The Bernoulli feedback queue with balking: stochastic order results and equilibrium joining rules

We consider customer joining behaviour for a system that consists of a FCFS queue with Bernoulli feedback. A consequence of the feedback characteristic is that the sojourn time of a customer already in the system depends on the joining decisions taken by future arrivals to the system. By establishing stochastic order results for coupled versions of the system, we prove the existence, and uniqueness, of Nash equilibrium joining policies, and show that these are characterized by (possibly randomized) threshold rules. We contrast the Nash rule with the socially optimizing joining rule that minimizes the long-term expected average sojourn time (or cost) per customer. The latter rule is characterized by a non-randomized threshold, and we show that the Nash rule admits at least as many customers into the system as the socially optimizing one.