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Example - Confidence Bounds on Failure Intensity Using the values of and estimated in the example given above, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity. Crow Bounds Given that the data is failure terminated, the Crow confidence bounds for the cumulative failure intensity at the 90% confidence level and for hours are: The Crow confidence bounds for the instantaneous failure intensity at the 90% confidence level and for hours are calculated by first estimating the bounds on the instantaneous MTBF.
Once these are calculated, take the inverse as shown below. Details on the confidence bounds for instantaneous MTBF are presented. The following figures display plots of the Crow confidence bounds for the cumulative and instantaneous failure intensity, respectively. Example - Confidence Bounds on MTBF Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above. Solution Fisher Matrix Bounds From the previous example: And for hours, the partial derivatives of the cumulative and instantaneous MTBF are: Therefore, the variances become: So, at 90% confidence level and hours, the Fisher Matrix confidence bounds are: The following two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBFs.
Confidence bounds can also be obtained on the parameters. For Fisher Matrix confidence bounds: and: For Crow confidence bounds: and: Multiple Systems When more than one system is placed on test during developmental testing, there are multiple data types which are available depending on the testing strategy and the format of the data.
The data types that allow for the analysis of multiple systems using the Crow-AMSAA (NHPP) model are given below:. Goodness-of-fit Tests For all multiple systems data types, the is available. For Multiple Systems (Concurrent Operating Times) and Multiple Systems with Dates, two additional tests are also available:. Multiple Systems (Known Operating Times) A description of Multiple Systems (Known Operating Times) is presented on the page.
Consider the data in the table below for two prototypes that were placed in a reliability growth test. Multiple Systems with Dates An overview of the Multiple Systems with Dates data type is presented on the page. While Multiple Systems with Dates requires a date for each event, including the start and end times for each system, once the equivalent single system is determined, the parameter estimation is the same as it is for Multiple Systems (Concurrent Operating Times). See for details. Grouped Data A description of Grouped Data is presented in the page. Parameter Estimation for Grouped Data For analyzing grouped data, we follow the same logic described previously for the model. If the equation from the section above is linearized: According to Crow, the likelihood function for the grouped data case, (where failures are observed and is the number of groups), is: And the MLE of based on this relationship is: where is the total number of failures from all the groups.
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The estimate of is the value that satisfies: See for details on how confidence bounds for grouped data are calculated. Chi-Squared Test A chi-squared goodness-of-fit test is used to test the null hypothesis that the Crow-AMSAA reliability model adequately represents a set of grouped data. This test is applied only when the data is grouped. The expected number of failures in the interval from to is approximated by: For each interval, shall not be less than 5 and, if necessary, adjacent intervals may have to be combined so that the expected number of failures in any combined interval is at least 5. Let the number of intervals after this recombination be, and let the observed number of failures in the new interval be.
Finally, let the expected number of failures in the new interval be. Then the following statistic is approximately distributed as a chi-squared random variable with degrees of freedom. The null hypothesis is rejected if the statistic exceeds the critical value for a chosen significance level. In this case, the hypothesis that the Crow-AMSAA model adequately fits the grouped data shall be rejected. Critical values for this statistic can be found in chi-squared distribution tables. Grouped Data Examples Example - Simple Grouped Consider the grouped failure times data given in the following table.
Solve for the Crow-AMSAA parameters using MLE. Grouped Failure Times Data Run Number Cumulative Failures End Time(hours) 1 2 200 5.298 28.072 0.693 3.673 2 3 400 5.991 35.898 1.099 6.582 3 4 600 6.397 40.921 1.386 8.868 4 11 3000 8.006 64.102 2.398 19.198 Sum = 25.693 168.992 5.576 38.321 Solution Using RGA, the value of, which must be solved numerically, is 0.6315. Using this value, the estimator of is: Therefore, the intensity function becomes: Example - Helicopter System A new helicopter system is under development. System failure data has been collected on five helicopters during the final test phase.
The actual failure times cannot be determined since the failures are not discovered until after the helicopters are brought into the maintenance area. However, total flying hours are known when the helicopters are brought in for service, and every 2 weeks each helicopter undergoes a thorough inspection to uncover any failures that may have occurred since the last inspection. Therefore, the cumulative total number of flight hours and the cumulative total number of failures for the 5 helicopters are known for each 2-week period. The total number of flight hours from the test phase is 500, which was accrued over a period of 12 weeks (six 2-week intervals). For each 2-week interval, the total number of flight hours and total number of failures for the 5 helicopters were recorded.
The grouped data set is displayed in the following table. Grouped Data for a New Helicopter System Interval Interval Length Failures in Interval 1 0 - 62 12 2 62 -100 6 3 100 - 187 15 4 187 - 210 3 5 210 - 350 18 6 350 - 500 16 Do the following:.
Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation. Calculate the confidence bounds on the cumulative and instantaneous MTBF using the Fisher Matrix and Crow methods. Solution. Using RGA, the value of, must be solved numerically.
Once has been estimated then the value of can be determined. The parameter values are displayed below: The can be obtained on the parameters and at the 90% confidence level by: and: Crow confidence bounds can also be obtained on the parameters and at the 90% confidence level, as: and:. The Fisher Matrix confidence bounds for the cumulative MTBF and the instantaneous MTBF at the 90% 2-sided confidence level and for hour are: and: The next two figures show plots of the Fisher Matrix confidence bounds for the cumulative and instantaneous MTBF. Observing the data during the fourth month (between 500 and 625 hours), 38 failures were reported. This number is very high in comparison to the failures reported in the other months.
A quick investigation found that a number of new data collectors were assigned to the project during this month. It was also discovered that extensive design changes were made during this period, which involved the removal of a large number of parts. It is possible that these removals, which were not failures, were incorrectly reported as failed parts. Based on knowledge of the system and the test program, it was clear that such a large number of actual system failures was extremely unlikely. The consensus was that this anomaly was due to the failure reporting. For this analysis, it was decided that the actual number of failures over this month is assumed to be unknown, but consistent with the remaining data and the Crow-AMSAA reliability growth model.
Considering the problem interval as the gap interval, we will use the data over the interval and over the interval The are the appropriate equations to estimate and because the failure times are known. In this case. The maximum likelihood estimates of and are: The next figure is a plot of the cumulative number of failures versus time.
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This plot is approximately linear, which also indicates a good fit of the model. Discrete Data The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called discrete or attribute data).
The following discrete data types are available:. Sequential. Grouped per Configuration. Mixed Sequential data and Grouped per Configuration are very similar as the parameter estimation methodology is the same for both data types. Mixed data is a combination of Sequential Data and Grouped per Configuration and is presented in. Grouped per Configuration Suppose system development is represented by configurations.
This corresponds to configuration changes, unless fixes are applied at the end of the test phase, in which case there would be configuration changes. Let be the number of trials during configuration and let be the number of failures during configuration. Then the cumulative number of trials through configuration, namely, is the sum of the for all, or: And the cumulative number of failures through configuration, namely, is the sum of the for all, or: The expected value of can be expressed as and defined as the expected number of failures by the end of configuration. Applying the learning curve property to implies: Denote as the probability of failure for configuration 1 and use it to develop a generalized equation for in terms of the. From the equation above, the expected number of failures by the end of configuration 1 is: Applying the equation again and noting that the expected number of failures by the end of configuration 2 is the sum of the expected number of failures in configuration 1 and the expected number of failures in configuration 2: By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis, is obtained, such that: In this equation, represents the trial number. Thus, an equation for the reliability (probability of success) for the configuration is obtained: Sequential Data From the section, the following equation is given: For the special case where for all, the equation above becomes a smooth curve, that represents the probability of failure for trial by trial data, or: When, this is the same as Sequential Data where systems are tested on a trial-by-trial basis. The equation for the reliability for the trial is: Parameter Estimation for Discrete Data This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data which includes Sequential data and Grouped per Configuration.
An example is presented illustrating these concepts. The estimation procedures provide maximum likelihood estimates (MLEs) for the model's two parameters,.
The MLEs for and allow for point estimates for the probability of failure, given by: And the probability of success (reliability) for each configuration is equal to: The likelihood function is: Taking the natural log on both sides yields: Taking the derivative with respect to and respectively, exact MLEs for and are values satisfying the following two equations.