Crow-AMSAA Parameter Estimation Example - Revision history

Lisa Hacker at 21:21, 18 September 2023

2023-09-18T21:21:10Z

← Older revision		Revision as of 21:21, 18 September 2023
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Crow-AMSAA (NHPP)\|~~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 18:34, 30 May 2014

2014-05-30T18:34:38Z

← Older revision		Revision as of 18:34, 30 May 2014
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	The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.		The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.

	[[Image:FIvsTimeExample1.png\|center\|~~400px\|Failure Intensity vs. Time plot~~]]		[[Image:FIvsTimeExample1.png\|center\|450px]]

Chris Kahn at 16:43, 14 May 2014

2014-05-14T16:43:22Z

← Older revision		Revision as of 16:43, 14 May 2014
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	A prototype of a system was tested with design changes incorporated during the test. The following table presents the data collected over the entire test. Find the Crow-AMSAA parameters and the intensity function using maximum likelihood estimators.		A prototype of a system was tested with design changes incorporated during the test. The following table presents the data collected over the entire test. Find the Crow-AMSAA parameters and the intensity function using maximum likelihood estimators.

	<center>'''Developmental ~~test data~~''' </center>		<center>'''Developmental Test Data''' </center>
	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	!Row		!Row

David J. Groebel at 00:13, 23 April 2014

2014-04-23T00:13:00Z

← Older revision		Revision as of 00:13, 23 April 2014
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	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Crow-AMSAA (NHPP)\|Reliability Growth and Repairable System Analysis Reference]]''.
	</noinclude>		</noinclude>

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	The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.		The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.

	[[Image:FIvsTimeExample1.png\|center\|~~500px~~\|Failure Intensity vs. Time plot]]		[[Image:FIvsTimeExample1.png\|center\|400px\|Failure Intensity vs. Time plot]]

Kate Racaza at 17:25, 31 January 2014

2014-01-31T17:25:07Z

Show changes

Kate Racaza at 23:35, 30 January 2014

2014-01-30T23:35:24Z

← Older revision		Revision as of 23:35, 30 January 2014
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	===Confidence Bounds on Cumulative and Instantaneous Failure Intensity===		====Confidence Bounds on Cumulative and Instantaneous Failure Intensity====
	Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the example given above, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.		Using the values of <math>\hat{\beta }\,\!</math> and <math>\hat{\lambda }\,\!</math> estimated in the example given above, calculate the 90% 2-sided confidence bounds on the cumulative and instantaneous failure intensity.

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	===Confidence Bounds on Cumulative and Instantaneous MTBF===		====Confidence Bounds on Cumulative and Instantaneous MTBF====
	Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.		Calculate the confidence bounds on the cumulative and instantaneous MTBF for the data from the example given above.

Kate Racaza at 23:33, 30 January 2014

2014-01-30T23:33:35Z

Show changes

David J. Groebel at 20:49, 30 January 2014

2014-01-30T20:49:55Z

← Older revision		Revision as of 20:49, 30 January 2014
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	For the failure terminated test, <math>{\beta}\,\!</math> is:		For the failure terminated test, <math>{\beta}\,\!</math> is:


	:<math>\begin{align}		:<math>\begin{align}
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	&=\frac{22}{22\ln 620-\underset{i=1}{\overset{22}{\mathop{\sum }}}\,\ln {{T}_{i}}} \\		&=\frac{22}{22\ln 620-\underset{i=1}{\overset{22}{\mathop{\sum }}}\,\ln {{T}_{i}}} \\
	\end{align}\,\!</math>		\end{align}\,\!</math>


	where:		where:


	:<math>\underset{i=1}{\overset{22}{\mathop \sum }}\,\ln {{T}_{i}}=105.6355\,\!</math>		:<math>\underset{i=1}{\overset{22}{\mathop \sum }}\,\ln {{T}_{i}}=105.6355\,\!</math>


	Then:		Then:


	:<math>\widehat{\beta }=\frac{22}{22\ln 620-105.6355}=0.6142\,\!</math>		:<math>\widehat{\beta }=\frac{22}{22\ln 620-105.6355}=0.6142\,\!</math>


	And for <math>{\lambda}\,\!</math> :		And for <math>{\lambda}\,\!</math> :


	:<math>\begin{align}		:<math>\begin{align}
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	& =\frac{22}{{{620}^{0.6142}}}=0.4239 \\		& =\frac{22}{{{620}^{0.6142}}}=0.4239 \\
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Therefore, <math>{{\lambda }_{i}}(T)\,\!</math> becomes:		Therefore, <math>{{\lambda }_{i}}(T)\,\!</math> becomes:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0217906\frac{\text{failures}}{\text{hr}}		= & 0.0217906\frac{\text{failures}}{\text{hr}}
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.		The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.


	[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]		[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]

Chris Kahn at 22:38, 29 January 2014

2014-01-29T22:38:29Z

← Older revision		Revision as of 22:38, 29 January 2014
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	For the failure terminated test, <math>{\beta}\,\!</math> is:		For the failure terminated test, <math>{\beta}\,\!</math> is:


	:<math>\begin{align}		:<math>\begin{align}
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	&=\frac{22}{22\ln 620-\underset{i=1}{\overset{22}{\mathop{\sum }}}\,\ln {{T}_{i}}} \\		&=\frac{22}{22\ln 620-\underset{i=1}{\overset{22}{\mathop{\sum }}}\,\ln {{T}_{i}}} \\
	\end{align}\,\!</math>		\end{align}\,\!</math>


	where:		where:


	:<math>\underset{i=1}{\overset{22}{\mathop \sum }}\,\ln {{T}_{i}}=105.6355\,\!</math>		:<math>\underset{i=1}{\overset{22}{\mathop \sum }}\,\ln {{T}_{i}}=105.6355\,\!</math>


	Then:		Then:


	:<math>\widehat{\beta }=\frac{22}{22\ln 620-105.6355}=0.6142\,\!</math>		:<math>\widehat{\beta }=\frac{22}{22\ln 620-105.6355}=0.6142\,\!</math>


	And for <math>{\lambda}\,\!</math> :		And for <math>{\lambda}\,\!</math> :


	:<math>\begin{align}		:<math>\begin{align}
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	& =\frac{22}{{{620}^{0.6142}}}=0.4239 \\		& =\frac{22}{{{620}^{0.6142}}}=0.4239 \\
	\end{align}\,\!</math>		\end{align}\,\!</math>


	Therefore, <math>{{\lambda }_{i}}(T)\,\!</math> becomes:		Therefore, <math>{{\lambda }_{i}}(T)\,\!</math> becomes:


	:<math>\begin{align}		:<math>\begin{align}
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	= & 0.0217906\frac{\text{failures}}{\text{hr}}		= & 0.0217906\frac{\text{failures}}{\text{hr}}
	\end{align}\,\!</math>		\end{align}\,\!</math>


	The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.		The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.


	[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]		[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]

David J. Groebel at 01:27, 25 January 2014

2014-01-25T01:27:48Z

← Older revision		Revision as of 01:27, 25 January 2014
Line 88:		Line 88:

	The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.		The next figure shows the plot of the failure rate. If no further changes are made, the estimated MTBF is <math>\tfrac{1}{0.0217906}\,\!</math> or 46 hours.


	[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]		[[Image:FIvsTimeExample1.png\|center\|500px\|Failure Intensity vs. Time plot]]