Analysis of discrete-time queues with general service demands and finite-support service capacities

In this paper, we study a non-classical discrete-time queueing model with variable service demands and variable service capacities. Specifically, we consider the case where the service capacities of the system, i.e., the numbers of work units that the system can perform during each slot, are independent from slot to slot and have an identical general distribution with finite support. New customers enter the system according to a general independent arrival process. The service demands of the customers, i.e., the numbers of work units of service that each customer requires from the system, are general independent. We present an analytical technique to analyze the behavior of this queueing system. The analysis leads to expressions for the probability generating functions and the moments of the unfinished work in the system, the queueing delay of an arbitrary customer and the number of customers in the system in steady state. We also derive approximations for the tail probabilities of the system content and the customer delay. Numerical results are included to illustrate the impact of the various model parameters on the system performance.