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 interbatch times, geometrically distributed service times, and infinite number of servers: the GIX/Geom/∞ queue. Under the assumptions of early arrival system (EAS) and late arrival system (LAS), 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.