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==T-NT Exponential==
#REDIRECT [[Temperature-NonThermal_Relationship#T-NT_Exponential]]
<br>
By setting  <math>m=L(U,V)</math>, the exponential  <math>pdf</math>  becomes:
 
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::<math>f(t,U,V)=\frac{{{U}^{n}}}{C}{{e}^{-\tfrac{B}{V}}}\cdot {{e}^{-\tfrac{{{U}^{n}}}{C}\left( {{e}^{-\tfrac{B}{V}}} \right)t}}</math>
 
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===T-NT Exponential Statistical Properties Summary===
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====Mean or MTTF====
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The mean,  <math>\overline{T},</math>  or Mean Time To Failure (MTTF) for the T-NT exponential model is given by:
 
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::<math>\begin{align}
  & \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt = & \int\limits_{0}^{\infty }t\cdot \frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt = & \frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}} 
\end{align}</math>
 
<br>
 
====Median====
<br>
The median,  <math>\breve{T},</math>
for the T-NT exponential model is given by:
 
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::<math>\breve{T}=\frac{1}{\lambda }0.693=0.693\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}</math>
 
====Mode====
<br>
The mode,  <math>\tilde{T},</math> 
for the T-NT exponential model is given by:
 
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::<math>\tilde{T}=0</math>
 
====Standard Deviation====
<br>
The standard deviation,  <math>{{\sigma }_{T}}</math> , for the T-NT exponential model is given by:
 
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::<math>{{\sigma }_{T}}=\frac{1}{\lambda }=m=\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}</math>
 
<br>
====T-NT Exponential Reliability Function====
<br>
The T-NT exponential reliability function is given by:
 
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::<math>R(T,U,V)={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
 
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This function is the complement of the T-NT exponential cumulative distribution function or:
 
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::<math>R(T,U,V)=1-Q(T,U,V)=1-\int_{0}^{T}f(T)dT</math>
 
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and,
 
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::<math>R(T,U,V)=1-\int_{0}^{T}\frac{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dT={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
 
<br>
 
====Conditional Reliability====
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The conditional reliability function for the T-NT exponential model is given by,
 
::<math>R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
 
<br>
====Reliable Life====
<br>
For the T-NT 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}},U,V)={{e}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
 
<br>
::<math>\ln [R({{t}_{R}},U,V)]{{=}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
 
<br>
:or:
 
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::<math>{{t}_{R}}=-\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}\ln [R({{t}_{R}},U,V)]</math>
 
<br>
 
===Parameter Estimation===
<br>
====Maximum Likelihood Estimation Method====
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Substituting the T-NT relationship into the exponential log-likelihood equation yields:
 
<br>
::<math>\begin{align}
  & \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{U_{i}^{n}}{C}{{e}^{-\tfrac{B}{{{V}_{i}}}}}\cdot {{e}^{-\tfrac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right){{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)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}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}}</math>
<br>
<br>
::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}}</math>
 
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and:
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• <math>{{F}_{e}}</math>  is the number of groups of exact times-to-failure data points.
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• <math>{{N}_{i}}</math>  is the number of times-to-failure data points in the  <math>{{i}^{th}}</math>  time-to-failure data group.
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• <math>B</math>  is the T-NT parameter (unknown, the first of three parameters to be estimated).
<br>
• <math>C</math>  is the second T-NT parameter (unknown, the second of three parameters to be estimated).
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• <math>n</math>  is the third T-NT parameter (unknown, the third of three parameters to be estimated).
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• <math>{{V}_{i}}</math>  is the temperature level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{U}_{i}}</math>  is the non-thermal stress level of the  <math>{{i}^{th}}</math>  group.
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• <math>{{T}_{i}}</math>  is the exact failure time of the  <math>{{i}^{th}}</math>  group.
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• <math>S</math>  is the number of groups of suspension data points.
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• <math>N_{i}^{\prime }</math>  is the number of suspensions in the  <math>{{i}^{th}}</math>  group of suspension data points.
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• <math>T_{i}^{\prime }</math>  is the running time of the  <math>{{i}^{th}}</math>  suspension data group.
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• <math>FI</math>  is the number of interval data groups.
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• <math>N_{i}^{\prime \prime }</math>  is the number of intervals in the i <math>^{th}</math>  group of data intervals.
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• <math>T_{Li}^{\prime \prime }</math>  is the beginning of the i <math>^{th}</math>  interval.
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• <math>T_{Ri}^{\prime \prime }</math>  is the ending of the i <math>^{th}</math>  interval.
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The solution (parameter estimates) will be found by solving for the parameters  <math>B,</math>  <math>C</math>  and  <math>n</math>  so that  <math>\tfrac{\partial \Lambda }{\partial B}=0,</math>  <math>\tfrac{\partial \Lambda }{\partial C}=0</math>  and  <math>\tfrac{\partial \Lambda }{\partial n}=0</math> .
<br>
 
{{TNT weibull}}
 
{{TNT Lognormal}}

Latest revision as of 05:47, 15 August 2012

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