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

We considerM transmitting stations sending packets to a single receiver over a slotted time-multiplexed link. For each phase consisting ofT consecutive slots, the receiver dynamically allocates these slots among theM transmitters. Our objective is to characterize policies that minimize the long-term average of the total number of messages awaiting service at theM 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 lengthT, 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 lengthT.