Complete analytic and computational analyses of the discrete-time bulk-arrival infinite-server system: GIX/Geom/∞

In this paper, we consider the discrete-time queueing system with bulk arrivals, generally distributed inter-batch times, geometrically distributed service times, and infinite number of servers: the GI^X/Geom/∞ queue. Under the assumptions of early arrival system and late arrival system, we derive the system size distributions at two different epochs – prearrival and random. In addition, simple relations among the binomial moments of the steady-state system-size distributions of the two systems are derived. Special cases, such as the GI/Geom/∞ or GI/D/∞ queues, are also presented. Some numerical results are included at the end.