Performance of discrete-time queueing systems

In this paper, a very general discrete-time queueing model with one single server and an infinite waiting room is studied. The number of arrivals during any discrete time-unit (henceforth called a ‘slot’) as well as the service time of each customer (expressed in slots) have general probability distributions. Both single and bulk arrival processes are considered. The system is analysed by using a two-dimensional state description. Results of the analysis include: the probability generating function of the number of customers in the system at various sets of observation epochs, and the probability generating functions of the unfinished work, the system time of the customers, and the busy period (expressed in slots). Also a number of fundamental relationships between the main performance measures of discrete-time queueing systems in general are derived. The results of the paper are applicable in the study of queueing phenomena in the area of digital communications systems.