A semi-Markovian model allowing for inhomogenities with respect to process time

A semi-Markovian model allowing for inhomogenities with respect to process time A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q ij ( t ), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ ij ( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i . Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N 2 integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Reliability Engineering and System Safety Elsevier

A semi-Markovian model allowing for inhomogenities with respect to process time

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Publisher
Elsevier
Copyright
Copyright © 2000 Elsevier Science Ltd
ISSN
0951-8320
eISSN
1879-0836
DOI
10.1016/S0951-8320(00)00044-2
Publisher site
See Article on Publisher Site

Abstract

A non-homogeneous semi-Markov process is considered as an approach to model reliability characteristics of components or small systems with complex test resp. maintenance strategies. This approach generalizes previous results achieved for ordinary inhomogeneous Markov processes. This paper focuses on the following topics to make the application of semi-Markovian models feasible: rather than transition probabilities Q ij ( t ), which are used in normal mathematical text books to define semi-Markov processes, transition rates λ ij ( ) are used, as is usual for ordinary Markov processes. These transition rates may depend on two types of time in general: on process time and on sojourn time in state i . Such transition rates can be followed from failure and repair rates of the underlying technical components, in much the same way, as this is known for ordinary Markov processes. Rather than immediately starting to solve the Kolmogorov equations, which would result in N 2 integral equations, a system of N integral equations for frequency densities of reaching states is considered. Once this system is solved, the initial value problem for state probabilities can be solved by straightforward integration. An example involving 14 states has been solved as an illustration using the approach.

Journal

Reliability Engineering and System SafetyElsevier

Published: Oct 1, 2000

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