Queue-length and waiting-time distributions of discrete-time GI(X)/Geom/1 queueing systems with early and late arrivals

In this paper, we give a unified approach to solving discrete-time GI(X)/Geom/1 queues with batch arrivals. The analysis has been carried out for early- and late-arrival systems using the supplementary variable technique. The distributions of numbers in systems at prearrival epochs have been expressed in terms of roots of associated characteristic equations. Furthermore, distributions at arbitrary as well as outside observer's observation epochs have been obtained using the relation derived in this paper. We also present delay analyses for both the systems. Numerical results are presented for various interarrival-time and batch-size distributions.