Confidence Bounds for Repairable Systems Analysis - Revision history

Lisa Hacker at 20:42, 18 September 2023

2023-09-18T20:42:16Z

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	{{template:RGA BOOK ~~SUB~~\|Appendix E\|Confidence Bounds for Repairable Systems Analysis}}		{{template:RGA BOOK\|Appendix E\|Confidence Bounds for Repairable Systems Analysis}}
	In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for [[Repairable Systems Analysis\|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.		In this appendix, we will present the two methods used in the RGA software to estimate the confidence bounds for [[Repairable Systems Analysis\|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.

David J. Groebel: /* Crow Bounds */

2014-03-17T21:27:26Z

Crow Bounds

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	\end{align}\,\!</math>		\end{align}\,\!</math>

	where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#~~Cumulative_Failure_Intensity~~\|cumulative failure intensity]].		where <math>CFI_L\,\!</math> and <math>CFI_U\,\!</math> are calculated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Crow_Bounds_4\|cumulative failure intensity]].

	===Instantaneous MTBF===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS		===Instantaneous MTBF===<!-- THIS SECTION HEADER IS LINKED FROM ANOTHER SECTION IN THIS

David J. Groebel: /* Crow Bounds */

2014-03-17T21:25:23Z

Crow Bounds

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	====Crow Bounds====		====Crow Bounds====
	The ~~Crow~~ cumulative failure intensity ~~confidence bounds~~ are given by:		The 2-sided confidence bounds on the cumulative failure intensity are given by:

	:<math>~~C{{(t)}_{L}}~~=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t}\,\!</math>		:<math>CFI_L=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot t \cdot S}\,\!</math>

	:<math>~~C{{(t)}_{U}}~~=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t}\,\!</math>		:<math>CFI_U=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot t \cdot S}\,\!</math>

			where:

			*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time <math>t\,\!</math>.
			*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
			*<math>t\,\!</math> = time at which calculations are being conducted.

	===Cumulative MTBF===		===Cumulative MTBF===

David J. Groebel: /* Crow Bounds */

2014-03-17T21:20:27Z

Crow Bounds

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	where:		where:

	*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to ~~time~~ time <math>t\,\!</math>.		*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time <math>t\,\!</math>.
	*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>		*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
	*<math>t\,\!</math> = time at which calculations are being conducted.		*<math>t\,\!</math> = time at which calculations are being conducted.

David J. Groebel: /* Crow Bounds */

2014-03-17T21:20:06Z

Crow Bounds

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	where:		where:

	*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time time t.		*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time time <math>t\,\!</math>.
	*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>		*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
	*<math>t\,\!</math> = time at which calculations are being conducted.		*<math>t\,\!</math> = time at which calculations are being conducted.

David J. Groebel: /* Crow Bounds */

2014-03-17T21:19:21Z

Crow Bounds

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	====Crow Bounds====		====Crow Bounds====
	:~~<math>\begin{array}{*{35}{l}}~~		The 2-sided confidence bounds on the cumulative number of failures are given by:
	~~{{N}_{L}}(T)=\tfrac{T}{\hat{\beta }}{{\lambda }_{i}}{{(T)}_{L}} \\~~
	~~{{N}_{U}}(T)=\tfrac{T}{\hat{\beta }}{{\lambda }_{i}}{{(T)}_{U}} \\~~
	~~\end{array}\,\!</math>~~

	~~where~~ <math>{{\~~lambda~~ }_{i}}{{(T)}_{L}}\,\!</math> ~~and~~ <math>{{\lambda }~~_{i~~}}{{~~(T)~~}_{U}}\,\!</math> ~~can be obtained using the equations given for the confidence bounds on [[Confidence_Bounds_for_Repairable_Systems_Analysis#Instantaneous_Failure_Intensity\|Instantaneous Failure Intensity]]~~.		:<math>N{{\left( t \right)}_{L}}=\frac{\chi _{\tfrac{\alpha }{2},2N}^{2}}{2\cdot S}\,\!</math>

			:<math>N{{\left( t \right)}_{U}}=\frac{\chi _{1-\tfrac{\alpha }{2},2N+2}^{2}}{2\cdot S}\,\!</math>

			where:

			*<math>N\,\!</math> = total number of failures across all systems. This is not the number of failures up to time time t.
			*<math>S=\frac{\left( \frac{N}{{\hat{\lambda }}} \right)}{{{t}^{{\hat{\beta }}}}}\,\!</math>
			*<math>t\,\!</math> = time at which calculations are being conducted.

	===Cumulative Failure Intensity===		===Cumulative Failure Intensity===

David J. Groebel: /* Crow Bounds */

2014-02-11T21:21:21Z

Crow Bounds

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	'''Failure Terminated'''		'''Failure Terminated'''

	~~Step 1: Calculate <math>({{\hat{R}}_{lower}},{{\hat{R}}_{upper}})=({{R}^{\tfrac{1}{{{p}_{1}}}}}~~,~~{{R}^{\tfrac{1}{{{p}_{~~2~~}}}}})\,\!</math>.~~		For failure terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

	~~Step 2~~: ~~Let~~ <math>R={{\hat{R}}_{~~lower~~}}\,\~~!</math> and solve numerically for <math>~~{{t}_{1}}\,\~~!</math> using <math>R=~~{{e}^{-[\~~hat~~{~~\lambda~~ }{{({{~~{\hat{t~~}}}~~_{1~~}}~~+d)~~}^{\~~hat~~{~~\beta~~ }~~}}-\hat~~{~~\lambda }\hat~~{t}_{1}~~^{\hat{\beta~~ }}]}}\,\!</math>.		:<math>\left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{p}_{1}}}}},{{R}^{\tfrac{1}{{{p}_{2}}}}}\, \right)\,\!</math>

	~~Step 3: Let~~ <math>~~R={{\hat{R}}_{upper}}~~\,\!</math> and ~~solve numerically for~~ <math>~~{{t}_{2}}~~\,\!</math> ~~using <math>R={{e}^{-~~[~~\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}~~]~~}}\,\!</math>~~.		where <math>p_1\,\!</math> and <math>p_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]] for failure terminated data.

	~~Step 4:~~ If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{~~lower~~}}={{t}_{1}}\,\!</math> and <math>{{t}_{~~upper~~}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{~~lower~~}}={{t}_{2}}\,\!</math> and <math>{{t}_{~~upper~~}}={{t}_{1}}\,\!</math>.		Let <math>R={{\hat{R}}_{L}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.

			Let <math>R={{\hat{R}}_{U}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.

			If <math>{{t}_{1}}<{{t}_{2}}\,\!</math> then <math>{{t}_{L}}={{t}_{1}}\,\!</math> and <math>{{t}_{U}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math> then <math>{{t}_{L}}={{t}_{2}}\,\!</math> and <math>{{t}_{U}}={{t}_{1}}\,\!</math>.

	'''Time Terminated'''		'''Time Terminated'''

	~~Step 1~~: ~~Calculate~~ <math>({{\hat{R}}_{~~lower~~}},{{\hat{R}}_{~~upper~~}})=({{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}})\,\!</math>.		For time terminated data, the 2-sided confidence bounds on time given reliability and mission time estimated by calculating:

			:<math>\left( {{{\hat{R}}}_{L}},{{{\hat{R}}}_{U}} \right)=\left( {{R}^{\tfrac{1}{{{\Pi }_{1}}}}},{{R}^{\tfrac{1}{{{\Pi }_{2}}}}}\, \right)</math>.

			where <math>\Pi_1\,\!</math> and <math>\Pi_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]] for time terminated data.

	~~Step 2:~~ Let <math>R={{\hat{R}}_{~~lower~~}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.		Let <math>R={{\hat{R}}_{L}}\,\!</math> and solve numerically for <math>{{t}_{1}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{1}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{1}^{\hat{\beta }}]}}\,\!</math>.

	~~Step 3:~~ Let <math>R={{\hat{R}}_{~~upper~~}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.		Let <math>R={{\hat{R}}_{U}}\,\!</math> and solve numerically for <math>{{t}_{2}}\,\!</math> using <math>R={{e}^{-[\hat{\lambda }{{({{{\hat{t}}}_{2}}+d)}^{\hat{\beta }}}-\hat{\lambda }\hat{t}_{2}^{\hat{\beta }}]}}\,\!</math>.

	~~Step 4:~~ If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{~~lower~~}}={{t}_{1}}\,\!</math> and <math>{{t}_{~~upper~~}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{~~lower~~}}={{t}_{2}}\,\!</math> and <math>{{t}_{~~upper~~}}={{t}_{1}}\,\!</math>.		If <math>{{t}_{1}}<{{t}_{2}}\,\!</math>. then <math>{{t}_{L}}={{t}_{1}}\,\!</math> and <math>{{t}_{U}}={{t}_{2}}\,\!</math>. If <math>{{t}_{1}}>{{t}_{2}}\,\!</math>. then <math>{{t}_{L}}={{t}_{2}}\,\!</math> and <math>{{t}_{U}}={{t}_{1}}\,\!</math>.

	===Mission Time Given Reliability and Time===		===Mission Time Given Reliability and Time===

David J. Groebel: /* Crow Bounds */

2014-02-11T21:05:05Z

Crow Bounds

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	'''Failure Terminated'''		'''Failure Terminated'''

	~~With~~ failure terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval ~~for~~ the current reliability at time <math>t\,\!</math> in a specified mission ~~time~~ <math>d\,\!</math> is:		For failure terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval on the current reliability at time <math>t\,\!</math> for a specified mission duration <math>d\,\!</math> is:

	:<math>({{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>		:<math>\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{p}_{2}}}}} \right)\,\!</math>

	where		where:

	:<math>\hat{R}(\~~tau~~ )={{e}^{-[\hat{\lambda }{{(t+\~~tau~~ )}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}\,\!</math>		*<math>\hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\!</math>
			*<math>p_1\,\!</math> and <math>p_2\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]] for failure terminated data.
	<math>~~{{p}_{1}}~~\,\!</math> and <math>~~{{p}_{2}}~~\,\!</math> ~~can be~~ obtained from ~~the equations for failure terminated data for~~ the confidence bounds on [[~~Confidence_Bounds_for_Repairable_Systems_Analysis~~#Instantaneous_MTBF\|~~Instantaneous~~ MTBF]].

	'''Time Terminated'''		'''Time Terminated'''

	~~With~~ time terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval ~~for~~ the current reliability at time <math>t\,\!</math> in a specified mission ~~time~~ <math>~~\tau~~ \,\!</math> is:		For time terminated data, the 100( <math>1-\alpha \,\!</math> )% confidence interval on the current reliability at time <math>t\,\!</math> for a specified mission duration <math>d\,\!</math> is:

	:<math>({{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{1}}}}},{{[\hat{R}(d)]}^{\tfrac{1}{{{p}_{2}}}}})\,\!</math>		:<math>\left( {{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{1}}}}},{{\left[ \hat{R}\left( d \right) \right]}^{\tfrac{1}{{{\Pi }_{2}}}}} \right)\,\!</math>

	where:		where:

	:<math>\hat{R}(d)={{e}^{-[\hat{\lambda }{{(t+d)}^{\hat{\beta }}}-\hat{\lambda }{{t}^{\hat{\beta }}}]}}\,\!</math>		*<math>\hat{R}\left( d \right)={{e}^{-\left[ \hat{\lambda }{{\left( t+d \right)}^{{\hat{\beta }}}}-\hat{\lambda }{{t}^{{\hat{\beta }}}} \right]}}\,\!</math>
			* <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> are obtained from the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]] for time terminated data.
	<math>{{p}_{1}}\,\!</math> and <math>{{p}_{2}}\,\!</math> ~~can be~~ obtained from ~~the equation for time terminated data for~~ the confidence bounds on [[~~Confidence_Bounds_for_Repairable_Systems_Analysis~~#Instantaneous_MTBF\|~~Instantaneous~~ MTBF]].

	===Time Given Reliability and Mission Time===		===Time Given Reliability and Mission Time===

David J. Groebel: /* Crow Bounds */

2014-02-11T18:20:22Z

Crow Bounds

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	====Crow Bounds====		====Crow Bounds====
	~~Step 1: Calculate~~ <math>~~{{\lambda }_{i}}~~(T)~~=\tfrac{1}{MTB{{F}_{i}}}~~\,\!</math>.		The 2-sided confidence bounds on time given instantaneous failure intensity <math>(IFI)\,\!</math> are estimated using the process for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Time_Given_Instantaneous_MTBF\|time given instantaneous MTBF]] where <math>IMTBF=\frac{1}{IFI}\,\!</math>.

	~~Step 2: Use~~ the ~~equations in~~ the [[~~Confidence_Bounds_for_Repairable_Systems_Analysis~~#Time_Given_Instantaneous_MTBF\|~~Bounds on Time Given Instantaneous~~ MTBF]] ~~section to calculate the bounds on time given the instantaneous failure intensity~~.

	===Reliability===		===Reliability===

David J. Groebel: /* Crow Bounds */

2014-02-11T18:19:32Z

Crow Bounds

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	====Crow Bounds====		====Crow Bounds====
	Step 1: Calculate the confidence bounds on the instantaneous MTBF. <!-- as presented in Section 5.5.2.[Per Kate: THE SECTION 5.5.2 IS ABOUT THE CHI-SQUARED TEST FOR GROUPED DATA. NEED TO ASK SME TO VERIFY THE CORRECT SECTION TO LINK TO.]-->

	~~Step 2: Calculate the bounds on time as follows.~~

	'''Failure Terminated'''		'''Failure Terminated'''

	:<math>\~~hat{T}={{(~~\~~frac{\lambda \beta \cdot MTB{{F}_{i}}}{c})}^{1~~/~~(1-\beta )}}~~\,\!</math>		Calculate the constants <math>p_1\,\!</math> and <math>p_2\,\!</math> using procedures described for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:

	~~So the lower an upper bounds on time are~~:		:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{1}}} \right)}^{\tfrac{1}{1-\beta }}}</math>

	:<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot ~~MTB{{F}_{i}}~~}{{{c}_{1}}})}^~~{1/(1-\beta )}}\,\!</math>~~		:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{p}_{2}}} \right)}^{\tfrac{1}{1-\beta }}}</math>

	~~:<math>{~~{\~~hat~~{~~T}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{c}_{2}}})~~}^{~~1/(~~1-\beta )}}~~\,\!~~</math>

	'''Time Terminated'''		'''Time Terminated'''

	:<math>~~\hat~~{~~T}=~~{{(\~~frac{\lambda \beta \cdot MTB{{F~~}_{i}}}{\Pi }~~)}^~~{~~1/(1-\beta )~~}}\,\!</math>		Calculate the constants <math>{{\Pi }_{1}}\,\!</math> and <math>{{\Pi }_{2}}\,\!</math> using procedures described for the confidence bounds on [[Confidence Bounds for Repairable Systems Analysis#Instantaneous_MTBF\|instantaneous MTBF]]. The lower and upper confidence bounds on time are then given by:

	So the lower and upper bounds on time are:

	:<math>{{\hat{T}}_{L}}={{(\frac{\lambda \beta \cdot ~~MTB{{F}_{i}}~~}{{{\Pi }_{1}}})}^{1/(1-\beta )}}\,\!</math>		:<math>{{\hat{t}}_{L}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{1}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>

	:<math>{{\hat{T}}_{U}}={{(\frac{\lambda \beta \cdot ~~MTB{{F}_{i}}~~}{{{\Pi }_{2}}})}^{1/(1-\beta )}}\,\!</math>		:<math>{{\hat{t}}_{U}}={{\left( \frac{\lambda \beta \cdot IMTBF}{{{\Pi }_{2}}} \right)}^{\tfrac{1}{1-\beta }}}\,\!</math>

	===Time Given Instantaneous Failure Intensity===		===Time Given Instantaneous Failure Intensity===