First two moment entropy maximisation approach for M/G/1 queues with second optional service and server breakdowns

We consider the M/G/1 queue with second optional service and server breakdowns. A customer leaves the system either after the first required service with probability (1 – θ ) or immediately goes for a second optional service with probability θ after the completion of the first required service. For this queueing model, it is rather difficult to obtain the steady-sate probability explicitly. We apply the maximum entropy approach to approximate the system size distributions by using the first and second moments of the system size. Accuracy comparisons between the two approximate solutions are conducted. Numerical results indicate that using the first moment approach is more accurate than using the second moment approach.