Maximum entropy approach for batch-arrival queue under NN policy with an un-reliable server and single vacation

We consider the M[x]/G/1M[x]/G/1 queueing system, in which the server operates NN policy and a single vacation. As soon as the system becomes empty the server leaves for a vacation of random length VV. When he returns from the vacation and the system size is greater than or equal to a threshold value NN, he starts to serve the waiting customers. If he finds fewer customers than NN. he waits in the system until the system size reaches or exceeds NN. 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 batch size, vacation time, service time and repair time distributions. We demonstrate that the maximum entropy approach is efficient enough for practical purpose and is a feasible method for approximating the solution of complex queueing systems.