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the average queue length is equivalent to minimizing the average packet delay in steady state. In fact, the cost function can be modi ed to a more general objective limt e[j(Q(t))] for some Schur convex and increasing functions [146] j : RK R. 11.5.2 Optimal Solution
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In this section, we focus on the optimal solution to the scheduling problem (Problem 11.3). Before we discuss the delay-optimal solution, we consider the following theorem regarding the throughput optimality [146] of a simple rate allocation and power allocation policy, namely, the Tse Hanly (TH) policy [132], in the fading multiaccess channel in Equation (11.23). Theorem 11.6 (TH Policy Is Throughput-Optimal) For any given realization of the channel state h, let PTH( h, m) and RTH( h, m) be the optimizing solution of the following optimization problem
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(PTH , R TH ) = arg max m k rk - b k pk
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where r C( h, p), p P (peak power constraint) and b are the Lagrange multipliers chosen to satisfy the average power constraint e[p] P. Then a throughput-optimal scheduling policy is given by PTH( h, a * q) and CTH( h, a * q), where a is any positive vector, q is the current queue length, and a * q is the vector whose component is given by aiqi. Proof Please refer to Appendix 11F. Using the proof of Theorem 11.6 in Appendix 11F and Theorem 11.8 in Appendix 11E, we can have an average delay bound for the TH policy as summarized in the following corollary. Corollory 11.1 (System Delay) For any arrival rate vector, l S, under the TH policy (PTH( h, v), CTH( h, v)) in Equation (11.24), the steady-state queue length has a nite mean Qk for all k and Aspx qr code 2d barcode printingon .net
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Proof Please refer to Appendix 11F. Let v = a * q be the vector of weighted queue sizes. In the case of one user K = 1, it can be shown [132] that (PTH( h, v), CTH( h, v)) reduces to the wellknown water lling scheme whereby more power is allocated to favorable channel states and less or no power is allocated to unfavorable channel states. Water lling is known to maximize throughput in a single-user channel with an in nitely backlogged transmitter. Theorem 11.6 shows that it is also throughput-optimal for a communication system with random packet arrivals. In the general case of multiple users K > 1 and unequal weighted queue sizes, several users typically transmit, and little can be said about the optimal power allocation policy. The optimal rate allocation policy, on the other hand, satis es a general principle known as the longest weighted queue highest possible rate (LWQHPR). Given any power allocation policy P, the LWQHPR is as de ned below De nition 11.7 [Longest Weighted Queue Highest Possible Rate (LWQHPR)] For any feasible power allocation policy P (satisfying both the peak and the average power constraints), the rate allocation policy (LWQHPR) is given by R* (H, Q) = arg for some ak > 0. In fact, following the same proof as inTheorem 11.6, for any given feasible power allocation policy P, the LWQHPR rate allocation policy is throughputoptimal and the delay bound in Corollary 11.1 holds. In fact, the LWQHPR policy is equivalent to the MMW scheduling policy introduced in De nition 11.5. The LWQHPR becomes the regular MW scheduler if ak = 1 for all k [1, . . . , K], and it is also known as the longest queue highest possible rate (LQHPR) policy [146]. Besides throughput optimality, we shall illustrate that for systems with symmetric fading, the LQHPR rate allocation policy is also strongly delay-optimal given any symmetric power allocation policy [146]. We shall rst introduce the following de nitions. De nition 11.8 (Symmetric Fading Process H(t)) A fading process H(t) is = (h1, . . . , hK), we have called symmetric if for any h Pr[H1 (t ) = h1 , . . . , H K (t ) = hK ] = Pr[H1 (t ) = hp (1) , . . . , H K (t ) = hp (K ) ] (11.28) max
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