Developing Effective Service Policies for Multiclass Queues with Abandonment: Asymptotic Optimality and Approximate Policy Improvement

Abstract

We study a single server queuing model with multiple classes and impatient customers. The goal is to determine a service policy to maximize the long-run reward rate earned from serving customers net of holding costs and penalties respectively due to customers waiting for and leaving before receiving service. We first show that it is without loss of generality to study a pure-reward model. Since standard methods can usually only compute the optimal policy for problems with up to three customer classes, our focus is to develop a suite of heuristic approaches, with a preference for operationally simple policies with good reward characteristics. One such heuristic is the Rμθ rule—a priority policy that ranks all customer classes based on the product of reward R, service rate μ, and abandonment rate θ. We show that the Rμθ rule is asymptotically optimal as customer abandonment rates approach zero and often performs well in cases where the simpler Rμ rule performs poorly. The paper also develops an approximate policy improvement method that uses simulation and interpolation to estimate the bias function for use in a dynamic programming recursion. For systems with two or three customer classes, our numerical study indicates that the best of our simple priority policies is near optimal in most cases; when it is not, the approximate policy improvement method invariably tightens up the gap substantially. For systems with five customer classes, our heuristics typically achieve within 4% of an upper bound for the optimal value, which is computed via a linear program that relies on a relaxation of the original system. The computational requirement of the approximate policy improvement method grows rapidly when the number of customer classes or the traffic intensity increases.