Equilibrium Strategies in an M n /M/1 Queue with Server Breakdowns and Delayed Repairs

Ophthalmic units use sophisticated equipment to enable accurate diagnosis of refractive errors. This equipment is subject to two types of breakdowns. One is simple breakdown which can be repaired in-house, and the other is complex breakdown which requires repair by the original equipment manufacturer (OEM) and results in a delay time between the server breakdown and the start of the repair. In this paper, we model this scenario as an M n /M/1 queuing system with two types of breakdowns and delayed repairs due to complex breakdowns, where the delay time and repair times for simple and complex breakdowns are generally distributed. We obtain the steady-state probabilities and provide the recursive formulas for the Laplace–Stieltjes transforms (LSTs) of conditional residual delay time and repair times given the system state. For the fully observable case, we derive the equilibrium joining strategies of customers who decide to join or balk based on their observation of the system state. Moreover, two numerical experiments are conducted to explore the equilibrium joining probabilities.