Repairable Systems Analysis - Revision history

Sharon Honecker: /* Cramér-von Mises Example */

2015-09-16T21:24:33Z

Cramér-von Mises Example

← Older revision		Revision as of 21:24, 16 September 2015
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	'''Step 4''': Calculate:		'''Step 4''': Calculate:

	:<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{(Z_{j}^{\overline{\beta }}-\tfrac{2j-1}{2M})}^{2}}=0.~~0611~~\,\!</math>		:<math>C_{M}^{2}=\tfrac{1}{12M}+\underset{j=1}{\overset{M}{\mathop{\sum }}}\,{{(Z_{j}^{\overline{\beta }}-\tfrac{2j-1}{2M})}^{2}}=0.0636\,\!</math>

	'''Step 5''': From the [[Crow-AMSAA (NHPP)#Critical_Values\|table of critical values for the Cramér-von Mises test]], find the critical value (CV) for <math>M=34\,\!</math> at a significance level <math>\alpha =0.10\,\!</math>. <math>CV=0.172\,\!</math>.		'''Step 5''': From the [[Crow-AMSAA (NHPP)#Critical_Values\|table of critical values for the Cramér-von Mises test]], find the critical value (CV) for <math>M=34\,\!</math> at a significance level <math>\alpha =0.10\,\!</math>. <math>CV=0.172\,\!</math>.

David J. Groebel at 19:51, 2 June 2014

2014-06-02T19:51:16Z

← Older revision		Revision as of 19:51, 2 June 2014
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	Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.		Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.

	[[File:Rga12.png\|center\|~~4500px~~\|Instantaneous Failure Intensity vs. Time plot.]]		[[File:Rga12.png\|center\|450px\|Instantaneous Failure Intensity vs. Time plot.]]

	Each component life <math>{{X}_{j}}\,\!</math> in the socket is governed by the same distribution <math>F(x)\,\!</math>.		Each component life <math>{{X}_{j}}\,\!</math> in the socket is governed by the same distribution <math>F(x)\,\!</math>.

David J. Groebel at 19:50, 2 June 2014

2014-06-02T19:50:51Z

← Older revision		Revision as of 19:50, 2 June 2014
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	Sometimes, the [[Crow Extended]] model, which was introduced in a previous chapter for analyzing developmental data, is also applied for fielded repairable systems. Applying the Crow Extended model on repairable system data allows analysts to project the system MTBF after reliability-related issues are addressed during the field operation. Projections are calculated based on the mode classifications (A, BC and BD). The calculation procedure is the same as the one for the developmental data, and is not repeated in this chapter.		Sometimes, the [[Crow Extended]] model, which was introduced in a previous chapter for analyzing developmental data, is also applied for fielded repairable systems. Applying the Crow Extended model on repairable system data allows analysts to project the system MTBF after reliability-related issues are addressed during the field operation. Projections are calculated based on the mode classifications (A, BC and BD). The calculation procedure is the same as the one for the developmental data, and is not repeated in this chapter.

	==Distribution Example===		===Distribution Example===
	Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.		Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.

David J. Groebel at 19:50, 2 June 2014

2014-06-02T19:50:37Z

Show changes

Chris Kahn: /* Confidence Bounds for Repairable Systems Analysis */

2014-04-23T21:38:12Z

Confidence Bounds for Repairable Systems Analysis

← Older revision		Revision as of 21:38, 23 April 2014
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	=Confidence Bounds for Repairable Systems Analysis=		=Confidence Bounds for Repairable Systems Analysis=
	The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher matrix approach is based on the Fisher information ~~mtrix~~ and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.		The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher matrix approach is based on the Fisher information matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.

	==Confidence Bounds Example==		==Confidence Bounds Example==

Chris Kahn: /* Confidence Bounds for Repairable Systems Analysis */

2014-04-23T21:35:45Z

Confidence Bounds for Repairable Systems Analysis

← Older revision		Revision as of 21:35, 23 April 2014
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	=Confidence Bounds for Repairable Systems Analysis=		=Confidence Bounds for Repairable Systems Analysis=
	The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher ~~Matrix~~ approach is based on the Fisher ~~Information Matrix~~ and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.		The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher matrix approach is based on the Fisher information mtrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.

	==Confidence Bounds Example==		==Confidence Bounds Example==

Chris Kahn: /* Confidence Bounds for Repairable Systems Analysis */

2014-04-23T21:33:55Z

Confidence Bounds for Repairable Systems Analysis

← Older revision		Revision as of 21:33, 23 April 2014
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	=Confidence Bounds for Repairable Systems Analysis=		=Confidence Bounds for Repairable Systems Analysis=
	The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. ~~See~~ [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.		The RGA software provides two methods to estimate the confidence bounds for repairable systems analysis. The Fisher Matrix approach is based on the Fisher Information Matrix and is commonly employed in the reliability field. The Crow bounds were developed by Dr. Larry Crow. See [[Confidence Bounds for Repairable Systems Analysis]] for details on how these confidence bounds are calculated.

	==Confidence Bounds Example==		==Confidence Bounds Example==

Chris Kahn at 22:46, 29 January 2014

2014-01-29T22:46:27Z

← Older revision		Revision as of 22:46, 29 January 2014
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	Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.		Visualize a socket into which a component is inserted at time 0. When the component fails, it is replaced immediately with a new one of the same kind. After each replacement, the socket is put back into an ''as good as new'' condition. Each component has a time-to-failure that is determined by the underlying distribution. It is important to note that a distribution relates to a single failure. The sequence of failures for the socket constitutes a random process called a ''renewal process''. In the illustration below, the component life is <math>{{X}_{j}}\,\!</math>, and <math>{{t}_{j}}\,\!</math> is the system time to the <math>{{j}^{th}}\,\!</math> failure.

	[[File:Rga12.png\|center\|~~300px~~\|Instantaneous Failure Intensity vs. Time plot.]]		[[File:Rga12.png\|center\|500px\|Instantaneous Failure Intensity vs. Time plot.]]

Kate Racaza: /* Crow Extended for Repairable Systems Analysis Example */

2014-01-29T19:10:17Z

Crow Extended for Repairable Systems Analysis Example

← Older revision		Revision as of 19:10, 29 January 2014
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	===Crow Extended for Repairable Systems Analysis Example===		===Crow Extended for Repairable Systems Analysis Example===
	The failures and fixes of two repairable systems in the field are recorded. Both systems started operating from time 0. System 1 ends at time = 504 and system 2 ends at time = 541. All the BD modes are fixed at the end of the test. A fixed effectiveness factor equal to 0.6 is used. Answer the following questions:		{{:Crow_Extended_Model_for_Repairable_Systems_Analysis_Example}}

	~~:1) Estimate the parameters of the Crow Extended model.~~
	~~:2) Calculate the projected MTBF after the delayed fixes.~~
	~~:3) If no fixes were performed for the future failures, what would be the expected number of failures at time 1,000?~~


	~~'''Solution'''~~
	~~:1) The next figure shows the estimated Crow Extended parameters.~~

	~~[[Image:rga13.14.png\|thumb\|center\|450px\|Crow Extended model for repairable systems.]]~~

	~~:2) The next figure shows the projected MTBF at time = 541 (i.e., the age of the oldest system).~~

	~~[[Image:rga13.15.png\|thumb\|center\|450px\|MTBFs from Crow Extended model.]]~~

	~~:3) The next figure shows the expected number of failures at time = 1,000.~~

	~~[[Image:rga13.16.png\|thumb\|center\|450px\|Cumulative number of failures at time = 1,000.]]~~

Kate Racaza: /* Optimum Overhaul Example */

2014-01-29T19:05:03Z

Optimum Overhaul Example

@@ Line 246: / Line 246: @@
 ===Optimum Overhaul Example===
-Field data have been collected for a system that begins its wearout phase at time zero. The start time for each system is equal to zero and the end time for each system is 10,000 miles. Each system is scheduled to undergo an overhaul after a certain number of miles. It has been determined that the cost of an overhaul is four times more expensive than a repair. The table below presents the data. Do the following:
+{{:Optimum_Overhaul_Example}}
 ===Crow Extended for Repairable Systems Analysis Example===