Interpreting Variable Failure Rate Data

Abstract

The bathtub curve in figure 3.2 showed that, in addition to random failures, there are distributions of increasing and decreasing failure rate. In these variable failure rate cases it is of little value to consider the actual failure rate since only Reliability and MTBF are meaningful. In chapter 3 we saw that:

$$R\left( t \right) = \exp \left[ { - \int_0^t {\lambda \left( t \right)dt} } \right]$$

Since the relationship between failure rate and time takes many forms, and depends on the device in question, the integral cannot be evaluated for the general case. Even if the variation of failure rate with time were known it might well be of such a complicated nature that the integration would prove far from simple. In practice it is found that the relationship can usually be described by the following three-parameter distribution known as the Weibull Distribution.

$$R\left( t \right) = \exp \left[ { - {{\left( {\frac{{t - \gamma }} {\eta }} \right)}^\beta }} \right]$$

In the constant failure rate case it was seen that statements of MTBF and reliability could be made from the failure rate parameter which completely defined the distribution. In the Weibull case the reliability function requires three parameters (γ, β, η). They do not have a physical meaning as does failure rate and must be treated as merely numbers which allow us to compute reliability and MTBF. In the special case of γ = 0 and β = 1 the expression reduces to the simple exponential case with η = MTBF. This is slightly misleading because in the general case η is not equal to MTBF.