Stationary Optimal Control of a Reflected Brownian Motion

The variational inequality approach is used to study the average cost per unit time (stationary) stochastic control problem for a Brownian Motion with reflection. This problem arises in the diffusion approximation of a queueing process, in the context of deciding between two operational modes, and when there is a holding cost on the size of the queue as well as costs of switching from one mode to the other. We generalize previous results obtained by Chernoff and Petkau. We also address the question of comparing the performance of the optimal stationary policy against that of any admissible strategy.