A Comparative Numerical Study of the Spectral Theory Approach of Nishimura and the Roots Method Based on the Analysis of BDMMAP/G/1 Queue

Abstract

This paper considers an infinite-buffer queuing system with birth-death modulated Markovian arrival process (BDMMAP) with arbitrary service time distribution. BDMMAP is an excellent representation of the arrival process, where the fractal behavior such as burstiness, correlation, and self-similarity is observed, for example, in ethernet LAN traffic systems. This model was first apprised by Nishimura (2003), and to analyze it, he proposed a twofold spectral theory approach. It seems from the investigations that Nishimura’s approach is tedious and difficult to employ for practical purposes. The objective of this paper is to analyze the same model with an alternative methodology proposed by Chaudhry et al. (2013) (to be referred to as CGG method). The CGG method appears to be rather simple, mathematically tractable, and easy to implement as compared to Nishimura’s approach. The Achilles tendon of the CGG method is the roots of the characteristic equation associated with the probability generating function (pgf) of the queue length distribution, which absolves any eigenvalue algebra and iterative analysis. Both the methods are presented in stepwise manner for easy accessibility, followed by some illustrative examples in accordance with the context.