Algorithmic discussions of distributions of numbers of busy channels for GI/Geom/m/m queues

The high speed multi-access communication channels such as BISDN and ATM operate on a discrete-time basis rather than on a continuous-time basis. The usual Erlang loss formula developed for continuous-time is not valid for discrete-time queues, though models such as these can be derived from the corresponding discrete-time models. In this paper, we consider the discrete-time GI/Geom/m/m queue. Two variants of this model, viz. late-arrival system with delayed access and early arrival system have been discussed and analyzed. Distributions of numbers of busy channels at various epochs have been obtained using the supplementary variable technique. The performance measures such as probability of loss, and average number of busy channels have been presented for interarrival distributions: geometric, deterministic, arbitrary, etc. It is hoped that the results obtained in this paper may provide useful information to designers of telecommunication systems, practitioners, and others.