Optimal Server Allocation in Batches

In this paper we consider the problem of allocating bandwidth to two transmitters/queues with arbitrary arrival processes, so as to minimize the total expected (discounted) holding cost of backlogged packets in the system over a finite (infinite) horizon. The bandwidth is in the form of time slots in a TDMA schedule. Allocation decisions are made based on the queue backlog information, which is delayed. In addition, the allocation is done in batches, in that a queue can be assigned any number of slots not exceeding the total number in a batch. In this paper we show that if the packet holding cost as a function of the packet backlog in the system is non-decreasing, supermodular and superconvex, then (1) the value function (or the cost to go) at each time slot will also satisfy these properties; (2) the optimal policy for assigning a single slot is of the threshold type; and (3) optimally allocating M slots at a time can be achieved by repeatedly using a policy that assigns each slot optimally given the previous allocations. Thus the problem of finding the optimal allocation strategy for a batch of slots reduces to that of optimally allocating a single slot, which is typically much easier to obtain. We provide sufficient conditions for the same results to hold in the case of discounted cost over an infinite horizon and in the case of average cost criterion. The above results are then applied to a special case where the holding costs are linear and equal for all queues, in which minimizing holding cost is equivalent to maximizing the system throughput.