Journal of Applied Mathematics and Stochastic Analysis
Volume 9 (1996), Issue 4, Pages 469-488

Descriptors for point processes based on runs: the Markovian arrival process

Marcel F. Neuts

Department of Systems and Industrial Engineering, The University of Arizona, Tucson 85721-0022, AZ, USA

Received 1 June 1996; Revised 1 September 1996

Copyright © 1996 Marcel F. Neuts. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper is part of a broader investigation of properties of a point process that can be identified by imagining that the process is involved in a competition for the generation of runs of events. The general purpose of that methodology is to quantify the prevalence of gaps and bursts in realizations of the process. The Markovian arrival process (MAP) is highly versatile in qualitative behavior and its analysis is numerically tractable by matrix-analytic methods. It can therefore serve well as a benchmark process in that investigation. In this paper, we consider the MAP and a regular grid competing for runs of lengths at least r1 and r2, respectively. A run of length r in one of the processes is defined as a string of r successive events occurring without an intervening event in the other process.

This article is dedicated to the memory of Roland L. Dobrushin.