Characterization of Time to Failure in Prognostics: Brief Tutorial Guide to Prognostics Professionals

As a random variable, the survival time or Time to Failure (TTF) of a certain component or system can be fully characterized by its probability density function (pdf) fT (t) or its Cumulative Distribution Function (CDF) FT (t). Moreover, it might be also identified by transform functions such as the Moment Generating Function (MGF) and the Characteristic Function (CF). In reliability engineering, additional specific equivalent characterizations are used including the reliability function (survival function) which is the Complementary Cumulative Distribution Function (CCDF), and the failure rate (hazard rate), which is the probability density function normalized w.r.t. reliability. In prognostics, a prominent emerging subfield of reliability engineering, the characterizing functions are still supplemented by other specifically tailored ones. Notable among these is the Mean Residual Life (MRL) (also know as the Remaining Useful Life (RUL)). The purpose of this paper is to compile and interrelate the most prominent among these characterizing functions and explore their important properties. The paper points out that there is currently a significant proliferation of characterizing functions emerged in various fields. It shows that, under mild conditions, the product and quotient of two characterizing functions are also characterizing functions. The choice of one characterizing function in a certain application is a matter of convenience and taste. Our survey is far from being a conclusive one as it is intended to be just a brief tutorial guide for prognostics scholars, especially beginners. We had to arbitrarily leave out many of the less known characterizing functions such as the aging intensity function, log-odds rate, and entropy-related functions.