Maximum entropy solutions for batch arrival queue with an un-reliable server and delaying vacations

We consider the M[x]/G/1 queueing system with server vacations characterized by a special feature, reflecting real-life situations, in which the server, upon finding an empty system at the end of a vacation, activates a timer of duration T and waits dormant in the system. If a batch arrives during the dormant period, the server starts providing his service for the waiting customers, but if no arrivals occur, the server waits no more and takes another vacation. The server is subject to breakdowns according to a Poisson process and his repair time obeys an arbitrary distribution. We use maximum entropy principle to derive the approximate formulas for the steady-state probability distributions of the queue length. We perform a comparative analysis between the approximate results with established exact results for various vacation time, timer duration, service time and repair time distributions. We demonstrate that the maximum entropy approach is accurate enough for practical purpose and is a useful method for solving complex queueing systems.