International Journal of Stochastic Analysis

International Journal of Stochastic Analysis / 1989 / Article

Open Access

Volume 2 |Article ID 531849 | https://doi.org/10.1155/S1048953389000043

Marcel F. Neuts, Ushio Sumita, Yoshitaka Takahashi, "Renewal characterization of Markov modulated Poisson processes", International Journal of Stochastic Analysis, vol. 2, Article ID 531849, 18 pages, 1989. https://doi.org/10.1155/S1048953389000043

Renewal characterization of Markov modulated Poisson processes

Abstract

A Markov Modulated Poisson Process (MMPP) M(t) defined on a Markov chain J(t) is a pure jump process where jumps of M(t) occur according to a Poisson process with intensity λi whenever the Markov chain J(t) is in state i. M(t) is called strongly renewal (SR) if M(t) is a renewal process for an arbitrary initial probability vector of J(t) with full support on P={i:λi>0}. M(t) is called weakly renewal (WR) if there exists an initial probability vector of J(t) such that the resulting MMPP is a renewal process. The purpose of this paper is to develop general characterization theorems for the class SR and some sufficiency theorems for the class WR in terms of the first passage times of the bivariate Markov chain [J(t),M(t)]. Relevance to the lumpability of J(t) is also studied.

Copyright © 1989 Hindawi Publishing Corporation. 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.


More related articles

 PDF Download Citation Citation
 Order printed copiesOrder
Views194
Downloads410
Citations

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.