The operating characteristic analysis on a general input queue with N policy and a startup time

This paper studies a single removable server in a finite capacity G/M/1/K queueing system with combined N policy and a exponential startup before each service period. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining interarrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential interarrival time distribution. The distributions of the number of customers in the queue at pre-arrival epochs and at arbitrary epochs, as well as the distributions of the waiting time (in the queue) and the busy period are established.