Remaining useful life in theory and practice

Remaining useful life (RUL) is nowadays in fashion, both in theory and applications. Engineers use it mostly when they have to decide whether to do maintenance, or to delay it, due to production requirements. Most often, it is assumed that in later life of an equipment (in wear-out period), the hazard function is increasing, and then the expected RUL, mu(t), is decreasing. We noticed that the standard deviation of RUL, sigma(t), is also decreasing, which was expected and known, but that the ratio sigma(t)/mu(t) is also increasing, which was a surprise. Initiated by this observation, we have proved that under some general conditions, which include Weibull distribution with shape parameter > 1, this is indeed the case. Even more, we have proved that the limiting distribution of standardized RUL is exponential, so that the variability of RUL is relatively large. The role of condition monitoring in the evaluation of RUL is discussed. Various models for RUL depending on covariates are considered.