On Optimal Balking Rules and Toll Charges in the <i>GI</i>/<i>M</i>/1 Queuing Process

Abstract

This paper considers a GI/M/1 queuing process with an associated linear cost-reward structure and stationary balking process, and, based on a probabilistic analysis of the system, it derives optimal joining rules for an individual arrival, as well as for the entire community of customers. For the infinite-horizon, average-reward criterion, it shows that, among all stationary policies, the optimal strategies are control-limit rules of the form: join if and only if the queue size is not greater than some specific number. However, it finds that, in general, exercising self-optimization does not optimize public good. Accordingly, the paper explores the idea of controlling the queue size by levying tolls—thus achieving the system's over-all-optimal economic performance. Finally, it analyzes a “competition” model in which customers face a service agency that is a profit-making organization, and shows it to be similar to the monopoly model of price theory.