Analyses of the discrete-time group-arrival queueing systems and their applications

Abstract

This dissertation analyzes two discrete-time queueing systems: $GI^X$/D/c and $GI^X /Geom/ ∞$, which are applicable to analyses of communication systems. An extended version of the simple relation between GI/D/1 and GI/D/c systems allows us to obtain the explicit results of the waiting-time distributions of the $GI^X$ /D/c model with a restriction on the arrival group size distribution. Simple computational analysis is also presented for the usefulness of practitioners working in areas, such as ATM switching elements and traffic concentrators. Additionally, under the assumptions of EAS (Early Arrival System) and LAS (Late Arrival System), we derive the system size distributions of the infinite-server $GI^X/Geom/∞$ system at two different epochs-prearrival and random. Simple relations among the binomial moments of the steady-state system-size distributions of the two systems are derived. As a special case of the infinite-server system, this dissertation obtains the results of the $GI^X/D/∞$ system without any restriction on the arrival group size distribution, which can be used as an approximation of the $GI^X$ /D/c system under light traffic. The conventional analyses of queueing systems obtain the results using transforms, such as PGF (Probability Generating Function) or LT (Laplace Transform), from which the aimed distribution is generally hard to extract. In contrast, for the usefulness of the practitioners, this dissertation gives the transform-free results of the waiting-time distribution of the discrete-time $GI^X$ /D/c systems. In getting the system-size distribution of the $GI^X/Geom/∞$ system, this dissertation uses the transform of binomial moment, from which we can easily get the explicit result of the distribution.