# A Compound Measure of Dependability for Continuous-Time Markov Models of Repairable Systems

## Abstract

In the previous chapter, we were looking at the distribution of \({M_{{A_1}}}({t_0})\)
the number of visits during [0, t_{0}], t_{0}> 0, to a subset A_{1} of the state space S by an irreducible finite Markov process Y. In the reliability context, \({M_{{A_1}}}({t_0})\) will become the number of repair periods M_{B} (t_{0}) if we put A_{1}= B, the set of system ‘down’ states. The cumulative distribution function of M_{B}(t_{0}) is of course available from Theorem 5.1. For reliability assessment it is desirable to consider several system characteristics simultaneously; in this chapter, M_{B}(t_{0}) will be supplemented by T_{G}(t_{0}), the total time spent by Y during [0, t_{0}] in the set of ‘good’ states G = A_{2} = S\A_{1}. More precisely, we shall consider here Pr{ T_{G}(t_{0}) > t, M_{B}(t_{0}) ≤ m } for t ∈ (0, t_{0}) and m ∈ {0, 1, … }, i.e., the probability that during [0, t_{0}] the system is ‘up’ for more than t units of time *and* that the number of ‘down’ periods does not exceed m.

### Keywords

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