A generalized SBBP/G/1 queue and its applications

In this paper, we consider a discrete-time single-server queueing system with a generalized switched batch Bernoulli arrival and a general service time processes (a generalized SBBP/G/1 queue). Customers arrive to the system in batches and service times of customers are generally distributed. The batch size and the service time distributions are governed by a discrete-time alternating renewal process with states 1 and 2. The arrival and the service processes in this system are semi-Markovian in the sense that their distributions depend not only on the state in the current slot but also on the state in the next slot in the alternating renewal process. Sojourn times in each state have a general distribution whose generating function is represented as a rational function. The purpose of this paper is two-fold. The first is to provide the analytical results in a fairly general assumption on discrete-time queues with two-state Markov modulation. The second is to show a rich applicability of the mathematical model to important queueing systems such as a discrete-time GI[X]/G/1 queue, a discrete-time queue with independent GI[X]/G and BBP/G input streams and a discrete-time queue with service interruptions. Using the semi-Markovian nature of the arrival and the service mechanisms in our model, we show that our model is readily applied to analyze those systems.