Time-average and asymptotically optimal flow control policies in networks with multiple transmitters

We consider M transmitting stations sending packets to a single receiver over a slotted time-multiplexed link. For each phase consisting of T consecutive slots, the receiver dynamically allocates these slots among the M transmitters. Our objective is to characterize policies that minimize the long-term average of the total number of messages awaiting service at the M transmitters. We establish necessary and sufficient conditions on the arrival processes at the transmitters for the existence of finite cost time-average policies; it is not enough that the average arrival rate is strictly less than the slot capacity. We construct a pure strategy that attains a finite average cost under these conditions. This in turn leads to the existence of an optimal time-average pure policy for each phase length T , and to upper and lower bounds on the cost this policy achieves. Furthermore, we show that such an optimal time-average policy has the same properties as those of optimal discounted policies investigated by the authors in a previous paper. Finally, we prove that in the absence of costs accrued by messages within the phase, there exists a policy such that the time-average cost tends toward zero as the phase length T →∞.