Spectral expansion method for QBD and QBD-M processes in performance modeling of computing and communication systems: A review

Abstract

Spectral Expansion Method (SEM), forming the major part of the doctoral thesis of the author, was developed at the Newcastle University (United Kingdom) during the early nineties. Though it started as an alternative to the Matrix Geometric Method (MGM), evaluations by the author and independent evaluations by others have clearly shown the superiority of SEM over the MGM in a number of aspects. SEM computes the steady state probabilities of the QBD (Quasi-Birth-Death) and the QBD-M (Quasi-simultaneous-Multiple-Births-Deaths) processes which are continuous time Markov chains (CTMC) occuring rather abundantly in modeling advanced computing systems, emerging telecommunication systems, high speed networks, flexible manufacturing systems and many other discrete event systems. The method has been used for performability computations in very diverse areas of ICT (information and Communication Technologies) with tremendous success. The applicability of this method is growing wider and becoming more crucial than before, with the advent of new paradigms such as computing clouds, server farms, IaaS (Infrastructure as a Service), broad band and high speed networks, mobile communications, Internet based service industry, the method is proven to be applicable and useful in all these. This paper would serve as a useful review of the SEM and as a tutorial for modelers and simulation technologists. © 2012 Springer-Verlag.