Known Operating Times - Duane Example - Revision history

Lisa Hacker at 21:21, 18 September 2023

2023-09-18T21:21:31Z

← 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 [~~[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 22:04, 3 June 2014

2014-06-03T22:04:42Z

← Older revision		Revision as of 22:04, 3 June 2014
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	Weibull++ can be utilized to provide a Reliability vs. Time plot which is shown below.		Weibull++ can be utilized to provide a Reliability vs. Time plot which is shown below.

	[[Image:rga4.18.png\|center\|450px]]</li>		[[Image:rga4.18.png\|center\|450px]]
			</li>
			</ol>

David J. Groebel at 22:18, 2 June 2014

2014-06-02T22:18:10Z

← Older revision		Revision as of 22:18, 2 June 2014
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	<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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	Do the following:		Do the following:

	#Estimate the Duane parameters.		#Estimate the Duane parameters.
	#Assume both units are tested for an additional 100 hours each. How many failures do you expect in that period?		#Assume both units are tested for an additional 100 hours each. How many failures do you expect in that period?
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	<li>The figure below shows the data entered into RGA along with the estimated Duane parameters.		<li>The figure below shows the data entered into RGA along with the estimated Duane parameters.

	[[Image:rga4.15.png~~\|thumb~~\|center\|~~400px\|Entered data and the estimated parameters.~~]]</li>		[[Image:rga4.15.png\|center\|600px]]</li>

	<li>The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.		<li>The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.

	[[Image:rga4.16.png~~\|thumb~~\|center\|~~400px\|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.~~]]</li>		[[Image:rga4.16.png\|center\|450px]]</li>

	<li>If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.		<li>If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.

	[[Image:rga4.17.png~~\|thumb~~\|center\|~~400px\|Calculate the instantaneous failure intensity at the end of the test.~~]]		[[Image:rga4.17.png\|center\|450px]]

	An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:		An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:
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	\end{align}\,\!</math>		\end{align}\,\!</math>

	Weibull++ can be utilized to provide a Reliability vs. Time plot~~. This~~ is shown ~~in the following figure~~.		Weibull++ can be utilized to provide a Reliability vs. Time plot which is shown below.

	[[Image:rga4.18.png~~\|thumb~~\|center\|~~400px\|Reliability vs. Time plot.~~]]</li>		[[Image:rga4.18.png\|center\|450px]]</li>

Chris Kahn at 23:20, 31 January 2014

2014-01-31T23:20:45Z

← Older revision		Revision as of 23:20, 31 January 2014
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	'''Solution'''		'''Solution'''

	1) The figure below shows the data entered into RGA along with the estimated Duane parameters.		<ol>
			<li>The figure below shows the data entered into RGA along with the estimated Duane parameters.

	[[Image:rga4.15.png\|thumb\|center\|400px\|Entered data and the estimated parameters.]]		[[Image:rga4.15.png\|thumb\|center\|400px\|Entered data and the estimated parameters.]]</li>

	2) The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.		<li>The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.

	[[Image:rga4.16.png\|thumb\|center\|400px\|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.]]		[[Image:rga4.16.png\|thumb\|center\|400px\|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.]]</li>

	3) If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.		<li>If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.

	[[Image:rga4.17.png\|thumb\|center\|400px\|Calculate the instantaneous failure intensity at the end of the test.]]		[[Image:rga4.17.png\|thumb\|center\|400px\|Calculate the instantaneous failure intensity at the end of the test.]]
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	Weibull++ can be utilized to provide a Reliability vs. Time plot. This is shown in the following figure.		Weibull++ can be utilized to provide a Reliability vs. Time plot. This is shown in the following figure.

	[[Image:rga4.18.png\|thumb\|center\|400px\|Reliability vs. Time plot.]]		[[Image:rga4.18.png\|thumb\|center\|400px\|Reliability vs. Time plot.]]</li>

David J. Groebel at 22:56, 30 January 2014

2014-01-30T22:56:06Z

← Older revision		Revision as of 22:56, 30 January 2014
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	\|29\|\| 2\|\| 755.8\|\| 573.6		\|29\|\| 2\|\| 755.8\|\| 573.6
	\|}		\|}


	'''Solution'''		'''Solution'''

	:1) The figure below shows the data entered into RGA along with the estimated Duane parameters.		1) The figure below shows the data entered into RGA along with the estimated Duane parameters.

	[[Image:rga4.15.png\|thumb\|center\|400px\|Entered data and the estimated parameters.]]		[[Image:rga4.15.png\|thumb\|center\|400px\|Entered data and the estimated parameters.]]

	:2) The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.		2) The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.

	[[Image:rga4.16.png\|thumb\|center\|400px\|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.]]		[[Image:rga4.16.png\|thumb\|center\|400px\|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.]]

	:3) If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.		3) If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.

	[[Image:rga4.17.png\|thumb\|center\|400px\|Calculate the instantaneous failure intensity at the end of the test.]]		[[Image:rga4.17.png\|thumb\|center\|400px\|Calculate the instantaneous failure intensity at the end of the test.]]
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	An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:		An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:

	::<math>\begin{align}		:<math>\begin{align}
	R(t)= & {{e}^{-\lambda t}} \\		R(t)= & {{e}^{-\lambda t}} \\
	= & {{e}^{-(0.0107)t}}		= & {{e}^{-(0.0107)t}}

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

2013-11-21T22:23:22Z

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

New page

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

Two identical systems were tested. Any design changes made to improve the reliability of these systems were incorporated into both systems when any system failed. A total of 29 failures occurred. The data set is given in the table below.

Do the following:

#Estimate the Duane parameters.
#Assume both units are tested for an additional 100 hours each. How many failures do you expect in that period?
#If testing/development were halted at this point, what would the reliability equation for this system be?

{|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
|+'''Developmental Test Data'''
!Failure Number
!Failed Unit
!Test Time Unit 1(hours)
!Test Time Unit 2 (hours)
|-
|1|| 1|| 0.2|| 2.0
|-
|2|| 2|| 1.7|| 2.9
|-
|3|| 2|| 4.5|| 5.2
|-
|4|| 2|| 5.8|| 9.1
|-
|5|| 2|| 17.3|| 9.2
|-
|6|| 2|| 29.3|| 24.1
|-
|7|| 1|| 36.5|| 61.1
|-
|8|| 2|| 46.3|| 69.6
|-
|9|| 1|| 63.6|| 78.1
|-
|10|| 2|| 64.4|| 85.4
|-
|11|| 1|| 74.3|| 93.6
|-
|12|| 1|| 106.6|| 103
|-
|13|| 2|| 195.2 ||117
|-
|14|| 2|| 235.1|| 134.3
|-
|15|| 1|| 248.7 ||150.2
|-
|16|| 2|| 256.8 ||164.6
|-
|17|| 2|| 261.1|| 174.3
|-
|18|| 2|| 299.4 ||193.2
|-
|19|| 1|| 305.3|| 234.2
|-
|20|| 1|| 326.9|| 257.3
|-
|21|| 1|| 339.2|| 290.2
|-
|22|| 1|| 366.1|| 293.1
|-
|23|| 2|| 466.4|| 316.4
|-
|24|| 1|| 504|| 373.2
|-
|25|| 1|| 510|| 375.1
|-
|26|| 2|| 543.2|| 386.1
|-
|27|| 2|| 635.4 ||453.3
|-
|28|| 1|| 641.2|| 485.8
|-
|29|| 2|| 755.8|| 573.6
|}

'''Solution'''

:1) The figure below shows the data entered into RGA along with the estimated Duane parameters.

[[Image:rga4.15.png|thumb|center|400px|Entered data and the estimated parameters.]]

:2) The current accumulated test time for both units is 1329.4 hours. If the process were to continue for an additional combined time of 200 hours, the expected cumulative number of failures at <math>T=1529.4\,\!</math> is 31.2695, as shown in the figure below. At T = 1329.4, the expected number of failures is 29.2004. Therefore, the expected number of failures that would be observed over the additional 200 hours is <math>31.2695-29.2004=2.0691\approx 2\,\!</math>.

[[Image:rga4.16.png|thumb|center|400px|The expected cumulative number of failures at <math>T=1529.4\,\!</math>.]]

:3) If testing/development were halted at this point, the system failure intensity would be equal to the instantaneous failure intensity at that time, or <math>\lambda =0.0107\,\!</math> failures/hour. See the following figure.

[[Image:rga4.17.png|thumb|center|400px|Calculate the instantaneous failure intensity at the end of the test.]]

An exponential distribution can be assumed since the value of the failure intensity at that instant in time is known. Therefore:

::<math>\begin{align}
R(t)= & {{e}^{-\lambda t}} \\
= & {{e}^{-(0.0107)t}}
\end{align}\,\!</math>

Weibull++ can be utilized to provide a Reliability vs. Time plot. This is shown in the following figure.

[[Image:rga4.18.png|thumb|center|400px|Reliability vs. Time plot.]]