Assessing Product Reliability
8.4. Reliability Data Analysis
8.4.5. How do you fit system repair rate models?
|This section covers estimating MTBF's and calculating upper and lower confidence bounds||The HPP
or exponential model is widely used
for two reasons:
For the HPP system model, as well as for the non repairable exponential population model, there is only one unknown parameter (or equivalently, the MTBF = 1/). The method used for estimation is the same for the HPP model and for the exponential population model.
|The best estimate of the MTBF is just "Total Time" divided by "Total Failures"||The estimate of the MTBF is
|Confidence Interval Factors multiply the estimated MTBF to obtain lower and upper bounds on the true MTBF||How
To Use the MTBF Confidence Interval Factors
|Confidence bound factor tables for 60, 80, 90 and 95% confidence||
|Formulas for confidence bound factors - even for "zero fails" case||Confidence bounds for the typical Type I censoring
situation are obtained from chi-square distribution tables or programs.
The formula for calculating confidence intervals is:
In this formula, Χ 2α/2,2r is a value that the chi-square statistic with 2r degrees of freedom is less than with probability α/2. In other words, the left-hand tail of the distribution has probability α/2. An even simpler version of this formula can be written using T = the total unit test time:
These bounds are exact for the case of one or more repairable systems on test for a fixed time. They are also exact when non repairable units are on test for a fixed time and failures are replaced with new units during the course of the test. For other situations, they are approximate.
When there are zero failures during the test or operation time, only a (one-sided) MTBF lower bound exists, and this is given by
MTBFlower = T/(-ln)The interpretation of this bound is the following: if the true MTBF were any lower than MTBFlower, we would have seen at least one failure during T hours of test with probability at least 1-α. Therefore, we are 100(1-α) % confident that the true MTBF is not lower than MTBFlower.
|Calculation of confidence limits||
A one-sided, lower 100(1-α/2) % confidence bound for the MTBF is given by
LOWER = 2T/G -1(1-α/2, [2(r+1)])
where T is the total unit or system test time, r is the total number of failures, and G(q,ν) is the Χ 2 distribution function with shape parameter ν.
A one-sided, upper 100(1-α/2) % confidence bound for the MTBF is given by
UPPER = 2T/G -1(α/2, [2r])
The two intervals together, (LOWER, UPPER), are a 100(1-α) % two-sided confidence interval for the true MTBF.
Please use caution when using CDF and inverse CDF functions in commercial software because some functions require left-tail probabilities and others require right-tail probabilities. In the left-tail case, α/2 is used for the upper bound because 2T is being divided by the smaller percentile, and 1-α/2 is used for the lower bound because 2T is divided by the larger percentile. For the right-tail case, 1-α/2 is used to compute the upper bound and α/2 is used to compute the lower bound. Our formulas for G -1(q,ν) assume the inverse CDF function requires left-tail probabilities.
|Example showing how to calculate confidence limits||A system was observed for two calendar months
of operation, during which time it was in operation for 800 hours and had
The MTBF estimate is 800/2 = 400 hours. A 90 %, two-sided confidence interval is given by (400×0.3177, 400×5.6281) = (127, 2251). The same interval could have been obtained using
LOWER = 1600/G -1(0.95,6)Note that 127 is a 95 % lower limit for the true MTBF. The customer is usually only concerned with the lower limit and one-sided lower limits are often used for statements of contractual requirements.
|Zero fails confidence limit calculation||What could we have said if the system had no failures? For a 95 % lower confidence limit on the true MTBF, we either use the 0 failures factor from the 90 % confidence interval table and calculate 800 × 0.3338 = 267, or we use T/(ln α) = 800/(ln 0.05) = 267.|