Template:TNT Exponential

T-NT Exponential
By setting $$m=L(U,V)$$, the exponential  $$pdf$$  becomes:


 * $$f(t,U,V)=\frac{C}{{e}^{-\tfrac{B}{V}}}\cdot {{e}^{-\tfrac{C}\left( {{e}^{-\tfrac{B}{V}}} \right)t}}$$

Mean or MTTF
The mean, $$\overline{T},$$  or Mean Time To Failure (MTTF) for the T-NT exponential model is given by:


 * $$\begin{align}

& \overline{T}= & \int\limits_{0}^{\infty }t\cdot f(t,U,V)dt = & \int\limits_{0}^{\infty }t\cdot \frac{C}{{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dt = & \frac{C} \end{align}$$

Median
The median, $$\breve{T},$$ for the T-NT exponential model is given by:


 * $$\breve{T}=\frac{1}{\lambda }0.693=0.693\frac{C}$$

Mode
The mode, $$\tilde{T},$$ for the T-NT exponential model is given by:


 * $$\tilde{T}=0$$

Standard Deviation
The standard deviation, $${{\sigma }_{T}}$$, for the T-NT exponential model is given by:


 * $${{\sigma }_{T}}=\frac{1}{\lambda }=m=\frac{C}$$

T-NT Exponential Reliability Function
The T-NT exponential reliability function is given by:


 * $$R(T,U,V)={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}$$

This function is the complement of the T-NT exponential cumulative distribution function or:


 * $$R(T,U,V)=1-Q(T,U,V)=1-\int_{0}^{T}f(T)dT$$

and,


 * $$R(T,U,V)=1-\int_{0}^{T}\frac{C}{{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}dT={{e}^{-\tfrac{T\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}$$

Conditional Reliability
The conditional reliability function for the T-NT exponential model is given by,


 * $$R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac={{e}^{-\tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}$$

Reliable Life
For the T-NT exponential model, the reliable life, or the mission duration for a desired reliability goal, $${{t}_{R}}$$, is given by:


 * $$R({{t}_{R}},U,V)={{e}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}$$


 * $$\ln [R({{t}_{R}},U,V)]{{=}^{-\tfrac{{{t}_{R}}\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}}}$$


 * or:


 * $${{t}_{R}}=-\frac{C}\ln [R({{t}_{R}},U,V)]$$

Maximum Likelihood Estimation Method
Substituting the T-NT relationship into the exponential log-likelihood equation yields:


 * $$\begin{align}

& \ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{U_{i}^{n}}{C}{{e}^{-\tfrac{B}}}\cdot {{e}^{-\tfrac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}}} \right){{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{U_{i}^{n}}{C}\left( {{e}^{-\tfrac{B}}} \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}$$

where:


 * $$R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}}}}}$$


 * $$R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}}}}}$$

and: •	 $${{F}_{e}}$$ is the number of groups of exact times-to-failure data points. •	 $${{N}_{i}}$$ is the number of times-to-failure data points in the  $${{i}^{th}}$$  time-to-failure data group. •	 $$B$$ is the T-NT parameter (unknown, the first of three parameters to be estimated). •	 $$C$$ is the second T-NT parameter (unknown, the second of three parameters to be estimated). •	 $$n$$ is the third T-NT parameter (unknown, the third of three parameters to be estimated). •	 $${{V}_{i}}$$ is the temperature level of the  $${{i}^{th}}$$  group. •	 $${{U}_{i}}$$ is the non-thermal stress level of the  $${{i}^{th}}$$  group. •	 $${{T}_{i}}$$ is the exact failure time of the  $${{i}^{th}}$$  group. •	 $$S$$ is the number of groups of suspension data points. •	 $$N_{i}^{\prime }$$ is the number of suspensions in the  $${{i}^{th}}$$  group of suspension data points. •	 $$T_{i}^{\prime }$$ is the running time of the  $${{i}^{th}}$$  suspension data group. •	 $$FI$$ is the number of interval data groups. •	 $$N_{i}^{\prime \prime }$$ is the number of intervals in the i $$^{th}$$  group of data intervals. •	 $$T_{Li}^{\prime \prime }$$ is the beginning of the i $$^{th}$$  interval. •	 $$T_{Ri}^{\prime \prime }$$ is the ending of the i $$^{th}$$  interval. The solution (parameter estimates) will be found by solving for the parameters $$B,$$   $$C$$  and  $$n$$  so that  $$\tfrac{\partial \Lambda }{\partial B}=0,$$   $$\tfrac{\partial \Lambda }{\partial C}=0$$  and  $$\tfrac{\partial \Lambda }{\partial n}=0$$.