On the queue length in the discrete cyclic-waiting system of Geo/G/1 type

Abstract

We consider a discrete time queueing system with geometrically distributed interarrival and general service times, with FCFS service discipline. The service of a customer is started at the moment of arrival (in case of free system) or at moments differing from it by the multiples of a given cycle time T (in case of occupied server or waiting queue). Earlier we investigated such system from the viewpoint of waiting time, actually we deal with the number of present customers. The functioning is described by means of an embedded Markov chain considering the system at moments just before starting the services of customers. We find the transition probabilities, the generating function of ergodic distribution and the stability condition. The model may be used to describe the transmission of optical signals.