < 2 2ro. Proof: Proposition 1 We consider te cases and separately. For, we ave p wf () = and η() = wic is an increasing function of. r o σ 2 (2 2ro 1), (26) For, (12), (13), and (25) imply η() = r o (1/2) log 2 (/ ( ) ). (27) σ 2 2 2ro 1 We define u() = r o 1/2 log 2 (/ ), so tat η() = ˆη(u()) were ˆη(u) = u σ 2 2 2u (28) 1 It is straigtforward to verify tat ˆη(u) is a strictly decreasing function of u for u. Since u() is a strictly decreasing function of and u() wen 22ro, it follows tat η() is an increasing function of for 2 2ro. We also employ te following proposition for te proof of Lemma 1. Proposition 2 For disjoint sets Ψ and Ψ, let f(x) be an arbitrary function suc tat f(x ) f(x ) for all x Ψ and x Ψ. For any nonnegative function g(x) satisfying Ψ g(x)dx = Ψ g(x)dx, we ave Ψ f(x)g(x)dx Ψ f(x)g(x)dx. Wit tese preliminaries, we now verify te claims of Lemma 1. (a) Equations (16), (17), and (19) imply E {p ()} = E {p ()} = P av. (b) From equations (17) and (18), te service and outage sets of p () are H s (p ()) = { b } and H o (p ()) = { < b } respectively. Terefore, H o (p ()) H s (p ()). (c) Let Ψ = H inv H inv so tat Ψ = H inv \Ψ and Ψ = H inv \Ψ are two disjoint sets and nonempty wen p () p (). Te average rate of p () can be expressed as R = R[p wf ()]f() d + (r o R[p wf()])f() d.(29) H inv Te rate contribution of te water filling component is R wf = R[p wf()]f() d. (3) Since H inv = Ψ Ψ, Definition 2 for te efficiency function η() allows us to write R = R wf + η()p res ()f() d (31) H inv = R wf + η()p res ()f() d Ψ + η()p res ()f() d. (32) Ψ Similarly, H inv = Ψ Ψ, so te average rate for p () can be expressed as R = R wf + η()p res()f() d Ψ + η()p res()f() d. (33) Ψ

7 7 Tus, R R = η()p res()f() d Ψ η()p res()f() d. (34) Ψ Note tat te construction of H inv implies Ψ Ψ. Tat is, for any Ψ and Ψ. By Proposition 1, η() is a strictly increasing function of for < 22ro. Tus, η( ) η( ). Furtermore, equation (19) implies ()f() d = ()f() d. (35) Ψ p res Ψ p res Terefore, te conditions of Proposition 2 are satisfied and we ave R R. (d) From equations (11), (15), (17) and (18), te service sets H s (p ()) and H s (p ()) are disjoint unions given by H s (p ()) = H wf H inv = H wf Ψ Ψ, (36) H s (p ()) = H wf H inv = H wf Ψ Ψ. (37) Tis implies Pr{H s (p ())} Pr{H s (p ())} (38) = Pr{Ψ } Pr{Ψ } (39) 1 = Ψ p res () p res ()f() d 1 Ψ p res ()p res ()f() d. (4) From equations (12) and (13), we observe tat 1/p res() is a increasing function of. Since Ψ Ψ, Proposition 2 implies Pr{H s (p ())} Pr{H s (p ())}. REFERENCES [1] S. Nanda, K. Balacandran, and S. Kumar, Adaptation tecniques in wireless packet data services, IEEE Communications Magazine, vol. 38, no. 1, pp , Jan. 2. [2] ETSI, General Packet Radio Service(GPRS); Mobile Station (MS)- Base Station System(BSS) Interface; Radio Link Control/Medium Access Control (RLC/MAC) Protocol (GSM 4.6), Tec. Spec., vol. 8, no. 1, Nov [3] P. Bender, P. Black, M. Grob, R. Padovani, N. Sindusayana, and A. Viterbi, CDMA/HDR: A Bandwidt-Efficient Hig-Speed Wireless Data Service for Nomadic Users, IEEE Communications Magazine, vol. 38, no. 7, pp. 7 77, July 2. [4] A. J. Goldsmit and P. Varaiya, Capacity of fading cannels wit cannel side information, IEEE Transactions on Information Teory, vol. 43, no. 6, pp , Nov [5] S. V. Hanly and D. N. C. Tse, Multiaccess fading cannels: part II: Delay-limited capacities, IEEE Transactions on Information Teory, vol. 44, no. 7, pp , Nov [6] E. Biglieri, J. Proakis, and S. Samai, Fading cannels: Informationteoretic and communications aspects, IEEE Transactions on Information Teory, vol. 44, no. 6, pp , Oct [7] R. Gallager, Information Teory and Reliable Communication, Jon Wiley and Sons, [8] T. Cover and J. Tomas, Elements of Information Teory, Jon Wiley Sons, Inc., [9] L. H. Ozarow, S. Samai, and A. D. Wyner, Information teoretic considerations for cellular mobile radio, IEEE Transactions on Veicular Tecnology, vol. 43, no. 2, pp , May [1] G. Caire, G. Taricco, and E. Biglieri, Optimum power control over fading cannels, IEEE Transactions on Information Teory, vol. 45, no. 5, pp , July [11] R. Negi, M. Carikar, and J. Cioffi, Minimum outage transmission over fading cannel wit delay constraint, in Proc. of te IEEE International Communication Conference(ICC 99), pp , June 2. [12] S. Verdú, A general formula for cannel capacity, IEEE Transactions on Information Teory, vol. 4, no. 4, pp , July [13] J. Luo, R. Yates, and P. Spasojević, Service outage based capacity and power allocation in parallel fading cannels, in IEEE International Symposium on Information Teory, pp. 18, June 22. [14] J. C. Clegg, Calculus of Variations, Jon Wiley Sons, Inc., [15] M. S. Bazaraa and C. M. Setty, Nonlinear programming teory and algoritms, Jon Wiley and Sons, LIST OF FIGURES 1 (a) Rate allocation R[ˆp()] for policy ˆp(), (b) te improved policy p () given by (16) wit water filling p wf () and residual power 1 ( H inv )p res (), (c) te new policy p () given by (17) wit water filling p wf () and residual power 1 ( H inv )p res () For optimum solution types I-IV, power policies are given on te left and corresponding rate allocation are on te rigt Comparison of service outage approac wit oter capacity notions in te Rayleig fading cannel, for a fixed ɛ = Outage probability acieved by te water filling allocation for different basic rate r Jiangong Luo received er B.S. and M.S. degrees in Electrical Engineering from University of Science and Tecnology of Cina, Hefei, Cina. Since 1997 se as been a P.D. candidate at WINLAB, Rutgers - te State University of New Jersey. Lang Lin (S 97) received is B.S. and M.Pil. degrees in electrical engineering from Beijing University, Cina, and te Hong Kong University of Science and Tecnology, respectively. Since 1997 e as been a P.D. candidate at WINLAB, Rutgers - te State University of New Jersey, New Jersey. Roy Yates received te B.S.E. degree in 1983 from Princeton University, and te S.M. and P.D. degrees in 1986 and 199 from M.I.T., all in Electrical Engineering. Since 199, e as been wit te Wireless Information Networks Laboratory (WINLAB) and te ECE department at Rutgers University. Presently, e serves as an Associate Director of WINLAB and Professor in te ECE Dept. He is a co-autor (wit David Goodman) of te text Probability and Stocastic Processes: A Friendly Introduction for Electrical and Computer Engineers, Jon Wiley and Sons, His researc interests include power control, interference suppression, and media access protocols for wireless communications systems. Predrag Spasojević (M ) received te Diploma of Engineering degree from te Scool of Electrical Engineering, University of Sarajevo, in 199; and Master of Science and Doctor of Pilosopy degrees in electrical engineering from Texas

8 A&M University, College Station, Texas, in 1992 and 1999, respectively. From 2 to 21, e as been wit WINLAB, Rutgers University, as Lucent Post-Doctoral Fellow, were e is currently an Assistant Professor in te Department of Electrical & Computer Engineering. His researc interests are in te general areas of communication teory and signal processing. 8