Jordan Representation of the Markovian Arrival Process of Order 2 and Moments Fitting

The Markovian arrival process can be represented in different ways. The most intuitive way is the Markovian representation with two transition rate matrices (D0, D1). Markovian arrival processes can be represented by the Laplace transform or the moments of the stationary intervals. In this study, we consider the Jordan representation specifically for the Markovian arrival process of order 2. The Jordan representation is also given in two matrices (E, R). However, the Jordan representation is minimal, whereas the Markovian representation is not. We present closed-form Jordan representations for the Markovian arrival process of order 2 in terms of moments and parameters of other representations. The transformation between the Jordan and other representations including the Markovian representation, the Laplace transform, and characteristic polynomial coefficients is also presented.