A recursive framework for time-dependent characteristics of tested and maintained standby units with arbitrary distributions for failures and repairs

The time-dependent unavailability and the failure and repair intensities of periodically tested aging standby system components are solved with recursive equations under three categories of testing and repair policies. In these policies, tests or repairs or both can be minimal or perfect renewals. Arbitrary distributions are allowed to times to failure as well as to repair and renewal durations. Major preventive maintenance is done periodically or at random times, e.g. when a true demand occurs. In the third option process renewal is done if a true demand occurs or when a certain mission time has expired since the previous maintenance, whichever occurs first. A practical feature is that even if a repair can renew the unit, it does not generally renew the alternating process. The formalism updates and extends earlier results by using a special backward-renewal equation method, by allowing scheduled tests not limited to equal intervals and accepting arbitrary distributions and multiple failure types and causes, including failures caused by tests, human errors and true demands. Explicit solutions are produced to integral equations associated with an age-renewal maintenance policy.