Total Time on Test Transforms

Abstract

The total time on test transform is essentially a quantile-based concept developed in the early 1970s. Apart from its applications in reliability problems, it has also been found useful in other areas like stochastic modelling, maintenance scheduling, risk assessment of strategies and energy sales. When several units are tested for studying their life lengths, some of the units would fail while others may survive the test period. The sum of all observed and incomplete life lengths is the total time on test statistic. As the number of units on test tends to infinity, the limit of this statistic is called the total time on test transform (TTT). The definitions and properties of these two concepts are discussed and the functional forms of TTT for several life distributions are presented in Table 5.1. We discuss the Lorenz curve, Bonferroni curve and the Leimkuhler curve which are closely related to the TTT. Identities connecting various curves, characterizations of distributions in terms of these curves and their relationships with various reliability functions are detailed subsequently. In view of the ageing classes in the quantile set-up introduced in Chap. 4, it is possible to characterize these classes in terms of TTT. Accordingly, we give necessary and sufficient conditions for IHR, IHRA, DMRL, NBU, NBUE, HNBUE, NBUHR, NBUHRA, IFHA*t 0, UBAE, DMRLHA, DVRL, and NBU-t 0 classes in terms of the total time on test transform. Another interesting property of the TTT is that it uniquely determines the lifetime distribution. There have been several generalizations of the TTT. We discuss these extensions and their properties, with special reference to the TTT of order n. Relationships between the reliability functions of the baseline model and those of the TTT of order n (which is also a quantile function) are described and then utilized to describe the pattern of ageing of the transformed distributions. Some life distributions are characterized. The discussion of the applications of TTT in modelling includes derivation of the L-moments and other descriptive measures of the original distribution. Some of the areas in reliability engineering that widely use TTT are preventive maintenance, availability, replacement problems and burn-in strategies.