Template:T-h exponential

T-H Exponential
By setting $$m=L(U,V)$$  in Eqn. (Temp-Hum) the exponential $$pdf$$  becomes:


 * $$f(t,V,U)=\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}\cdot {{e}^{-\tfrac{t}{A}\cdot {{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}$$

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


 * $$\overline{T}=\mathop{}_{0}^{\infty }t\cdot f(t,V,U)dt$$

Substituting Eqn. (t-h exp pdf) yields:


 * $$\begin{align}

& \overline{T}= & \mathop{}_{0}^{\infty }t\cdot \frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dt \\ & = & A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}} \end{align}$$

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


 * $$\breve{T}=0.693\cdot A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}$$

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


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

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


 * $${{\sigma }_{T}}=A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}$$

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


 * $$R(T,V,U)={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}$$

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


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

and:


 * $$R(T,V,U)=1-\mathop{}_{0}^{T}\frac{1}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}{{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}dT={{e}^{-\tfrac{T}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}$$

Conditional Reliability
The conditional reliability function for the T-H exponential model is given by:


 * $$R(T,t,V,U)=\frac{R(T+t,V,U)}{R(T,V,U)}=\frac={{e}^{-\tfrac{t}{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}$$

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


 * $$R({{t}_{R}},V,U)={{e}^{-\tfrac{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}}}$$


 * $$\ln [R({{t}_{R}},V,U)]=-\frac{A}{{e}^{-\left( \tfrac{\phi }{V}+\tfrac{b}{U} \right)}}$$

or:


 * $${{t}_{R}}=-A{{e}^{\tfrac{\phi }{V}+\tfrac{b}{U}}}\ln [R({{t}_{R}},V,U)]$$

Maximum Likelihood Estimation Method
Substituting the T-H model into the exponential log-likelihood equation yields:


 * $$\begin{align}

& \ln (L)= & \Lambda =\underset{i=1}{\overset{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{A}{{e}^{-\left( \tfrac{\phi }+\tfrac{b} \right)}}\cdot {{e}^{-\tfrac{A}\cdot {{e}^{-\left( \tfrac{\phi }+\tfrac{b} \right)}}}} \right] \\ & & -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{A}{{e}^{-\left( \tfrac{\phi }+\tfrac{b} \right)}}+\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:


 * $$R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}$$


 * $$R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{A}{{e}^{-\left( \tfrac{\phi }+\tfrac{b}{U_{i}^{\prime \prime }} \right)}}}}$$

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. •	 $$A$$ is the T-H parameter (unknown, the first of three parameters to be estimated). •	 $$\phi $$ is the second T-H parameter (unknown, the second of three parameters to be estimated). •	 $$b$$ is the third T-H 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 relative humidity 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. •	 .. 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 $$A,$$   $$\phi $$  and  $$b$$  so that  $$\tfrac{\partial \Lambda }{\partial A}=0,$$   $$\tfrac{\partial \Lambda }{\partial \phi }=0$$  and  $$\tfrac{\partial \Lambda }{\partial b}=0$$.