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 |
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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 This kind of reliability test is often called a
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. |
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Use the Test length Table to determine how
long to test |
Test Length Guide Table
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.
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. |
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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. |