An M/G/1 queue with second optional service and server breakdowns

An M/G/1 with second optional service and unreliable server is studied in this paper. We assume that customers arrive to the system according to a Poisson process with rate λ. All demand the first “essential” service, whereas only some of them demand the second “optional” service. The service times of the first essential service are i.i.d. random variables, and that of the second optional service are i.i.d. exponential random variables. We assume that the server has a service-phase dependent, exponentially distributed life time as well as a service-phase dependent, generally distributed repair time. Using a supplementary variable method, we obtain the transient and the steady-state solutions for both queueing and reliability measures of interest.