Transient Behavior Analysis of a Finite Capacity Queue with Working Breakdowns and Server

Abstract—This study considers the M/M/1/N queue with working breakdowns and server vacations. We assume that the arrivals occur according to a Poisson process. The server is subject to breakdowns and repairs while in operation. At failure times, the server still works at a lower service rate rather than completely stopping service. When there are no customers in the system, the server takes multiple vacations. Service, repair, and vacation times are exponentially distributed. We apply the Markov process theory to present the differential-difference equations of the queueing system. A numerical technique based on the fourth-order Runge–Kutta method is used to compute the transient state probabilities. Using the transient solutions, we develop some system performance measures, such as the time-dependent expected number of customers in the system, the time-dependent expected waiting time in the system, etc. Finally, a sensitivity analysis is performed to assess the effects of various parameters on the performance measures of the system through numerical results.