On the Mean Time between Failures for Repairable Systems

Much of the recent work on modeling repairable systems involves Poisson processes with nonconstant intensity functions, viz, nonhomogeneous Poisson processes. Since times between failures are not identically distributed when the process is nonhomogeneous, it is not clear what concept should take the place of the mean time between failures in assessing the reliability of a repairable system. A number of alternate concepts can be found in the literature. We investigate the relationship between two of the most frequently considered alternatives: the reciprocal of the intensity function, and the mean waiting time from t until the next failure. Theorem 1 states a necessary and sufficient condition for the mean time until the next failure to be asymptotically proportional to the reciprocal of the intensity function. Some examples, including the familiar log-linear and power-intensity processes satisfy this condition. A monotonicity property is also established between these two concepts which could be used to obtain conservative statistical confidence limits for the mean time until the next failure, based on results which are already available for the intensity function of the power-intensity process. However, further study of concepts such as the rate of convergence would be needed in order to determine the degree of approximation of the nominal confidence level to the actual level.