Counting Process of MAP(3)s and Moment Fittings

Moments of stationary intervals and those of the counting process can be used for moment fittings of the point processes. As for the Markovian arrival processes, the moments of stationary intervals are given as a polynomial function of parameters whereas the moments of the counting process involve exponential terms. Therefore, moment fittings are more complicated with the counting process than with stationary intervals. However, in queueing network analysis, cross-correlation between point processes can be modeled more conveniently with counting processes than with stationary intervals. A Laplace-Stieltjies transform of the stationary intervals of MAP(3)s is recently proposed in minimal number of parameters. We extend the results and present the Laplace transform of the counting process of MAP(3)s. We also show how moments of the counting process such as index of dispersions for counts, IDC, and limiting IDC can be used for moment fittings. Examples of exact MAP(3) moment fittings are also presented on the basis of moments of stationary intervals and those of the counting process.