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(Created page with '==Eyring-Exponential== <br> The <math>pdf</math> of the 1-parameter exponential distribution is given by: <br> ::<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot t}}</math> <br> …')
 
 
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==Eyring-Exponential==
#REDIRECT [[Eyring_Relationship#Eyring-Exponential]]
 
<br>
The <math>pdf</math> of the 1-parameter exponential distribution is given by:
 
<br>
::<math>f(t)=\lambda \cdot {{e}^{-\lambda \cdot 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}\cdot {{e}^{-\tfrac{t}{m}}}</math>
 
<br>
The Eyring-exponential model  <math>pdf</math>  can then be obtained by setting  <math>m=L(V)</math>  in Eqn. (eyring):
 
<br>
::<math>m=L(V)=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
<br>
and substituting for  <math>m</math>  in Eqn. (pdfexpm2):
 
<br>
::<math>f(t,V)=V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}\cdot t}}</math>
<br>
===Eyring-Exponential Statistical Properties Summary===
====Mean or MTTF====
<br>
The mean, <math>\overline{T},</math> or Mean Time To Failure (MTTF) for the Eyring-exponential is given by:
 
<br>
::<math>\begin{align}
  & \overline{T}= & \mathop{}_{0}^{\infty }t\cdot f(t,V)dt=\mathop{}_{0}^{\infty }t\cdot V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-tV{{e}^{\left( A-\tfrac{B}{V} \right)}}}}dt \\
& = & \frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}} 
\end{align}</math>
====Median====
<br>
 
The median, <math>\breve{T},</math>
for the Eyring-exponential model is given by:
 
<br>
 
::<math>\breve{T}=0.693\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
 
<br>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math>
for the Eyring-exponential model is <math>\tilde{T}=0.</math>
<br>
====Standard Deviation====
<br>
The standard deviation,  <math>{{\sigma }_{T}}</math>, for the Eyring-exponential model is given by:
<br>
::<math>{{\sigma }_{T}}=\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}</math>
<br>
====Eyring-Exponential Reliability Function====
<br>
The Eyring-exponential reliability function is given by:
 
<br>
::<math>R(T,V)={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
<br>
This function is the complement of the Eyring-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}V{{e}^{\left( A-\tfrac{B}{V} \right)}}{{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}dT={{e}^{-T\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
 
====Conditional Reliability====
<br>
The conditional reliability function for the Eyring-exponential model is given by:
<br>
::<math>R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-t\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
<br>
====Reliable Life====
<br>
For the Eyring-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}^{-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}}}</math>
<br>
::<math>\ln [R({{t}_{R}},V)]=-{{t}_{R}}\cdot V\cdot {{e}^{\left( A-\tfrac{B}{V} \right)}}</math>
<br>
:or:
<br>
::<math>{{t}_{R}}=-\frac{1}{V}{{e}^{-\left( A-\tfrac{B}{V} \right)}}\ln [R({{t}_{R}},V)]</math>
 
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
<br>
 
The complete exponential log-likelihood function of the Eyring model is composed of two summation portions:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ {{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{e}^{-{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}\cdot {{T}_{i}}}} \right] \\
&  & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\cdot {{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}\cdot 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}^{-T_{Li}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}}</math>
 
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-T_{Ri}^{\prime \prime }{{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}}</math>
 
<br>
:and:
<br>
• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
<br>
• <math>{{N}_{i}}</math>  is the number of times-to-failure in the  <math>{{i}^{th}}</math>  time-to-failure data group.
<br>
• <math>{{V}_{i}}</math>  is the stress level of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>A</math>  is the Eyring parameter (unknown, the first of two parameters to be estimated).
<br>
• <math>B</math>  is the second Eyring parameter (unknown, the second of two parameters to be estimated).
<br>
• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
<br>
• <math>S</math>  is the number of groups of suspension data points.
<br>
• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
<br>
• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
<br>
• <math>FI</math>  is the number of interval data groups.
<br>
• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the i <math>^{th}</math>  group of data intervals.
<br>
• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
<br>
• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
<br>
The solution (parameter estimates) will be found by solving for the parameters  <math>\widehat{A}</math>  and  <math>\widehat{B}</math>  so that  <math>\tfrac{\partial \Lambda }{\partial A}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial B}=0</math>  where:
 
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial A}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( 1-{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{T}_{i}} \right)-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{V}_{i}}\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}T_{i}^{\prime } \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\left( T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right){{V}_{i}}{{e}^{A-\tfrac{B}{{{V}_{i}}}}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} 
\end{align}</math>
 
<br>
::<math>\begin{align}
  & \frac{\partial \Lambda }{\partial B}= & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left[ {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}{{T}_{i}}-\frac{1}{{{V}_{i}}} \right]+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\cdot {{e}^{\left( A-\tfrac{B}{{{V}_{i}}} \right)}}T_{i}^{\prime } \\
&  & \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\left( T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime } \right){{e}^{A-\tfrac{B}{{{V}_{i}}}}}}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} 
\end{align}</math>
 
<br>

Latest revision as of 22:58, 16 August 2012

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