Template:Test-fix-find-test rga

Test-Fix-Find-Test

Traditional reliability growth models provide assessments for two types of testing and corrective action strategies: test-fix-test and test-find-test. In test-fix-test, failure modes are found during testing and corrective actions for these modes are incorporated during the test. Data from this type of test can be modeled appropriately with the Crow-AMSAA model, described in Chapter 5. In test-find-test, modes are found during testing but all of the corrective actions are delayed and incorporated after the completion of the test. Data from this type of test can be modeled appropriately with the Crow-AMSAA Projection model, described in Section 9.2. However, a common strategy involves a combination of these two approaches, namely some corrective actions are incorporated during the test and some corrective actions are delayed and incorporated at the end of the test. This strategy is referred to as test-fix-find-test. Data from this test can be modeled appropriately with the Crow Extended reliability growth model, which is described next.

Recall that B failure modes are all failure modes that will receive a corrective action. In order to provide the assessment and management metric structure for corrective actions during and after a test, two types of B modes are defined. BC failure modes are corrected during the test and BD failure modes are delayed until the end of the test. Type A failure modes are defined as before; i.e. those failure modes that will not receive a corrective action, either during or at the end of the test.

Development of the Crow Extended Model

Let [math]\displaystyle{ {{\lambda }_{BD}} }[/math] denote the constant failure intensity for the BD failure modes and let [math]\displaystyle{ h(t|BD) }[/math] denote the first occurrence function for the BD failure modes. In addition, as before, let [math]\displaystyle{ K }[/math] be the number of BD failure modes, let [math]\displaystyle{ {{d}_{i}} }[/math] be the effectiveness factor for the [math]\displaystyle{ {{i}^{th}} }[/math] BD failure mode and let [math]\displaystyle{ \overline{d} }[/math] be the average effectiveness factor.
The Crow Extended model projected failure intensity is given by:

[math]\displaystyle{ {{\lambda }_{EM}}={{\lambda }_{CA}}-{{\lambda }_{BD}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\overline{d}h(T|BD) }[/math]

where [math]\displaystyle{ {{\lambda }_{CA}}=\lambda \beta {{T}^{\beta -1}} }[/math] is the achieved failure intensity at time [math]\displaystyle{ T }[/math] .
The Crow Extended model projected MTBF is:

[math]\displaystyle{ {{M}_{EM}}=1/{{\lambda }_{EM}} }[/math]

This is the MTBF after the delayed fixes have been implemented. Under the extended reliability growth model, the demonstrated failure intensity before the delayed fixes is the first term, [math]\displaystyle{ {{\lambda }_{CA}} }[/math] . The demonstrated MTBF at time [math]\displaystyle{ T }[/math] before the delayed fixes is given by:

[math]\displaystyle{ {{M}_{CA}}\text{ }={{[{{\lambda }_{CA}}]}^{-1}} }[/math]

If you assume that there are no delayed corrective actions (BD modes) then the model reduces to the special case of the Crow-AMSAA model (the first term only in Eqn. (extendl)) and the achieved MTBF equals the projection. That is, there is no jump. If you assume that there are no corrective actions during the test (BC modes) then the model reduces to the test-find-test scenario described in the previous section.  Estimation of the Crow Extended Model In the general estimation of the Crow Extended model, it is required that all failure times during the test are known. Furthermore, the ID of each A, BC and BD failure mode needs to be entered.
The estimate of the projected failure intensity for the Crow Extended model is given by:

[math]\displaystyle{ {{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T}+\overline{d}\widehat{h}(T|BD) }[/math]

where [math]\displaystyle{ {{N}_{i}} }[/math] is the total number of failures for the [math]\displaystyle{ {{i}^{th}} }[/math] BD mode and [math]\displaystyle{ {{d}_{i}} }[/math] is the corresponding assigned EF. In order to obtain the first term, [math]\displaystyle{ {{\widehat{\lambda }}_{CA}} }[/math] , fit all of the data (regardless of mode classification) to the Crow-AMSAA model to estimate [math]\displaystyle{ \widehat{\beta } }[/math] and [math]\displaystyle{ \widehat{\lambda } }[/math] , thus:

[math]\displaystyle{ {{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}} }[/math]

The remaining terms are analyzed with the Crow Extended model, which is applied to only the BD data.

[math]\displaystyle{ {{\widehat{\lambda }}_{BD}}=\frac{{{N}_{BD}}}{T} }[/math]

[math]\displaystyle{ \begin{align} & \widehat{h}(T|BD)= & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ & = & \frac{M{{\widehat{\beta }}_{BD}}}{T} \end{align} }[/math]

[math]\displaystyle{ {{\widehat{\beta }}_{BD}} }[/math] is the unbiased estimated of [math]\displaystyle{ \beta }[/math] for the Crow-AMSAA model based on the first occurrence of [math]\displaystyle{ M }[/math] distinct BD modes. 
The structure for the Crow Extended model includes the following special data analysis cases:

1. Test-fix-test with no failure modes known or with BC failure modes known. With this type of data, the Crow Extended model will take the form of the traditional Crow-AMSAA analysis described previously.

2. Test-find-test with BD failure modes known. With this type of data, the Crow Extended model will take the form of the Crow-AMSAA Projection analysis described in Section 9.2.

3. Test-fix-find-test with BC and BD failure modes known. With this type of data, the full capabilities of the Crow Extended model will be applied, as described in the following sections.

Reliability Growth Potential and Maturity Metrics

The growth potential and some maturity metrics for the Crow Extended model are calculated as follows.
• Initial system MTBF and failure intensity are given by:

[math]\displaystyle{ {{\widehat{M}}_{I}}=\frac{\Gamma \left( 1+\tfrac{1}{\widehat{\beta }} \right)}{{{\widehat{\lambda }}^{\tfrac{1}{\widehat{\beta }}}}} }[/math]

and:

[math]\displaystyle{ {{\widehat{\lambda }}_{I}}={{[{{\widehat{M}}_{I}}]}^{-1}} }[/math]

where [math]\displaystyle{ \widehat{\beta } }[/math] and [math]\displaystyle{ \widehat{\lambda } }[/math] are the estimators of the Crow-AMSAA model for all data regardless of the failure mode classification (i.e. A, BC or BD).
• A mode failure intensity and MTBF are given by:

[math]\displaystyle{ {{\widehat{\lambda }}_{A}}=\frac{{{N}_{A}}}{T} }[/math]

[math]\displaystyle{ {{\widehat{M}}_{A}}={{[{{\widehat{\lambda }}_{A}}]}^{-1}} }[/math]

• Initial BD mode failure intensity are given by:

[math]\displaystyle{ {{\widehat{\lambda }}_{BD}}=\frac{{{N}_{BD}}}{T} }[/math]

• BC mode initial failure intensity and MTBF are given by:

[math]\displaystyle{ {{\widehat{\lambda }}_{I(BC)}}={{\widehat{\lambda }}_{I}}-{{\widehat{\lambda }}_{A}}-{{\widehat{\lambda }}_{BD}} }[/math]

[math]\displaystyle{ {{\widehat{M}}_{I(BC)}}={{[{{\widehat{\lambda }}_{I(BC)}}]}^{-1}} }[/math]

• Failure intensity [math]\displaystyle{ h(T|BC) }[/math] and instantaneous MTBF [math]\displaystyle{ M(T|BC) }[/math] for new BC failure modes at the end of test time [math]\displaystyle{ T }[/math] are given by:

[math]\displaystyle{ \widehat{h}(T|BC)=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}} }[/math]

[math]\displaystyle{ \widehat{M}(T|BC)={{[\widehat{h}(T|BC)]}^{-1}} }[/math]

where [math]\displaystyle{ \widehat{\beta } }[/math] and [math]\displaystyle{ \widehat{\lambda } }[/math] are the estimators of the Crow-AMSAA model for the first occurrence of distinct BC modes.
• Average effectiveness factor for BC failure modes is given by:

[math]\displaystyle{ {{\widehat{d}}_{BC}}=\frac{\left[ \tfrac{N_{BC}^{\left( \tfrac{1}{{{{\hat{\beta }}}_{BC}}} \right)}}{\Gamma \left( 1+\tfrac{1}{{{{\hat{\beta }}}_{BC}}} \right)} \right]-{{N}_{BC}}}{\left[ \tfrac{N_{BC}^{\left( \tfrac{1}{{{{\hat{\beta }}}_{BC}}} \right)}}{\Gamma \left( 1+\tfrac{1}{{{{\hat{\beta }}}_{BC}}} \right)} \right]-{{M}_{BC}}} }[/math]

where [math]\displaystyle{ {{N}_{BC}} }[/math] is the total number of observed BC modes, [math]\displaystyle{ {{M}_{BC}} }[/math] is the number of unique BC modes and [math]\displaystyle{ {{\hat{\beta }}_{BC}} }[/math] is the MLE for the first occurrence of distinct BC modes. If [math]\displaystyle{ {{\hat{\beta }}_{BC}}\ge 1 }[/math] then [math]\displaystyle{ {{\widehat{d}}_{BC}} }[/math] equals zero.
• Growth potential failure intensity and growth potential MTBF are given by:

[math]\displaystyle{ {{\widehat{\lambda }}_{GP}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} }[/math]

[math]\displaystyle{ {{\widehat{M}}_{GP}}={{[{{\widehat{\lambda }}_{GP}}]}^{-1}} }[/math]