Template:Bounds on instantaneous mtbf rsa - Revision history

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

2012-08-27T00:33:21Z

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

← Older revision		Revision as of 00:33, 27 August 2012
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	~~====Bounds on Instantaneous MTBF====~~		#REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Bounds_on_Instantaneous_MTBF]]
	~~=====Fisher Matrix Bounds=====~~
	~~The instantaneous MTBF, <math>{{m}_{i}}(t)</math> , must be positive, thus <math>\ln {{m}_{i}}(t)</math> is approximately treated as being normally distributed.~~

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


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

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

	~~:where:~~

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


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


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

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


	~~=====Crow Bounds=====~~
	~~''Failure Terminated Data''~~
	~~<br>~~
	~~To calculate the bounds for failure terminated data, consider the following equation:~~

	::<math>G(\mu \|n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx</math>

	Find the values <math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> by finding the solution <math>c</math> to <math>G({{n}^{2}}/c\|n)=\xi </math> for <math>\xi =\tfrac{\alpha }{2}</math> and <math>\xi =1-\tfrac{\alpha }{2}</math> , respectively. If using the biased parameters, <math>\hat{\beta }</math> and <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:

	~~::<math>\begin{align}~~
	~~& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\~~
	~~& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}}~~
	~~\end{align}</math>~~

	where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters, <math>\bar{\beta }</math> and <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:

	~~::<math>\begin{align}~~
	~~& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\~~
	~~& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}}~~
	~~\end{align}</math>~~

	~~where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .~~
	~~<br>~~
	~~<br>~~
	~~''Time Terminated Data''~~
	~~<br>~~
	~~To calculate the bounds for time terminated data, consider the following equation where <math>{{I}_{1}}(.)</math> is the modified Bessel function of order one:~~

	~~::<math>H(x\|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}</math>~~

	Find the values <math>{{\Pi }_{1}}</math> and <math>{{\Pi }_{2}}</math> by finding the solution <math>x</math> to <math>H(x\|k)=\tfrac{\alpha }{2}</math> and <math>H(x\|k)=1-\tfrac{\alpha }{2}</math> in the cases corresponding to the lower and upper bounds, respectively. <br>
	Calculate <math>\Pi =\tfrac{{{n}^{2}}}{4{{x}^{2}}}</math> for each case. If using the biased parameters, <math>\hat{\beta }</math> and <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:

	~~::<math>\begin{align}~~
	~~& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\~~
	~~& {{~~[~~MTB{{F}_{i}}~~]~~}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}}~~
	~~\end{align}</math>~~

	where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters, <math>\bar{\beta }</math> and <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:

	~~::<math>\begin{align}~~
	~~& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\~~
	~~& {{[MTB{{F}_{i}}~~]~~}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}}~~
	~~\end{align}</math>~~

	~~where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .~~
	~~<br>~~
	~~<br>~~

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

2012-01-10T22:39:59Z

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

New page

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

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

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

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

:where:

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

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

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

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

=====Crow Bounds=====
''Failure Terminated Data''
 
To calculate the bounds for failure terminated data, consider the following equation:

::<math>G(\mu |n)=\mathop{}_{0}^{\infty }\frac{{{e}^{-x}}{{x}^{n-2}}}{(n-2)!}\underset{i=0}{\overset{n-1}{\mathop \sum }}\,\frac{1}{i!}{{\left( \frac{\mu }{x} \right)}^{i}}\exp (-\frac{\mu }{x})\,dx</math>

Find the values <math>{{p}_{1}}</math> and <math>{{p}_{2}}</math> by finding the solution <math>c</math> to <math>G({{n}^{2}}/c|n)=\xi </math> for <math>\xi =\tfrac{\alpha }{2}</math> and <math>\xi =1-\tfrac{\alpha }{2}</math> , respectively. If using the biased parameters, <math>\hat{\beta }</math> and <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:

::<math>\begin{align}
& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{p}_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{p}_{2}}
\end{align}</math>

where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters, <math>\bar{\beta }</math> and <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:

::<math>\begin{align}
& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-2}{N} \right)\cdot {{p}_{2}}
\end{align}</math>

where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .
 
 
''Time Terminated Data''
 
To calculate the bounds for time terminated data, consider the following equation where <math>{{I}_{1}}(.)</math> is the modified Bessel function of order one:

::<math>H(x|k)=\underset{j=1}{\overset{k}{\mathop \sum }}\,\frac{{{x}^{2j-1}}}{{{2}^{2j-1}}(j-1)!j!{{I}_{1}}(x)}</math>

Find the values <math>{{\Pi }_{1}}</math> and <math>{{\Pi }_{2}}</math> by finding the solution <math>x</math> to <math>H(x|k)=\tfrac{\alpha }{2}</math> and <math>H(x|k)=1-\tfrac{\alpha }{2}</math> in the cases corresponding to the lower and upper bounds, respectively. 
Calculate <math>\Pi =\tfrac{{{n}^{2}}}{4{{x}^{2}}}</math> for each case. If using the biased parameters, <math>\hat{\beta }</math> and <math>\hat{\lambda }</math> , then the upper and lower confidence bounds are:

::<math>\begin{align}
& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot {{\Pi }_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot {{\Pi }_{2}}
\end{align}</math>

where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> . If using the unbiased parameters, <math>\bar{\beta }</math> and <math>\bar{\lambda }</math> , then the upper and lower confidence bounds are:

::<math>\begin{align}
& {{[MTB{{F}_{i}}]}_{L}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{1}} \\
& {{[MTB{{F}_{i}}]}_{U}}= & MTB{{F}_{i}}\cdot \left( \frac{N-1}{N} \right)\cdot {{\Pi }_{2}}
\end{align}</math>

where <math>MTB{{F}_{i}}=\tfrac{1}{\hat{\lambda }\hat{\beta }{{t}^{\hat{\beta }-1}}}</math> .

Template:Bounds on instantaneous mtbf rsa - Revision history

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

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

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