Abstract
We propose new easily computable bounds for different quantities which are solutions of Markov renewal equations linked to some continuous-time semi-Markov process (SMP). The idea is to construct two new discrete-time SMP which bound the initial SMP in some sense. The solution of a Markov renewal equation linked to the initial SMP is then shown to be bounded by solutions of Markov renewal equations linked to the two discrete time SMP. Also, the bounds are proved to converge. To illustrate the results, numerical bounds are provided for two quantities from the reliability field: mean sojourn times and probability transitions.
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Mercier, S. Numerical Bounds for Semi-Markovian Quantities and Application to Reliability. Methodol Comput Appl Probab 10, 179–198 (2008). https://doi.org/10.1007/s11009-007-9035-5
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DOI: https://doi.org/10.1007/s11009-007-9035-5
Keywords
- Continuous and discrete time homogeneous semi-Markov processes
- Markov renewal equations
- Numerical algorithms