Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Time Given Instantaneous MTBF

2012-08-27T00:36:04Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Time Given Instantaneous MTBF

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	~~====Bounds on Time Given Instantaneous MTBF====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Time_Given_Instantaneous_MTBF]]
	~~=====Fisher Matrix Bounds=====~~
	~~The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as being normally distributed.~~

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

	~~The confidence bounds on the time are given by:~~

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

	~~:where:~~

	::<math>Var(\widehat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })</math>

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


	~~::<math>\widehat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}</math>~~


	~~::<math>\begin{align}~~
	~~& \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}[\frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}~~] \\
	~~& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}~~
	~~\end{align}</math>~~

	~~<br>~~
	~~=====Crow Bounds=====~~
	~~Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.~~
	~~<br>~~
	~~Step 2: Calculate the bounds on time as follows.~~
	~~<br>~~
	~~<br>~~
	~~''Failure Terminated Data''~~

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


	~~So the lower an upper bounds on time are:~~


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


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


	~~''Time Terminated Data''~~

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


	~~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}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}</math>~~

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous MTBF==== =====Fisher Matrix Bounds===== The time, $T$ , must be positive, thus $\ln T$ is approximately treated as…'

2012-01-10T22:44:07Z

Created page with '====Bounds on Time Given Instantaneous MTBF==== =====Fisher Matrix Bounds===== The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as…'

New page

====Bounds on Time Given Instantaneous MTBF====
=====Fisher Matrix Bounds=====
The time, <math>T</math> , must be positive, thus <math>\ln T</math> is approximately treated as being normally distributed.

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

The confidence bounds on the time are given by:

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

:where:

::<math>Var(\widehat{T})={{\left( \frac{\partial T}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial T}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda })+2\left( \frac{\partial T}{\partial \beta } \right)\left( \frac{\partial T}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })</math>

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

::<math>\widehat{T}={{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}</math>

::<math>\begin{align}
& \frac{\partial T}{\partial \beta }= & {{\left( \lambda \beta \cdot MTB{{F}_{i}} \right)}^{1/(1-\beta )}}[\frac{1}{{{(1-\beta )}^{2}}}\ln (\lambda \beta \cdot MTB{{F}_{i}})+\frac{1}{\beta (1-\beta )}] \\
& \frac{\partial T}{\partial \lambda }= & \frac{{{(\lambda \beta \cdot MTB{{F}_{i}})}^{1/(1-\beta )}}}{\lambda (1-\beta )}
\end{align}</math>

<br>
=====Crow Bounds=====
Step 1: Calculate the confidence bounds on the instantaneous MTBF as presented in Section 5.5.2.
<br>
Step 2: Calculate the bounds on time as follows.
<br>
<br>
''Failure Terminated Data''

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

So the lower an upper bounds on time are:

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

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

''Time Terminated Data''

::<math>\hat{T}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{\Pi })}^{1/(1-\beta )}}</math>

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}}_{U}}={{(\frac{\lambda \beta \cdot MTB{{F}_{i}}}{{{\Pi }_{2}}})}^{1/(1-\beta )}}</math>

Template:Bounds on time given instantaneous mtbf rsa - Revision history

Richard House: Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Time Given Instantaneous MTBF

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous MTBF==== =====Fisher Matrix Bounds===== The time, T , must be positive, thus \ln T is approximately treated as…'

Nicolette Young: Created page with '====Bounds on Time Given Instantaneous MTBF==== =====Fisher Matrix Bounds===== The time, $T$ , must be positive, thus $\ln T$ is approximately treated as…'