A Constrained Optimization Problem for a Two-Class Queueing Model

We discuss dynamic server control in a two-class service system under a constraint on the number of high-priority customers. A class of randomized threshold policies is defined, and is proven to contain an optimal policy in the case without abandonments. The proof of optimality is then used to construct heuristic policies for the case of low-priority abandonments, which we test numerically. The experiments we run suggest that, even when abandonments are introduced, these classes of policies outperform priority policies, and, in some cases, are near-optimal. Both cases are considered under the average cost criterion. This paper makes two primary contributions. First, the proof of optimality in the case without abandonments provides an alternative method for proving the existence of optimal policies satisfying certain structural results without relying on leveraging properties of the value function. The second contribution lies in the practicality of the heuristic policies. While implementing a stationary policy generally relies on the cumbersome task of observing the number of customers of each class, the threshold structure of the heuristic policies only requires focus on a particular aspect of the system, such as total number of customers.