A minimal representation of Markov arrival processes and a moments matching method

The paper investigates the problem of minimal representation of Markov arrival processes of order nn (MAP(nn)). The minimal representation of MAPs is crucial for developing effective fitting methods. It seems that all existing MAP fitting methods are based on the D0, D1 representation which is known to be redundant. We present the minimal number of parameters to define a MAP(nn) and provide a numerical moments-matching method based on a minimal representation.The discussion starts with a characterization of phase type (PH) distributions and then the analysis of MAPs follows a similar pattern. This characterization contains essential results on the identity of stationary behaviour of MAPs and on the number of parameters required to describe the stationary behaviour.The proposed moments matching method is also applicable for PH distributions. In this case it is a unique method that fits a general PH distribution of order nn based on 2n−12n−1 parameters.