Reliability Analysis of a Repairable Single Unit Under General Age-Dependence

The case is considered of a single two-state unit which undergoes a symmetric alternating process of failure and restoration. It is admitted that under general circumstances aging may be described in terms of total elapsed calendar-time, total accumulated on-time (or down-time) and, possibly, upon the number of transitions. Thus it is of utmost importance to be able to calculate and find explicit analytic expressions for the p.d.f. of total on-time (or total down-time) at a given time instant. The solution of the problem relies upon a set of integral equations. This set can be easily reduced to a set of partial differential equations. The solutions appear to be rather simple and manageable for a number of cases of practical interest. Finally, relations are established with the Chapman-Kolmogorov equations describing the non-homogeneous Markov repair process.