On Cold Standby Repairable Systems with a Random Change Point in Failure and/or Repair Times

This paper is devoted to studying two repairable systems consisting of one active and one standby component for both cases, when the failure times of the units have discrete and continuous distributions. In Model I the failure times of the units do not have the same distribution, but it is assumed that a change occurs in the distribution of the failure times due to an environmental effect and hence this distribution changes after a random number of failures. Similarly, in Model II, it is assumed that the repair times of the units do not have the same distribution, but this changes after a random number of repairs. The two systems under consideration fail if either a damage size upon the failure of the active component is larger than a positive threshold (the repair limit) or the repair time of the failed unit exceeds the lifetime of the active unit, whichever happens first. The lifetimes of these systems are represented as compound random variables. For the continuous (discrete) time case, the Laplace transform (the probability generating function) of the system’s lifetime is obtained as well as its Mean Time to Failure. By assuming that the random change point has a discrete phase-type distribution, several explicit results for reliability characteristics of the systems are obtained. Under particular cases for the distributions of damages, of failure times and of repair times, it is shown that the Laplace transform (probability generating function) of system’s lifetime is rational, and the reliability evaluation of the systems are performed via well-known distributional properties of the matrix-exponential (matrix-geometric) distributions. To illustrate our results, some numerical examples, both for the discrete and the continuous case, are given. Finally, using the proposed model, an actuarial extension for the classical risk model is also discussed.