Strategic Customer Behavior in an M/M/1 Feedback Queue with General Payoffs

We consider an M/M/1 feedback queue in which service attempts may fail, requiring the customer to rejoin the queue. Arriving customers act strategically, deciding whether to join the queue based on a threshold strategy that depends on the number of customers present. Their decisions balance the expected service reward against the costs associated with waiting, while accounting for the behavior of others. This model was first analyzed by Fackrell, Taylor and Wang (2021), who assumed that waiting costs were a linear function of the time in the system. They showed that increasing the reward for successful service or allowing reneging can paradoxically make all customers worse off. In this paper, we adopt a different setting in which waiting does not incur direct costs, but service rewards are subject to discounting over time. We show that under this assumption, paradoxical effects can still arise. Furthermore, we develop a numerical method to recover the sojourn time distribution under a threshold strategy and demonstrate how this can be used to derive equilibrium strategies under other payoff metrics.