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| ==T-NT Exponential==
| | #REDIRECT [[Temperature-NonThermal_Relationship#T-NT_Exponential]] |
| <br>
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| By setting <math>m=L(U,V)</math>, the exponential <math>pdf</math> becomes:
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| <br>
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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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| <br>
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| ===T-NT Exponential Statistical Properties Summary===
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| <br>
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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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| <br>
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| ::<math>\begin{align}
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| & \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt \\
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| & = & \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 \\
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| & = & \frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}
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| \end{align}</math>
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| ====Median====
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| The median, <math>\breve{T},</math>
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| 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>
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| ====Mode====
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| The mode, <math>\tilde{T},</math>
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| for the T-NT exponential model is given by:
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| ::<math>\tilde{T}=0</math>
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| ====Standard Deviation====
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| 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>
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| ====T-NT Exponential Reliability Function====
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| 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-\mathop{}_{0}^{T}f(T)dT</math>
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| and,
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| ::<math>R(T,U,V)=1-\mathop{}_{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>
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| ====Conditional Reliability====
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| The conditional reliability function for the T-NT exponential model is given by,
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| ::<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>
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| ====Reliable Life====
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| 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:
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| ::<math>R({{t}_{R}},U,V)={{e}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
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| <br>
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| ::<math>\ln [R({{t}_{R}},U,V)]{{=}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}</math>
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| <br>
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| :or:
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| ::<math>{{t}_{R}}=-\frac{C}{{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}\ln [R({{t}_{R}},U,V)]</math>
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| ===Parameter Estimation===
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| ====Maximum Likelihood Estimation Method====
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| <br>
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| Substituting the T-NT relationship into the exponential log-likelihood equation yields:
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| <br>
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| ::<math>\begin{align}
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| & \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] \\
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| & & -\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 }]
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| \end{align}</math>
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| <br>
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| :where:
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| ::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}}}}</math>
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| <br>
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| <br>
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| ::<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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| <br>
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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).
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| • <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> .
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| <br>
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| {{TNT weibull}}
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| {{TNT Lognormal}}
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