Reliability analysis of an M/G/1 repairable queueing system with vacations and patient server

This paper considers an M/G/1 repairable queueing system with vacations and patient server. It is assumed that the service station may break down during service period. Whenever a busy period ends, the server is allowed to take a vacation. When the server returns from vacation and finds an idle system, he/she waits for a patience period within the system. If a customer arrives during this period, a new busy period starts at once. Otherwise, a new vacation begins immediately at the completion epoch of the patience period. Using the total probability decomposition technique and the Laplace transform tool, some crucial reliability indices of the service station are derived, including the distribution of the time to first failure, the transient and steady-state unavailability, the mean number of failures during time interval (0, t] and the steady-state failure frequency. The results of our study indicate that the reliability measures obtained satisfy the stochastic decomposition property. Finally, some numerical examples are provided to illustrate the influence of various system parameters on these reliability indices.