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the rst example, we have illustrated that channel state information would be useful in scheduling to exploit multiuser diversity. System capacity can be substantially increased by giving users with good CSI higher priorities. However, if the scheduler adapts to the CSI only, there is no guarantee on the delay performance. In the second example, we have illustrated the importance of buffercondition-aware scheduling strategy, especially if our objective is to minimize the average packet delay. In the nal example, we have illustrated that the conventional power water- lling solution may not be truly optimal when buffer status is taken into account. Hence, it is important to have a cross-layer scheduling strategy that adapts to both the channel state information and the buffer conditions. This requires a combined framework based on information theory (for the physical layer) and queuing theory (for the queue dynamics). In Section 11.4.2, we consider one important design objective of the crosslayer scheduling adapting to both the CSI and queue state, namely, the stability of the system. We de ne the stability region as the set of packet arrival rate vectors l (Poisson arrivals) where the length of the buffers in the system remains bounded (using an appropriately chosen scheduler). We have characterized the stability region of multiuser systems with a time-invariant physical layer as well as a stochastic physical layer. For instance, the stability region is given by the convex hull of the capacity region of the underlying physical layer. For example, the stability regions of the time-invariant physical layer and the stochastic physical layer are given by int(C ) and int( e [C (H)]), respectively. Finally, we elaborate on the speci c scheduler designs that can maintain system stability. Finally, in Section 11.5, we consider another important cross-layer design objective to minimize the packet delay. The delay-optimal scheduler design is cast into an optimal control problem, and the solution is given by the LQHPR policy, which schedules packets according to load balancing rules.
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APPENDIX 11A: PROOF OF THEOREM 11.1 ON STABILITY REGION OF DETERMINISTIC PHYSICAL LAYER Necessary Condition. Let Qi(t) be the queue size of user i at time t, and let Q(t) = (Qi(t)) be the vector of queue sizes. Let r(t) C be the scheduled rate vector within the time invariant capacity region and A(t) be the vector of bits arrival for the K users at time t. We have Q(t ) = A(0) + A(1) + + A(t - 1) - tv (11.32)
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where v = 1/t (r(0) + r(1) + . . . + r(t - 1)). Since v belongs to the set S of the convex convex combinations of the feasible rate vectors r(t) C, its expected value belongs to the convex combination. Thus, it follows from (11.32) that l - E[q(t)]/t S. By taking the limit as t goes to in nity, we conclude that l S, since S is a closed set.
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We now show that l is strictly dominated by a vector in S. The total bit arrival of user i during the interval [0, t - 1] has a Poisson distribution with mean lit. Let Bi be the event that it exceeds lit + l i t . The probability of Bi is at least an absolute constant c. It follows from (11.32) that e [Q(t ) B1 , B2 , . . . , BK ] tl + tl - te [v B1 , . . . , BK ] (11.33)
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The left side of (11.33) is upper-bounded by E[Q(t)]/cK, which is bounded by a constant independent of t. By choosing t suf ciently large, we conclude the r is strictly dominated by the vector e[v|B1, . . . , BK], which belongs to S. Suf ciency Condition. We show that if l = e[A] is strictly dominated by a vector in S, then there is a scheduling algorithm for which the system is h h ergodic. Assume that (1 + e)l S n=1anrn with a 0 and S n=1an = 1, where rn is the nth rate vector in the feasible rate region C. At each timestep, schedule rate rn with probability an. Since the arrival rate of user i is li and the departure rate, when queue i is nonempty, is at least (1 + e)li, the system is ergodic and the expected queue size of each link is O(1/e). (see the paper by Kahale and Wright [64] for details).
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APPENDIX 11B: PROOF OF THEOREM 11.2 ON STABILITY REGION OF STOCHASTIC PHYSICAL LAYER Necessary Condition. The random process describing the behavior of the entire system is given by S(t) S(t ) = {(U i 1 (t ), . . . , U iQi (t ) (t )), m(t ), i = 1, . . . , K} where Qi(t) is the queue length of user i at time t and Uik(t) is the current delay of the kth packet in the queue of user i at time t. Let Di(t) = min(Qi(t), mim(t)) be the number of packets for user i served at time t. Hence, we have Qi (t + 1) = Qi (t ) - Di (t ) + Ai (t + 1) " i By stability of the system, we mean that the Markov chain S(t) is stable.7 Consider a scheduling policy static service split (SSS) parameterized by a aj(S ). When the system is in state S, the SSS rule chooses for rate vector rjm with probability aj(S). Assume that the stability of the system is achievable. For any xed timeslot t, we have
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A Markov chain S(t) is stable if the set of positive recurrent states is nonempty and contains a nite subset that is reached with probability 1 from any initial state [71]. In this context the term stability implies the existence of a stationary probability distribution. In addition, if all positive recurrent states are connected, the stationary distribution is unique.
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