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We study moment properties of counting processes for certain subclasses of Markovian arrival processes (MAPs). We focus on the class of Markovian transition counting processes (MTCPs) which counts all the transitions of the background Markov chain. While Markov modulated Poisson processes (MMPPs) are often used to model bursty traffic, MTCPs deliver similar modelling capability and reduce the mathematical and computational complexity. To support the suggested use of MTCPs as alternatives to MMPPs, we establish an equivalence in terms of first and second moments of counts with a sub-class of MMPPs which we refer to as slow MMPPs (SMMPPs). For this we construct matched MTCPs (MMTCPs), where an MMTCP is parameterized with the same basic parameters as an SMMPP. In the course of this study, we summarize and derive general results about moment matrices of MAPs by using deviation matrix representations in novel forms that have not appeared in the existing literature.
Asanjarani et al. (Thu,) studied this question.