Introduction to Matrix Analytic Methods in Queues 1

7Matrix-Analytic Methods (Discrete-Time)

Matrix-analytic methods (MAM) were developed by Neuts in the late 1970s. To explain what these methods constitute, we use Neuts’s own words. In the invited article published by the European Journal of Operational Research, Neuts (1984) passionately says, “The history of the matrix methods (so called for brevity) is short, but worth telling. I tackled a number of models involving embedded Markov renewal processes, evidently with some measure of success, since the papers were published in noted journals and some academic recognition came my way. It privately bothered me that, as the papers grew longer and the analysis more complex, the explicit or qualitative results in them became fewer and fewer.” He continues further, “In the history of mathematics, a similarity of formalism has always indicated similarity of structure and an ultimate level of understanding is that of unifying structure.”

Stochastic modeling occurs naturally in many walks of life. The mathematical tools needed to solve a specific problem vary depending on the application of the stochastic model. The telecommunications area first adopted MAM soon after their introduction. It is well known that the self-similarity property is seen in Ethernet traffic, WWW traffic, signaling traffic, multi-media traffic and other high-speed network traffic, and hence Poisson/exponential distributions are not well suited for modeling such traffic. The benefit of using Markov-modulated Poisson ...