The number of working periods of a repairable Markov system during a finite time interval

In reliability analysis, continuous parameter homogeneous irreducible finite Markov processes are used to model repairable systems with time-independent transition rates between individual states. The state space is then partitioned into the set of up states and the set of down states. The number of completed repair events during a finite time interval is an important (undiscounted) cost measure for such a system; it can be expressed in terms of the number of working periods during the same time interval. This paper derives a closed-form expression for the PMF of this latter quantity. The tool used is a recent result on the joint distribution of sojourn times in finite Markov processes. The MatLab implementation of the Markov model of a 2-unit parallel power transmission system is used to demonstrate the utility of the formula. The quantity examined is the number of completed repairs during a finite time interval. The method is viable in this case whereas the more usual randomization technique is not