An $M^{[X]}/G/1$ queue with optional service and working breakdown

In this study, a batch arrival single service queue with two stages of service (second stage is optional) and working breakdown is investigated. When the system is in operation, it may breakdown at any time. During breakdown period, instead of terminating the service totally, it continues at a slower rate. We find the time-dependent probability generating functions in terms of their Laplace transforms and derive explicitly the corresponding steady state results. Furthermore, numerous measures indicating system performances, such as the average queue size and the average queue waiting time, has been obtained. Some of the numerical results and graphical representations were also presented.