Chapter 2 Basic Reliability Concepts

The basic reliability functions that can be used to model lifetime data and explain the failure patterns are the topics of discussion in this chapter. We begin with the conventional hazard rate defined as the ratio of the probability mass function to the survival function. This is followed up by an alternative hazard function introduced to overcome certain limitations of the conventional rate. Properties of both these hazard rates and their interrelationships are discussed. Then, the concept of residual life distribution and its characteristics like the mean, variance and moments are discussed. Various identities connecting the hazard rates, mean residual life function and various residual functions are derived, and some special relationships are employed for characterizing discrete life distributions. We then work out two problems to demonstrate how the characteristic properties enable the identification of the life distribution. Also, the role of partial moments in the context of reliability modelling is examined. Various concepts in reversed time has been of interest in reliability and related areas. Accordingly, reversed hazard rate, reversed residual mean life and reversed variance life are all defined and their interrelationships and characterizations based on them are reviewed. In the case of finite range distributions, it is shown that all the concepts in reversed time can assume constant values and these are related to the reversed lack of memory property characteristic of the reversed geometric law. Along with the traditional reliability functions, the notion of odds functions can also play a role in reliability modelling and analysis. We explain the relevant results in this connection. The log-odds functions and rates and their applications are also studied. Mixture distributions and weighted distributions also appear as models in certain situations, and the hazard rates and reversed hazard rates for these two cases are derived and are subsequently used to characterize certain lifetime distributions.