Duane Graphical Method Example - Revision history

Lisa Hacker at 21:29, 18 September 2023

2023-09-18T21:29:03Z

← Older revision		Revision as of 21:29, 18 September 2023
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	<!-- IF YOU EDIT THE DATA SET IN THIS EXAMPLE, YOU MUST ALSO EDIT THE DATA SET AND RESULTS SHOWN ON THE FOLLOWING PAGES: Duane Linear Regression Example and Duane Confidence Bounds Example. -->		<!-- IF YOU EDIT THE DATA SET IN THIS EXAMPLE, YOU MUST ALSO EDIT THE DATA SET AND RESULTS SHOWN ON THE FOLLOWING PAGES: Duane Linear Regression Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [~~[Duane Model\|~~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 22:15, 2 June 2014

2014-06-02T22:15:14Z

← Older revision		Revision as of 22:15, 2 June 2014
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	<!-- IF YOU EDIT THE DATA SET IN THIS EXAMPLE, YOU MUST ALSO EDIT THE DATA SET AND RESULTS SHOWN ON THE FOLLOWING PAGES: Duane Linear Regression Example and Duane Confidence Bounds Example. -->		<!-- IF YOU EDIT THE DATA SET IN THIS EXAMPLE, YOU MUST ALSO EDIT THE DATA SET AND RESULTS SHOWN ON THE FOLLOWING PAGES: Duane Linear Regression Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference ~~book~~]]''.		''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference]]''.

	</noinclude>		</noinclude>
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	<li>Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.		<li>Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.

	[[Image:rga4.2.png\|center\|~~400px\|Cumulative MTBF plot~~]]		[[Image:rga4.2.png\|center\|450px]]
	<center>Cumulative MTBF plot</center>		<center>Cumulative MTBF plot</center>

	[[Image:rga4.3.png\|center\|~~400px\|Instantaneous MTBF plot~~]]		[[Image:rga4.3.png\|center\|450px]]
	<center>Instantaneous MTBF plot</center>		<center>Instantaneous MTBF plot</center>

	By changing the x-axis scaling, you are able to extend the line to <math>T=1\,\!</math>. You can get the value of <math>b\,\!</math> from the graph by positioning the cursor at the point where the line meets the y-axis. Then read the value of the y-coordinate position at the bottom left corner. In this case, <math>b\,\!</math> is approximately 14 hours. The next figure illustrates this.		By changing the x-axis scaling, you are able to extend the line to <math>T=1\,\!</math>. You can get the value of <math>b\,\!</math> from the graph by positioning the cursor at the point where the line meets the y-axis. Then read the value of the y-coordinate position at the bottom left corner. In this case, <math>b\,\!</math> is approximately 14 hours. The next figure illustrates this.

	[[Image:rga4.4.png\|center\|~~400px\|Cumulative MTBF plot for <math>b\approx 14\,\!</math> at <math>T=1\,\!</math>.~~]]		[[Image:rga4.4.png\|center\|450px]]
	<center>Cumulative MTBF plot for <math>b \approx 14\,\!</math> at <math>T=1\,\!</math></center>		<center>Cumulative MTBF plot for <math>b \approx 14\,\!</math> at <math>T=1\,\!</math></center>

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	Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\hat{\alpha }=0.3733\,\!</math>.		Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\hat{\alpha }=0.3733\,\!</math>.
	</li>		</li>


	<li>Now the equation for the cumulative MTBF growth curve is:		<li>Now the equation for the cumulative MTBF growth curve is:
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	</li>		</li>


	<li>Using the following equation for the instantaneous MTBF, or		<li>Using the following equation for the instantaneous MTBF, or

	:<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>		:<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>

	~~:the~~ equation for the instantaneous MTBF growth curve is:		The equation for the instantaneous MTBF growth curve is:

	:<math>\begin{align}		:<math>\begin{align}

Chris Kahn at 16:15, 14 May 2014

2014-05-14T16:15:58Z

← Older revision		Revision as of 16:15, 14 May 2014
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	Do the following:		Do the following:
	~~:1)~~ Plot the cumulative MTBF growth curve.		#Plot the cumulative MTBF growth curve.
	~~:2)~~ Write the equation of this growth curve.		#Write the equation of this growth curve.
	~~:3)~~ Write the equation of the instantaneous MTBF growth model.		#Write the equation of the instantaneous MTBF growth model.
	~~:4)~~ Plot the instantaneous MTBF growth curve.		#Plot the instantaneous MTBF growth curve.

	'''Solution'''		'''Solution'''
			<ol>
	1) Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.		<li>Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.

	[[Image:rga4.2.png\|center\|400px\|Cumulative MTBF plot]]		[[Image:rga4.2.png\|center\|400px\|Cumulative MTBF plot]]
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	= & 0.3711		= & 0.3711
	\end{align}\,\!</math>		\end{align}\,\!</math>
			</li>

	2. Substituting the second set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0,\,\!</math> into the equation yields:
			<li>Substituting the second set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0,\,\!</math> into the equation yields:

	:<math>\begin{align}		:<math>\begin{align}
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	Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\hat{\alpha }=0.3733\,\!</math>.		Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\hat{\alpha }=0.3733\,\!</math>.
			</li>

	3. Now the equation for the cumulative MTBF growth curve is:
			<li>Now the equation for the cumulative MTBF growth curve is:

	:<math>{{\hat{m}}_{c}}=14\cdot {{T}^{\text{ }0.3733}}\,\!</math>		:<math>{{\hat{m}}_{c}}=14\cdot {{T}^{\text{ }0.3733}}\,\!</math>
			</li>
	4. Using the following equation for the instantaneous MTBF, or

			<li>Using the following equation for the instantaneous MTBF, or

	:<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>		:<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>
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	[[Image:rga4.5.png\|center\|400px\|Cumulative and Instantaneous MTBF vs. Time plot.]]		[[Image:rga4.5.png\|center\|400px\|Cumulative and Instantaneous MTBF vs. Time plot.]]
	<center>Cumulative and Instantaneous MTBF vs. Time plot</center>		<center>Cumulative and Instantaneous MTBF vs. Time plot</center></li>
			</ol>

David J. Groebel at 22:52, 30 January 2014

2014-01-30T22:52:10Z

← Older revision		Revision as of 22:52, 30 January 2014
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	'''Solution'''		'''Solution'''

	:1) Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.		1) Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.


	[[Image:rga4.2.png\|center\|400px\|Cumulative MTBF plot]]		[[Image:rga4.2.png\|center\|400px\|Cumulative MTBF plot]]
	<center>Cumulative MTBF plot</center>		<center>Cumulative MTBF plot</center>


	[[Image:rga4.3.png\|center\|400px\|Instantaneous MTBF plot]]		[[Image:rga4.3.png\|center\|400px\|Instantaneous MTBF plot]]
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	[[Image:rga4.4.png\|center\|400px\|Cumulative MTBF plot for <math>b\approx 14\,\!</math> at <math>T=1\,\!</math>.]]		[[Image:rga4.4.png\|center\|400px\|Cumulative MTBF plot for <math>b\approx 14\,\!</math> at <math>T=1\,\!</math>.]]
	<center>Cumulative MTBF plot for <math>b \approx 14\,\!</math> at <math>T=1\,\!</math></center>		<center>Cumulative MTBF plot for <math>b \approx 14\,\!</math> at <math>T=1\,\!</math></center>


	Another way of determining <math>b\,\!</math> is to calculate <math>\alpha \,\!</math> by using two points on the fitted straight line and substituting the corresponding <math>{{\hat{m}}_{c}}\,\!</math> and <math>T\,\!</math> values into:		Another way of determining <math>b\,\!</math> is to calculate <math>\alpha \,\!</math> by using two points on the fitted straight line and substituting the corresponding <math>{{\hat{m}}_{c}}\,\!</math> and <math>T\,\!</math> values into:

	::<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{{{c}_{2}}}} \right)-\ln \left( {{{\hat{m}}}_{{{c}_{1}}}} \right)}{\ln \left( {{T}_{_{2}}} \right)-\ln \left( {{T}_{_{1}}} \right)}\,\!</math>		:<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{{{c}_{2}}}} \right)-\ln \left( {{{\hat{m}}}_{{{c}_{1}}}} \right)}{\ln \left( {{T}_{_{2}}} \right)-\ln \left( {{T}_{_{1}}} \right)}\,\!</math>

	Then substitute this <math>\alpha \,\!</math> and choose a set of values for <math>{{\hat{m}}_{{{c}_{1}}}}\,\!</math> and <math>{{T}_{_{1}}}\,\!</math> into the cumulative MTBF equation, <math>{{\hat{m}}_{c}}=b{{T}^{\alpha }}\,\!</math>, and solve for <math>b\,\!</math>. The slope of the line, <math>\alpha \,\!</math>, may also be found from the linearized form of the cumulative MTBF equation, or :		Then substitute this <math>\alpha \,\!</math> and choose a set of values for <math>{{\hat{m}}_{{{c}_{1}}}}\,\!</math> and <math>{{T}_{_{1}}}\,\!</math> into the cumulative MTBF equation, <math>{{\hat{m}}_{c}}=b{{T}^{\alpha }}\,\!</math>, and solve for <math>b\,\!</math>. The slope of the line, <math>\alpha \,\!</math>, may also be found from the linearized form of the cumulative MTBF equation, or :

	::<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{c}} \right)-\ln (b)}{\ln (T)-\ln (1)}\,\!</math>		:<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{c}} \right)-\ln (b)}{\ln (T)-\ln (1)}\,\!</math>

	Using the cumulative MTBF plot for the example, at <math>{{T}_{_{1}}}=200\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{1}}}}=100\,\!</math> hours, and <math>{{T}_{_{2}}}=3,500\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{2}}}}=300\,\!</math> hours. From the cumulative MTBF plot for <math>b=14\,\!</math> hours when <math>T=1\,\!</math>, substituting the first set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0\,\!</math>, into the equation yields:		Using the cumulative MTBF plot for the example, at <math>{{T}_{_{1}}}=200\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{1}}}}=100\,\!</math> hours, and <math>{{T}_{_{2}}}=3,500\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{2}}}}=300\,\!</math> hours. From the cumulative MTBF plot for <math>b=14\,\!</math> hours when <math>T=1\,\!</math>, substituting the first set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0\,\!</math>, into the equation yields:

	::<math>\begin{align}		:<math>\begin{align}
	{{\alpha }_{1}}= & \frac{\ln (100)-\ln (14)}{\ln (200)-\ln (1)} \\		{{\alpha }_{1}}= & \frac{\ln (100)-\ln (14)}{\ln (200)-\ln (1)} \\
	= & 0.3711		= & 0.3711
	\end{align}\,\!</math>		\end{align}\,\!</math>

			2. Substituting the second set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0,\,\!</math> into the equation yields:

	~~:2. Substituting the second set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0,\,\!</math> into the equation yields:~~		:<math>\begin{align}

	::<math>\begin{align}
	{{\alpha }_{2}}= & \frac{\ln (300)-\ln (14)}{\ln (3,500)-\ln (1)} \\		{{\alpha }_{2}}= & \frac{\ln (300)-\ln (14)}{\ln (3,500)-\ln (1)} \\
	= & 0.3755		= & 0.3755
	\end{align}\,\!</math>		\end{align}\,\!</math>

			Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\hat{\alpha }=0.3733\,\!</math>.

	~~Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\widehat{\alpha }=0.3733\,\!</math>.~~		3. Now the equation for the cumulative MTBF growth curve is:

	:3. Now the equation for the cumulative MTBF growth curve is:

	::<math>{{\hat{m}}_{c}}=14\cdot {{T}^{\text{ }0.3733}}\,\!</math>		:<math>{{\hat{m}}_{c}}=14\cdot {{T}^{\text{ }0.3733}}\,\!</math>

	:4. Using the following equation for the instantaneous MTBF, or		4. Using the following equation for the instantaneous MTBF, or

	::<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>		:<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>

	::the equation for the instantaneous MTBF growth curve is:		:the equation for the instantaneous MTBF growth curve is:

	::<math>\begin{align}		:<math>\begin{align}
	{{\hat{m}}_{i}}=\frac{1}{1-0.3733} \cdot {{14T}^{0.3733}}		{{\hat{m}}_{i}}=\frac{1}{1-0.3733} \cdot {{14T}^{0.3733}}
	\end{align}\,\!</math>		\end{align}\,\!</math>

Kate Racaza at 21:19, 21 November 2013

2013-11-21T21:19:00Z

← Older revision		Revision as of 21:19, 21 November 2013
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			<!-- IF YOU EDIT THE DATA SET IN THIS EXAMPLE, YOU MUST ALSO EDIT THE DATA SET AND RESULTS SHOWN ON THE FOLLOWING PAGES: Duane Linear Regression Example and Duane Confidence Bounds Example. -->
	<noinclude>{{Banner RGA Examples}}		<noinclude>{{Banner RGA Examples}}
	''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.		''This example appears in the [[Duane Model\|Reliability Growth and Repairable System Analysis Reference book]]''.

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

2013-11-21T20:20:52Z

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

New page

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

</noinclude>
A complex system's reliability growth is being monitored and the data set is given in the table below.

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Cumulative Test Hours and the Corresponding Observed Failures for the Complex System'''
|-
!Point Number
!Cumulative Test Time(hours)
!Cumulative Failures
!Cumulative MTBF(hours)
!Instantaneous MTBF(hours)
|-
|1||200 ||2||100.0||100
|-
|2||400 ||3||133.0||200
|-
|3||600 ||4||150.0||200
|-
|4||3,000||11||273.0||342.8
|}

Do the following:
:1) Plot the cumulative MTBF growth curve.
:2) Write the equation of this growth curve.
:3) Write the equation of the instantaneous MTBF growth model.
:4) Plot the instantaneous MTBF growth curve.

'''Solution'''

:1) Given the data in the second and third columns of the above table, the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, values are calculated in the fourth column. The information in the second and fourth columns are then plotted. The first figure below shows the cumulative MTBF while the second figure below shows the instantaneous MTBF. It can be seen that a straight line represents the MTBF growth very well on log-log scales.

[[Image:rga4.2.png|center|400px|Cumulative MTBF plot]]
<center>Cumulative MTBF plot</center>

[[Image:rga4.3.png|center|400px|Instantaneous MTBF plot]]
<center>Instantaneous MTBF plot</center>

By changing the x-axis scaling, you are able to extend the line to <math>T=1\,\!</math>. You can get the value of <math>b\,\!</math> from the graph by positioning the cursor at the point where the line meets the y-axis. Then read the value of the y-coordinate position at the bottom left corner. In this case, <math>b\,\!</math> is approximately 14 hours. The next figure illustrates this.

[[Image:rga4.4.png|center|400px|Cumulative MTBF plot for <math>b\approx 14\,\!</math> at <math>T=1\,\!</math>.]]
<center>Cumulative MTBF plot for <math>b \approx 14\,\!</math> at <math>T=1\,\!</math></center>

Another way of determining <math>b\,\!</math> is to calculate <math>\alpha \,\!</math> by using two points on the fitted straight line and substituting the corresponding <math>{{\hat{m}}_{c}}\,\!</math> and <math>T\,\!</math> values into:

::<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{{{c}_{2}}}} \right)-\ln \left( {{{\hat{m}}}_{{{c}_{1}}}} \right)}{\ln \left( {{T}_{_{2}}} \right)-\ln \left( {{T}_{_{1}}} \right)}\,\!</math>

Then substitute this <math>\alpha \,\!</math> and choose a set of values for <math>{{\hat{m}}_{{{c}_{1}}}}\,\!</math> and <math>{{T}_{_{1}}}\,\!</math> into the cumulative MTBF equation, <math>{{\hat{m}}_{c}}=b{{T}^{\alpha }}\,\!</math>, and solve for <math>b\,\!</math>. The slope of the line, <math>\alpha \,\!</math>, may also be found from the linearized form of the cumulative MTBF equation, or :

::<math>\alpha =\frac{\ln \left( {{{\hat{m}}}_{c}} \right)-\ln (b)}{\ln (T)-\ln (1)}\,\!</math>

Using the cumulative MTBF plot for the example, at <math>{{T}_{_{1}}}=200\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{1}}}}=100\,\!</math> hours, and <math>{{T}_{_{2}}}=3,500\,\!</math> hours, <math>{{\hat{m}}_{{{c}_{2}}}}=300\,\!</math> hours. From the cumulative MTBF plot for <math>b=14\,\!</math> hours when <math>T=1\,\!</math>, substituting the first set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0\,\!</math>, into the equation yields:

::<math>\begin{align}
{{\alpha }_{1}}= & \frac{\ln (100)-\ln (14)}{\ln (200)-\ln (1)} \\
= & 0.3711
\end{align}\,\!</math>

:2. Substituting the second set of values, <math>b=14\,\!</math> hours and <math>\ln 1=0,\,\!</math> into the equation yields:

::<math>\begin{align}
{{\alpha }_{2}}= & \frac{\ln (300)-\ln (14)}{\ln (3,500)-\ln (1)} \\
= & 0.3755
\end{align}\,\!</math>

Averaging these two <math>\alpha \,\!</math> values yields a better estimate of <math>\widehat{\alpha }=0.3733\,\!</math>.

:3. Now the equation for the cumulative MTBF growth curve is:

::<math>{{\hat{m}}_{c}}=14\cdot {{T}^{\text{ }0.3733}}\,\!</math>

:4. Using the following equation for the instantaneous MTBF, or

::<math>{{m}_{i}} = \frac{1}{1-\alpha }{{{\hat{m}}}_{c}},:\ \ \alpha \not{=}1 \,\!</math>

::the equation for the instantaneous MTBF growth curve is:

::<math>\begin{align}
{{\hat{m}}_{i}}=\frac{1}{1-0.3733} \cdot {{14T}^{0.3733}}
\end{align}\,\!</math>

The above equation is plotted in the instantaneous MTBF plot shown in the example and in the cumulative and instantaneous MTBF vs. Time plot. In the following figure, you can see that a parallel shift upward of the cumulative MTBF, <math>{{\hat{m}}_{c}}\,\!</math>, line by a distance of <math>\tfrac{1}{1-\alpha }\,\!</math> gives the instantaneous MTBF, or the <math>{{\hat{m}}_{i}}\,\!</math>, line.

[[Image:rga4.5.png|center|400px|Cumulative and Instantaneous MTBF vs. Time plot.]]
<center>Cumulative and Instantaneous MTBF vs. Time plot</center>