PROOF OF THEOREM 11.6 ON THROUGHPUT OPTIMALITY OF TSE-HANLY POLICY

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(11.40)

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then there exists a steady-state distribution for Q(t) with bounded rst moments Q such that SkgkQ q. Please see References 104 and 88 for the proof.

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APPENDIX 11F: PROOF OF THEOREM 11.6 ON THROUGHPUT OPTIMALITY OF TSE HANLY POLICY

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2 We choose the Lyapunov function V(q) = Skakqk. Conditions 1 and 3 (From Appendix 11E) are straightforward from the assumptions of nite expected arrivals and nonzero probability of no arrivals to all the queues. Hence, we shall focus on condition 2. For condition 2, we rst show that

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2 k = W log 2 1 + H k Pk . This follows to queue k in the tth timeslot, and C 2 sz directly from the de nition of the Lyapunov function since Qk(t + T) = (Qk(t) + Ak(t) - rk(t)T)+ and [(x)+]2 x2 as well as by the peak power constraint pk(t) Pk and the peak fading constraint |Hk(t)|2 |Hk|2. , m) and RTH( h, m) be the power control and rate control policies Let PTH( h given by Equation (11.24) for the weight m, respectively. We have r k = e h [RTH( h, m)] satis es Sk mkr Sk mkr k for all r e[C], where e[C] is the average k capacity region of theMAC channel over ergodic fading realizations. De ne mk = akqk for some positive constants ak, P* ( h, q) = PTH( h, m), and R*( h, q) , m). Since PTH( h, m) satis es the power constraints P and P in every = RTH( h queue state q, we see that P*( h, q) also satis es the average and peak power constraints and hence is feasible. Under P* and R* policies, we have

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for all r C . For any l S = int C , there exists d = (d, . . . , d) such that l + d C . Thus we have

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CROSS-LAYER SCHEDULING DESIGN BASED ON QUEUEING THEORY

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(11.41)

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Choose any e > 0 and de ne the compact region as q +e K L = q R + : a k qk 2Td k Then for any q L, the right side of Equation (11.41) is strictly less than -e. Therefore, by Theorem 11.8 in Appendix 11E, there exists a steady-state distribution for the queue length Q(t). Furthermore, the inequality from Equation (11.41) can be combined with the last part of Theorem 11.8 to provide the delay guarantee as in Corollary 11.1.

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EXERCISES 1. [Optimality of Power Water-Filling] Consider a point to point fast fading channel with channel outputs given by: Y = HX + Z where H is the complex Gaussian channel fading coef cient, X is the complex lowpass equivalent transmit signal and Z is the complex white Gaussian noise. Assume that an encoding frame spans across an ergodic realization of the channel fading H(t) and the trans2 mitter has an average power constraint E X = P0 . (a) Derive the optimal power control policy for delay insensitive user to maximize the ergodic capacity. (b) From (a), explain whether the power control algorithm is to compensate the effect of fading or not In commercial systems like IS95 (CDMA), the power control is to compensate the effect of channel fading (increase power when fading is poor and decrease power when the fading is good). Is this consistent with the power adaptation policy in (a) Explain the reason if there is a difference. (c) Derive the optimal power control policy for a point to point fading channel for minimizing the user s delay for delay sensitive user.

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[Importance of stable scheduler] Assume all users have data in the buffer at all times. Consider a system with 2 users. Arrival rate of user A is 94 bits per time slot; while arrival

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