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Convergence of the number of failed components in a Markov system with nonidentical components. (English) Zbl 0998.60076
Authors’ summary: For most repairable systems, the number $N\left(t\right)$ of failed components at time $t$ appears to be a good quality parameter, so it is critical to study this random function. Here the components are assumed to be independent and both their lifetime and their repair time are exponentially distributed. Moreover, the system is considered new at time 0. Our aim is to compare the random variable $N\left(t\right)$ with $N\left(\infty \right)$, especially in terms of total variation distance. This analysis is used to prove a cut-off phenomenon in the same way as B. Ycart [ESAIM, Probab. Stat. 3, 89-106 (1999; Zbl 0932.60077)] but without the assumption of identical components.
##### MSC:
 60J27 Continuous-time Markov processes on discrete state spaces 60K10 Applications of renewal theory 62N05 Reliability and life testing (survival analysis) 90B25 Reliability, availability, maintenance, inspection, etc. (optimization)
##### Keywords:
Markov system; availability; cut-off; total variation