Properties of optimal resource sharing in a delay channel

In this paper we consider the problem of allocating bandwidth/server to two user transmitters/queues with arbitrary arrival processes, to minimize the total expected holding cost of backlogged packets in the system over a finite horizon. However, the queue backlog information is delayed due to communication delay in the channel. In addition, the bandwidth allocation is done in batches, so that a queue can be assigned any number of slots not exceeding the total number in a batch. This problem is motivated by channel allocation in a communication system involving large propagation delay, e.g., a typical satellite data communication scenario. Our principal interest in this paper is to investigate whether the optimal assignment of a batch of slots can be achieved by sequentially using a strategy that is optimal in assigning a single slot, which is typically much easier to find. In this paper we show that if the cost c(x), as a function of the packet backlog x in the system, is non-decreasing, supermodular and superconvex, then (1) the value function 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.