Availability of continuous service and computing long-run MTBF and reliability for Markov systems (Q2748554)

This largely expository article discusses repairable systems modelled by a continuous-time Markov chain \(X\) whose finite state-space is partitioned into an operating set \(O\) and a tailed set \(F\). Alongside a review of standard theory, focusing on the respective sojourn times in \(O\) and \(F\) at steady state, two availability measures relevant to safety critical applications are studied: (a) the availability of continuous service, i.e. \(P(X(s)\in O\), \(t\leq s\leq t+\Delta t\mid X(0)=i)\), (b) the availability of reliable service, i.e. \(P(X(t)\in O')\), where \(O'\) comprises operating states whose reliability function at \(\Delta t\) exceeds a fixed threshold. With the aid of results from \textit{J. Mi} [Prob. Eng. Inf. Sci. 13, 359--375 (1999; Zbl 0978.60099)], the measure (a) is used to derive formulas for the long-run MTBF and MTTR.