Template:Crow discrete reliability growth model camsaa - Revision history

Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

2012-08-23T12:37:39Z

← Older revision		Revision as of 12:37, 23 August 2012
Line 1:		Line 1:
	~~==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 camsaa-cd}}~~

	~~{{maximum likelihood estimators camsaa-cd}}~~

	~~{{mixed data camsaa~~-~~cd}}~~

Nicolette Young: /* Mixed Data */

2012-01-09T17:12:33Z

Mixed Data

← Older revision		Revision as of 17:12, 9 January 2012
Line 6:		Line 6:
	{{maximum likelihood estimators camsaa-cd}}		{{maximum likelihood estimators camsaa-cd}}

	~~===Mixed Data===~~		{{mixed data camsaa-cd}}
	~~<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.~~

Nicolette Young: /* Maximum Likelihood Estimators */

2012-01-09T17:11:58Z

Maximum Likelihood Estimators

@@ Line 4: / Line 4: @@
 {{model development camsaa-cd}}
-===Maximum Likelihood Estimators===
+{{maximum likelihood estimators camsaa-cd}}
 ===Mixed Data===

Nicolette Young: /* Model Development */

2012-01-09T17:08:45Z

Model Development

@@ Line 2: / Line 2: @@
 The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
 <br>
-===Model Development===
+{{model development camsaa-cd}}
 ===Maximum Likelihood Estimators===

Nicolette Young: Created page with '==Crow Discrete Reliability Growth Model== The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
===Mode…'

2012-01-09T17:07:53Z

Created page with '==Crow Discrete Reliability Growth Model== The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data). ===Mode…'

New page

==Crow Discrete Reliability Growth Model==
The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
 
===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:
 
::<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'''
 
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.
 
• Configuration 1 experienced 5 failures,
 
• Configuration 2 experienced 3 failures,
 
• Configuration 3 experienced 4 failures and
 
• Configuration 4 experienced 4 failures.
 
 
:1) Estimate the parameters of the Crow-AMSAA model using maximum likelihood estimation.
:2) Estimate the unreliability and reliability by configuration.
 
 
'''Solution'''
 
: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.
 
<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
|}
 
[[Image:rga5.16.png|thumb|center|400px|Estimated unreliability by configuration.]]
 
[[Image:rga5.17.png|thumb|center|400px|Estimated reliability by configuration.]]
 

===Mixed Data===
 
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.
 
'''Example 9'''
 
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.
 
{|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
|}

 
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.
 
[[Image:rga5.18.png|thumb|center|400px|Instantaneous unreliability at the end of the test.]]
 
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:
 
: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 .
 
 
====Bounds on Average Reliability for Mixed Data====
The process to calculate the average reliability confidence bounds for mixed data is as follows:
 
: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.

Template:Crow discrete reliability growth model camsaa - Revision history

Lisa Hacker: Redirected page to Crow-AMSAA - NHPP

Nicolette Young: /* Mixed Data */

Nicolette Young: /* Maximum Likelihood Estimators */

Nicolette Young: /* Model Development */

Nicolette Young: Created page with '==Crow Discrete Reliability Growth Model== The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data). ===Mode…'

Nicolette Young: Created page with '==Crow Discrete Reliability Growth Model== The Crow-AMSAA model can be adapted for the analysis of success/failure data (also called "discrete" or "attribute" data).
===Mode…'