In this paper, a discrete-time single-server queueing system with an infinite waiting room, referred to as the G’(G’)/Geo/1 model, i.e., a system with general interarrival-time distribution, general arrival bulk-size distribution and geometrical service times, is studied. A method of analysis based on integration along contours in the complex plane is presented. Using this technique, analytical expressions are obtained for the probability generating functions of the system contents at various observation epochs and of the delay and waiting time of an arbitrary customer, assuming a first-come-first-served queueing discipline, under the single restriction that the probability generating function for the interarrival-time distribution be rational. Furthermore, treating several special cases the authors rediscover a number of well-known results, such as Hunger’s result for the G/Geo/1 model. Finally, as an illustration of the generality of the analysis, it is applied to the derivation of the waiting time and the delay of the more general G’(G’)/G/1 model and the system contents of a multi-server buffer-system with independent arrivals and random output interruptions.
