Quantile-Based Reliability Concepts

There are several functions in reliability theory used to describe the patterns of failure in different mechanisms or systems as a function of age. The functional forms of many of these concepts characterize the life distribution and therefore enable the identification of the appropriate model. In this chapter, we discuss these basic concepts, first using the distribution function approach and then introduce their analogues in terms of quantile functions. Various important concepts introduced here include the hazard rate, mean residual life, variance residual life, percentile residual life, coefficient of variation of residual life, and their counterparts in reversed time. The expressions for all these functions for standard life distributions are given in the form of tables to facilitate easy reference. Formulas for the determination of the distribution from these functions, their characteristic properties and characterization theorems for different life distributions by relationships between various functions are reviewed. Many of the quantile functions in the literature do not have closed-form expressions for their distributions, and they have to be evaluated numerically. This renders analytic manipulation of these reliability functions based on the distribution function rather difficult. Accordingly, we introduce equivalent definitions and properties of the traditional concepts in terms of quantile functions. This leads to hazard quantile function, mean residual quantile function and so on. The interrelationships between these functions are presented along with characterizations. Various examples given in the sequel illustrate how the quantile based reliability functions can be found directly from the quantile functions of life distributions. Expressions of such functions for standard life distributions can also be read from the tables provided in each case.