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==Arrhenius-Exponential==
#REDIRECT [[Arrhenius_Relationship#Arrhenius-Exponential]]
<br>
The  <math>pdf</math>  of the 1-parameter exponential distribution is given by:
 
<br>
::<math>f(t)=\lambda {{e}^{-\lambda t}}</math>
 
<br>
It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail in Chapter 5) is given by:
 
 
<br>
::<math>\lambda =\frac{1}{m}</math>
 
<br>
thus:
 
 
<br>
::<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}</math>
 
<br>
The Arrhenius-exponential model  <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (arrhenius).
<br>
Therefore:
 
<br>
::<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}</math>
 
<br>
Substituting for  <math>m</math>  in Eqn. (pdfexpm) yields a  <math>pdf</math>  that is both a function of time and stress or:
 
 
<br>
::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}</math>
 
===Arrhenius-Exponential Statistical Properties Summary===
<br>
====Mean or MTTF====
<br>
The mean,  <math>\overline{T},</math>  or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,
 
 
<br>
::<math>\begin{align}
  & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt \\
& = & C{{e}^{\tfrac{B}{V}}} 
\end{align}</math>
 
<br>
====Median====
<br>
The median, <math>breve{T},</math>
of the Arrhenius-exponential model is given by:
 
<br>
::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}</math>
 
 
<br>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math> 
of the Arrhenius-exponential model is given by:
 
<br>
::<math>\tilde{T}=0</math>
 
<br>
 
====Standard Deviation====
The standard deviation,  <math>{{\sigma }_{T}}</math> , of the Arrhenius-exponential model is given by:
 
<br>
::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}</math>
 
====Arrhenius-Exponential Reliability Function====
<br>
 
The Arrhenius-exponential reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
<br>
This function is the complement of the Arrhenius-exponential cumulative distribution function or:
 
<br>
::<math>R(T,V)=1-Q(T,V)=1-\mathop{}_{0}^{T}f(T,V)dT</math>
 
<br>
and:
 
<br>
::<math>R(T,V)=1-\mathop{}_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
<br>
 
====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by,
 
::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
====Reliable Life====
<br>
For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal,  <math>{{t}_{R}},</math>  is given by:
 
<br>
::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}</math>
 
 
 
<br>
::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}</math>
 
<br>
or:
 
<br>
::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]</math>
 
===Parameter Estimation===
 
====Maximum Likelihood Estimation Method====
<br>
The log-likelihood function for the exponential distribution is as shown next:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda T_{i}^{\prime } \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })] 
\end{align}</math>
<br>
where:
 
<br>
::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}</math>
 
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}</math>
 
<br>
and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
 
• <math>{{N}_{i}}</math>  is the number of times-to-failure in the  <math>{{i}^{th}}</math>  time-to-failure data group.
 
• <math>\lambda </math>  is the failure rate parameter (unknown).
 
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
 
• <math>S</math>  is the number of groups of suspension data points.
 
• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
 
• <math>T_{i}^{\prime }</math>  is the time of the  <math>{{i}^{th}}</math>  suspension data group.
 
• <math>FI</math>  is the number of interval data groups.
 
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the i <math>^{th}</math>  group of data intervals.
 
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
 
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
<br>
Substituting the Arrhenius-exponential model into the log-likelihood function yields:
 
<br>
::<math>\begin{align}
  & \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}T_{i}^{\prime }+\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }]
\end{align}</math>
 
 
<br>
where:
 
<br>
::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}</math>
 
 
<br>
The solution (parameter estimates) will be found by solving for the parameters  <math>\widehat{B},</math>  <math>\widehat{C}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial B}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial C}=0</math> , where:
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} 
\end{align}</math>
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} 
\end{align}</math>
 
<br>

Latest revision as of 23:57, 15 August 2012

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