Test-Find-Test Data Example - Revision history

Lisa Hacker at 21:30, 18 September 2023

2023-09-18T21:30:22Z

← Older revision		Revision as of 21:30, 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 18:23, 30 May 2014

2014-05-30T18:23:13Z

← Older revision		Revision as of 18:23, 30 May 2014
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	</noinclude>		</noinclude>

	Consider the data in the first table below. A system was tested for <math>T=400\,\!</math> hours. There were a total of <math>N=42\,\!</math> 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 <math>{{N}_{A}}=10\,\!</math> A mode failures and <math>{{N}_{BD}}=32\,\!</math> BD mode failures. In addition, there are <math>M=16\,\!</math> 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 <math>{{j}^{th}}\,\!</math> observed distinct BD mode is denoted by <math>{{N}_{j}}\,\!</math>, and the total number of BD failures during the test is <math>{{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\!</math>. These values and effectiveness factors are given in the second table. Do the following:		Consider the data in the first table below. A system was tested for <math>T=400\,\!</math> hours. There were a total of <math>N=42\,\!</math> 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 <math>{{N}_{A}}=10\,\!</math> A mode failures and <math>{{N}_{BD}}=32\,\!</math> BD mode failures. In addition, there are <math>M=16\,\!</math> 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 <math>{{j}^{th}}\,\!</math> observed distinct BD mode is denoted by <math>{{N}_{j}}\,\!</math>, and the total number of BD failures during the test is <math>{{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\!</math>. These values and effectiveness factors are given in the second table

			Do the following:

	#Determine the projected MTBF and failure intensity.		#Determine the projected MTBF and failure intensity.

David J. Groebel at 18:22, 30 May 2014

2014-05-30T18:22:59Z

← Older revision		Revision as of 18:22, 30 May 2014
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	</noinclude>		</noinclude>

	Consider the data in the first table below. A system was tested for <math>T=400\,\!</math> hours. There were a total of <math>N=42\,\!</math> 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 <math>{{N}_{A}}=10\,\!</math> A mode failures and <math>{{N}_{BD}}=32\,\!</math> BD mode failures. In addition, there are <math>M=16\,\!</math> 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 <math>{{j}^{th}}\,\!</math> observed distinct BD mode is denoted by <math>{{N}_{j}}\,\!</math>, and the total number of BD failures during the test is <math>{{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\!</math>. These values and effectiveness factors are given in the second table.		Consider the data in the first table below. A system was tested for <math>T=400\,\!</math> hours. There were a total of <math>N=42\,\!</math> 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 <math>{{N}_{A}}=10\,\!</math> A mode failures and <math>{{N}_{BD}}=32\,\!</math> BD mode failures. In addition, there are <math>M=16\,\!</math> 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 <math>{{j}^{th}}\,\!</math> observed distinct BD mode is denoted by <math>{{N}_{j}}\,\!</math>, and the total number of BD failures during the test is <math>{{N}_{BD}}=\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{N}_{j}}\,\!</math>. These values and effectiveness factors are given in the second table. Do the following:

	Do the following:

	#Determine the projected MTBF and failure intensity.		#Determine the projected MTBF and failure intensity.

David J. Groebel at 18:22, 30 May 2014

2014-05-30T18:22:20Z

← Older revision		Revision as of 18:22, 30 May 2014
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	The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.		The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.

	[[Image:rga9.3.png\|center\|~~400px\|Demonstrated, projected and growth potential MTBF.~~]]		[[Image:rga9.3.png\|center\|450px]]

	[[Image:rga9.4.png\|center\|~~400px\|Demonstrated, projected and growth potential failure intensity.~~]]		[[Image:rga9.4.png\|center\|450px]]
	</li>		</li>
	</ol>		</ol>

Chris Kahn at 23:37, 23 April 2014

2014-04-23T23:37:17Z

← Older revision		Revision as of 23:37, 23 April 2014
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	#Determine the demonstrated MTBF and failure intensity.		#Determine the demonstrated MTBF and failure intensity.

	<div style="width~~=300px align~~="~~center";~~ float: left;">		<div style="margin-left:auto; margin-right: auto; width: 590px;">
			<div style="float: left;">
	{\|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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	</div>		</div>

	<div style="~~width=300px align="center";~~ float: ~~left~~;">		<div style="float: right;">
	{\|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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	\|16\|\| 1\|\| 395.2\|\| .46		\|16\|\| 1\|\| 395.2\|\| .46
	\|}		\|}
			</div>
	</div>		</div>
	<p style="clear: both;"></p>		<p style="clear: both;"></p>

Chris Kahn at 23:15, 23 April 2014

2014-04-23T23:15:45Z

← Older revision		Revision as of 23:15, 23 April 2014
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	</div>		</div>

	<div style="width~~: 200px~~; float: left;">		<div style="width=300px align="center"; float: left;">
	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|-		\|-

David J. Groebel at 23:19, 22 April 2014

2014-04-22T23:19:19Z

← Older revision		Revision as of 23:19, 22 April 2014
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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 ~~book~~]]''.		''This example appears in the [[Crow_Extended\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

David J. Groebel at 23:17, 22 April 2014

2014-04-22T23:17:54Z

← Older revision		Revision as of 23:17, 22 April 2014
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	Do the following:		Do the following:

	1) Determine the projected MTBF and failure intensity.		#Determine the projected MTBF and failure intensity.
			#Determine the growth potential MTBF and failure intensity.
			#Determine the demonstrated MTBF and failure intensity.

	~~2) Determine the growth potential MTBF and failure intensity.~~		<div style="width=300px align="center"; float: left;">

	~~3) Determine the demonstrated MTBF and failure intensity.~~

	<div style="width: 300px; float: left;">
	{\|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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	'''Solution'''		'''Solution'''

	1) The maximum likelihood estimates of <math>{{\beta }_{BD}}\,\!</math> and <math>{{\lambda }_{BD}}\,\!</math> are determined to be:		<ol>
			<li>The maximum likelihood estimates of <math>{{\beta }_{BD}}\,\!</math> and <math>{{\lambda }_{BD}}\,\!</math> are determined to be:


	:<math>\begin{align}		:<math>\begin{align}
	{{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\		{{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>


	The unbiased estimate of <math>\beta \,\!</math> is:		The unbiased estimate of <math>\beta \,\!</math> is:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.7472		= & 0.7472
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0661		= & 0.0661
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The projected MTBF is:		The projected MTBF is:


	:<math>M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\!</math>		:<math>M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\!</math>

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


	:<math>\begin{align}		:<math>\begin{align}
	{{N}_{A}}/T=0.0250		{{N}_{A}}/T=0.0250
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\!</math>		:<math>\frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\!</math>


	The growth potential failure intensity is estimated by:		The growth potential failure intensity is estimated by:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0446		= & 0.0446
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The growth potential MTBF is:		The growth potential MTBF is:


	:<math>M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\!</math>		:<math>M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\!</math>
			</li>

			<li>The demonstrated failure intensity and MTBF are estimated by:

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


	:<math>\begin{align}		:<math>\begin{align}
	{{\widehat{\lambda }}_{D}}(T) = & \frac{{{N}_{A}}+{{N}_{BD}}}{T} \\		{{\widehat{\lambda }}_{D}}(T) = & \frac{{{N}_{A}}+{{N}_{BD}}}{T} \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
	M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\		M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\
	= & 9.5238		= & 9.5238
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.		The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.

	[[Image:rga9.3.png\|center\|400px\|Demonstrated, projected and growth potential MTBF.]]		[[Image:rga9.3.png\|center\|400px\|Demonstrated, projected and growth potential MTBF.]]


	[[Image:rga9.4.png\|center\|400px\|Demonstrated, projected and growth potential failure intensity.]]		[[Image:rga9.4.png\|center\|400px\|Demonstrated, projected and growth potential failure intensity.]]
			</li>
			</ol>

Chris Kahn at 00:00, 30 January 2014

2014-01-30T00:00:56Z

← Older revision		Revision as of 00:00, 30 January 2014
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	1) The maximum likelihood estimates of <math>{{\beta }_{BD}}\,\!</math> and <math>{{\lambda }_{BD}}\,\!</math> are determined to be:		1) The maximum likelihood estimates of <math>{{\beta }_{BD}}\,\!</math> and <math>{{\lambda }_{BD}}\,\!</math> are determined to be:


	:<math>\begin{align}		:<math>\begin{align}
	{{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\		{{{\hat{\beta }}}_{BD}} = & \frac{M}{\underset{i=1}{\overset{M}{\mathop{\sum }}}\,\ln (\tfrac{T}{{{X}_{i}}})} \\
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	\end{align}\,\!</math>		\end{align}\,\!</math>


	The unbiased estimate of <math>\beta \,\!</math> is:		The unbiased estimate of <math>\beta \,\!</math> is:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.7472		= & 0.7472
	\end{align}\,\!</math>		\end{align}\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0661		= & 0.0661
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The projected MTBF is:		The projected MTBF is:


	:<math>M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\!</math>		:<math>M\widehat{T}B{{F}_{P}}={{[r(T)]}^{-1}}=15.127\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
	{{N}_{A}}/T=0.0250		{{N}_{A}}/T=0.0250
	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\!</math>		:<math>\frac{1}{T}\underset{i=1}{\overset{16}{\mathop \sum }}\,(1-{{d}_{i}}){{N}_{i}}=0.0196\,\!</math>


	The growth potential failure intensity is estimated by:		The growth potential failure intensity is estimated by:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0446		= & 0.0446
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The growth potential MTBF is:		The growth potential MTBF is:


	:<math>M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\!</math>		:<math>M\widehat{T}B{{F}_{GP}}={{[{{\widehat{r}}_{GP}}]}^{-1}}=22.4467\,\!</math>


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


	:<math>\begin{align}		:<math>\begin{align}
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	\end{align}\,\!</math>		\end{align}\,\!</math>


	:<math>\begin{align}		:<math>\begin{align}
	M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\		M\widehat{T}B{{F}_{D}} = & {{[{{\widehat{\lambda }}_{D}}(T)]}^{-1}} \\
	= & 9.5238		= & 9.5238
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.		The first chart below shows the demonstrated, projected and growth potential MTBF. The second shows the demonstrated, projected and growth potential failure intensity.

David J. Groebel at 18:24, 27 January 2014

2014-01-27T18:24:51Z

← Older revision		Revision as of 18:24, 27 January 2014
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	[[Image:rga9.3.png\|center\|400px\|Demonstrated, projected and growth potential MTBF.]]		[[Image:rga9.3.png\|center\|400px\|Demonstrated, projected and growth potential MTBF.]]


	[[Image:rga9.4.png\|center\|400px\|Demonstrated, projected and growth potential failure intensity.]]		[[Image:rga9.4.png\|center\|400px\|Demonstrated, projected and growth potential failure intensity.]]