Chapter List

6.1 Introduction

The notion of failure rate is crucial in reliability and survival analysis. However, obtaining the failure rate in many practical situations is often not so simple, as the structure of the system to be considered, for instance, can be rather complex, or the process of the failure development cannot be described in a simple way. In these cases a proper model can help a lot in deriving reliability characteristics. In this chapter we consider several models that can be effectively used for deriving and analyzing the corresponding failure rate, and eventually the survival function. The general approach developed eventually boils down to constructing an equivalent failure rate for different settings (survival in the plane, multiple availability on demand, mixtures of distributions). It can be used for other applications as well.

Denote by T ?0 a lifetime random variable and assume that the corresponding cumulative distribution function (cdf) F(t) is absolutely continuous. Then the following exponential formula exists:

where ? (t), t ?0, is the failure rate. In many instances a conventional statistical analysis of the overall random variable T presents certain difficulties, as the corresponding data can be scarce ( e.g. the failure times of the highly reliable...