8. Assessing Product Reliability
8.3. Reliability Data Collection
8.3.1. How do you plan a reliability assessment test?

## 8.3.1.1. Exponential life distribution (or HPP model) tests

Using an exponential (or HPP) model to test whether a system meets its MTBF requirement is common in industry

Exponential tests are common in industry for verifying that tools, systems or equipment are meeting their reliability requirements for Mean Time Between Failure (MTBF). The assumption is that the system has a constant failure (or repair) rate, which is the reciprocal of the MTBF. The waiting time between failures follows the exponential distribution model.

A typical test situation might be: a new complex piece of equipment or tool is installed in a factory and monitored closely for a period of several weeks to several months. If it has no more than a pre-specified number of failures during that period, the equipment "passes" its reliability acceptance test.

This kind of reliability test is often called a Qualification Test or a Product Reliability Acceptance Test (PRAT). Contractual penalties may be invoked if the equipment fails the test. Everything is pegged to meeting a customer MTBF requirement at a specified confidence level.

How Long Must You Test A Piece of Equipment or a System In order to Assure a Specified MTBF at a Given Confidence?

You start with a given MTBF objective, say $$M$$, and a confidence level, say $$100(1-\alpha)$$. You need one more piece of information to determine the test length: how many fails do you want to allow and still "pass" the equipment? The more fails allowed, the longer the test required. However, a longer test allowing more failures has the desirable feature of making it less likely a good piece of equipment will be rejected because of random "bad luck" during the test period.

The recommended procedure is to iterate on $$r$$ = the number of allowable fails until a larger $$r$$ would require an unacceptable test length. For any choice of $$r$$, the corresponding test length is quickly calculated by multiplying $$M$$ (the objective) by the factor in the table below corresponding to the $$r$$-th row and the desired confidence level column.

For example, to confirm a 200-hour MTBF objective at 90 % confidence, allowing up to 4 failures on the test, the test length must be 200 × 7.99 = 1598 hours. If this is unacceptably long, try allowing only 3 fails for a test length of 200 × 6.68 = 1336 hours. The shortest test would allow no fails and last 200 × 2.3 = 460 hours. All these tests guarantee a 200-hour MTBF at 90 % confidence, when the equipment passes. However, the shorter test are much less "fair" to the supplier in that they have a large chance of failing a marginally acceptable piece of equipment.

Use the Test length Table to determine how long to test
Test Length Guide Table
 NUMBER OF FAILURES ALLOWED FACTOR FOR GIVEN CONFIDENCE LEVELS $$r$$ 50% 60% 75% 80% 90% 95% 0 .693 .916 1.39 1.61 2.30 3.00 1 1.68 2.02 2.69 2.99 3.89 4.74 2 2.67 3.11 3.92 4.28 5.32 6.30 3 3.67 4.18 5.11 5.52 6.68 7.75 4 4.67 5.24 6.27 6.72 7.99 9.15 5 5.67 6.29 7.42 7.90 9.28 10.51 6 6.67 7.35 8.56 9.07 10.53 11.84 7 7.67 8.38 9.68 10.23 11.77 13.15 8 8.67 9.43 10.80 11.38 13.00 14.43 9 9.67 10.48 11.91 12.52 14.21 15.70 10 10.67 11.52 13.02 13.65 15.40 16.96 15 15.67 16.69 18.48 19.23 21.29 23.10 20 20.68 21.84 23.88 24.73 27.05 29.06

The formula to calculate the factors in the table is the following. $$\mbox{FAC } = 0.5 \, \chi_{\alpha; \, 2(r+1)}^2$$ with $$\chi_{\alpha; \, 2(r+1)}^2$$ denoting the upper $$100(1-\alpha)$$ percentile of the Chi-Square distribution with $$2(r+1)$$ degrees of freedom.

Example: A new factory tool must meet a 400-hour MTBF requirement at 80 % confidence. You have up to two months of 3-shift operation to decide whether the tool is acceptable. What is a good test plan?

Two months of around-the-clock operation, with some time off for maintenance and repairs, amounts to a maximum of about 1300 hours. The 80 % confidence factor for $$r$$ = 1 is 2.99, so a test of 400 × 2.99 = about 1200 hours (with up to 1 fail allowed) is the best that can be done.

Shorten required test times by testing more than one system NOTE: Exponential test times can be shortened significantly if several similar tools or systems can be put on test at the same time. Test time means the same as "tool hours" and one tool operating for 1000 hours is equivalent (as far as the exponential model is concerned) to 2 tools operating for 500 hours each, or 10 tools operating for 100 hours each. Just count all the fails from all the tools and the sum of the test hours from all the tools.