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, t₀], t₀> 0, to a subset A₁ 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 (t₀) if we put A₁= B, the set of system ‘down’ states. The cumulative distribution function of M_B(t₀) 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(t₀) will be supplemented by T_G(t₀), the total time spent by Y during [0, t₀] in the set of ‘good’ states G = A₂ = S\A₁. More precisely, we shall consider here Pr{ T_G(t₀) > t, M_B(t₀) ≤ m } for t ∈ (0, t₀) and m ∈ {0, 1, … }, i.e., the probability that during [0, t₀] the system is ‘up’ for more than t units of time and that the number of ‘down’ periods does not exceed m.