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==Crow Discrete Reliability Growth Model==
#REDIRECT [[Crow-AMSAA - NHPP]]
The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
<br>
===Model Development===
Suppose system development is represented by  <math>i</math>  configurations. This corresponds to  <math>i-1</math>  configuration changes, unless fixes are applied at the end of the test phase, in which case there would be  <math>i</math>  configuration changes. Let  <math>{{N}_{i}}</math>  be the number of trials during configuration  <math>i</math>  and let  <math>{{M}_{i}}</math>  be the number of failures during configuration  <math>i</math> . Then the cumulative number of trials through configuration  <math>i</math> , namely  <math>{{T}_{i}}</math> , is the sum of the  <math>{{N}_{i}}</math>  for all  <math>i</math> , or:
 
::<math>{{T}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{N}_{i}}</math>
 
And the cumulative number of failures through configuration  <math>i</math> , namely  <math>{{K}_{i}}</math> , is the sum of the  <math>{{M}_{i}}</math>  for all  <math>i</math> , or:
 
::<math>{{K}_{i}}=\underset{}{\overset{}{\mathop \sum }}\,{{M}_{i}}</math>
 
The expected value of  <math>{{K}_{i}}</math>  can be expressed as  <math>E[{{K}_{i}}]</math>  and defined as the expected number of failures by the end of configuration  <math>i</math> . Applying the learning curve property to  <math>E[{{K}_{i}}]</math>  implies:
 
::<math>E\left[ {{K}_{i}} \right]=\lambda T_{i}^{\beta }</math>
 
Denote  <math>{{f}_{1}}</math>  as the probability of failure for configuration 1 and use it to develop a generalized equation for  <math>{{f}_{i}}</math>  in terms of the  <math>{{T}_{i}}</math>  and  <math>{{N}_{i}}</math> . From Eqn. (expectedn), the expected number of failures by the end of configuration 1 is:
 
::<math>E\left[ {{K}_{1}} \right]=\lambda T_{1}^{\beta }={{f}_{1}}{{N}_{1}}</math>
 
::<math>\therefore {{f}_{1}}=\frac{\lambda T_{1}^{\beta }}{{{N}_{1}}}</math>
 
Applying Eqn. (expectedn) 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:
 
::<math>\begin{align}
  & E\left[ {{K}_{2}} \right]= & \lambda T_{2}^{\beta } \\
& = & {{f}_{1}}{{N}_{1}}+{{f}_{2}}{{N}_{2}} \\
& = & \lambda T_{1}^{\beta }+{{f}_{2}}{{N}_{2}} 
\end{align}</math>
 
::<math>\therefore {{f}_{2}}=\frac{\lambda T_{2}^{\beta }-\lambda T_{1}^{\beta }}{{{N}_{2}}}</math>
 
By this method of inductive reasoning, a generalized equation for the failure probability on a configuration basis,  <math>{{f}_{i}}</math> , is obtained, such that:
 
::<math>{{f}_{i}}=\frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}}</math>
 
For the special case where  <math>{{N}_{i}}=1</math>  for all  <math>i</math> , Eqn. (dfi) becomes a smooth curve,  <math>{{g}_{i}}</math> , that represents the probability of failure for trial by trial data, or:
 
::<math>{{g}_{i}}=\lambda \cdot {{i}^{\beta }}-\lambda \cdot {{\left( i-1 \right)}^{\beta }}</math>
 
In Eqn. (dfi1),  <math>i</math>  represents the trial number. Thus using Eqn. (dfi), an equation for the reliability (probability of success) for the  <math>{{i}^{th}}</math>  configuration is obtained:
 
::<math>{{R}_{i}}=1-{{f}_{i}}</math>
 
And using Eqn. (dfi1), the equation for the reliability for the  <math>{{i}^{th}}</math>  trial is:
 
::<math>{{R}_{i}}=1-{{g}_{i}}</math>
 
===Maximum Likelihood Estimators===
This section describes procedures for estimating the parameters of the Crow-AMSAA model for success/failure data. An example is presented illustrating these concepts. The estimation procedures described below provide maximum likelihood estimates (MLEs) for the model's two parameters,  <math>\lambda </math>  and  <math>\beta </math> . The MLEs for  <math>\lambda </math>  and  <math>\beta </math>  allow for point estimates for the probability of failure, given by:
 
::<math>{{\hat{f}}_{i}}=\frac{\hat{\lambda }T_{i}^{{\hat{\beta }}}-\hat{\lambda }T_{i-1}^{{\hat{\beta }}}}{{{N}_{i}}}=\frac{\hat{\lambda }\left( T_{i}^{{\hat{\beta }}}-T_{i-1}^{{\hat{\beta }}} \right)}{{{N}_{i}}}</math>
 
And the probability of success (reliability) for each configuration  <math>i</math>  is equal to:
 
::<math>{{\hat{R}}_{i}}=1-{{\hat{f}}_{i}}</math>
 
The likelihood function is:
<br>
::<math>\underset{i=1}{\overset{k}{\mathop \prod }}\,\left( \begin{matrix}
  {{N}_{i}}  \\
  {{M}_{i}}  \\
\end{matrix} \right){{\left( \frac{\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{M}_{i}}}}{{\left( \frac{{{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta }}{{{N}_{i}}} \right)}^{{{N}_{i}}-{{M}_{i}}}}</math>
 
Taking the natural log on both sides yields:
 
::<math>\begin{align}
  & \Lambda = & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ \ln \left( \begin{matrix}
  {{N}_{i}}  \\
  {{M}_{i}}  \\
\end{matrix} \right)+{{M}_{i}}\left[ \ln (\lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] \\
&  & +\underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ ({{N}_{i}}-{{M}_{i}})\left[ \ln ({{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta })-\ln {{N}_{i}} \right] \right] 
\end{align}</math>
 
Taking the derivative with respect to  <math>\lambda </math>  and  <math>\beta </math>  respectively, exact MLEs for  <math>\lambda </math>  and  <math>\beta </math>  are values satisfying the following two equations:
 
::<math>\begin{align}
  & \underset{i=1}{\overset{K}{\mathop \sum }}\,{{H}_{i}}\times {{S}_{i}}= & 0 \\
& \underset{i=1}{\overset{K}{\mathop \sum }}\,{{U}_{i}}\times {{S}_{i}}= & 0 
\end{align}</math>
 
:where:
 
::<math>\begin{align}
  & {{H}_{i}}= & \underset{i=1}{\overset{K}{\mathop \sum }}\,\left[ T_{i}^{\beta }\ln {{T}_{i}}-T_{i-1}^{\beta }\ln {{T}_{i-1}} \right] \\
& {{S}_{i}}= & \frac{{{M}_{i}}}{\left[ \lambda T_{i}^{\beta }-\lambda T_{i-1}^{\beta } \right]}-\frac{{{N}_{i}}-{{M}_{i}}}{\left[ {{N}_{i}}-\lambda T_{i}^{\beta }+\lambda T_{i-1}^{\beta } \right]} \\
& {{U}_{i}}= & T_{i}^{\beta }-T_{i-1}^{\beta }\, 
\end{align}</math>
 
'''Example 8'''
<br>
A one-shot system underwent reliability growth development testing for a total of 68 trials. Delayed corrective actions were incorporated after the 14th, 33rd and 48th trials. From trial 49 to trial 68, the configuration was not changed.
<br>
• Configuration 1 experienced 5 failures,
<br>
• Configuration 2 experienced 3 failures,
<br>
• Configuration 3 experienced 4 failures and
<br>
• Configuration 4 experienced 4 failures.
<br>
<br>
:1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
:2) Estimate the unreliability and reliability by configuration.
<br>
<br>
'''Solution'''
<br>
:1) The solution of Eqns. (solution1) and (solution2) provides for  <math>\lambda </math>  and  <math>\beta </math>  corresponding to 0.5954 and 0.7801, respectively.
:2) Table 5.6 displays the results of Eqns. (ffffi) and (rrrri).
Figures 4fig816 and 4fig817 show plots of the estimated unreliability and reliability by configuration.
<br>
<center>Table 5.6 - Estimated failure probability and reliability by configuration</center>
{|style= align="center" border="2"
!Configuration(<math>i</math>)
!Estimated Failure Probability
!Estimated Reliability
|-
|1|| 0.333|| 0.667
|-
|2|| 0.234|| 0.766
|-
|3|| 0.206|| 0.794
|-
|4|| 0.190|| 0.810
|}
<br>
[[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
<br>
[[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]
<br>
 
===Mixed Data===
<br>
In the RGA Software, the Discrete Data > Mixed Data option gives a data sheet that can have input data that is either configuration in groups or individual trial by trial, or a mixed combination of individual trials and configurations of more than one trial. The calculations use the same mathematical methods described in section 5.3 for the Crow-AMSAA grouped data.
<br>
'''Example 9'''
<br>
Table 5.7 shows the number of fai
<math>\widehat{\beta }=0.7950</math>
lures of each interval of trials and the cumulative number of trials in each interval for a reliability growth test. For example, the first row of Table 5.7 indicates that for an interval of 14 trials, 5 failures occurred.
<br>
{|style= align="center" border="2"
|+'''Table 5.7 - Mixed data for Example 9'''
!Failures in Interval
!Cumulative Trials
|-
|5|| 14
|-
|3|| 33
|-
|4|| 48
|-
|0|| 52
|-
|1|| 53
|-
|0|| 57
|-
|1|| 58
|-
|0|| 62
|-
|1|| 63
|-
|0|| 67
|-
|1|| 68
|}
 
<br>
Using RGA 7, the parameters of the Crow-AMSAA model are estimated as follows:
 
:and:
 
::<math>\widehat{\lambda }=0.5588</math>
 
As we have seen, the Crow-AMSAA instantaneous failure intensity,  <math>{{\lambda }_{i}}(T)</math> , is defined as:
 
::<math>{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0</math>
 
Using the above parameter estimates, we can calculate the or instantaneous unreliability at the end of the test, or  <math>T=68.</math>
 
::<math>{{R}_{i}}(68)=0.5588\cdot 0.7950\cdot {{68}^{0.7950-1}}=0.1871</math>
 
This result that can be obtained from the Quick Calculation Pad (QCP), for  <math>T=68,</math>  as seen in Figure Mixednst.FI.
<br>
[[Image:rga5.18.png|thumb|center|400px|Instantaneous unreliability at the end of the test.]]
<br>
The instantaneous reliability can then be calculated as:
 
::<math>{{R}_{inst}}=1-0.1871=0.8129</math>
 
The average unreliability is calculated as:
 
::<math>\text{Average Unreliability }({{t}_{1,}}{{t}_{2}})=\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}</math>
 
and the average reliability is calculated as:
 
::<math>\text{Average Reliability }({{t}_{1,}}{{t}_{2}})=1-\frac{\lambda t_{2}^{\beta }-\lambda t_{1}^{\beta }}{{{t}_{2}}-{{t}_{1}}}</math>
 
====Bounds on Average Failure Probability for Mixed Data====
The process to calculate the average unreliability confidence bounds for mixed data is as follows:
<br>
:1) Calculate the average failure probability    .
:2) There will exist a  <math>{{t}^{*}}</math>  between  <math>{{t}_{1}}</math>  and  <math>{{t}_{2}}</math>  such that the instantaneous unreliability at  <math>{{t}^{*}}</math>  equals the average unreliability    . The confidence intervals for the instantaneous unreliability at  <math>{{t}^{*}}</math>  are the confidence intervals for the average unreliability    .
<br>
<br>
====Bounds on Average Reliability for Mixed Data====
The process to calculate the average reliability confidence bounds for mixed data is as follows:
<br>
:1) Calculate confidence bounds for average unreliability    as described above.
:2) The confidence bounds for reliability are 1 minus these confidence bounds for average unreliability.

Latest revision as of 12:37, 23 August 2012

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