The characteristic polynomial and the Laplace representations of MAP(2)s

For the Markovian arrival process of order n, MAP(n), the matrix-based representation is well known. However, the redundancy of parameters complicates the description of moments and moment fittings. In this article, we propose more fundamental representations for MAP(2)s based on characteristic polynomials and Laplace–Stieljes transform. These minimal, unique, and real-valued representations provide a unified framework for moment fittings. That is, moments of stationary intervals and the counting process can be written in four parameters. We present six different ways of exact moment fitting for MAP(2)s. The transformation from our minimal representations to Markovian representation in six rate parameters is also given in closed form by exploiting canonical forms of MAP(2)s.