Maximum entropy analysis to the N policy M/G/1 queueing system with server breakdowns and general startup times

We study a single removable and unreliable server in the N policy M/G/1 queueing system with general startup times where arrivals form a Poisson process and service times are generally distributed. When N customers are accumulated in the system, the server is immediately turned on but is temporarily unavailable to the waiting customers. He needs a startup time before providing service until the system becomes empty. 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 distributions, such as exponential (M), fc-stage Erlang (Ek), and deterministic (D). We demonstrate that the maximum entropy approach is accurate enough for practical purposes and is a useful method for solving complex queueing systems.