A new approach to the cumulative operational time for semi-Markov models of repairable systems

Semi-Markovian reliability models of repairable systems are considered here, whose state space is partitioned into the set of up-, and the set of down-states. The cdf of the cumulative operational time over a finite time interval [0,t] is represented in terms of the work-mission-availability and a system of integral equations is shown to hold for the latter. The equations are of the convolution type which then allows closed form expressions to be established for both the work-mission-availability and the cdf of the cumulative operational time. The semi-Markov model of a two-unit system is examined numerically by solving the resulting integral equations with the two-point trapezoidal rule. The results are compared with those from simulation and an earlier solution scheme based on (non-convolution) integral equations for the cdf of the cumulative operational time.