Analysis of a discrete-time queue with general independent arrivals, general service demands and fixed service capacity

This paper analyzes a single-server discrete-time queueing model with general independent arrivals, where the service process of the server is characterized in two steps. Specifically, the model assumes that (1) each customer represents a random, arbitrarily distributed, amount of work for the server, the service demand, and (2) the server disposes of a fixed number of work units that can be executed per slot, the service capacity. For this non-classical queueing model, we obtain explicit closed-form results for the probability generating functions (pgf's) of the unfinished work in the system (expressed in work units) and the queueing delay of an arbitrary customer (expressed in time slots). Deriving the pgf of the number of customers in the system turns out to be hard, in general. Nevertheless, we derive this pgf explicitly in a number of special cases, i.e., either for geometrically distributed service demands, and/or for Bernoulli arrivals or geometric arrivals. The obtained results show that the tail distributions of the unfinished work, the customer delay and the system content all exhibit a geometric decay, with semi-analytic formulas for the decay rates available. Another interesting conclusion is that, for a given system load, the mean customer delay converges to constant limiting values when the service capacity per slot goes to infinity, and either the mean arrival rate or the mean service demand increases proportionally. Accurate approximative analytical expressions are available for these limiting values.