The MAP/G/1 queueing system has been considered to study the effects of high burstiness and strong correlation between interarrival times. This type of queueing system can be easily found in broadband-integrated services digital networks (B-ISDNs) based on Asynchronous Transfer Mode (ATM) which is expected to be a suitable transfer mode to integrate various kinds of traffics such as voice, data, image, motion video, and so on.
In the analysis of queueing systems with MAPs, the matrix analytic method and Markov renewal theory have been employed until recent years. Note that the matrix analytic method allows no more than one random variable to have the countably infinite state space for the description of system dynamics. However, in most real systems we have models which require two or more random variables, each of which is defined in either a countably infinite space (e.g., the number of customers in the queue) or a continuous space (e.g., the unfinished work in the system, or the elapsed/remaining service time) to describe them properly. For those models the matrix analytic method does not give any help in analysis, and hence a new analytic method should be needed.
In this thesis we present how to apply the supplementary variable method to analyze the MAP/G/1 queueing system and its variants which can be applicable for ATM networks. The supplementary variable method is known as a simple and convenient method to derive the double transform of the queue length and the supplementary variable used. To solve the matrix differential equations arising in queueing systems with MAPs when we use the supplementary variable method, we need to extend the notion of Laplace transform for positive real numbers $s$ to that of Laplace transform for matrices S.
In chapter 3, we consider a packet voice/data multiplexer which is modeled by the MAP/G/1 queueing system where a single server queue is fed by an MAP and served in first-come-first-served (FCFS) manner, with general ...