Queue size distribution and capacity optimum design for N‐policy Geo(λ1,λ2,λ3)/G/1 queue with with setup time and variable input rate

In this paper we consider a discrete‐time $Geo/G/1$ queue with $N$ ‐policy and setup time, where the customers’ input rate varies according to the server’s status: idle, setup and busy states. By using the total probability decomposition technique, we study the transient and equilibrium properties of the queue length from the beginning of the arbitrary initial state, and obtain both the recursion expressions of the $z$ ‐transformation of the transient queue length distribution and the recursion expressions of the steady state queue length distribution at arbitrary time epoch $n^{+}$ . The results obtained in this paper indicate that the queue length distribution in equilibrium no longer follows the stochastic decomposition discipline. The important relations between the steady state queue length distributions at different time epochs $(n^{‐},n,n^{+})$ are discovered. Finally, by numerical examples, we discuss the sensitivity of the steady state queue length distribution towards system parameters, and illustrate the application of the expressions for the steady state queue length distribution in the system capacity design.