Confidence Bounds for Repairable Systems Analysis

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In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for Repairable Systems Analysis. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. 

Beta

Fisher Matrix Bounds

The parameter must be positive, thus is approximately treated as being normally distributed.

All variance can be calculated using the Fisher Information Matrix. is the natural log-likelihood function.

Crow Bounds

Calculate the conditional maximum likelihood estimate of  :

The Crow 2-sided 100% confidence bounds on are:

Lambda

Fisher Matrix Bounds

The parameter must be positive, thus is approximately treated as being normally distributed. These bounds are based on:

The approximate confidence bounds on are given as:

where .

The variance calculation is the same the equations given in the confidence bounds on Beta.

Crow Bounds

Failure Terminated

The confidence bounds on for failure terminated data are calculated using:

where:

  • = total number of failures.
  • = number of systems.
  • = end time for the qth system.

Time Terminated

The confidence bounds on for time terminated data are calculated using:

where:

  • = total number of failures.
  • = number of systems.
  • = end time for the qth system.

Cumulative Number of Failures

Fisher Matrix Bounds

The cumulative number of failures, . must be positive, thus is approximately treated as being normally distributed.

where:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on the cumulative number of failures are given by:

where:

  • = total number of failures across all systems. This is not the number of failures up to time .
  • = time at which calculations are being conducted.

Cumulative Failure Intensity

Fisher Matrix Bounds

The cumulative failure intensity, must be positive, thus is approximately treated as being normally distributed.

The approximate confidence bounds on the cumulative failure intensity are then estimated using:

where:

and:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on the cumulative failure intensity are given by:

where:

  • = total number of failures across all systems. This is not the number of failures up to time .
  • = time at which calculations are being conducted.

Cumulative MTBF

Fisher Matrix Bounds

The cumulative MTBF, . must be positive, thus is approximately treated as being normally distributed.

The approximate confidence bounds on the cumulative MTBF are then estimated from:

where:

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on the cumulative MTBF are given by:

where and are calculated using the process for the confidence bounds on cumulative failure intensity.

Instantaneous MTBF

Fisher Matrix Bounds

The instantaneous MTBF, . must be positive, thus is approximately treated as being normally distributed.

The approximate confidence bounds on the instantaneous MTBF are then estimated from:

where:

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

Crow Bounds

Failure Terminated

For failure terminated data and the 2-sided confidence bounds on instantaneous MTBF , consider the following equation:

Find the values and by finding the solution and for the lower and upper bounds, respectively.

If using the biased parameters, and , then the upper and lower confidence bounds are:

where .

If using the unbiased parameters, and , then the upper and lower confidence bounds are:

where .

Time Terminated

Consider the following equation where is the modified Bessel function of order one:

Find the values and by finding the solution to and in the cases corresponding to the lower and upper bounds, respectively. Calculate for each case.

If using the biased parameters, and , then the upper and lower confidence bounds are:

where .

If using the unbiased parameters, and , then the upper and lower confidence bounds are:

where .

Instantaneous Failure Intensity

Fisher Matrix Bounds

The instantaneous failure intensity, . must be positive, thus is approximately treated as being normally distributed.

The approximate confidence bounds on the instantaneous failure intensity are then estimated from:

where and:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on the instantaneous failure intensity are given by:

where and are calculated using the process presented for the confidence bounds on the instantaneous MTBF.

Time Given Cumulative Failure Intensity

Fisher Matrix Bounds

The time, . must be positive, thus is approximately treated as being normally distributed.

The confidence bounds on the time are given by:

where:

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on time given cumulative failure intensity are given by:

Then estimate, the number of failures, , such that:

The lower and upper confidence bounds on time are then estimated using:

Time Given Cumulative MTBF

Fisher Matrix Bounds

The time, . must be positive, thus is approximately treated as being normally distributed.

The confidence bounds on the time are given by:

where:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on time given cumulative MTBF are estimated using the process for the confidence bounds on time given cumulative failure intensity where .

Time Given Instantaneous MTBF

Fisher Matrix Bounds

The time, . must be positive, thus is approximately treated as being normally distributed.

The confidence bounds on the time are given by:

where:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

Failure Terminated

Calculate the constants and using procedures described for the confidence bounds on instantaneous MTBF. The lower and upper confidence bounds on time are then given by:

Time Terminated

Calculate the constants and using procedures described for the confidence bounds on instantaneous MTBF. The lower and upper confidence bounds on time are then given by:

Time Given Instantaneous Failure Intensity

Fisher Matrix Bounds

These bounds are based on:

The confidence bounds on the time are given by:

where:

The variance calculation is the same as the calculations given in the confidence bounds on Beta.

Crow Bounds

The 2-sided confidence bounds on time given instantaneous failure intensity are estimated using the process for the confidence bounds on time given instantaneous MTBF where .

Reliability

Fisher Matrix Bounds

These bounds are based on:

The confidence bounds on reliability are given by:

The variance calculation is the same as the calculations in the confidence bounds on Beta.

Crow Bounds

Failure Terminated

For failure terminated data, the 100( )% confidence interval on the current reliability at time for a specified mission duration is:

where:

  • and are obtained from the confidence bounds on instantaneous MTBF for failure terminated data.

Time Terminated

For time terminated data, the 100( )% confidence interval on the current reliability at time for a specified mission duration is:

where:

  • and are obtained from the confidence bounds on instantaneous MTBF for time terminated data.

Time Given Reliability and Mission Time

Fisher Matrix Bounds

The time, . must be positive, thus is approximately treated as being normally distributed.

The confidence bounds on time are calculated by using:

where:

is calculated numerically from:

The variance calculations are done by:

Crow Bounds

Failure Terminated

For failure terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

where and are obtained from the confidence bounds on instantaneous MTBF for failure terminated data.

Let and solve numerically for using .

Let and solve numerically for using .

If then and . If then and .

Time Terminated

For time terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

.

where and are obtained from the confidence bounds on instantaneous MTBF for time terminated data.

Let and solve numerically for using .

Let and solve numerically for using .

If . then and . If . then and .

Mission Time Given Reliability and Time

Fisher Matrix Bounds

The mission time, . must be positive, thus is approximately treated as being normally distributed.

The confidence bounds on mission time are given by using:

where:

Calculate from:

The variance calculations are done by:

Crow Bounds

Failure Terminated

Step 1: Calculate .

Step 2: Let and solve for such that:

Step 3: Let and solve for such that:

Step 4: If . then and . If . then and .

Time Terminated

Step 1: Calculate .

Step 2: Let and solve for using the same equation given for the failure terminated data.

Step 3: Let and solve for using the same equation given for the failure terminated data.

Step 4: If . then and . If . then and .

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