In such cases you define a confidence level between 0 and 100 and then compute the lower bound chisquare value using two degrees of freedom The equation for calculating MTTF for zero failures is given by However the results can be simplified using t ID: 25631 Download Pdf

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In such cases you define a confidence level between 0 and 100 and then compute the lower bound chisquare value using two degrees of freedom The equation for calculating MTTF for zero failures is given by However the results can be simplified using t

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Technical Brief from Relex Software Corporation Page 1 of 4 Calculating MTTF When You Have Zero Failures How do you calculate the MTTF (Mean Time To Failure) when you have zero failures? If you use the stan- dard equation for MTTF, which is the ratio of total testing time and the number of failures, you get an answer of infinity. In such cases, you define a confidence level between 0 and 100 and then compute the lower bound chi-square value using two degrees of freedom. The equation for calculating MTTF for zero failures is given by: However, the results can be simplified

using the relationship between the exponential and Chi-square distributions. Chi-Square Fundamentals The Chi-square ( ) distribution is derived from the Normal distribution. It is the distribution of a sum of squares of a number ( ) of independent standard normal variables (a mean of 0 and a variance of 1). The probability density function (PDF), , of the distribution is: ........................................................................................ (1) Here, is a positive integer, and is the gamma function. It should be noted that the shape parameter is usually known as the degrees

of freedom of the distribution. This distribution arises in many areas of statistics, including reliability applications. Particularly, it is used for assessing the goodness-of-fit of models and tests of significance (trend analysis and independency of variables for example), finding confi- dence intervals, etc. We can find closed-form expressions for the cumulative distribution function (CDF) of the distribution when the degrees of freedom ( ) is even. When , then it reduces to the exponential distribution with rate parameter of (i.e., a mean of 2). This relationship can be used to find the

confidence intervals of MTTF and the failure rate when there are 0 failures. Confidence Intervals The Chi-square ( ) distribution can be used to find the confidence intervals on the failure rate ( ) and the Mean Time to Failure (MTTF) of the exponential distributions. The distribution can also be applied to the Weibull distribution when the shape parameter ( ) is known (WeiBayes method). This feature focuses on confidence intervals when the underlying failure distribution is exponential. esult Total Equivalent Time Chi-Square Value for (1 Confidence Leve l)

--------------------------------------------------------------------------------------------------------------------- --- esult Total Equivalent Time Natural Log of (1 - Confidence Leve l) ------------------------------------------------------------------------------------------------------- --- fx () () () ----------------------------------- ------------ , x 12

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Technical Brief from Relex Software Corporation Page 2 of 4 The estimate for failure rate ( ) is calculated as the ratio of the number of failures and total testing time . Alternately, the estimate of MTTF is

calculated as the ratio of the total testing time and the number of fail- ures . Even though it is the best available estimate, it by itself gives no measure of precision or risk. If the critical decisions to be made depend on the true value of , we should ask the following important ques- tions: Can the true be as high as ? Are we confident it is no worse than ? And, how much better than might the true be? When critical decisions are being made, confidence intervals should be used. For type I (time-censored) or type II (failure-censored at the failure) data, factors based on the

distribution can be derived and used as multipliers of to obtain the upper and lower ends of the confidence interval. The remarkable thing about these multipliers is that they depend only on the number of failures observed during the test (or in the field). Suppose we want a confidence interval for , where is the risk we are willing to accept that our interval does not contain the true value of . For example, corresponds to a 90% interval. We can calculate a lower bound for and an upper bound for . These two numbers give the desired confidence interval. When , this method sets a lower 5% bound

and an upper 95% bound, having between them a 90% chance of containing the true . However, in some applications, we may be interested in only one-sided bounds. Because MTTF is the reciprocal of the failure rate, the lower and upper bounds for MTTF can be obtained by using the upper and lower bounds of the failure rate: ........................................................................................................... (2) ............................................................................................................. (3) Zero Failure Calculations If no failure is observed

during a test on units over a duration of , then or . This cannot realistically be true because we may have had a small or restricted test. Moreover, this estimate does not take into account the number on test and the test duration. Had the test been continued, it is highly likely that a failure would eventually have occurred. An upper level estimate for both one-sided and two-sided confidence limits can be obtained with using the following equation: ............................................................................... (4) However, the lower limit for the two-sided confidence limit

cannot be obtained with because it is defined only for . It is possible to relax this limitation by conservatively assuming that a failure occurs in the very next instant. Then, can be used to evaluate the lower two-sided confidence limit. This conservative modification, although sometimes used to allow a complete statistical analysis, lacks firm statistical basis. .2 100 1 () 0.1 100 () 100 1 () 0.1 TTF lower bound upper boun ----------------------------- --- TTF upper bound lower boun ---------------------------- --- 0nT == TTF nT 0 == r0 2r 100 () 2r ---------------------------------- 2r 1

() 100 1 () () 2r -------------------------------------------------------- r0 r0 r1

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Technical Brief from Relex Software Corporation Page 3 of 4 Therefore, when there are no failures, we can find only an upper confidence limit for . Alterna- tively, we can find only a lower confidence limit for MTTF. The probability statement corre- sponding to the upper confidence limit on is as follows (based on the above approximation): ............................................................................................................... (5)

............................................................................................................................... ................... (6) Therefore, the upper confidence limit for is given by: ......................................................................................................................... (7) However, as mentioned in Chi-Square Fundamentals, a Chi-square ( ) distribution with two degrees of freedom is equivalent to the exponential distribution with a mean of 2. Therefore, after simplifications, the confidence limit for is given by:

............................................................................................................................... .(8) If there is only one unit being tested, then the upper confidence limit on is: ............................................................................................................................... .(9) Similarly, the lower confidence limit for MTTF is given by: ............................................................................................................................ (10 If there is only one unit being tested, then the lower confidence

limit on MTTF is: .............................................................................................................................(1 1) As shown in the above equations, Chi-square tables are not needed for evaluating these confidence limits. The simple formula obtained using the exponential distribution can be used instead. The 50% zero failure estimate is often used as a point estimate for . This should be interpreted very care- fully. It is a value of that makes the likelihood of obtaining zero failures in the given test similar to the chance of having a coin that has been

flipped landing heads side up. We are not really 50% confident of anything; we have just picked a that will produce zero failures 50% of the time. The following table presents the lower bounds for MTTF at various confidence levels for a one-unit test when no failures are observed during the testing period ( ). 100 1 () 100 1 () 21001 () () ------------------------------------- --- --- 100 1 () 100 1 () 21001 () 2nT -------------------------- --- 100 1 () () ln nT ------------- --- 100 1 () () ln ------------- --- 100 1 () TTF 100 nT () ln ------------- --- TTF 100 () ln ------------- ---

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Technical Brief from Relex Software Corporation Page 4 of 4 The Relex Reliability Prediction and Reliability Block Diagram (RBD) modules and the Relex FRACAS Management System all support and automate the use of confidence intervals for finding MTTF even when zero failures exist. If you would like additional information about how these calculations are imple- mented in Relex software products, please email info@relexsoftware.com Confidence Level Lower Bound on MTTF 1 99.4992 5 19.4957 10 9.4912 25 3.4761 50 1.4427 75 0.7213 90 0.4343 95 0.3338 99 0.2171

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