Analysis of a discrete-time $$Geo^{\lambda _1,\lambda _2}/G/1$$ G e o λ 1 , λ 2 / G / 1 queue with N-policy and D-policy

Abstract

This paper explores a discrete-time Geo / G / 1 queueing system with N-policy and D-policy wherein customers arrive at the system in variable input rates according to the states of the server. When the system becomes empty, the server stays idle until the number of waiting customers reaches N or the sum of the service times of the waiting customers in the system reaches or exceeds a predetermined positive integer D, whichever happens first (referred to as Min(N, D)-policy). Employing the renewal theory, the total probability decomposition law and the probability generating function technique we obtain the probability generating function for the transient queue size distribution at time epoch \(n^+\). Also, the explicit recursive formulas of the steady-state queue length distribution at time epochs \(n^+\), n, \(n^-\) and outside observer’s time epoch are derived, respectively. Finally, some numerical experiments are conducted to investigate the sensitivity of system performance measures and the optimization of system capacity.