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

2012-08-27T00:32:38Z

Redirected page to RGA Models for Repairable Systems Analysis#Bounds on Cumulative MTBF

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	~~====Bounds on Cumulative MTBF====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Cumulative_MTBF]]
	~~=====Fisher Matrix Bounds=====~~
	~~The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is approximately treated as being normally distributed.~~

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

	~~The approximate confidence bounds on the cumulative MTBF are then estimated from:~~


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

	~~:where:~~

	~~::<math>{{\widehat{m}}_{c}}(t)=\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}</math>~~


	~~::<math>\begin{align}~~
	& Var({{\widehat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
	~~& & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(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 {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}\ln (t) \\~~
	~~& \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}}{{t}^{1-\widehat{\beta }}}~~
	~~\end{align}</math>~~


	~~=====Crow Bounds=====~~
	~~To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:~~

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


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

	~~:Then~~

	~~::<math>\begin{align}~~
	~~& {{~~[~~MTB{{F}_{c}}~~]~~}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\~~
	~~& {{[MTB{{F}_{c}}~~]~~}_{U}}= & \frac{1}{C{{(t)}_{L}}}~~
	~~\end{align}</math>~~

Nicolette Young: Created page with '====Bounds on Cumulative MTBF==== =====Fisher Matrix Bounds===== The cumulative MTBF, ${{m}_{c}}(t)$ , must be positive, thus $\ln {{m}_{c}}(t)$ is appro…'

2012-01-10T22:39:18Z

Created page with '====Bounds on Cumulative MTBF==== =====Fisher Matrix Bounds===== The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is appro…'

New page

====Bounds on Cumulative MTBF====
=====Fisher Matrix Bounds=====
The cumulative MTBF, <math>{{m}_{c}}(t)</math> , must be positive, thus <math>\ln {{m}_{c}}(t)</math> is approximately treated as being normally distributed.

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

The approximate confidence bounds on the cumulative MTBF are then estimated from:

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

:where:

::<math>{{\widehat{m}}_{c}}(t)=\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}</math>

::<math>\begin{align}
& Var({{\widehat{m}}_{c}}(t))= & {{\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial {{m}_{c}}(t)}{\partial \lambda } \right)}^{2}}Var(\widehat{\lambda }) \\
& & +2\left( \frac{\partial {{m}_{c}}(t)}{\partial \beta } \right)\left( \frac{\partial {{m}_{c}}(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 {{m}_{c}}(t)}{\partial \beta }= & -\frac{1}{\widehat{\lambda }}{{t}^{1-\widehat{\beta }}}\ln (t) \\
& \frac{\partial {{m}_{c}}(t)}{\partial \lambda }= & -\frac{1}{{{\widehat{\lambda }}^{2}}}{{t}^{1-\widehat{\beta }}}
\end{align}</math>

=====Crow Bounds=====
To calculate the Crow confidence bounds on cumulative MTBF, first calculate the Crow cumulative failure intensity confidence bounds:

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

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

:Then

::<math>\begin{align}
& {{[MTB{{F}_{c}}]}_{L}}= & \frac{1}{C{{(t)}_{U}}} \\
& {{[MTB{{F}_{c}}]}_{U}}= & \frac{1}{C{{(t)}_{L}}}
\end{align}</math>

Template:Bounds on cumulative mtbf rsa - Revision history

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

Nicolette Young: Created page with '====Bounds on Cumulative MTBF==== =====Fisher Matrix Bounds===== The cumulative MTBF, {{m}_{c}}(t) , must be positive, thus \ln {{m}_{c}}(t) is appro…'

Nicolette Young: Created page with '====Bounds on Cumulative MTBF==== =====Fisher Matrix Bounds===== The cumulative MTBF, ${{m}_{c}}(t)$ , must be positive, thus $\ln {{m}_{c}}(t)$ is appro…'