Statistics for Maintenance Modelling: Continuous Lifetime Distributions 2

# Summary¤

Although particular lifetime distributions are sometimes chosen on theoretical grounds, in the more uncertain environment away from the laboratory, the complex interactions of the various factors contributing to failure means that choosing a distribution based on its shape is usually more appropriate. In this case, the number of parameters a model has and the ease of manipulation of the probability density, reliability and hazard functions are also factors - although the support provided by Excel makes this less important than it was. The probability density functions for the different distributions considered in this and the previous session, often only differ by the size of the right hand tail - it is usually the hazard functions that are more distinctive.

The main characteristics of the distributions considered in this session are:

- Normal - symmetric bell shaped probability density function, increasing hazard function (and increasing rate of change of the hazard function).
- Lognormal - the probability density function has a range of shapes from the normal's bell shape to a no left tail/long right tail shape.
- Gamma - models the situation where several individual failures have to occur before there is a system failure. For N > 1, the hazard function is increasing but the rate of increase is decreasing.
- Logistic - this leads to an s shaped increasing hazard function. However the use of the logistic distribution for reliability modelling is relatively rare, more usually the loglogistic distribution is used.
- Loglogistic - the loglogistic's hazard function is either decreasing or peaked depending on whether 1 or not.
- Pareto - this has a reciprocal hazard function and is suitable for modelling situations when there is no hard right hand cut off.