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| ===Confidence Bounds for Repairable Systems Analysis===
| | #REDIRECT [[RGA_Models_for_Repairable_Systems_Analysis#Confidence_Bounds_for_Repairable_Systems_Analysis]] |
| {{bounds on beta rsa}}
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| {{bounds on lambda rsa}}
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| {{bounds on growth rate rsa}}
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| {{bounds on cumulative mtbf rsa}}
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| {{bounds on instantaneous mtbf rsa}}
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| {{bounds on cumulative failure intensity rsa}}
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| {{bounds on instantaneous failure intensity rsa}}
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| {{bounds on time given cumulative mtbf rsa}}
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| {{bounds on time given instantaneous mtbf rsa}}
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| {{bounds on time given cumulative failure intensity rsa}}
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| {{bounds on time given instantaneous failure intensity rsa}}
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| {{bounds on reliability rsa}}
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| {{bounds on time given reliability and mission time rsa}}
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| {{bounds on mission time given reliability and time rsa}}
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| ====Bounds on Cumulative Number of Failures====
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| =====Fisher Matrix Bounds=====
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| 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.
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| ::<math>\frac{\ln (\widehat{N}(t))-\ln (N(t))}{\sqrt{Var\left[ \ln \widehat{N}(t) \right]}}\sim N(0,1)</math>
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| ::<math>N(t)=\widehat{N}(t){{e}^{\pm {{z}_{\alpha }}\sqrt{Var(\widehat{N}(t))}/\widehat{N}(t)}}</math>
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| :where:
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| ::<math>\widehat{N}(t)=\widehat{\lambda }{{t}^{\widehat{\beta }}}</math>
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| <br>
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| ::<math>\begin{align}
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| & 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 }) \\
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| & & +2\left( \frac{\partial N(t)}{\partial \beta } \right)\left( \frac{\partial N(t)}{\partial \lambda } \right)cov(\widehat{\beta },\widehat{\lambda })
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| \end{align}</math>
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| The variance calculation is the same as Eqns. (var1), (var2) and (var3).
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| <br>
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| ::<math>\begin{align}
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| & \frac{\partial N(t)}{\partial \beta }= & \hat{\lambda }{{t}^{\widehat{\beta }}}\ln (t) \\
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| & \frac{\partial N(t)}{\partial \lambda }= & t\widehat{\beta }
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| \end{align}</math>
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| <br>
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| =====Crow Bounds=====
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| ::<math>\begin{array}{*{35}{l}}
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| {{N}_{L}}(T)=\tfrac{T}{\widehat{\beta }}{{\lambda }_{i}}{{(T)}_{L}} \\
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| {{N}_{U}}(T)=\tfrac{T}{\widehat{\beta }}{{\lambda }_{i}}{{(T)}_{U}} \\
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| \end{array}</math>
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| where <math>{{\lambda }_{i}}{{(T)}_{L}}</math> and <math>{{\lambda }_{i}}{{(T)}_{U}}</math> can be obtained from Eqn. (inr).
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| <br>
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| <br>
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| =====Example 3=====
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| 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.
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| <br>
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| ''Solution''
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| <br>
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| The maximum likelihood estimates of <math>\widehat{\lambda }</math> and <math>\widehat{\beta }</math> from Example 1 are:
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| ::<math>\begin{align}
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| & \widehat{\beta }= & 0.45300 \\
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| & \widehat{\lambda }= & 0.36224
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| \end{align}</math>
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| From Eq. (reliability), the mission reliability at <math>t=2000</math> for mission time <math>d=40</math> is:
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| ::<math>\begin{align}
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| & \widehat{R}(t)= & {{e}^{-\left[ \lambda {{\left( t+d \right)}^{\beta }}-\lambda {{t}^{\beta }} \right]}} \\
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| & = & 0.90292
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| \end{align}</math>
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| 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:
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| ::<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>
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| ::<math>\begin{align}
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| & {{[\widehat{R}(t)]}_{L}}= & 0.83711 \\
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| & {{[\widehat{R}(t)]}_{U}}= & 0.94392
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| \end{align}</math>
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| The Crow confidence bounds for the mission reliability are:
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| ::<math>\begin{align}
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| & {{[\widehat{R}(t)]}_{L}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{1}}}}} \\
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| & = & {{[0.90292]}^{\tfrac{1}{0.71440}}} \\
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| & = & 0.86680 \\
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| & {{[\widehat{R}(t)]}_{U}}= & {{[\widehat{R}(\tau )]}^{\tfrac{1}{{{\Pi }_{2}}}}} \\
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| & = & {{[0.90292]}^{\tfrac{1}{1.6051}}} \\
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| & = & 0.93836
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| \end{align}</math>
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| Figures ConfReliFish and ConfRelCrow show the Fisher Matrix and Crow confidence bounds on mission reliability for mission time <math>d=40</math> .
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| [[Image:rga13.3.png|thumb|center|300px|Conditional Reliability vs. Time plot with Fisher Matrix confidence bounds.]]
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| <br>
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| <br>
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| [[Image:rga13.4.png|thumb|center|300px|Conditional Reliability vs. Time plot with Crow confidence bounds.]] | |
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| <br>
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