## Characterizing RUL loss rate

by Prof. Pierre Dersin
3rd March, 2021

The “RUL loss rate”, or time derivative or the RUL (remaining useful life), measures the speed at which an asset’s condition degrades and it therefore is getting closer to failure in the absence of any preventive maintenance action. The average RUL loss rate is the derivative of the “mean residual life” (MRL) ; therefore understanding the latter’s properties is of potential interest for maintenance policy optimization. First we study a special class of time-to-failure distributions: those for which the MRL is a linear function of time, i.e. the average RUL loss rate is constant. This class contains very well known special cases: the exponential distribution, for which the MRL is constant (equal to the MTTF), and therefore the RUL loss rate is equal to 0; at, at the other extreme, what we call the Dirac distribution, for which the asset’s lifetime is deterministic, and the loss rate is equal to 1: for every hour that passes, the remaining useful life decreases by one hour. In general, for that family of distributions, which we characterize explicitly, the average RUL loss rate takes a value between 0 and 1. For instance, the uniform distribution is characterized by an average RUL loss rate of one half. It is shown that, for that special family, the average RUL loss rate can be obtained explicitly as a (decreasing) function of the coefficient of variation of the time to failure. A closed-form expression for the confidence interval for the RUL is obtained. Then those results are generalised in two directions:

1) by introducing a nonlinear time transformation that allows for transposing to some classes of time-to-failure distributions ( such as Weibull or gamma) the results obtained in the special case.

2) by considering concurrent degradation modes; for instance, the combination of a mode with constant failure rate (exponential distribution) and a mode with deterministic life time (Dirac distribution);

Implications for maintenance policy are discussed.