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Analysis of Markov renewal shock models
- Part of
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- Journal:
- Journal of Applied Probability / Volume 32 / Issue 3 / September 1995
- Published online by Cambridge University Press:
- 14 July 2016, pp. 821-831
- Print publication:
- September 1995
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- Article
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A trivariate stochastic process
is considered, describing a sequence of random shocks {Xn} at random intervals {Yn} with random system state {Jn}. The triviariate stochastic process satisfies a Markov renewal property in that the magnitude of shocks and the shock intervals are correlated pairwise and the corresponding joint distributions are affected by transitions of the system state which occur after each shock according to a Markov chain. Of interest is a system lifetime terminated whenever a shock magnitude exceeds a prespecified level z. The distribution of system lifetime, its moments and a related exponential limit theorem are derived explicitly. A similar transform analysis is conducted for a second type of system lifetime with system failures caused by the cumulative magnitude of shocks exceeding a fixed level z.
is considered, describing a sequence of random shocks {