Optimal insertion of customers with waiting time targets

Abstract

The insertion of randomly-arriving high-priority customers (HCs) into existing queues leads to longer waiting time of regular or scheduled customers. However, given their different levels of priority, not all HCs need to be served immediately. To reduce the waiting time for regular customers without affecting the service quality for HCs, this paper addresses the optimal insertion problem of HCs by considering their waiting time targets assigned based on their levels of priority. A finite-horizon Markov decision process model is proposed to minimize the total waiting cost incurred by both regular customers and HCs. The marginal waiting penalty is constant for regular customers and non-decreasing for HCs. Several properties are observed: the optimal control policy is proved to be a state-dependent threshold policy; the marginal effect of each state variable on the optimal action is bounded by 1; and, in some meaningful cases, the threshold proves to be state-independent. Based on these properties, several heuristic policies are proposed to solve large-size problems. Numerical experiments are performed to validate the structure of the optimal control policies and compare the performances of the heuristic policies. These results imply that the optimal control policy significantly outperforms the traditional HC-first policy. The best heuristic policy, characterized by a threshold policy derived through policy iteration, performs within 0.44% of an optimal policy for most cases, which offers insights into the near-optimal control for large-size problems.