Template:TNT weibull

T-NT Weibull
By setting $$\eta =L(U,V)$$, the T-NT Weibull model is given by:


 * $$f(t,U,V)=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t\cdot {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}}}$$

Mean or MTTF
The mean, $$\overline{T}$$, for the T-NT Weibull model is given by:
 * $$\overline{T}=\frac{C}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)$$

where $$\Gamma \left( \tfrac{1}{\beta }+1 \right)$$  is the gamma function evaluated at the value of  $$\left( \tfrac{1}{\beta }+1 \right)$$.

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


 * $$\breve{T}=\frac{C}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}$$

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


 * $$\tilde{T}=\frac{C}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}$$

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


 * $${{\sigma }_{T}}=\frac{C}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}$$

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


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

Conditional Reliability Function
The T-NT Weibull conditional reliability function at a specified stress level is given by:


 * $$R(T,t,U,V)=\frac{R(T+t,U,V)}{R(T,U,V)}=\frac$$

or:
 * $$R(T,t,U,V)={{e}^{-\left[ {{\left( \tfrac{\left( T+t \right){{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }}-{{\left( \tfrac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta }} \right]}}$$

Reliable Life
For the T-NT Weibull model, the reliable life, $${{T}_{R}}$$, of a unit for a specified reliability and starting the mission at age zero is given by:


 * $${{T}_{R}}=\frac{C}{{\left\{ -\ln \left[ R\left( {{T}_{R}},U,V \right) \right] \right\}}^{\tfrac{1}{\beta }}}$$

T-NT Weibull Failure Rate Function
The T-NT Weibull failure rate function, $$\lambda (T)$$, is given by:


 * $$\lambda \left( T,U,V \right)=\frac{f\left( T,U,V \right)}{R\left( T,U,V \right)}=\frac{\beta {{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C}{{\left( \frac{T{{U}^{n}}{{e}^{-\tfrac{B}{V}}}}{C} \right)}^{\beta -1}}$$

Maximum Likelihood Estimation Method
Substituting the T-NT relationship into the Weibull log-likelihood function yields:


 * $$\begin{align}

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


 * $$R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C}U_{i}^{\prime \prime n}{{e}^{-\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}}}$$

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. •	 $$\beta $$ is the Weibull shape parameter (unknown, the first of four parameters to be estimated). •	 $$B$$ is the first T-NT parameter (unknown, the second of four parameters to be estimated). •	 $$C$$ is the second T-NT parameter (unknown, the third of four parameters to be estimated). •	 $$n$$ is the third T-NT parameter (unknown, the fourth of four 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 $$B,$$ $$C,$$ $$n$$  and  $$\beta $$  so that  $$\tfrac{\partial \Lambda }{\partial B}=0,$$   $$\tfrac{\partial \Lambda }{\partial C}=0,$$   $$\tfrac{\partial \Lambda }{\partial n}=0$$  and  $$\tfrac{\partial \Lambda }{\partial \beta }=0$$.