Optimal control of a broadcasting server

A stochastic control problem motivated by broadcast applications is considered in this paper. A natural queueing model abstraction in which each service to a queue clears all the customers at once is adopted, which can also be considered as a batch processing queueing model with infinite batch size. Each broadcast can be charged a non-negative cost. In addition, there is a cost whose rate is given as a function of the number of customers waiting in the system at any point. For any cost rate which is a convex function in the number of customers, it is shown that the optimal control is of the threshold type in order to minimize the infinite horizon discounted cost. This result complements the existing literature on batch processing queueing models that have typically only considered monotone costs. For a system with two classes of customers where each service can clear all customers of any given class, with monotone waiting costs and zero service costs, we show that the optimal control can be represented as a double-switch curve in the two dimensional state space. The structure of the optimal policy for multiple queues is a natural next question, and an interesting future direction is to explore the performance of simple index policies.