Template:Confidence bounds for rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Confidence Bounds for Repairable Systems Analysis

2012-08-27T00:29:46Z

Redirected page to RGA Models for Repairable Systems Analysis#Confidence Bounds for Repairable Systems Analysis

← Older revision		Revision as of 00:29, 27 August 2012
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	~~===Confidence Bounds for Repairable Systems Analysis===~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Confidence_Bounds_for_Repairable_Systems_Analysis]]
	~~{{bounds on beta rsa}}~~

	~~{{bounds on lambda rsa}}~~

	~~{{bounds on growth rate rsa}}~~

	~~{{bounds on cumulative mtbf rsa}}~~

	~~{{bounds on instantaneous mtbf rsa}}~~

	~~{{bounds on cumulative failure intensity rsa}}~~

	~~{{bounds on instantaneous failure intensity rsa}}~~

	~~{{bounds on time given cumulative mtbf rsa}}~~

	~~{{bounds on time given instantaneous mtbf rsa}}~~

	~~{{bounds on time given cumulative failure intensity rsa}}~~

	~~{{bounds on time given instantaneous failure intensity rsa}}~~

	~~{{bounds on reliability rsa}}~~

	~~{{bounds on time given reliability and mission time rsa}}~~

	~~{{bounds on mission time given reliability and time rsa}}~~

	~~{{bounds on cumulative number of failures rsa}}~~

Nicolette Young: /* Bounds on Cumulative Number of Failures */

2012-01-10T22:49:11Z

Bounds on Cumulative Number of Failures

← Older revision		Revision as of 22:49, 10 January 2012
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	{{bounds on mission time given reliability and time rsa}}		{{bounds on mission time given reliability and time rsa}}

	~~====Bounds~~ on ~~Cumulative Number of Failures====~~		{{bounds on cumulative number of failures rsa}}
	~~=====Fisher Matrix Bounds=====~~
	~~The~~ cumulative number of failures~~, <math>N(t)</math> , must be positive, thus <math>\ln \left( N(t) \right)</math> is approximately treated as being normally distributed.~~

	~~::<math>\frac{\ln (\widehat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \widehat{N}(t) \right]}}\sim N(0,1)</math>~~


	~~::<math>N(t)=\widehat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{N}(t))}/\widehat{N}(t)}}</math>~~


	~~:where:~~

	~~::<math>\widehat{N}(t)=\widehat{\lambda }{{t}^{\widehat{\beta }}}</math>~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& Var(\widehat{N}(t))= & {{\left( \frac{\partial N(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial N(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\~~
	~~& & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })~~
	~~\end{align}</math>~~


	~~The variance calculation is the same as Eqns. (var1), (var2) and (var3).~~

	~~<br>~~
	~~::<math>\begin{align}~~
	~~& \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\widehat{\beta }}}\ln (t) \\~~
	~~& \frac{\partial N(t)}{\partial \lambda }= & t\widehat{\beta }~~
	~~\end{align}</math>~~

	~~<br>~~
	~~=====Crow Bounds=====~~
	~~::<math>\begin{array}{*{35}{l}}~~
	~~{{N}_{L}}(T)=\tfrac{T}{\widehat{\beta }}{{\lambda }_{i}}{{(T)}_{L}} \\~~
	~~{{N}_{U}}(T)=\tfrac{T}{\widehat{\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 from Eqn. (inr).~~
	~~<br>~~
	~~<br>~~
	~~=====Example 3=====~~
	Using the data from Example 1, calculate the mission reliability at <math>t=2000</math> hours and mission time <math>d=40</math> hours along with the confidence bounds at the 90% confidence level.
	~~<br>~~
	~~''Solution''~~
	~~<br>~~
	~~The maximum likelihood estimates of <math>\widehat{\lambda }</math> and <math>\widehat{\beta }</math> from Example 1 are:~~


	~~::<math>\begin{align}~~
	~~& \widehat{\beta }= & 0.45300 \\~~
	~~& \widehat{\lambda }= & 0.36224~~
	~~\end{align}</math>~~


	~~From Eq. (reliability), the mission reliability at <math>t=2000</math> for mission time <math>d=40</math> is:~~

	~~::<math>\begin{align}~~
	~~& \widehat{R~~}~~(t)= & {{e~~}~~^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\~~
	~~& = & 0.90292~~
	~~\end{align}</math>~~


	~~At the 90% confidence level and <math>T=2000</math> hours, the Fisher Matrix confidence bounds for the mission reliability for mission time <math>d=40</math> are given by:~~

	~~::<math>CB=\frac{\widehat{R}(t)}{\widehat{R}(t)+(1-\widehat{R}(t)){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{R}(t))}/\left[ \widehat{R}(t)(1-\widehat{R}(t)) \right]}}}</math>~~


	~~::<math>\begin{align}~~
	~~& {{[\widehat{R}(t)]}_{L}}= & 0.83711 \\~~
	~~& {{[\widehat{R}(t)]}_{U}}= & 0.94392~~
	~~\end{align}</math>~~


	~~The Crow confidence bounds for the mission reliability are:~~

	~~::<math>\begin{align}~~
	~~& {{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{1}}}}} \\~~
	~~& = & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\~~
	~~& = & 0.86680 \\~~
	~~& {{[\widehat{R}(t)]}_{U}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{2}}}}} \\~~
	~~& = & {{[0.90292]}^{\tfrac{1}{1.6051}}} \\~~
	~~& = & 0.93836~~
	~~\end{align}</math>~~


	~~Figures ConfReliFish and ConfRelCrow show the Fisher Matrix and Crow confidence bounds on mission reliability for mission time <math>d=40</math> .~~


	~~[[Image:rga13.3.png\|thumb\|center\|300px\|Conditional Reliability vs. Time plot with Fisher Matrix confidence bounds.]]~~
	~~<br>~~
	~~<br>~~
	~~[[Image:rga13.4.png\|thumb\|center\|300px\|Conditional Reliability vs. Time plot with Crow confidence bounds.]]~~

	~~<br>~~

Nicolette Young: /* Bounds on Mission Time Given Reliability and Time */

2012-01-10T22:48:35Z

Bounds on Mission Time Given Reliability and Time

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 {{bounds on time given reliability and mission time rsa}}
-====Bounds on Mission Time Given Reliability and Time====
+{{bounds on mission time given reliability and time rsa}}
 ====Bounds on Cumulative Number of Failures====
 =====Fisher Matrix Bounds=====

Nicolette Young: /* Bounds on Time Given Reliability and Mission Time */

2012-01-10T22:47:27Z

Bounds on Time Given Reliability and Mission Time

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 {{bounds on reliability rsa}}
-====Bounds on Time Given Reliability and Mission Time====
+{{bounds on time given reliability and mission time rsa}}
 ====Bounds on Mission Time Given Reliability and Time====
 =====Fisher Matrix Bounds=====

Nicolette Young: /* Bounds on Reliability */

2012-01-10T22:46:49Z

Bounds on Reliability

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 {{bounds on time given instantaneous failure intensity rsa}}
-====Bounds on Reliability====
+{{bounds on reliability rsa}}
 ====Bounds on Time Given Reliability and Mission Time====

Nicolette Young: /* Bounds on Time Given Instantaneous Failure Intensity */

2012-01-10T22:46:16Z

Bounds on Time Given Instantaneous Failure Intensity

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 {{bounds on time given cumulative failure intensity rsa}}
-====Bounds on Time Given Instantaneous Failure Intensity====
+{{bounds on time given instantaneous failure intensity rsa}}
 ====Bounds on Reliability====
 =====Fisher Matrix Bounds=====

Nicolette Young: /* Bounds on Time Given Cumulative Failure Intensity */

2012-01-10T22:44:42Z

Bounds on Time Given Cumulative Failure Intensity

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 {{bounds on time given instantaneous mtbf rsa}}
-====Bounds on Time Given Cumulative Failure Intensity====
+{{bounds on time given cumulative failure intensity rsa}}
 ====Bounds on Time Given Instantaneous Failure Intensity====

Nicolette Young: /* Bounds on Time Given Instantaneous MTBF */

2012-01-10T22:43:49Z

Bounds on Time Given Instantaneous MTBF

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 {{bounds on time given cumulative mtbf rsa}}
-====Bounds on Time Given Instantaneous MTBF====
+{{bounds on time given instantaneous mtbf rsa}}
 ====Bounds on Time Given Cumulative Failure Intensity====

Nicolette Young: /* Bounds on Time Given Cumulative MTBF */

2012-01-10T22:43:11Z

Bounds on Time Given Cumulative MTBF

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 {{bounds on instantaneous failure intensity rsa}}
-====Bounds on Time Given Cumulative MTBF====
+{{bounds on time given cumulative mtbf rsa}}
 ====Bounds on Time Given Instantaneous MTBF====

Nicolette Young: /* Bounds on Instantaneous Failure Intensity */

2012-01-10T22:42:08Z

Bounds on Instantaneous Failure Intensity

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 {{bounds on cumulative failure intensity rsa}}
-====Bounds on Instantaneous Failure Intensity====
+{{bounds on instantaneous failure intensity rsa}}
 ====Bounds on Time Given Cumulative MTBF====