Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Cumulative Number of Failures

2012-08-27T00:40:16Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Cumulative Number of Failures

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	~~====Bounds on Cumulative Number of Failures====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Cumulative_Number_of_Failures]]
	~~=====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: Created page with '====Bounds on Cumulative Number of Failures==== =====Fisher Matrix Bounds===== The cumulative number of failures, $N(t)$ , must be positive, thus $\ln \left( N(…'$

2012-01-10T22:49:26Z

Created page with '====Bounds on Cumulative Number of Failures==== =====Fisher Matrix Bounds===== The cumulative number of failures, <math>N(t)</math> , must be positive, thus <math>\ln \left( N(…'

New page

====Bounds on Cumulative Number of Failures====
=====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>

 
::<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).

 
::<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>

 
=====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).
 
 
=====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.
 
''Solution''
 
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.]]
 
 
[[Image:rga13.4.png|thumb|center|300px|Conditional Reliability vs. Time plot with Crow confidence bounds.]]

Template:Bounds on cumulative number of failures rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Cumulative Number of Failures

Nicolette Young: Created page with '====Bounds on Cumulative Number of Failures==== =====Fisher Matrix Bounds===== The cumulative number of failures, N(t) , must be positive, thus \ln \left( N(…'

Nicolette Young: Created page with '====Bounds on Cumulative Number of Failures==== =====Fisher Matrix Bounds===== The cumulative number of failures, $N(t)$ , must be positive, thus $\ln \left( N(…'$