Template:Bounds on time given instantaneous mtbf rsa: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
(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…')
 
 
Line 1: Line 1:
====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>

Latest revision as of 00:36, 27 August 2012

Redirect to:

Retrieved from "https://www.reliawiki.com/index.php?title=Template:Bounds_on_time_given_instantaneous_mtbf_rsa&oldid=33923"