A Single Vacation Model G/M/1/K with N Threshold Policy

This paper studies a G/M/1/K queueing system, where the server applies an N policy and takes a single vacation when the system is empty. We provide a recursive method, using the supplementary variable technique and treating the supplementary variable as the remaining inter-arrival time, to develop the steady-state probability distributions of the number of customers in the system. The method is illustrated analytically for exponential inter-arrival time distribution. Hereby, we establish 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 and the busy period. AMS (2000) subject classification. Primary 60K25, 90B22