Algorithmic analysis of the discrete time GIX/GY/1 queueing system

In this paper, we study the discrete time GIX/GY/1GIX/GY/1 queueing system. First, some general results are obtained for the stability condition, stationary distributions of the queue lengths and waiting times. Then we show that, for some practical situations, the queueing system of interest has the properties of both the M/G/1M/G/1 and GI/M/1GI/M/1 non-skip-free Markov chains described by [H.R. Gail, S.L. Hantler, B.A. Taylor, Non-skip-free M/G/1M/G/1 and GI/M/1GI/M/1 types of Markov chains, Advances in Applied Probability, 29 (3) (1997) 733–758]. For such cases, the queueing system can be analyzed as a QBD process, a M/G/1M/G/1 type Markov chain, or a GI/M/1GI/M/1 type Markov chain with finite blocks after re-blocking. Computational procedures are developed for computing a number of performance measures for the queueing system of interest. In addition, we study a GI/M/1GI/M/1 type Markov chain associated with the age process of the customers in service.