Wait-Time-Based Control of Single-Server Queues

Abstract

In this paper, we propose a dynamic control strategy that specifies the service time and admission probability to manage a queuing system in which each arriving customer can decide whether or not to join the queue. We present the general modeling framework and optimality conditions in terms of a Hamilton-Jacobian-Bellman equation to maximize a long-run average reward rate over an infinite horizon. To demonstrate its applicability, we first cast the model of Lin et al. (2023) into this framework. We further consider a make-to-order system in which customers have private information on their wait time sensitivities. We characterize the structure of the optimal mechanism and confirm the intuition that when the system congestion increases, the provider should only admit more patient customers while offering shorter service times.

Keywords: Queueing System, Optimal Control, Mechanism Design, HJB Equations, Wait-time