Markovian arrivals in stochastic modeling: a survey and some new results (invited article with discussion: Rafael Pérez-Ocón, Miklos Telek and Yiqiang Q. Zhao).

*(English)*Zbl 1209.60001From the introduction: This survey paper is aimed on providing information on Markovian arrival processes, putting emphasis on the discussion of extensions and variants of the batch Markovian arrival processes (BMAPs), as well as on the wide use of this class of processes in applications. Following the leads in this paper and the guidance provided by the bibliographical notes, readers can get access to the background materials where technical details and proofs are available. In Section 2, we first introduce the BMAP and the continuous phase type distribution. A number of important particular cases, the basic properties and descriptors of the BMAP, as well as some applications in queueing, reliability and inventory models are presented in subsequent sections. In Section 3, we consider a number of generalizations and variants of the BMAP including the discrete counterpart, the marked Markovian arrival process, the HetSigma approach, the Markov-additive processes of arrivals and the block-structured state-dependent event (BSDE) approach. The consideration of these extensions and variants enriches the methodology and enhances the versatility of the arrival processes in different directions. Based on the fact that the BSDE approach allows us to deal with modulated non-homogeneous settings, but keeping the dimensionality of the underlying matrices tractable, Section 4 applies this approach to the susceptible-infected-susceptible epidemic model. We conclude the survey with a few bibliographical notes.

Reviewer: Oleg K. Zakusilo (Kyïv)

##### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |

60Kxx | Special processes |

60J28 | Applications of continuous-time Markov processes on discrete state spaces |