Bernoulli schedule vacation queue with batch arrivals and random system breakdowns having general repair time distribution

We analyse a single server queue with general service time distribution, random system breakdowns and Bernoulli schedule server vacations where after a service completion, the server may decide to leave the system with probability p, or to continue serving customers with probability 1−p. It is assumed that the customers arrive to the system in batches of variable size, but served one by one. If the system breaks down, it enters a repair process immediately. It is assumed that the repair time has general distribution, while the vacation time has exponential distribution. The purpose is to find the steady-state results in explicit and closed form in terms of the probability-generating functions for the number of customers in the queue, the average number of customers and the average waiting time in the queue. Some special cases of interest are discussed and a numerical illustration is provided.