On Markovian and rational arrival processes. I

This article is the first part of a review carried out within the framework of the RFBR project No. 19-17-50126. The purpose of this review is to get the interested readers familiar with the basics of the theory of Markovian arrival processes to facilitate the application of these models in practice and, if necessary, to study them in detail. In the first part of the review, the properties of general Markovian arrival processes are presented and their relationship with Markov additive processes and Markov renewal processes is shown. In the second part of the review, the important for applications subclasses of Markovian arrival processes, i. e., simple and batch arrival processes of homogeneous and heterogeneous arrivals, are considered. After that, it is shown how the properties of Markovian arrival processes are associated with the product form of stationary distributions of Markov systems. In *conclusion, matrix-exponential distributions and rational arrival processes are discussed that expand the capabilities of Markovian arrival processes for modeling complex systems, while preserving the convenience of analyzing them using computations.