Semi-Markov failure rates processes
type: Item
instance of: scholarly article
title: Semi-Markov failure rates processes
zbMATH Open document ID: 1227.60104
author: Franciszek Grabski
published in: Applied Mathematics and Computation
publication date: 2011-07-22T00:00:00Z
review text: Let \(\{\lambda(u):u\geq 0\}\) be a semi-Markov process defined by a Markov renewal kernel. The author studies the survival function \(R\) defined as the expectation of \(\exp \left(-\int_0^t\lambda(u)du\right)\), for \(t \geq 0\). Note that, \(R(t)\) is considered to be the reliability of an object with a random failure rate. In the main theorem of the paper, it is proved that the associated conditional reliability functions satisfy a system of specific renewal equations. The solution of this system is unique in the class of measurable and uniformly bounded functions. This theorem is highlighted in the case of a finite or countable state space. Also, the particular case of a Furry-Yule process is treated. In this way, the author extends some results due to \textit{I. Kopocińska} and \textit{B. Kopociński} [Zastosow. Mat. 17, 5--14 (1980; Zbl 0471.60089)] and \textit{F. Grabski} [Appl. Math. Comput. 135, No.~1, 1--16 (2003; Zbl 1021.60070)].
reviewed by: Eugen Paltanea
Mathematics Subject Classification ID: 60K15
Mathematics Subject Classification ID: 60K20
zbMATH DE Number: 5931348
zbMATH Keywords: failure rate
zbMATH Keywords: Markov renewal process
zbMATH Keywords: semi-Markov process
zbMATH Keywords: Furry-Yule process
MaRDI profile type: Publication
full work available at URL: j.amc.2011.04.055
OpenAlex ID: W2074277294
cites work: Markov renewal theory
cites work: The reliability of an object with semi-Markov failure rate
cites work: Q3924956
cites work: Q3334750
cites work: Q4147381
cites work: Q4746614
cites work: Semi-Markov processes and reliability
cites work: Q4494276
DOI: 10.1016/J.AMC.2011.04.055
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description: scientific article; zbMATH DE number 5931348