A batch arrival unreliable server queue with two phases of service and Bernoulli vacation schedule under randomised vacation policy

This paper deals with an M^X/G/1 Bernoulli vacation queue with two phases of service in which an unreliable server operates a randomised vacation policy with at most M consecutive vacations. If the system is empty by the end of the M^th vacation, the server becomes idle in the system until at least one customer arrives at the queue. Assume that the server may meet an unpredictable breakdown according to a Poisson process and the repair time has a general distribution. For such a system, we derive the joint distribution of state of the server and queue size, the pgf of the stationary queue size distribution at a departure epoch, LST of busy period distribution and waiting time distribution along with some important performance measures of the model. Finally, we develop a cost model to determine optimal randomised policy along with some numerical illustration.