Crow Extended

In a previous chapter, we discussed the most widely used traditional reliability growth tracking model, Crow-AMSAA (NHPP). Using this model for reliability growth analysis assumes that the corrective actions for the observed failure modes are incorporated during the test (test-fix-test). However, in actual practice, fixes may be delayed until after the completion of the test (test-find-test) or some fixes may be implemented during the test while others are delayed (test-fix-find-test). At the end of a test phase, two reliability estimates are of concern: demonstrated reliability and projected reliability. The demonstrated reliability, which is based on data generated during the test phase, is an estimate of the system reliability for its configuration at the end of the test phase. The projected reliability measures the impact of the delayed fixes at the end of the current test phase.

Most of the reliability growth literature has been concerned with procedures and models for calculating the demonstrated reliability and very little attention has been paid to techniques for reliability projections. The procedure for making reliability projections utilizes engineering assessments of the effectiveness of the delayed fixes for each observed failure mode. These effectiveness factors are then used with the data generated during the test phase to obtain a projected estimate for the updated configuration by adjusting the number of failures observed during the test phase. The process of estimating the projected reliability is accomplished using the Crow Extended model. The Crow Extended model allows for a flexible growth strategy that can include corrective actions performed during the test, as well as delayed corrective actions. The test-find-test and test-fix-find-test scenarios are simply subsets of the Crow Extended model.

Background
When a system is tested and failure modes are observed, management can make one of two possible decisions, either to fix or not fix the failure mode. Therefore, the management strategy places failure modes into two categories: A modes and B modes. A modes are all failure modes such that when seen during the test no corrective action will be taken. This accounts for all modes for which management determines that it is not economically or otherwise justified to take 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 modes are corrected during the test and the corrective actions for BD modes are delayed until the end of the test. The management strategy is defined by how the corrective actions, if any, will be implemented. In summary, the classifications are defined as follows:


 * A indicates that no corrective action was performed or will be performed (management chooses not to address for technical, financial or other reasons).


 * BC indicates that the corrective action was implemented during the test. The analysis assumes that the effect of the corrective action was experienced during the test (as with other test-fix-test reliability growth analyses).


 * BD indicates that the corrective action will be delayed until after the completion of the current test.

The following picture shows an example of data entered for the Crow Extended model.

As you can see, each failure is indicated with A, BC or BD in the Classification column. In addition, any number or text can be used to specify the mode. In this example, numbers were used in the Mode column for simplicity, but you could just as easily use Seal Leak, or whatever designation you deem appropriate for identifying the failure mode.

Reliability growth is achieved by decreasing the failure intensity. The failure intensity for the A failure modes will not change. Therefore, reliability growth can only be achieved by decreasing the BC and BD mode failure intensity. It is also clear that, in general, the only part of the BD mode failure intensity that can be decreased is that which has been seen during testing, since the failure intensity due to BD modes that were unseen during testing still remains. BC failure modes are corrected during test and the BC failure intensity will not change any more at the end of test.

It is very important to note that once a BD failure mode is in the system it is rarely totally eliminated by a corrective action. After a BD mode has been found and fixed, a certain percentage of the failure intensity will be removed, but a certain percentage of the failure intensity will generally remain. For each BD mode, an effectiveness factor (EF) is required to estimate how effective you will be in eliminating the failure intensity due to the failure mode. The EF is the fractional decrease in a mode's failure intensity after a corrective action has been made and must be a value between 0 and 1. A study on EFs showed that an average EF $$d\,\!$$ was about 70 percent. Therefore, typically about 30 percent, (i.e., 100 $$(1-d)\,\!$$ percent), of the BD mode failure intensity will remain in the system after all of the corrective actions have been implemented. However, individual EFs for the failure modes may be larger or smaller than the average. The next figure displays the RGA software's Effectiveness Factor window where the effectiveness factors for each unique BD failure mode can be specified.



Test-Find-Test
Test-find-test is the case where all corrective actions are delayed until after the test. Therefore, there are no BC modes when analyzing test-find-test data. This scenario is also called the Crow-AMSAA Projection model, but for the purposes of the RGA software it is simply a special case of the Crow Extended model.

Suppose a system is subjected to development testing for a period of time, $$T\,\!$$. The system can be considered as consisting of two types of failure modes: A modes and BD modes. It is assumed that all BD modes are in series and fail independently according to the exponential distribution. Also assume that the rate of occurrence of A modes follows an exponential distribution with failure intensity $${{\lambda }_{A}}\,\!$$. The system MTBF is constant throughout the test phase since all of the corrective actions are delayed until after the completion of the test. After the delayed fixes have been implemented, the system MTBF will then jump to a higher value.

Let $$K\,\!$$ denote the total number of BD modes in the system and let $${{\lambda }_{i}}\,\!$$ denote the failure intensity for the $${{i}^{th}}\,\!$$ BD mode, such that $$i = 1,2,\ldots ,K\,\!$$. Then, at time equal to zero, the system failure intensity $$r(0)\,\!$$ is:


 * $$\begin{align}

r(0)={{\lambda }_{A}}+{{\lambda }_{BD}} \end{align}\,\!$$

where:
 * $${{\lambda }_{BD}}=\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{\lambda }_{i}}\,\!$$.

During the test $$(0,T)\,\!$$, a random number of $$M\,\!$$ distinct BD modes will be observed, such that $$M\le K\,\!$$. Denote the effectiveness factor (EF) for the $${{i}^{th}}\,\!$$ BD mode as $${{d}_{i}}\,\!$$, $$i = 1,2,\ldots ,K\,\!$$. The effectiveness factor $${{d}_{i}}\,\!$$ is the percent decrease in $${{\lambda }_{i}}\,\!$$ after a corrective action has been made for the $${{i}^{th}}\,\!$$ BD mode. That is, the corrective action for the $${{i}^{th}}\,\!$$ BD mode removes $$100\times {{d}_{i}}\,\!$$ percent of the failure rate and $$100\times (1-{{d}_{i}})\,\!$$ percent remains. The failure intensity for the $${{i}^{th}}\,\!$$ BD failure mode after a corrective action is $$(1-{{d}_{i}}){{\lambda }_{i}}\,\!$$. If corrective actions are taken on the $$M\,\!$$ BD modes observed by time $$T\,\!$$, then the system failure intensity is reduced from $$r(0)\,\!$$ to:


 * $$\begin{align}

r\left( T \right) = & {{\lambda }_{A}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,\left( 1-{{d}_{i}} \right){{\lambda }_{i}}+({{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop \sum }}\,{{\lambda }_{i}}) \\ = & {{\lambda }_{A}}+{{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}{{\lambda }_{i}} \end{align}\,\!$$

where:


 * $$\underset{i=1}{\overset{M}{\mathop{\sum }}}\,(1-{{d}_{i}}){{\lambda }_{i}}\,\!$$ is the failure intensity for the $$M\,\!$$ modes after the corrective actions


 * $$({{\lambda }_{BD}}-\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{\lambda }_{i}})\,\!$$ is the remaining failure intensity for all unseen BD modes

All $$M\,\!$$ BD modes observed by test time $$T\,\!$$ may not be fixed by time $$T\,\!$$ so the actual failure intensity at time $$T\,\!$$ may not be $$r(T)\,\!$$. However, $$r(T)\,\!$$ can be viewed as the achieved failure intensity at time $$T\,\!$$ if all fixes were updated and incorporated into the system. All of the fixes for the BD modes found during the test are incorporated as delayed fixes at the end of the test phase. Therefore, the system failure intensity is constant at $$r(0)={{\lambda }_{A}}+{{\lambda }_{BD}}\,\!$$ through the test phase and will then jump to a lower value $$r(T)\,\!$$ after the delayed fixes have been implemented. Let $${{N}_{A}}\,\!$$ and $${{N}_{BD}}\,\!$$ be the total number of A and BD failures observed during the test $$(0,T)\,\!$$ and let $$N={{N}_{A}}+{{N}_{BD}}\,\!$$. In addition, there are $$M\,\!$$ distinct BD modes observed during the test. After implementing the $$M\,\!$$ fixes, the failure intensity for the system at time $$T\,\!$$ (after the jump) is given by the function $$r(T)\,\!$$.

$$r(0)\,\!$$ is actually the demonstrated failure intensity, which is based on actual system performance of the hardware tested and not of some future configuration. A demonstrated reliability value should be determined at the end of each test phase. The demonstrated failure intensity is:


 * $${{\widehat{\lambda }}_{D}}(T)=r(0)=\frac{{{N}_{A}}+{{N}_{BD}}}{T}\,\!$$

The demonstrated MTBF is given by:


 * $$M\widehat{T}B{{F}_{D}}={{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}}\,\!$$

The detailed procedure for estimating $$r(T)\,\!$$ is given in Crow [20] and is reviewed here.

Let $$E[\cdot ]\,\!$$ denote the expected value:


 * $$E[r(T)]={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,{{d}_{i}}{{\lambda }_{i}}{{e}^{-{{\lambda }_{i}}T}}\,\!$$

Under realistic assumptions $$E[r(T)]\,\!$$ also may be expressed as:


 * $$E[r(T)]={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\overline{d}h(T)\,\!$$

where $$\overline{d}\,\!$$ is the mean effectiveness factor and $$h(T)\,\!$$ is the instantaneous rate at which a new BD mode will occur at time $$T\,\!$$. The maximum likelihood estimate for the $$h(T)\,\!$$ is:


 * $$h(T)={{\lambda }_{BD}}{{\beta }_{BD}}{{T}^{{{\beta }_{BD}}-1}}\,\!$$

And, $$\overline{d}h(T)\,\!$$ is the bias term, such that:


 * $$B(T)=\overline{d}h(T)\,\!$$

Estimation of Bias Term
Let $${{X}_{1}}<{{X}_{2}}<\ldots <{{X}_{M}}0\,\!$$ is estimated by:


 * $$h(t)={{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{t}^{{{\widehat{\beta }}_{BD}}-1}}\,\!$$

In particular, the maximum likelihood estimate for the rate of occurrence for the distinct BD modes at time $$T\,\!$$ is:


 * $$\begin{align}

\widehat{h}(T) = & {{\widehat{\lambda }}_{BD}}{{\widehat{\beta }}_{BD}}{{T}^{{{\widehat{\beta }}_{BD}}-1}} \\ = & \frac{M{{\widehat{\beta }}_{BD}}}{T} \end{align}\,\!$$

Furthermore, the maximum likelihood estimate of the bias term $$B(T)\,\!$$ is given by:


 * $$B(T)=\overline{d}\frac{M{{\widehat{\beta }}_{BD}}}{T}\,\!$$

The unbiased estimate of $${{\beta }_{BD}}\,\!$$ is:


 * $${{\bar{\beta }}_{BD}}=\frac{M-1}{M}{{\hat{\beta }}_{BD}}\,\!$$

Thus the unbiased estimate of the bias term is given by:


 * $$B(T)=\overline{d}\frac{M{{{\bar{\beta }}}_{BD}}}{T}\,\!$$

The mean $$\overline{d}\,\!$$ is given by:


 * $$\overline{d}=\frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}\,\!$$

Therefore, the projected failure intensity $$r(T)\,\!$$ is then estimated at the end of the test phase by:


 * $$\widehat{r}(T)=\left( \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right)+\overline{d}\left( \frac{M}{T}{{\overline{\beta }}_{BD}} \right)\,\!$$

The projected MTBF is:


 * $$M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}\,\!$$

Reliability Growth Potential
The failure intensity $$r(T)\,\!$$ will depend on the management strategy that determines the classification of the A and BD failure modes. The engineering effort applied to the corrective actions determines the effectiveness factors. In addition, $$r(T)\,\!$$ depends on $$h(t)\,\!$$, which is the rate at which problem failure modes are being seen during testing. $$h(t)\,\!$$ drives the opportunity to take corrective actions based on the seen failure modes and it is an important factor in the overall reliability growth rate. The reliability growth potential is the limiting value of $$r(T)\,\!$$ as $$T\,\!$$ increases. This limit is the maximum MTBF that can be attained with the current management strategy. The maximum MTBF will be attained when all $$K\,\!$$ BD modes have been observed and fixed with EFs $${{d}_{i}}\,\!$$. In terms of failure intensity, the growth potential is expressed by the following equation:


 * $${{r}_{GP}}={{\lambda }_{A}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}\,\!$$

In terms of the MTBF, the growth potential is given by:


 * $$\begin{align}

MTB{{F}_{GP}}=1/{{r}_{GP}} \end{align}\,\!$$

The procedure for estimating the growth potential is as follows. Suppose that the system is tested for a period of time $$T\,\!$$ and that $$N\,\!$$ failures have been observed. According to the management strategy, $${{N}_{A}}\,\!$$ of these failures are A modes and $${{N}_{BD}}\,\!$$ of these failures are BD modes. For the BD modes, there will be $$M\,\!$$ distinct fixes. As before, $${{N}_{i}}\,\!$$ is the total number of failures for the $${{i}^{th}}\,\!$$ BD mode and $${{d}_{i}}\,\!$$ is the corresponding assigned EF. From this data, the growth potential failure intensity is estimated by:


 * $${{\widehat{r}}_{GP}}(T)=\left( \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right)\,\!$$

The growth potential MTBF is estimated by:


 * $$M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}\,\!$$

Example: Test-Find-Test Data
Consider the data in the first table below. A system was tested for $$T=400\,\!$$ hours. There were a total of $$N=42\,\!$$ failures and all corrective actions will be delayed until after the end of the 400 hour test. Each failure has been designated as either an A failure mode (the cause will not receive a corrective action) or a BD mode (the cause will receive a corrective action). There are $${{N}_{A}}=10\,\!$$ A mode failures and $${{N}_{BD}}=32\,\!$$ BD mode failures. In addition, there are $$M=16\,\!$$ distinct BD failure modes, which means 16 distinct corrective actions will be incorporated into the system at the end of test. The total number of failures for the $${{j}^{th}}\,\!$$ observed distinct BD mode is denoted by $${{N}_{j}}\,\!$$ and the total number of BD failures during the test is $${{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\!$$. These values and effectiveness factors are given in the second table.


 * Determine the projected MTBF and failure intensity.
 * Determine the growth potential MTBF and failure intensity.
 * Determine the demonstrated MTBF and failure intensity.



Solution


 * 1)	The maximum likelihood estimates of $${{\beta }_{BD}}\,\!$$ and $${{\lambda }_{BD}}\,\!$$ are determined to be:


 * $$\begin{align}

{{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T})} \\ = & 0.7970 \\ 	 {{{\hat{\lambda }}}_{BD}} = & 0.1350 \end{align}\,\!$$ The unbiased estimate of $$\beta \,\!$$ is:


 * $$\begin{align}

{{\overline{\beta }}_{BD}} = & \frac{M-1}{M}{{{\hat{\beta }}}_{BD}} \\ = & 0.7472 \end{align}\,\!$$

Based on the test data, $$\overline{d}=\tfrac{1}{M}\underset{i=1}{\overset{M}{\mathop{\sum }}}\,{{d}_{i}}= 0.72125\,\!$$. Therefore, $$B(T)=\overline{d}\tfrac{M{{\overline{\beta }}_{BD}}}{T}=0.0215\,\!$$. The projected failure intensity due to incorporating the 16 corrective actions is:


 * $$\begin{align}

r(T) = & \left( \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right)+\overline{d}\left( \frac{M}{T}{{\overline{\beta }}_{BD}} \right) \\ = & 0.0661 \end{align}\,\!$$

The projected MTBF is:


 * $$M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\!$$


 * 2)	To estimate the maximum reliability that can be attained with this management strategy, use the following calculations.


 * $$\begin{align}

{{N}_{A}}/T=0.0250 \end{align}\,\!$$


 * $$\frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\!$$

The growth potential failure intensity is estimated by:


 * $$\begin{align}

{{\widehat{r}}_{GP}}(T) = & \left( \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right) \\ = & 0.0250+0.0196 \\  = & 0.0446  \end{align}\,\!$$

The growth potential MTBF is:


 * $$M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\!$$


 * 3)	The demonstrated failure intensity and MTBF are estimated by:


 * $$\begin{align}

{{\widehat{\lambda }}_{D}}(T) = & \frac{{{N}_{A}}+{{N}_{BD}}}{T} \\ = & \frac{42}{400} \\ = & 0.1050 	\end{align}\,\!$$
 * $$\begin{align}

M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\ = & 9.5238 	\end{align}\,\!$$ The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.





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, which was discussed in a previous chapter. 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, which was described above in the Test-Find-Test section. However, a common strategy involves a combination of these two approaches, where 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 $${{\lambda }_{BD}}\,\!$$ denote the constant failure intensity for the BD failure modes and let $$h(t|BD)\,\!$$ denote the first occurrence function for the BD failure modes. In addition, as before, let $$K\,\!$$ be the number of BD failure modes, let $${{d}_{i}}\,\!$$ be the effectiveness factor for the $${{i}^{th}}\,\!$$ BD failure mode and let $$\overline{d}\,\!$$ be the average effectiveness factor.

The Crow Extended model projected failure intensity is given by:


 * $${{\lambda }_{EM}}={{\lambda }_{CA}}-{{\lambda }_{BD}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}}){{\lambda }_{i}}+\overline{d}h(T|BD)\,\!$$

where $${{\lambda }_{CA}}=\lambda \beta {{T}^{\beta -1}}\,\!$$ is the achieved failure intensity at time $$T\,\!$$.

The Crow Extended model projected MTBF is:


 * $$\begin{align}

{{M}_{EM}}=1/{{\lambda }_{EM}} \end{align}\,\!$$

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, $${{\lambda }_{CA}}\,\!$$. The demonstrated MTBF at time $$T\,\!$$ before the delayed fixes is given by:


 * $${{M}_{CA}}\text{ }={{[{{\lambda }_{CA}}]}^{-1}}\,\!$$

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 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:


 * $${{\widehat{\lambda }}_{EM}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T}+\overline{d}\widehat{h}(T|BD)\,\!$$

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


 * $${{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}\,\!$$

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


 * $${{\widehat{\lambda }}_{BD}}=\frac{T}\,\!$$


 * $$\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}\,\!$$

$${{\widehat{\beta }}_{BD}}\,\!$$ is the unbiased estimated of $$\beta \,\!$$ for the Crow-AMSAA model based on the first occurrence of $$M\,\!$$ distinct BD modes.

The structure for the Crow Extended model includes the following special data analysis cases:


 * 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 in a previous chapter.
 * 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 previously in the Test-Find-Test section.
 * 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:
 * $${{\widehat{M}}_{I}}=\frac{\Gamma \left( 1+\tfrac{1}{\widehat{\beta }} \right)}\,\!$$


 * and:


 * $${{\widehat{\lambda }}_{I}}={{[{{\widehat{M}}_{I}}]}^{-1}}\,\!$$

where $$\widehat{\beta }\,\!$$ and $$\widehat{\lambda }\,\!$$ 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:
 * $${{\widehat{\lambda }}_{A}}=\frac{T}\,\!$$


 * $${{\widehat{M}}_{A}}={{[{{\widehat{\lambda }}_{A}}]}^{-1}}\,\!$$


 * Initial BD mode failure intensity are given by:
 * $${{\widehat{\lambda }}_{BD}}=\frac{T}\,\!$$


 * BC mode initial failure intensity and MTBF are given by:
 * $${{\widehat{\lambda }}_{I(BC)}}={{\widehat{\lambda }}_{I}}-{{\widehat{\lambda }}_{A}}-{{\widehat{\lambda }}_{BD}}\,\!$$


 * $${{\widehat{M}}_{I(BC)}}={{[{{\widehat{\lambda }}_{I(BC)}}]}^{-1}}\,\!$$


 * Failure intensity $$h(T|BC)\,\!$$ and instantaneous MTBF $$M(T|BC)\,\!$$ for new BC failure modes at the end of test time $$T\,\!$$ are given by:


 * $$\widehat{h}(T|BC)=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}\,\!$$


 * $$\widehat{M}(T|BC)={{[\widehat{h}(T|BC)]}^{-1}}\,\!$$

where $$\widehat{\beta }\,\!$$ and $$\widehat{\lambda }\,\!$$ 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:
 * $${{\widehat{d}}_{BC}}=\frac{\left[ \tfrac{N_{BC}^{\left( \tfrac{1} \right)}}{\Gamma \left( 1+\tfrac{1} \right)} \right]-{{N}_{BC}}}{\left[ \tfrac{N_{BC}^{\left( \tfrac{1} \right)}}{\Gamma \left( 1+\tfrac{1} \right)} \right]-{{M}_{BC}}}\,\!$$

where $${{N}_{BC}}\,\!$$ is the total number of observed BC modes, $${{M}_{BC}}\,\!$$ is the number of unique BC modes and $${{\hat{\beta }}_{BC}}\,\!$$ is the MLE for the first occurrence of distinct BC modes. If $${{\hat{\beta }}_{BC}}\ge 1\,\!$$ then $${{\widehat{d}}_{BC}}\,\!$$ equals zero.


 * Growth potential failure intensity and growth potential MTBF are given by:
 * $${{\widehat{\lambda }}_{GP}}={{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T}\,\!$$


 * $${{\widehat{M}}_{GP}}={{[{{\widehat{\lambda }}_{GP}}]}^{-1}}\,\!$$

Failure Mode Management Strategy
Management controls the resources for corrective actions. Consequently, the effectiveness factors are part of the management strategy. For the BD mode failure intensity that has been seen during development testing, 100 $$d\,\!$$ percent will be removed and 100 $$(1-d)\,\!$$ percent will remain in the system. Therefore, after the corrective actions have been made, the current system instantaneous failure intensity consists of the failure intensity due to the A modes plus the failure intensity for the unseen BC modes, plus the failure intensity for the unseen BD modes, plus the failure intensity for the BD modes that have been seen. The following pie chart shows how the system's instantaneous failure intensity can be broken down into its individual pieces based on the current failure mode strategy.



Keep in mind that the individual components of the system's instantaneous failure intensity will depend on the classifications defined in the data. For example, if BC modes are not present within the data then the BC mode MTBF will not be a part of the overall system MTBF. The individual pieces of the pie, as shown in the above figure, are calculated using the following equations.

Let:


 * $$\hat{r}(T)=\hat{\lambda }\hat{\beta }{{T}^{\hat{\beta }-1}}\,\!$$

where $$T\,\!$$ is the test time and $$\hat{\beta }\,\!$$ and $$\hat{\lambda }\,\!$$ are the maximum likelihood estimates of the Crow-AMSAA model for all of the data. $$\hat{\beta }\,\!$$ is the biased estimate of $$\beta \,\!$$. Therefore:


 * $$\hat{\beta }=\frac{N}{\underset{i=1}{\overset{N}{\mathop{\sum }}}\,\ln \left( \tfrac{T} \right)}\,\!$$


 * $$\hat{\lambda }=\frac{N}\,\!$$

where $$N\,\!$$ is the total number of failures and $${{X}_{i}}\,\!$$ is the $${{i}^{th}}\,\!$$ time-to-failure. Let the successive failures $$0<{{X}_{1}}<{{X}_{2}}<\ldots <{{X}_{3}}<{{X}_{N}}\,\!$$ be partitioned into the A mode failures ( $${{N}_{A}}\,\!$$ ), BC first occurrence failures ( $${{N}_{BCF}}\,\!$$ ), BC remaining failures ( $${{N}_{BCR}}\,\!$$ ), BD first occurrence failure ( $${{N}_{BDF}}\,\!$$ ) and the BD remaining failures ( $${{N}_{BDR}}\,\!$$ ). For continuous data, each portion of the pie chart due to each of the modes is calculated as follows:


 * A modes
 * $$A=\left( \frac{T} \right)\left[ \underset{i=1}{\overset{\mathop \sum }}\,\ln \left( \frac{T} \right) \right]\hat{r}(T)\,\!$$


 * BC modes unseen


 * $$B{{C}_{unseen}}=\left( \frac{T} \right)\left[ \underset{i=1}{\overset{\mathop \sum }}\,\ln \left( \frac{T} \right) \right]\hat{r}(T)\,\!$$


 * BC modes seen


 * $$B{{C}_{seen}}=\left( \frac{T} \right)\left[ \underset{i=1}{\overset{\mathop \sum }}\,\ln \left( \frac{T} \right) \right]\hat{r}(T)\,\!$$


 * BD modes unseen


 * $$B{{D}_{unseen}}=\left( \frac{T} \right)\left[ \underset{i=1}{\overset{\mathop \sum }}\,\ln \left( \frac{T} \right) \right]\hat{r}(T)\,\!$$


 * BD modes seen


 * $$B{{D}_{seen}}=\left( \frac{T} \right)\left[ \underset{i=1}{\overset{\mathop \sum }}\,\ln \left( \frac{T} \right) \right]\hat{r}(T)\,\!$$


 * BD modes remain


 * $$\begin{align}

B{{D}_{remain}} = & \left( 1-\frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}} \right)\cdot B{{D}_{seen}} \\ = & \left( 1-\overline{d} \right)\cdot B{{D}_{seen}} \end{align}\,\!$$


 * BD modes removed


 * $$\begin{align}

B{{D}_{removed}} = & \frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}\cdot B{{D}_{seen}} \\ = & \overline{d}\cdot B{{D}_{seen}} \end{align}\,\!$$

For grouped data, from Crow-AMSAA (NHPP) the maximum likelihood estimates of $$\beta \,\!$$ and $$\lambda \,\!$$ are calculated such that the following equations are satisfied:


 * $$\underset{i=1}{\overset{K}{\mathop \sum }}\,{{N}_{i}}\left[ \frac{t_{i}^\ln ({{t}_{i}})-t_{i-1}^\ln ({{t}_{i-1}})}{t_{i}^-t_{i-1}^}-\ln T \right]=0\,\!$$


 * $$\hat{\lambda }=\frac{N}{T_{K}^}\,\!$$

where $$K\,\!$$ is the number of groups and $$N=\underset{i=1}{\overset{K}{\mathop{\sum }}}\,{{N}_{i}}\,\!$$.


 * A modes
 * $$A=\left( \frac{T} \right)\left[ {{N}_{A}}\ln (T)-\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac\left( \frac{t_{i}^\ln (t_{i}^)-t_{i-1}^\ln (t_{i-1}^)}{t_{i}^-t_{i-1}^}-1 \right) \right]\hat{r}(T)\,\!$$


 * BC modes unseen


 * $$B{{C}_{unseen}}=\left( \frac{T} \right)\left[ {{N}_{BCF}}\ln (T)-\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac\left( \frac{t_{i}^\ln (t_{i}^)-t_{i-1}^\ln (t_{i-1}^)}{t_{i}^-t_{i-1}^}-1 \right) \right]\hat{r}(T)\,\!$$


 * BC modes seen


 * $$B{{C}_{seen}}=\left( \frac{T} \right)\left[ {{N}_{BCR}}\ln (T)-\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac\left( \frac{t_{i}^\ln (t_{i}^)-t_{i-1}^\ln (t_{i-1}^)}{t_{i}^-t_{i-1}^}-1 \right) \right]\hat{r}(T)\,\!$$


 * BD modes unseen


 * $$B{{D}_{unseen}}=\left( \frac{T} \right)\left[ {{N}_{BDF}}\ln (T)-\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac\left( \frac{t_{i}^\ln (t_{i}^)-t_{i-1}^\ln (t_{i-1}^)}{t_{i}^-t_{i-1}^}-1 \right) \right]\hat{r}(T)\,\!$$


 * BD modes seen


 * $$B{{D}_{seen}}=\left( \frac{T} \right)\left[ {{N}_{BDR}}\ln (T)-\underset{i=1}{\overset{K}{\mathop \sum }}\,\frac\left( \frac{t_{i}^\ln (t_{i}^)-t_{i-1}^\ln (t_{i-1}^)}{t_{i}^-t_{i-1}^}-1 \right) \right]\hat{r}(T)\,\!$$


 * BD modes remain


 * $$\begin{align}

B{{D}_{remain}} = & \left( 1-\frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}} \right)\cdot B{{D}_{seen}} \\ = & \left( 1-\overline{d} \right)\cdot B{{D}_{seen}} \end{align}\,\!$$


 * BD modes removed


 * $$\begin{align}

B{{D}_{removed}} = & \frac{1}{M}\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}\cdot B{{D}_{seen}} \\ = & \overline{d}\cdot B{{D}_{seen}} \end{align}\,\!$$

Example: Test-Fix-Find-Test Data
Consider the data given in the first table below. There were 56 total failures and $$T=400\,\!$$. The effectiveness factors of the unique BD modes are given in the second table. Determine the following:


 * Calculate the demonstrated MTBF and failure intensity.
 * Calculate the projected MTBF and failure intensity.
 * What is the rate at which unique BD modes are being generated during this test?
 * If the test continues for an additional 50 hours, what is the minimum number of new unique BD modes expected to be generated?



Solution


 * 1)	In order to obtain $${{\widehat{\lambda }}_{CA}}\,\!$$, use the traditional Crow-AMSAA model for test-fix-test to fit all 56 data points, regardless of the failure mode classification to get:


 * $$\begin{align}

\widehat{\beta }= & 0.91026 \\ \widehat{\lambda }= & 0.23969 \end{align}\,\!$$ Thus the achieved or demonstrated failure intensity is estimated by:
 * $$\begin{align}

{{\widehat{\lambda }}_{CA}} = & \widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}} \\ = & 0.23969\times 0.91026\times {{400}^{(0.91026-1)}} \\ = & 0.12744 	\end{align}\,\!$$ The achieved or demonstrated MTBF, $${{M}_{CA}}\,\!$$, is the system reliability attained at the end of test, $$T=400\,\!$$, and is estimated by:


 * $${{\widehat{M}}_{CA}}={{[{{\widehat{\lambda }}_{CA}}]}^{-1}}=7.84708\,\!$$


 * 2)	For this data set, $$M=16\,\!$$ and $$T=400\,\!$$.


 * $${{\widehat{\lambda }}_{BD}}=\frac{T}=\frac{32}{400}=0.08\,\!$$


 * $$\overline{d}=\underset{i=1}{\overset{M}{\mathop \sum }}\,{{d}_{i}}/M=0.72125\,\!$$


 * $$\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}/T=0.01955\,\!$$

Calculate $$\hat{\beta }\,\!$$ and $$\hat{\lambda }\,\!$$ of the BD modes using Eqns. (Extend1) and (Extend2):
 * $$\begin{align}

{{{\hat{\beta }}}_{BD}}= & 0.74715 \\ {{{\hat{\lambda }}}_{BD}} = & 0.18197 \end{align}\,\!$$ Then:
 * $$\overline{d}\widehat{h}(T|BD)=0.0215\,\!$$

Therefore:
 * $$\begin{align}

43)	  {{\widehat{\lambda }}_{EM}} = & {{\widehat{\lambda }}_{CA}}-{{\widehat{\lambda }}_{BD}}+\underset{i=1}{\overset{K}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T}+\overline{d}\widehat{h}(T|BD) \\ 	  = & 0.12744-0.08+0.0196+0.0215 \\ 	  = & 0.08854  	\end{align}\,\!$$ The Crow Extended model projected MTBF is:
 * $$\begin{align}

{{\widehat{M}}_{EM}} = & {{[{{\widehat{\lambda }}_{EM}}]}^{-1}} \\ = & 11.29418 	\end{align}\,\!$$ Consequently, based on the Crow Extended model and the data in Tables 9.3 and 9.4, the MTBF grew to 7.85 as a result of the corrective actions for the BC failure modes during the test. The MTBF then jumped to 11.29 after the test as a result of the delayed corrective actions for the BD failure modes. The management strategy can be summarized by the Failure Mode Strategy plot shown next.



This pie chart shows that 9.48% of the system's failure intensity has been left in (A modes), 31.81% of the failure intensity due to the BC modes has not been seen yet and 13.40% was removed during the test (BC modes - seen). In addition, 33.23% of the failure intensity due to the BD modes has not been seen yet, 3.37% will remain in the system since the corrective actions will not be completely effective at eliminating the identified failure modes and 8.72% will be removed after the delayed corrective actions.


 * 3)	The rate at which unique BD modes are being generated is equal to $$h{{(T|BD)}^{-1}}\,\!$$, where:


 * $$\begin{align}

h{{(T|BD)}^{-1}} = & \frac{1} \\ = & \frac{T}{M{{\widehat{\beta }}_{BD}}} \\ = & 33.4605 \end{align}\,\!$$


 * 4)	Unique BD modes are being generated every 33.4605 hours. If the test continues for another 50 hours, then at least one new unique BD mode would be expected to be seen from this additional testing.

As shown in the next figure, the MTBF of each individual failure mode can be plotted and the failure modes with the lowest MTBF can be identified. These are the failure modes that cause the majority of the system failures.



Confidence Bounds
The RGA software provides two methods to estimate the confidence bounds for the Crow Extended model when applied to developmental testing data. 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.

See Crow Extended Confidence Bounds for details on how these confidence bounds are calculated. This appendix contains calculations for:


 * Bounds on Demonstrated Failure Intensity
 * Bounds on Demonstrated MTBF
 * Bounds on Projected Failure Intensity
 * Bounds on Projected MTBF
 * Bounds on Growth Potential Failure Intensity
 * Bounds on Growth Potential MTBF

Confidence Bounds Example
Calculate the 2-sided 90% confidence bounds on the demonstrated, projected and growth potential failure intensity for the data in Table 9.1.

Solution

The estimated demonstrated failure intensity is $${{\widehat{\lambda }}_{D}}(T)=\tfrac{{{N}_{A}}+{{N}_{B}}}{T}=0.1050\,\!$$. Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


 * $$\begin{align}

{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{2}-\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{4}} \\ = & 0.08152 \end{align}\,\!$$


 * $$\begin{align}

{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{D}}(T)+\frac{2}+\sqrt{{{{\hat{\lambda }}}_{D}}(T){{C}^{2}}+\frac{4}} \\ = & 0.13525 \end{align}\,\!$$

The Crow confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


 * $$\begin{align}

{{[{{\lambda }_{D}}(T)]}_{L}} = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,1-\alpha /2)}^{2}}{2N} \\ = & 0.07985 \\ {{[{{\lambda }_{D}}(T)]}_{U}}  = & {{\widehat{\lambda }}_{D}}(T)\frac{\chi _{(2N,\alpha /2)}^{2}}{2N} \\ = & 0.13299 \end{align}\,\!$$

The projected failure intensity is. Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:


 * $$\begin{align}

{{[{{{\hat{\lambda }}}_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.04902 \end{align}\,\!$$


 * $$\begin{align}

{{[{{{\hat{\lambda }}}_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.08915 \end{align}\,\!$$

The Crow confidence bounds for the projected failure intensity are:


 * $$\begin{align}

{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{4}} \\ = & 0.04807 \\ {{[{{\lambda }_{P}}(T)]}_{U}}  = & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{4}} \\ = & 0.09090 \end{align}\,\!$$

The growth potential failure intensity is $$\widehat{r}_{GP} (T) = \left (\frac{N_A}{T} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} \right ) = 0.04455 \,\!$$.

Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:


 * $$\begin{align}

{{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{4}} \\ = & 0.03020 \\ {{r}_{U}}  = & {{{\hat{r}}}_{GP}}+\frac{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{4}} \\ = & 0.0656 \end{align}\,\!$$

The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential failure intensity.



The following figure shows these bounds based on the Crow method.



Another Confidence Bounds Example
Calculate the 2-sided confidence bounds at the 90% confidence level on the demonstrated, projected and growth potential MTBF for the data in Table 9.3. Solution For this example, there are A, BC and BD failure modes, so the estimated demonstrated failure intensity, $${{\hat{\lambda }}_{D}}(T)\,\!$$, is simply the Crow-AMSAA model applied to all A, BC, and BD data.


 * $${{\hat{\lambda }}_{D}}(T)={{\widehat{\lambda }}_{CA}}=\widehat{\lambda }\widehat{\beta }{{T}^{\widehat{\beta }-1}}=0.12744\,\!$$

Therefore, the demonstrated MTBF is:


 * $$MTB{{F}_{D}}={{[{{\hat{\lambda }}_{D}}(T)]}^{-1}}=7.84708\,\!$$

Based on this value, the Fisher Matrix confidence bounds for the demonstrated failure intensity at the 90% confidence level are:


 * $$\begin{align}

{{[{{\lambda }_{D}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\ = & 0.09339 \end{align}\,\!$$


 * $$\begin{align}

{{[{{\lambda }_{D}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{CA}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{CA}}(T))}/{{{\hat{\lambda }}}_{i}}(T)}} \\ = & 0.17390 \end{align}\,\!$$

The Fisher Matrix confidence bounds for the demonstrated MTBF at the 90% confidence level are:


 * $$\begin{align}

MTB{{F}_} = & \frac{1} \\ = & 5.75054 \\ MTB{{F}_}  = & \frac{1} \\ = & 10.70799 \end{align}\,\!$$

The Crow confidence bounds for the demonstrated MTBF at the 90% confidence level are:


 * $$\begin{align}

MTB{{F}_} = & \frac{1} \\ = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,\alpha /2)}{2N}} \\ = & 5.6325 \\ MTB{{F}_} = & \frac{1} \\ = & \frac{1}{{{\widehat{\lambda }}_{D}}(T)\tfrac{{{\chi }^{2}}(2N,1-\alpha /2)}{2N}} \\ = & 10.8779 \end{align}\,\!$$

The projected failure intensity is $$\hat{\lambda}_P (T) = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) = 0.0885 \,\!$$. Based on this value, the Fisher Matrix confidence bounds at the 90% confidence level for the projected failure intensity are:


 * $$\begin{align}

{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.0681 \end{align}\,\!$$


 * $$\begin{align}

{{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T){{e}^{-{{z}_{\alpha }}\sqrt{Var({{{\hat{\lambda }}}_{P}}(T))}/{{{\hat{\lambda }}}_{P}}(T)}} \\ = & 0.1152 \end{align}\,\!$$

The Fisher Matrix confidence bounds for the projected MTBF at the 90% confidence level are:


 * $$\begin{align}

MTB{{F}_} = & \frac{1} \\ = & 8.6818 \\ MTB{{F}_}  = & \frac{1} \\ = & 14.6926 \end{align}\,\!$$

The Crow confidence bounds for the projected failure intensity are:


 * $$\begin{align}

{{[{{\lambda }_{P}}(T)]}_{L}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}-\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot \ \,{{C}^{2}}+\frac{4}} \\ = & 0.0672 \\ {{[{{\lambda }_{P}}(T)]}_{U}} = & {{{\hat{\lambda }}}_{P}}(T)+\frac{2}+\sqrt{{{{\hat{\lambda }}}_{P}}(T)\cdot {{C}^{2}}+\frac{4}} \\ = & 0.1166 \end{align}\,\!$$

The Crow confidence bounds for the projected MTBF at the 90% confidence level are:


 * $$\begin{align}

MTB{{F}_} = & \frac{1} \\ = & 8.5743 \\ MTB{{F}_}  = & \frac{1} \\ = & 14.8769 \end{align}\,\!$$

The growth potential failure intensity is $$\widehat{\lambda}_{GP} = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} = 0.0670 \,\!$$. $$\hat{\lambda}_P (T) = \widehat{\lambda}_{CA} - \widehat{\lambda}_{BD} + \sum_{i=1}^M (1-d_i) \tfrac{N_i}{T} + \bar{d}\widehat{h}(T|BD) = 0.0885 \,\!$$.Based on this value, the Fisher Matrix and Crow confidence bounds at the 90% confidence level for the growth potential failure intensity are:


 * $$\begin{align}

{{r}_{L}} = & {{{\hat{r}}}_{GP}}+\frac{2}-\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{4}} \\ = & 0.0488 \\ {{r}_{U}}  = & {{{\hat{r}}}_{GP}}+\frac{2}+\sqrt{{{{\hat{r}}}_{GP}}{{C}^{2}}+\frac{4}} \\ = & 0.0919 \end{align}\,\!$$

The Fisher Matrix and Crow confidence bounds for the growth potential MTBF at the 90% confidence level are:


 * $$\begin{align}

MTB{{F}_{G{{P}_{L}}}} = & \frac{1} \\ = & 10.8790 \\  MTB{{F}_{G{{P}_{U}}}}  = & \frac{1} \\ = & 20.4855 \end{align}\,\!$$

The figure below shows the Fisher Matrix confidence bounds at the 90% confidence level for the demonstrated, projected and growth potential MTBF.



The next figure shows these bounds based on the Crow method.

Grouped Data
Parameter estimation for grouped data is the same as described previously in the Crow-AMSAA (NHPP) chapter. The equations used to estimate the parameters of the Crow Extended model are presented next. For test-find-test data, the maximum likelihood estimates of $${{\lambda }_{BD}}\,\!$$ and $${{\beta }_{BD}}\,\!$$ are calculated using the first occurrences of the BD modes such that:


 * $$\underset{i=1}{\overset{k}{\mathop \sum }}\,{{n}_{i}}\left[ \frac{T_{i}^{\widehat{\beta }}\ln {{T}_{i}}-T_{i-1}^{\widehat{\beta }}\ln {{T}_{i-1}}}{T_{i}^{\widehat{\beta }}-T_{i-1}^{\widehat{\beta }}}-\ln {{T}_{k}} \right]=0\,\!$$


 * $$\widehat{\lambda }=\frac{n}{T_{k}^{\widehat{\beta }}}\,\!$$

where $${{n}_{i}}\,\!$$ is the number of distinct BD modes within the $${{i}^{th}}\,\!$$ interval. For test-fix-find-test data, the maximum likelihood estimates of $${{\lambda }_{BC}}\,\!$$ and $${{\beta }_{BC}}\,\!$$ are estimated in the same manner using the first occurrences of the BC modes.

Confidence Bounds for Grouped Data

 * Parameters: The confidence bounds on the parameters for the Crow Extended model for grouped data are calculated using the same as the procedure presented in the Crow-AMSAA (NHPP) chapter.
 * Failure Intensity and MTBF:
 * If there are no BC modes, the confidence bounds on the demonstrated failure intensity and MTBF, projected failure intensity and MTBF and growth potential failure intensity and MTBF are the same as the procedure presented for non-grouped data.
 * If there are BC modes, then the confidence bounds on the demonstrated failure intensity and MTBF are the same as the procedure presented in Crow-AMSAA (NHPP) and the confidence bounds on the projected failure intensity and MTBF and growth potential failure intensity and MTBF are the same as fornon-grouped data.
 * Time: The confidence bounds on time are the same as the procedure presented in Crow-AMSAA (NHPP).

Mixed Data
The Crow Extended model also can be applied for discrete data from one-shot (success/failure) testing. In the RGA software, the Discrete Data > Mixed Data option gives a data sheet that can accommodate data from tests where a single unit is tested for each successive configuration (individual trial-by-trial), where multiple units are tested for each successive configuration (configurations in groups) or a combination of both. This data sheet can be analyzed with either the Crow-AMSAA (NHPP) model or the Crow Extended model. Corrective actions cannot take place at the time of failure for discrete data. With that in mind, the mixed data type does not allow for BC modes. For discrete data there are only A or BD modes. In terms of practical applications, think of a growth test for missile systems. Since these are one-shot items, the fixes to failure modes are delayed until at least the next trial. Note that for calculation purposes it is required to have at least three failures in the first interval. If that is not the case, then the data set needs to be grouped before calculating. The RGA software performs this operation in the background.

Example
A one-shot system underwent reliability growth testing for a total of 20 trials. The test was performed as a combination of groups of units with the same configuration and individual trials. The following table shows the data set. The "Failures in Interval" column specifies the number of failures that occurred in each interval and the "Cumulative Trials" column specifies the cumulative number of trials at the end of that interval. In other words, the first three rows contain the data from the first trial, in which 8 units with the same configuration were tested and 3 failures (with different failure modes) were observed. The next row contains data from the second trial, in which 2 units with the same configuration were tested and no failures occurred. And so on.

The table also gives the classifications of the failure modes. There are 5 BD modes. The average effectiveness factor for the BD modes is 0.7. Do the following:
 * 1)	Calculate the demonstrated reliability at the end of the test.


 * 2)	Calculate the growth potential reliability.

Solution

Based on the equations presented in Crow-AMSAA (NHPP), the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:


 * $$\widehat{\beta }=0.8572\,\!$$


 * and:


 * $$\widehat{\lambda }=0.4602\,\!$$

However, because there are only A or BD modes for mixed data, there is no growth during the test. In other words, the hypothesis for the $$\widehat{\beta }\,\!$$ parameter is that $$\widehat{\beta }=1.\,\!$$

From Crow-AMSAA (NHPP), we know that:


 * $$\widehat{\lambda }=\frac{n}{T_{n}^{\widehat{\beta }}}\,\!$$

or, if $$\widehat{\beta }=1\,\!$$, this becomes:


 * $$\begin{align}

\widehat{\lambda }= & \frac{n} \\ = & \frac{6}{20} \\ = & 0.3 \end{align}\,\!$$

As we have seen, the Crow-AMSAA instantaneous failure intensity, $${{\lambda }_{i}}(T)\,\!$$, is defined as:


 * $$\begin{align}

{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0 \end{align}\,\!$$

Using trials instead of time, and accommodating for $$\widehat{\beta }=1\,\!$$, we can calculate the instantaneous failure probability at the end of the test, or $$T=20\,\!$$ :


 * $${{Q}_{i}}(20)=\widehat{\lambda }=0.3\,\!$$

So the instantaneous reliability at the end of the test, or demonstrated reliability, is:


 * $$\begin{align}

{{R}_{i}}(20)= & 1-{{Q}_{i}}(20) \\ = & 1-0.3 \\  = & 0.7  \end{align}\,\!$$

The next figure shows the data sheet as calculated in the RGA software.

Based on Eqn. (extendedGP), the growth potential unreliability is:


 * $$\begin{align}

{{\widehat{Q}}_{GP}}(T)= & \left( \frac{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{T} \right) \\ = & \underset{i=1}{\overset{M}{\mathop \sum }}\,(1-0.7)\frac{T} \\ = & 0.3*(\frac{1+1+1+1+1+1)}{20} \\ = & 0.09 \end{align}\,\!$$

So the growth potential reliability is:


 * $$\begin{align}

{{\widehat{R}}_{GP}}(T) = & 1-{{\widehat{Q}}_{GP}}(T) \\ = & 1-0.09 \\  = & 0.91  \end{align}\,\!$$

The figures below show the calculation of the growth potential reliability for the mixed data using the RGA software's QCP, followed by the growth potential plot.





Multiple Systems with Event Codes
The Multiple Systems with Event Codes data type is used to analyze the failure data from a reliability growth test in which a number of systems are tested concurrently and the implemented fixes are tracked during the test phase. With this data type, all of the systems under test are assumed to have the same system hours at any given time. The Crow Extended model is used for this data type so all the underlying assumptions regarding the Crow Extended model apply. As such, this data type is only applicable to data within a single test phase.

As previously presented, the failure mode classifications for the Crow Extended model are defined as follows:


 * A indicates that no corrective action was performed or will be performed (management chooses not to address for technical, financial or other reasons).
 * BC indicates that the corrective action was implemented during the test. The analysis assumes that the effect of the corrective action was experienced during the test (as with other test-fix-test reliability growth analyses).
 * BD indicates that the corrective action will be delayed until after the completion of the current test.

Therefore, implemented fixes can only be applied to BC modes since all BD modes are assumed to be delayed until the end of the test. For each BC mode, there must be a separate entry in the data set that records the time when the fix was implemented during the test.

Event Codes
The Multiple Systems with Event Codes data type analyzed with the Crow Extended model uses a column to indicate the types of events that occurred during a test phase. Within RGA, event codes are entered within the Event column of the Multiple Systems with Event Codes data sheets.

The possible event codes that can be used in the analysis are:

I: denotes that a certain BC failure mode has been corrected at the specific time; in other words, a fix has been implemented. For this data type, each BC mode must have an associated I event. The I event is essentially a timestamp for when the fix was implemented during the test.

Q: indicates that the failure was due to a quality issue. An example of this might be a failure caused by a bolt not being tightened down properly. You have the option to decide whether or not to include quality issues in the analysis. This option can be specified by checking or clearing the Include Q Events checkbox under Event Code Options on the Analysis tab.

P: indicates that the failure was due to a performance issue. You can determine whether or not to include performance issues in the analysis. This option can be specified by checking or clearing the Include P Events checkbox under Event Code Options on the Analysis tab.

X: indicates that you wish to exclude the data point from the analysis. An "X" can be placed in front of any existing event code (e.g., XF to exclude a particular failure time) or entered by itself. This row of data will then not be included in the analysis.

S: indicates the system start time. This event code is only selectable in the Normal View.

F: indicates a failure time.

E: indicates the system end time. This event code is only selectable in the Normal View.

The analysis is based on the equivalent system that combines the operating hours of all the systems.

Equivalent Single System
In order to analyze a Multiple Systems with Event Codes data sheet, the data are converted into a Crow Extended equivalent single system. The implemented fixes (I events) are taken into account when building the equivalent single system from the data for multiple systems.

The basic assumptions and constraints for the use of this data type are listed below:


 * Failure modes are assumed to be independent of each other and with respect to the system configuration. The same applies to their related implemented fixes (I events). As such, each mode and its related implemented fixes (I events) are examined separately in terms of their impact to the system configuration.
 * If there are BC modes in the data set, there must be at least 3 unique BC modes to analyze the data (together with implemented fixes for each one of them).
 * If there are BD modes in the data set, there must be at least 3 unique BD modes to analyze the data.
 * To be consistent with the definition of BC modes in the Crow Extended model, every BC mode must have at least one implemented fix (I event) on at least one system.
 * Implemented fixes (I events) cannot be delayed to a later phase, because the Crow Extended model applies to a single phase only.

The following are the basic rules for calculating the equivalent single system on which the Crow Extended model is applied. Note that the list is not exhaustive since there is an infinite number of scenarios that can occur. These rules cover the most common scenarios. The main concept is to add the time that each system was tested under the same configuration.


 * 1) To get to the equivalent single system, each failure time for A modes and BD modes is calculated by adding the time that each system was tested under the same configuration. In practice this means multiplying the failure time in the system by the number of total systems under test. For example, if we have 4 total systems and system 2 has a BD1 mode at time 30, the BD1 mode failure time in the equivalent single system will be $$30*4=120\,\!$$. If system 3 had another BD1 mode at time 40, then that would yield another BD1 mode in the equivalent single system at time $$40*4=160\,\!$$. These calculations are done assuming that the start time for the systems are at time zero. If the start time is different than zero, then that time would have to be subtracted from the failure time on each system. For example, if system 1 started at time S=10 and there was a failure at time 30, the equivalent system time would be $$(30-10)*4=80\,\!$$.
 * 2) Each failure time for a BC mode that occurred before an implemented fix (I event) for that mode is also calculated by multiplying the failure time in the system by the number of total systems in test, as described above.
 * 3) The implemented fix (I event) time in the equivalent single system is calculated by adding the test time invested in each system before that I event takes place. It is the total time that the system has spent at the same configuration in terms of that specific mode.
 * 4) If the same BC mode occurs in another system after a fix (I event) has been implemented in one or more systems, the failure time in the equivalent single system is calculated by adding the test time for that BC mode, and one of the following for each of the other systems:
 * 5) If a BC mode occurs in a system that has already seen an I event for that mode, then you add the time up to the I event.
 * 6) If the I events occurred later than the BC failure time or those systems did not have any I events for that mode, then you add the time of the BC failure.
 * 7) If the same BC mode occurs in the same system after a fix (I event) has been implemented in one or more systems, the failure time in the equivalent single system is calculated by adding the test time of each system after that I event was implemented to the I event time in the equivalent single system, or zero if an I event was not present in that system.

Example: Equivalent Single System
Consider the data set shown in the following figure. The data sheet needs to be converted to the Crow Extended equivalent single system. This example is used to demonstrate the application of the five rules mentioned above.



Solution

The first step to create the equivalent single system is to isolate each failure mode and its implemented fixes independently from each other. The numbered items that follow represent an example application for each one of the five rules mentioned above and are presented in the same numbering sequence.


 * 1. The next figure illustrates the application of rule #1 for mode BD1. The mode in the equivalent single system is calculated as $$(75+75+75)=225\,\!$$ or $$75*3=225.\,\!$$




 * While the next figure illustrates the application of rule #1 for mode A110. The mode in the equivalent single system is calculated as $$(280+280+280)=840\,\!$$ or $$280*3=840.\,\!$$




 * 2. The next figure illustrates the application of rule #2 for the first occurrence of the mode BC10 in system 1. The mode in the equivalent single system is calculated as $$(150+150+150)=450\,\!$$ or $$150*3=450.\,\!$$




 * 3. The next figure illustrates the application of rule #3 for implemented fixes (I events) of the mode BC10. In the graph the I events are symbolized by having the letter "I" before the naming of the mode, in this case IBC10 for the implemented fix of mode BC10. The IBC10 in the equivalent single system is calculated as $$(200+175+200)=575\,\!$$.




 * 4. The next figure illustrates the application of rule #4 for the mode BC20 in system 1, which occurs after a fix for the same mode was implemented in system 2. The mode in the equivalent single system is calculated as $$(350+300+350)=1000.\,\!$$




 * 5. The next figure illustrates the application of rule #5 for the second occurrence of the mode BC10 in system 1, which occurs after an implemented fix (I event) had occurred for that mode in the same system. The mode in the equivalent single system is calculated as $$575+(175+200+175)=1125.\,\!$$



After having transferred the data set to the Crow Extended equivalent single system, the data set is analyzed using the Crow Extended model as presented in this chapter. The last figure shows the growth potential MTBF plot.



Transferring Data to an Equivalent Single System
RGA provides the capability to transfer a Multiple Systems with Event Codes data sheet to various other data types. The following picture shows the available data types that the data sheet can be converted into. When selecting to transfer to an equivalent single system, the data sheet is converted to a Crow Extended - Continuous Evaluation data sheet.



The Crow Extended - Continuous Evaluation model is designed for analyzing data across multiple test phases, while considering the data for all phases as one data set. Familiarity with this model is necessary for the discussion presented in this section.

When using the Crow Extended - Continuous Evaluation model to transfer the data sheet from Multiple Systems with Event Codes to an equivalent single system, the following rules are used (in addition to the five basic rules presented earlier for calculating the equivalent single system):


 * BD modes in the Crow Extended data sheet become BD modes in the equivalent single system of the Crow Extended - Continuous Evaluation data sheet.
 * BC modes in the Crow Extended data sheet become BD modes in the equivalent single system of the Crow Extended - Continuous Evaluation data sheet. These BD modes will have associated implemented fixes (I events). Implemented fixes (I events) for BC modes in the Crow Extended data sheet become implemented fixes (I events) for the converted BD modes in the equivalent single system of the Crow Extended - Continuous Evaluation data sheet.
 * If an implemented fix (I event) occurred at the same time as the failure and was implemented at that exact time across all systems, then this becomes a BC mode in the equivalent single system. If the fixes (I events) were not all implemented at the same time or if the fix was not implemented on all systems at the failure time, then this becomes a BD mode in the equivalent single system.

The next figure shows the transferred equivalent single system Crow Extended - Continuous Evaluation data sheet from the Multiple Systems with Event Codes data sheet for the data from the Equivalent Single System example given above.



Iteration Method for Naming Repeated Modes
When recording modes for transfer from the Multiple Systems with Event Codes to a Crow Extended -Continuous Evaluation equivalent single system, it is recommended to consider using an iteration method to name subsequent recurrences of the same mode. This will help alleviate any issues with the conversion of the definitions of the modes from the Crow Extended model to the Crow Extended - Continuous Evaluation model. For example, if the first occurrence of a mode is BC25, then the second occurrence is suggested to be named as BC25.1. The reasoning behind this recommendation is that in the case that BC25 in the Multiple Systems with Event Codes data sheet has received implemented fixes (I events) at the same time that the failure occurred, in all systems, then this mode will be translated as a BC mode in the Crow Extended - Continuous Evaluation equivalent single system. The next recurring failure would also be treated as a BC mode but in reality it did not have an implemented fix (I event) at the time of failure.

For example, consider the data set shown in the following figure, which represents one system only for simplicity. Notice that the modes BC25, BC35 and BC45 received implemented fixes at the time of failure. Based on that, when they get transferred to the Crow Extended - Continuous Evaluation equivalent single system, they will be considered as BC modes. The subsequent failures of the modes 25, 35, and 45 will also be converted to BC modes, when in reality they had implemented fixes (I events) at a later time.

The RGA software will display a warning if you try to convert this data sheet without using iterations.

The next figure shows the same data sheet with the use of iterations for the modes 25, 35 and 45. The subsequent failures are named as BC25.1, BC35.1 and BC45.1.



This way, the conversion to the Crow Extended - Continuous Evaluation model occurs in a valid fashion, since although the original BC modes are converted to BC25, BC35 and BC45, the subsequent failures are converted to BD25.1, BD35.1 and BD45.1 together with their respective implemented fixes (I events). This is shown in the next figure below. Note that the use of iterations is recommended only when transferring to the Crow Extended - Continuous Evaluation equivalent single system, it is not necessary when using the Multiple Systems with Event Codes data sheet that is calculated with the Crow Extended model.



Adjusting the Failure Mode Management Strategy
Three systems were subjected to a reliability growth test to evaluate the prototype of a new product. Once the test was completed a failure analysis was done and, based on this, a management strategy was able to be defined. It was determined that all corrective actions will be delayed until after the test. The tables below shows the collected data set and the associated effectiveness factors for the unique BD modes. The prototype is required to meet a projected MTBF goal of 55 hours. Do the following:


 * 1)	Estimate the parameters of the Crow Extended model.
 * 2)	Based on the current management strategy what is the projected MTBF?
 * 3)	If the projected MTBF goal is not met, alter the current management strategy to meet this requirement with as little adjustment as possible and without changing the EFs of the existing BD modes. Assume an EF = 0.7 for any newly assigned BD modes.



Solution
 * 1)	The next figure shows the estimated Crow Extended parameters.




 * 2)	There are a couple of ways to calculate the projected MTBF, but the easiest is via the Quick Calculation Pad (QCP), as shown next.




 * 3)	From the previous question, the projected MTBF is estimated to be 53.9390 hours, which is below the goal of 55 hours. To reach our goal, or to see if we can even get there, the management strategy must be changed. The effectiveness factors for the existing BD modes cannot be changed, however it is possible to change an A mode to a BD mode. But which A mode(s) should be changed? To answer this question, you can view the Individual Mode Failure Intensity plot with just the A modes displayed as shown next. As you can see from the plot, failure mode A45 has the highest failure intensity. Therefore, among the A modes this particular failure mode is having the greatest negative effect in regards to the system MTBF.



So change A45 to BD45. Be sure to change all instances of A45 to a BD mode. Enter an effectiveness factor for BD45 equal to 0.7 and recalculate the parameters of the Crow Extended model. Now go back to the QCP to calculate the projected MTBF as shown below. The projected MTBF is now estimated to be 55.5903 hours. Based on the change in the management strategy, the projected MTBF goal is now expected to be met.



Estimating the Failure Intensity Remaining After Fixes
A reliability growth test was conducted for 200 hours. Some of the corrective actions were applied during the test while others were delayed until after the test was completed. The tables below give the data set and the effectiveness factors for the BD modes. Do the following:


 * 1)	Estimate the parameters of the Crow Extended model.
 * 2)	Determine the average effectiveness factor of the BC modes using the Function Wizard.
 * 3)	What percent of the failure intensity will be left in the system due to the BD modes after implementing the delayed fixes?



Solution
 * 1)	The next figure shows the estimated parameters of the Crow Extended model.




 * 2)	After inserting a General Spreadsheet into the Folio, the Function Wizard can be accessed via the Tools menu. Once the Function Wizard is loaded, select Average Effectiveness Factor from the list of available functions and under Avg. Eff. Factor select BC modes as shown next. Click OK and the result will be placed into the General Spreadsheet. The average effectiveness factor for the BC modes is 0.6983.




 * 3)	The percent of the failure intensity left in the system due to the BD modes can be determined using the Failure Mode Strategy plot as shown next. Therefore, the percent of the failure intensity left in the system due to the BD modes is 1.79%.



Determining if Design Will Meet MTBF Goal
Two prototypes of a new design are tested simultaneously. Whenever a failure was observed for one unit, the current operating time of the other unit was also recorded. The test was terminated after 300 hours. All of the design changes for the prototypes were delayed until after completing the test and the data set is given in the table below. Assume a fixed effectiveness factor equal to 0.7. The MTBF goal for the new design is 30 hours. Do the following:


 * 1)	Estimate the parameters of the Crow Extended model.
 * 2)	What is the projected MTBF and growth potential?
 * 3)	Under the current management strategy, is it even possible to reach the MTBF goal of 30 hours?

Solution
 * 1)	The next figure shows the estimated Crow Extended parameters.




 * 2)	One possible method to calculate the projected MTBF and growth potential is to use the Quick Calculation Pad, but you can also view these two values at the same time by viewing the Growth Potential MTBF plot, which is displayed next. From the plot, the projected MTBF is equal to 16.87 hours and the growth potential is equal to 18.63 hours.




 * 3)	The current projected MTBF and growth potential MTBF are both well below the required goal of 30 hours. To check if this goal can even be reached, you can set the effectiveness factor equal to 1. In other words, if all of the corrective actions were to remove the failure modes completely then what would be the projected and growth potential MTBF? After changing the fixed effectiveness factor to 1, the parameters are recalculated and the Growth Potential plot is refreshed. The refreshed plot is shown in below. Even if you assume an effectiveness factor equal to 1, the growth potential is still only 27.27 hours. Based on the current design process, it will not be possible to reach the MTBF goal of 30 hours. Therefore, there are basically two options: start a new design stage or reduce the required MTBF goal.