A General Formula for Expected Number of Failures

An equipment is considered which is subject to repairable failures on the time interval [0, t]. At the time of each failure, repair is performed which requires length of time $T$ to perform, and as the result of each repair, the virtual age of the equipment decreases by time period u. Minimal repair is obtained by $u=0$ and complete repair / replacement by setting $u=t$ (all age accrued up to the point of failure). A mathematical model was derived for the expected number of failures with partial repairs which require a certain time period to perform. The model derived is a two-dimensional integral equation. It was also shown that in the special cases of minimal repairs and failure replacements the general model could be reduced to the well-known special formulas. A computer method was also suggested for finding solutions and a simple approximating rule was introduced to approximate the solutions. A Weibull numerical example illustrated the theoretical developments.