Duane Model - Revision history

David J. Groebel: /* Other Bounds */

2018-08-08T19:07:15Z

Other Bounds

← Older revision		Revision as of 19:07, 8 August 2018
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	:<math>C{{B}_{{{m}_{c}}}}={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var\left[ \ln ({{{\hat{m}}}_{c}}) \right]}}}\,\!</math>		:<math>C{{B}_{{{m}_{c}}}}={{\hat{m}}_{c}}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var\left[ \ln ({{{\hat{m}}}_{c}}) \right]}}}\,\!</math>

			:<math>Var\left[ \ln ({{{\hat{m}}}_{c}}) \right]=\frac{\sum\limits_{i=1}^{n}{{{\left( \ln {{{\hat{m}}}_{c}}({{t}_{i}})-\ln {{m}_{c}}({{t}_{i}}) \right)}^{2}}}}{n-2}\cdot \left( \frac{1}{n}+\frac{{{\left( \ln (t)-\frac{\sum\limits_{i=1}^{n}{\ln ({{t}_{i}})}}{n} \right)}^{2}}}{\sum\limits_{i=1}^{n}{{{\left( \ln ({{t}_{i}})-\frac{\sum\limits_{i=1}^{n}{\ln ({{t}_{i}})}}{n} \right)}^{2}}}} \right)\,\!</math>

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

David J. Groebel: /* Parameter Estimation */

2015-01-05T22:48:00Z

Parameter Estimation

← Older revision		Revision as of 22:48, 5 January 2015
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	The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.		The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.

	One of the parameters of the Duane model is <math>\alpha\,\!</math>. The second parameter can be represented as <math>A\,\!</math> or <math>b\,\!</math> where <math>A=\frac{1}{b}\,\!</math>. There is an option within the Application Setup that allows ~~the user~~ to determine whether to display <math>A\,\!</math> or <math>b\,\!</math> for the Duane model. All formulation within this reference uses the parameter <math>b\,\!</math>.		One of the parameters of the Duane model is <math>\alpha\,\!</math>. The second parameter can be represented as <math>A\,\!</math> or <math>b\,\!</math> where <math>A=\frac{1}{b}\,\!</math>. There is an option within the Application Setup that allows you to determine whether to display <math>A\,\!</math> or <math>b\,\!</math> for the Duane model. All formulation within this reference uses the parameter <math>b\,\!</math>.

	[[Image:DuaneShowAorb.png\|center\|450px]]		[[Image:DuaneShowAorb.png\|center\|450px]]

David J. Groebel: /* Parameter Estimation */

2015-01-05T22:31:37Z

Parameter Estimation

← Older revision		Revision as of 22:31, 5 January 2015
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	The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.		The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.

	One of the parameters of the Duane model is <math>alpha\,\!</math>. The second parameter can be represented as <math>A\,\!</math> or <math>b\,\!</math> where <math>A=\frac{1}{b}\,\!</math>. There is an option within the Application Setup that allows the user to determine whether to display <math>A\,\!</math> or <math>b\,\!</math> for the Duane model. All formulation within this reference uses the parameter <math>b\,\!</math>.		One of the parameters of the Duane model is <math>\alpha\,\!</math>. The second parameter can be represented as <math>A\,\!</math> or <math>b\,\!</math> where <math>A=\frac{1}{b}\,\!</math>. There is an option within the Application Setup that allows the user to determine whether to display <math>A\,\!</math> or <math>b\,\!</math> for the Duane model. All formulation within this reference uses the parameter <math>b\,\!</math>.

	[[Image:DuaneShowAorb.png\|center\|450px]]		[[Image:DuaneShowAorb.png\|center\|450px]]

David J. Groebel at 22:29, 5 January 2015

2015-01-05T22:29:28Z

← Older revision		Revision as of 22:29, 5 January 2015
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	The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.		The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.

	One of the parameters of the Duane model is alpha. The second parameter can be represented as A or b where A = 1/b. There is an option within the Application Setup that allows the user to determine whether to display A or b for the Duane model.		One of the parameters of the Duane model is <math>alpha\,\!</math>. The second parameter can be represented as <math>A\,\!</math> or <math>b\,\!</math> where <math>A=\frac{1}{b}\,\!</math>. There is an option within the Application Setup that allows the user to determine whether to display <math>A\,\!</math> or <math>b\,\!</math> for the Duane model. All formulation within this reference uses the parameter <math>b\,\!</math>.

	[[Image:DuaneShowAorb.png\|center\|450px]]		[[Image:DuaneShowAorb.png\|center\|450px]]

David J. Groebel: /* Parameter Estimation */

2015-01-05T22:22:25Z

Parameter Estimation

← Older revision		Revision as of 22:22, 5 January 2015
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	==Parameter Estimation==		==Parameter Estimation==
	The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.		The Duane model is a two parameter model. Therefore, to use this model as a basis for predicting the reliability growth that could be expected in an equipment development program, procedures must be defined for estimating these parameters as a function of equipment characteristics. Note that, while these parameters can be estimated for a given data set using curve-fitting methods, there exists no underlying theory for the Duane model that could provide a basis for ''a priori'' estimation.

			One of the parameters of the Duane model is alpha. The second parameter can be represented as A or b where A = 1/b. There is an option within the Application Setup that allows the user to determine whether to display A or b for the Duane model.

			[[Image:DuaneShowAorb.png\|center\|450px]]

	===Graphical Method===		===Graphical Method===

Chris Kahn at 16:12, 14 May 2014

2014-05-14T16:12:53Z

← Older revision		Revision as of 16:12, 14 May 2014
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	[8]]]. While analyzing the data, it was observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper, as shown next.		[8]]]. While analyzing the data, it was observed that the cumulative MTBF versus cumulative operating time followed a straight line when plotted on log-log paper, as shown next.

	[[Image:rga4.1.png\|center\|~~500px~~\|Cumulative MTBF vs. Cumulative Test Time postulated by Duane.]]		[[Image:rga4.1.png\|center\|450px\|Cumulative MTBF vs. Cumulative Test Time postulated by Duane.]]

	Based on that observation, Duane developed his model as follows. If <math>N(T)\,\!</math> is the number of failures by time <math>T\,\!</math>, the observed mean (average) time between failures, <math>MTB{{F}_{c}},\,\!</math> at time <math>T\,\!</math> is:		Based on that observation, Duane developed his model as follows. If <math>N(T)\,\!</math> is the number of failures by time <math>T\,\!</math>, the observed mean (average) time between failures, <math>MTB{{F}_{c}},\,\!</math> at time <math>T\,\!</math> is:
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	As shown in these derivations, the instantaneous failure intensity improvement line is obtained by shifting the cumulative failure intensity line down, parallel to itself, by a distance of <math>(1-\alpha )\,\!</math>. Similarly, the current or instantaneous MTBF growth line is obtained by shifting the cumulative MTBF line up, parallel to itself, by a distance of <math>\tfrac{1}{1-\alpha }\,\!</math>, as illustrated in the figure below.		As shown in these derivations, the instantaneous failure intensity improvement line is obtained by shifting the cumulative failure intensity line down, parallel to itself, by a distance of <math>(1-\alpha )\,\!</math>. Similarly, the current or instantaneous MTBF growth line is obtained by shifting the cumulative MTBF line up, parallel to itself, by a distance of <math>\tfrac{1}{1-\alpha }\,\!</math>, as illustrated in the figure below.

	[[Image:rga4.1.png\|center\|~~500px~~\|Cumulative MTBF vs. Cumulative Test Time postulated by Duane.]]		[[Image:rga4.1.png\|center\|450px\|Cumulative MTBF vs. Cumulative Test Time postulated by Duane.]]

	==Parameter Estimation==		==Parameter Estimation==

Chris Kahn: /* Graphical Method */

2014-05-14T16:06:26Z

Graphical Method

← Older revision		Revision as of 16:06, 14 May 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Sample ~~slope values~~ for ~~various equipment~~'''		\|+'''Sample Slope Values for Various Equipment'''
	\|-		\|-
	!Equipment		!Equipment

Chris Kahn: /* Graphical Method */

2014-05-14T16:06:08Z

Graphical Method

← Older revision		Revision as of 16:06, 14 May 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Sample values ~~for the slope ( <math>\alpha\,\!</math> )~~ for various equipment'''		\|+'''Sample slope values for various equipment'''
	\|-		\|-
	!Equipment		!Equipment

Chris Kahn: /* Graphical Method */

2014-05-14T16:04:30Z

Graphical Method

← Older revision		Revision as of 16:04, 14 May 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Sample values for the slope ( <math>\alpha \,\!</math> ) for various equipment'''		\|+'''Sample values for the slope ( <math>\alpha\,\!</math> ) for various equipment'''
	\|-		\|-
	!Equipment		!Equipment

Chris Kahn: /* Graphical Method */

2014-05-14T15:58:27Z

Graphical Method

← Older revision		Revision as of 15:58, 14 May 2014
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	{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"		{\|border="1" align="center" style="border-collapse: collapse;" cellpadding="5" cellspacing="5"
	\|+'''Sample values for the slope ( <math>\alpha )\,\!</math> for various equipment'''		\|+'''Sample values for the slope ( <math>\alpha \,\!</math> ) for various equipment'''
	\|-		\|-
	!Equipment		!Equipment