14.1 INTRODUCTION

The reliability bathtub curve (RBTC) is discussed in Section 4.4.3, and a relatively elementary method for its construction is presented there. In this chapter, mathematical models covering the whole RBTC or parts thereof are presented, their characteristics are analyzed, and the parameters of these mathematical models are determined for actual life data.

14.2 MODEL 1

The failure rate function for this model is given by [1]

(14.1)

the pdf by

and the reliability function by

(14.2)

This distribution may be thought of as a truncated extreme-value distribution with a Weibull-type parameterization rather than the usual location-scale parameterization. It can represent increasing, decreasing, and bathtub failure rates as shown in Fig. 14.1. The author [1] has claimed that this distribution has two properties which may cause it to be more applicable in certain cases than other more familiar models. First, the failure rate is exponentially increasing for large T, and second it may be bathtub shaped when ? = 0.5. The above properties indicate that this model may be used in certain cases where the product may be quite reliable and possibly even improve for some period of time and then fail rather quickly after it begins to wear-out or deteriorate. But it can be seen from Fig. 14.1 that when ? = 0.5 the ?( T) curve is not quite bathtub shaped. To estimate the parameters, taking the natural logarithm of both sides of Eq. (14.2) yields

(14.3)

Figure 14.1: Failure rate...