How to Justify Using the Exponential Distribution

Do you check assumptions? Not all assumptions are equal as some may lead you to a costly decision.

We regularly make assumptions about the uniformity of material, the consistency of part to part performance, and many other engineering elements of a design or process. We have to simply the problems we face in order to work out solutions and make decisions.

We rely on models and formulas that simply the world around us. If done well, the assumptions we make help us focus on the significant contributors that will influence how we create designs or select vendors. When the uncertainty that our models and assumptions are correct, we check assumptions. When the cost of making a wrong decision is high, we check assumptions.

In reliability engineering we often confront the assumption that the failures will occur as described by an exponential distribution. In some, I would say rare, situations assuming the exponential distribution is appropriate. In others it is not.

Common Practice

If you are using vendor data in the form of MTBF, MTTF, or failure rate without any other information, you are assuming the exponential distribution.

Many common reliability test design guidelines and standards assume an exponential distribution and may not even state it’s an assumption. Any test which tallies the total time on test certainly is assuming the exponential distribution.

It is a common practice to tally the total time items have been in service or under test and divide by the number of failures that have occurred. Once again, assuming the exponential distribution.

Unfortunately this is a common assumption.

Impact of Assumptions that are wrong

We know that assumptions are both necessary and risky. The risk includes the chance of error, or a poor decision, of increase costs or expense. We also take a personal risk to our credibility and trustworthiness as faulty assumptions come to light.

Using the exponential distribution and associated calculations may lead to either grossly under or over estimating required spares at great cost either way. It may cause us to set aside too much or too little warranty accruals, again either way is troublesome to corporations.

When the exponential distribution is inaccurate it leads to doubt and mistrust of reliability engineering. Once that trust is lost, it is very difficult to be effective as a reliability professional.

Consider the assumptions of your calculations and the supporting information for decisions. When the risks warranty, check the assumptions. Let’s consider couple basic ways to check the exponential distribution assumption.

Engineering Considerations

If the failure mechanism has either a decreasing failure rate over time, or exhibits a wear out pattern, then the assumption of exponential is not valid. Of course there are cases where the change if failure rate over time is insignificant and the exponential would be fine, still you should check.

When purchasing expensive or critical equipment, and both vendors provide only an MTBF value. Ask more questions, like what is the expected failure mechanism(s) and the time to failure pattern. Keep in mind that items like fans, motors, bearings, compressors, all have moving parts and will wear out. New technology and items from a new production line or facility will often have some portion failing early and overall showing a decreasing failure rate.

Think though the product and how similar products have performed in the past. If using the exponential distribution assumption in the past was a mistake it probably is today, also a mistake.

Decision Making Considerations

When it’s important, check the assumptions.

When it’s a new product launch, new market launch, new technology launch, when it’s your career and reputation on the line, check your assumptions.

As a general practice as least ask the questions about the assumptions. What is the supporting evidence that exponential is suitable? What is the impact of getting wrong based on a poor assumptions?

Statistical Tests

When you have data, or have similar enough data (measurements based on the same process, technology, product family, etc.) then you can use statical tools to determine if the assumed exponential distribution is valid or not.

The simplest is to plot the data using Weibull plotting paper or fit the data to a Weibull cumulative density function (CDF) to get an estimate of the beta parameter (slope). If beta is equal or very close to 1.0, then there is evidence that the exponential distribution is suitable. The Weibull distribution is equivalent to the exponential when beta equals 1.

You could also use a software package to fit the data to a Weibull distribution and check the confidence bounds about the beta parameter. If the bound include the value of 1.0, then there is some evidence that the exponential distribution is suitable. Keep in mind that the beta parameter is difficult to accurately estimate and may have large confidence bound even though the actual distribution has a slope significantly other than 1.0.

Another approach is a goodness-of-fit test. Like an hypothesis test, we set the null hypothesis to the data is comes from an exponential distribution, with the alternative hypothesis that the exponential is not a good fit. Fit, here meaning the curve described by the exponential distribution does not adequately describe the actual pattern the data described. Like in linear regression, it’s obvious when you view a plot and the line isn’t near most of the data points.

For the serious reader, check out two goodness-of-fit tests described in the NIST Engineering Statistics Handbook.

1. Chi-square test

2. Anderson-Darling test

With these tests you can evaluate the chance the fit is more likely to have come from a distribution other then the exponential. Which in my experience is most of the time.

Summary

Check your assumptions.

Ask for or run experiments to get data to adequately check your assumptions.

And, if all this is too much work, simply avoid using the exponential distribution.