On the Counting Process for a Class of Markovian Arrival Processes with an Application to a Queueing System

Abstract

In this paper, we consider the counting process for a class of Markovian arrival processes (MAPs). We assume that the representing matrices in such MAPs are expanded in terms of matrix representations of the standard generators in the Lie algebra of the special linear group. The primary purpose of this paper is to construct an explicit solution of the time-dependent distribution and factorial moments of the number of arrival events in (0,t] of the counting process for this class of MAPs. Our construction relies on the Baker--Hausdorff lemma and the specific structure of the representing matrices. To investigate the efficiency of CPU usage with the explicit solution, we have conducted numerical experiments on computing the time-dependent distribution of the counting process through the explicit solution and uniformization-based method. We show that the CPU times required to compute the time-dependent distribution of the number of arrival events in (0,t] through the explicit solution have little sensitivity to t, while the consumption of CPU times with the uniformization-based method becomes greater as t increases. For illustrative purposes, we present a system performance analysis of a queueing system for possible use in automatic call distribution (ACD) systems. As an application of the explicit solution, we use it to express the waiting time distribution of the queueing system. Some numerical examples are also given with comparisons to computer simulations.