Strategic Customer Behavior in an $M/M/1$ Feedback Queue

TL;DR

This paper analyzes customer decision-making in an $M/M/1$ feedback queue, identifying unique equilibrium strategies and stability, and examining how reneging affects customer behavior and system performance.

Contribution

It introduces a comprehensive game-theoretic analysis of customer strategies in feedback queues, including equilibrium existence, stability, and effects of reneging.

Findings

Existence of a unique symmetric Nash equilibrium threshold strategy.

The equilibrium strategy is evolutionarily stable.

Allowing reneging increases customer incentives to join but may reduce overall payoff.

Abstract

We investigate the behavior of equilibria in an $M/M/1$ feedback queue where price and time sensitive customers are homogeneous with respect to service valuation and cost per unit time of waiting. Upon arrival, customers can observe the number of customers in the system and then decide to join or to balk. Customers are served in order of arrival. After being served, each customer either successfully completes the service and departs the system with probability $q$, or the service fails and the customer immediately joins the end of the queue to wait to be served again until she successfully completes it. We analyse this decision problem as a noncooperative game among the customers. We show that there exists a unique symmetric Nash equilibrium threshold strategy. We then prove that the symmetric Nash equilibrium threshold strategy is evolutionarily stable. Moreover, if we relax the strategy restrictions by allowing customers to renege, in the new Nash equilibrium, customers have a greater incentive to join. However, this does not necessarily increase the equilibrium expected payoff, and for some parameter values, it decreases it.