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(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(…')
 
 
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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>

Latest revision as of 00:40, 27 August 2012

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