Mixed Data - Crow Extended Example - Revision history

Lisa Hacker at 20:53, 18 September 2023

2023-09-18T20:53:04Z

← Older revision		Revision as of 20:53, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow_Extended\|~~Reliability ~~Growth and Repairable System Analysis Reference]~~]''.		''This example appears in the [https://help.reliasoft.com/reference/reliability_growth_and_repairable_system_analysis Reliability growth reference]''.
	</noinclude>		</noinclude>

David J. Groebel at 18:21, 30 May 2014

2014-05-30T18:21:40Z

← Older revision		Revision as of 18:21, 30 May 2014
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	The next figure shows the data sheet as calculated in the RGA software.		The next figure shows the data sheet as calculated in the RGA software.

	[[Image:rga9.12.png\|center\|~~400px\|Mixed Data calculated with the Crow Extended Model.~~]]		[[Image:rga9.12.png\|center\|600px]]
	</li>		</li>
	<li>The growth potential unreliability is:		<li>The growth potential unreliability is:
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	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.		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.

	[[Image:rga9.13.png\|center\|~~400px\|Calculating the gowth potential reliaibility for Crow Extended mixed data using the QCP.~~]]		[[Image:rga9.13.png\|center\|450px]]


	[[Image:rga9.14.png\|center\|~~400px\|Growth potential for the Crow Extended mixed data of example 5.~~]]</li>		[[Image:rga9.14.png\|center\|450px]]</li>
	</ol>		</ol>

Chris Kahn at 17:46, 14 May 2014

2014-05-14T17:46:28Z

← Older revision		Revision as of 17:46, 14 May 2014
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	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:		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.		#Calculate the demonstrated reliability at the end of the test.
			#Calculate the growth potential reliability.
	2) Calculate the growth potential reliability.

	'''Solution'''		'''Solution'''
			<ol><li>
	Based on the equations presented in [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]], the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:		Based on the equations presented in [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]], the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:

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	[[Image:rga9.12.png\|center\|400px\|Mixed Data calculated with the Crow Extended Model.]]		[[Image:rga9.12.png\|center\|400px\|Mixed Data calculated with the Crow Extended Model.]]
			</li>
	The growth potential unreliability is:		<li>The growth potential unreliability is:

	:<math>\begin{align}		:<math>\begin{align}
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	[[Image:rga9.14.png\|center\|400px\|Growth potential for the Crow Extended mixed data of example 5.]]		[[Image:rga9.14.png\|center\|400px\|Growth potential for the Crow Extended mixed data of example 5.]]</li>
			</ol>

David J. Groebel at 23:20, 22 April 2014

2014-04-22T23:20:28Z

← Older revision		Revision as of 23:20, 22 April 2014
Line 1:		Line 1:
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow_Extended\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow_Extended\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

Kate Racaza at 17:59, 4 February 2014

2014-02-04T17:59:57Z

← Older revision		Revision as of 17:59, 4 February 2014
Line 45:		Line 45:

	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:		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.		1) Calculate the demonstrated reliability at the end of the test.

David J. Groebel at 18:18, 27 January 2014

2014-01-27T18:18:13Z

← Older revision		Revision as of 18:18, 27 January 2014
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	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.		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.


	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
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	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:		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.		1) Calculate the demonstrated reliability at the end of the test.

	~~:2) Calculate the growth potential reliability~~.

			2) Calculate the growth potential reliability.

	'''Solution'''		'''Solution'''
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	Based on the equations presented in [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]], the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:		Based on the equations presented in [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]], the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:

	::<math>\widehat{\beta }=0.8572\,\!</math>		:<math>\widehat{\beta }=0.8572\,\!</math>

	and:		and:

	::<math>\widehat{\lambda }=0.4602\,\!</math>		:<math>\widehat{\lambda }=0.4602\,\!</math>


	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 <math>\hat{\beta }\,\!</math> parameter is that <math>\hat{\beta }=1.\,\!</math>		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 <math>\hat{\beta }\,\!</math> parameter is that <math>\hat{\beta }=1.\,\!</math>
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	From the [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] model, we know that:		From the [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] model, we know that:

			:<math>\hat{\lambda }=\frac{n}{T_{n}^{\hat{\beta }}}\,\!</math>
	::<math>\hat{\lambda }=\frac{n}{T_{n}^{\hat{\beta }}}\,\!</math>


	or, if <math>\hat{\beta }=1\,\!</math>, this becomes:		or, if <math>\hat{\beta }=1\,\!</math>, this becomes:

	::<math>\begin{align}		:<math>\begin{align}
	\hat{\lambda }= & \frac{n}{{{T}_{n}}} \\		\hat{\lambda }= & \frac{n}{{{T}_{n}}} \\
	= & \frac{6}{20} \\		= & \frac{6}{20} \\
	= & 0.3		= & 0.3
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	::<math>\begin{align}		:<math>\begin{align}
	{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0		{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	::<math>{{Q}_{i}}(20)=\widehat{\lambda }=0.3\,\!</math>		:<math>{{Q}_{i}}(20)=\widehat{\lambda }=0.3\,\!</math>


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

	::<math>\begin{align}		:<math>\begin{align}
	{{R}_{i}}(20)= & 1-{{Q}_{i}}(20) \\		{{R}_{i}}(20)= & 1-{{Q}_{i}}(20) \\
	= & 1-0.3 \\		= & 1-0.3 \\
	= & 0.7		= & 0.7
	\end{align}\,\!</math>		\end{align}\,\!</math>


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

	[[Image:rga9.12.png\|center\|400px\|Mixed Data calculated with the Crow Extended Model.]]		[[Image:rga9.12.png\|center\|400px\|Mixed Data calculated with the Crow Extended Model.]]


	The growth potential unreliability is:		The growth potential unreliability is:

	::<math>\begin{align}		:<math>\begin{align}
	{{\widehat{Q}}_{GP}}(T)= & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) \\		{{\widehat{Q}}_{GP}}(T)= & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) \\
	= & \underset{i=1}{\overset{M}{\mathop \sum }}\,(1-0.7)\frac{{{N}_{i}}}{T} \\		= & \underset{i=1}{\overset{M}{\mathop \sum }}\,(1-0.7)\frac{{{N}_{i}}}{T} \\
Line 113:		Line 103:
	= & 0.09		= & 0.09
	\end{align}\,\!</math>		\end{align}\,\!</math>


	So the growth potential reliability is:		So the growth potential reliability is:

	::<math>\begin{align}		:<math>\begin{align}
	{{\widehat{R}}_{GP}}(T) = & 1-{{\widehat{Q}}_{GP}}(T) \\		{{\widehat{R}}_{GP}}(T) = & 1-{{\widehat{Q}}_{GP}}(T) \\
	= & 1-0.09 \\		= & 1-0.09 \\
	= & 0.91		= & 0.91
	\end{align}\,\!</math>		\end{align}\,\!</math>


	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.		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.


	[[Image:rga9.13.png\|center\|400px\|Calculating the gowth potential reliaibility for Crow Extended mixed data using the QCP.]]		[[Image:rga9.13.png\|center\|400px\|Calculating the gowth potential reliaibility for Crow Extended mixed data using the QCP.]]

Chris Kahn at 21:51, 22 January 2014

2014-01-22T21:51:28Z

← Older revision		Revision as of 21:51, 22 January 2014
Line 53:		Line 53:
	'''Solution'''		'''Solution'''

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

	::<math>\widehat{\beta }=0.8572\,\!</math>		::<math>\widehat{\beta }=0.8572\,\!</math>
Line 64:		Line 64:
	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 <math>\hat{\beta }\,\!</math> parameter is that <math>\hat{\beta }=1.\,\!</math>		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 <math>\hat{\beta }\,\!</math> parameter is that <math>\hat{\beta }=1.\,\!</math>

	From the [[Crow-AMSAA - NHPP\|Crow-AMSAA (NHPP)]] model, we know that:		From the [[Crow-AMSAA (NHPP)\|Crow-AMSAA (NHPP)]] model, we know that:

Kate Racaza: Created page with '{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. A one-shot sy…'

2014-01-20T19:53:28Z

Created page with '<noinclude>{{Banner RGA Examples}} ''This example appears in the Reliability Growth and Repairable System Analysis Reference book''. </noinclude> A one-shot sy…'

New page

<noinclude>{{Banner RGA Examples}}
''This example appears in the [[Crow_Extended|Reliability Growth and Repairable System Analysis Reference book]]''.
</noinclude>

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.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|-
|colspan="4" style="text-align:center"|'''Mixed Data'''
|-
!Failures in Interval
!Cumulative Trials
!Classification
!Mode
|-
|1||8||BD||1
|-
|1||8||BD||2
|-
|1||8||BD||3
|-
|0||10|| ||
|-
|0||11|| ||
|-
|0||12|| ||
|-
|1||13||BD||2
|-
|0||14|| ||
|-
|0||15|| ||
|-
|1||16||BD||4
|-
|0||17|| ||
|-
|0||18|| ||
|-
|0||19|| ||
|-
|1||20||BD||5
|}

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|Crow-AMSAA (NHPP)]], the parameters of the Crow-AMSAA (NHPP) model are estimated as follows:

::<math>\widehat{\beta }=0.8572\,\!</math>

and:

::<math>\widehat{\lambda }=0.4602\,\!</math>

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 <math>\hat{\beta }\,\!</math> parameter is that <math>\hat{\beta }=1.\,\!</math>

From the [[Crow-AMSAA - NHPP|Crow-AMSAA (NHPP)]] model, we know that:

::<math>\hat{\lambda }=\frac{n}{T_{n}^{\hat{\beta }}}\,\!</math>

or, if <math>\hat{\beta }=1\,\!</math>, this becomes:

::<math>\begin{align}
\hat{\lambda }= & \frac{n}{{{T}_{n}}} \\
= & \frac{6}{20} \\
= & 0.3
\end{align}\,\!</math>

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

::<math>\begin{align}
{{\lambda }_{i}}(T)=\lambda \beta {{T}^{\beta -1}},\text{with }T>0,\text{ }\lambda >0\text{ and }\beta >0
\end{align}\,\!</math>

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

::<math>{{Q}_{i}}(20)=\widehat{\lambda }=0.3\,\!</math>

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

::<math>\begin{align}
{{R}_{i}}(20)= & 1-{{Q}_{i}}(20) \\
= & 1-0.3 \\
= & 0.7
\end{align}\,\!</math>

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

[[Image:rga9.12.png|center|400px|Mixed Data calculated with the Crow Extended Model.]]

The growth potential unreliability is:

::<math>\begin{align}
{{\widehat{Q}}_{GP}}(T)= & \left( \frac{{{N}_{A}}}{T}+\underset{i=1}{\overset{M}{\mathop \sum }}\,(1-{{d}_{i}})\frac{{{N}_{i}}}{T} \right) \\
= & \underset{i=1}{\overset{M}{\mathop \sum }}\,(1-0.7)\frac{{{N}_{i}}}{T} \\
= & 0.3*(\frac{1+1+1+1+1+1)}{20} \\
= & 0.09
\end{align}\,\!</math>

So the growth potential reliability is:

::<math>\begin{align}
{{\widehat{R}}_{GP}}(T) = & 1-{{\widehat{Q}}_{GP}}(T) \\
= & 1-0.09 \\
= & 0.91
\end{align}\,\!</math>

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.

[[Image:rga9.13.png|center|400px|Calculating the gowth potential reliaibility for Crow Extended mixed data using the QCP.]]

[[Image:rga9.14.png|center|400px|Growth potential for the Crow Extended mixed data of example 5.]]