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Arrhenius Relationship

2021-10-29T16:32:06Z

Cruz Daniel: changed undefined phi term in log likelihood equation under maximum likelihood section to use reliability term instead - suggestion from Sayyed Vazirizade at University of Arizona

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The Arrhenius life-stress model (or relationship) is probably the most common life-stress relationship utilized in accelerated life testing. It has been widely used when the stimulus or acceleration variable (or stress) is thermal (i.e., temperature). It is derived from the Arrhenius reaction rate equation proposed by the Swedish physical chemist Svandte Arrhenius in 1887.

===Formulation===
The Arrhenius reaction rate equation is given by:

::<math>R(T)=A{{e}^{-\tfrac{{{E}_{a}}}{k\cdot T}}}\,\!</math>

where:

*<math>R\,\!</math> is the speed of reaction.

*<math>A\,\!</math> is an unknown nonthermal constant.

*<math>{{E}_{a}}\,\!</math> is the activation energy <math>(\text{eV})\,\!</math>.

*<math>k\,\!</math> is the Boltzmann's constant <math>(8.6173303\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}})\,\!</math>.

*<math>T\,\!</math> is the absolute temperature <math>(\text{K})\,\!</math>.

The activation energy is the energy that a molecule must have to participate in the reaction. In other words, the activation energy is a measure of the effect that temperature has on the reaction.

The Arrhenius life-stress model is formulated by assuming that life is proportional to the inverse reaction rate of the process, thus the Arrhenius life-stress relationship is given by:

::<math>L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

where:

*<math>L\,\!</math> represents a quantifiable life measure, such as mean life, characteristic life, median life, or <math>B(x)\,\!</math> life, etc.

*<math>V\,\!</math> represents the stress level (formulated for temperature and '''temperature values in absolute units, degrees Kelvin or degrees Rankine''').

*<math>C\,\!</math> is one of the model parameters to be determined, <math>(C>0)\,\!</math>.

*<math>B\,\!</math> is another model parameter to be determined.

[[Image:ALTA6.1.png|center|400px|Graphical look at the Arrhenius life-stress relationship (linear scale) for a different life characteristics, assuming a Weibull distribution.]]

Since the Arrhenius is a physics-based model derived for temperature dependence, it is used for temperature accelerated tests. For the same reason, temperature values must be in absolute units (Kelvin or Rankine), even though the Arrhenius equation is unitless.

===Life Stress Plots===
The Arrhenius relationship can be linearized and plotted on a Life vs. Stress plot, also called the Arrhenius plot. The relationship is linearized by taking the natural logarithm of both sides in the Arrhenius equation or:

::<math>ln(L(V))=ln(C)+\frac{B}{V}\,\!</math>

[[Image:ALTA6.2.png|center|400px|Arrhenius plot for Weibull life distribution.]]

In the linearized Arrhenius equation, <math>\ln (C)\,\!</math> is the intercept of the line and <math>B\,\!</math> is the slope of the line. Note that the inverse of the stress, and not the stress, is the variable. In the above figure, life is plotted versus stress and not versus the inverse stress. This is because the linearized Arrhenius equation was plotted on a reciprocal scale. On such a scale, the slope <math>B\,\!</math> appears to be negative even though it has a positive value. This is because <math>B\,\!</math> is actually the slope of the reciprocal of the stress and not the slope of the stress. The reciprocal of the stress is decreasing as stress is increasing ( <math>\tfrac{1}{V}\,\!</math> is decreasing as <math>V\,\!</math> is increasing). The two different axes are shown in the next figure.

[[Image:ALTA6.3.png|center|400px|An illustration of both reciprocal and non-reciprocal scales.]]

The Arrhenius relationship is plotted on a reciprocal scale for practical reasons. For example, in the above figure it is more convenient to locate the life corresponding to a stress level of 370K than to take the reciprocal of 370K (0.0027) first, and then locate the corresponding life.
The shaded areas shown in the above figure are the imposed at each test stress level. From such imposed ''pdfs'' one can see the range of the life at each test stress level, as well as the scatter in life. The next figure illustrates a case in which there is a significant scatter in life at each of the test stress levels.

[[Image:ALTA6.4.png|center|400px|An example of scatter in life at each test stress level.]]

===Activation Energy and the Parameter ''B'' ===
Depending on the application (and where the stress is exclusively thermal), the parameter <math>B\,\!</math> can be replaced by:

::<math>B=\frac{{{E}_{a}}}{k}=\frac{\text{activation energy}}{\text{Boltzman}{{\text{n}}^{\prime }}\text{s constant}}=\frac{\text{activation energy}}{8.617385\times {{10}^{-5}}\text{eV}{{\text{K}}^{-1}}}\,\!</math>

Note that in this formulation, the activation energy <math>{{E}_{a}}\,\!</math> must be known a priori. If the activation energy is known then there is only one model parameter remaining, <math>C.\,\!</math> Because in most real life situations this is rarely the case, all subsequent formulations will assume that this activation energy is unknown and treat <math>B\,\!</math> as one of the model parameters. <math>B\,\!</math> has the same properties as the activation energy. In other words, <math>B\,\!</math> is a measure of the effect that the stress (i.e. temperature) has on the life. The larger the value of <math>B,\,\!</math> the higher the dependency of the life on the specific stress. Parameter <math>B\,\!</math> may also take negative values. In that case, life is increasing with increasing stress. An example of this would be plasma filled bulbs, where low temperature is a higher stress on the bulbs than high temperature.

[[Image:ALTA6.5.png|center|400px|Behavior of the parameter ''B''.]]

===Acceleration Factor===
Most practitioners use the term acceleration factor to refer to the ratio of the life (or acceleration characteristic) between the use level and a higher test stress level or:

::<math>{{A}_{F}}=\frac{{{L}_{USE}}}{{{L}_{Accelerated}}}\,\!</math>

For the Arrhenius model this factor is:

::<math>{{A}_{F}}=\frac{{{L}_{USE}}}{{{L}_{Accelerated}}}=\frac{C\text{ }{{e}^{\tfrac{B}{{{V}_{u}}}}}}{C\text{ }{{e}^{\tfrac{B}{{{V}_{A}}}}}}=\frac{\text{ }{{e}^{\tfrac{B}{{{V}_{u}}}}}}{\text{ }{{e}^{\tfrac{B}{{{V}_{A}}}}}}={{e}^{\left( \tfrac{B}{{{V}_{u}}}-\tfrac{B}{{{V}_{A}}} \right)}}\,\!</math>

Thus, if <math>B\,\!</math> is assumed to be known a priori (using an activation energy), the assumed activation energy alone dictates this acceleration factor!

=Arrhenius-Exponential=
The ''pdf'' of the 1-parameter exponential distribution is given by:

::<math>\begin{align}
f(t)=\lambda {{e}^{-\lambda t}}
\end{align}\,\!</math>

It can be easily shown that the mean life for the 1-parameter exponential distribution (presented in detail [[Distributions used in Accelerated Testing#The Exponential Distribution|here]]) is given by:

::<math>\lambda =\frac{1}{m}\,\!</math>

thus:

::<math>f(t)=\frac{1}{m}{{e}^{-\tfrac{t}{m}}}\,\!</math>

The Arrhenius-exponential model ''pdf'' can then be obtained by setting <math>m=L(V)\,\!</math>:

Therefore:

::<math>m=L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

Substituting for <math>m\,\!</math> yields a ''pdf'' that is both a function of time and stress or:

::<math>f(t,V)=\frac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot {{e}^{-\tfrac{1}{C{{e}^{\tfrac{B}{V}}}}\cdot t}}\,\!</math>

==Arrhenius-Exponential Statistical Properties Summary==
====Mean or MTTF====
The mean, <math>\overline{T},\,\!</math> or Mean Time To Failure (MTTF) of the Arrhenius-exponential is given by,

::<math>\begin{align}
\overline{T}=\int_{0}^{\infty }t\cdot f(t,V)dt=\int_{0}^{\infty }t\cdot \frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}dt =\ C{{e}^{\tfrac{B}{V}}}
\end{align}\,\!</math>

====Median====
The median, <math>\breve{T}\,\!</math> of the Arrhenius-exponential model is given by:

::<math>\breve{T}=0.693\cdot C{{e}^{\tfrac{B}{V}}}\,\!</math>

====Mode====
The mode, <math>\tilde{T},\,\!</math> of the Arrhenius-exponential model is given by:

::<math>\tilde{T}=0\,\!</math>

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}}\,\!</math>, of the Arrhenius-exponential model is given by:

::<math>{{\sigma }_{T}}=C{{e}^{\tfrac{B}{V}}}\,\!</math>

====Arrhenius-Exponential Reliability Function====
The Arrhenius-exponential reliability function is given by:

::<math>R(T,V)={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

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

::<math>R(T,V)=1-Q(T,V)=1-\int_{0}^{T}f(T,V)dT\,\!</math>

and:

::<math>R(T,V)=1-\int_{0}^{T}\frac{1}{C{{e}^{\tfrac{B}{V}}}}{{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}dT={{e}^{-\tfrac{T}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

====Conditional Reliability====
The Arrhenius-exponential conditional reliability function is given by:

::<math>R((t|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-\lambda (T+t)}}}{{{e}^{-\lambda T}}}={{e}^{-\tfrac{t}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

====Reliable Life====
For the Arrhenius-exponential model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},\,\!</math> is given by:

::<math>R({{t}_{R}},V)={{e}^{-\tfrac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}}}\,\!</math>

::<math>\ln [R({{t}_{R}},V)]=-\frac{{{t}_{R}}}{C{{e}^{\tfrac{B}{V}}}}\,\!</math>

or:

::<math>{{t}_{R}}=-C{{e}^{\tfrac{B}{V}}}\ln [R({{t}_{R}},V)]\,\!</math>

==Parameter Estimation==
====Maximum Likelihood Estimation Method====
The log-likelihood function for the exponential distribution is as shown next:

::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \lambda {{e}^{-\lambda {{T}_{i}}}} \right]-\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\lambda 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>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-\lambda T_{Li}^{\prime \prime }}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-\lambda T_{Ri}^{\prime \prime }}}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>\lambda \,\!</math> is the failure rate parameter (unknown).

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

Substituting the Arrhenius-exponential model into the log-likelihood function yields:

::<math>\begin{align}
\Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{e}^{-\tfrac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{T}_{i}}}} \right] -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}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>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-\tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-\tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}}}\,\!</math>

The solution (parameter estimates) will be found by solving for the parameters <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}-\frac{C}{{{V}_{i}}} \right)+\frac{1}{C}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime })C{{V}_{i}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= & \frac{1}{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{T}_{i}}}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}}-1 \right)+\frac{1}{{{C}^{2}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{T_{i}^{\prime }}{{{e}^{\tfrac{B}{{{V}_{i}}}}}} \overset{FI}{\mathop{\underset{i=1}{\mathop{+\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{T_{Li}^{\prime \prime }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime }R_{Ri}^{\prime \prime }}{(R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }){{C}^{2}}{{e}^{\tfrac{B}{{{V}_{i}}}}}}
\end{align}\,\!</math>

=Arrhenius-Weibull=

The ''pdf'' for the 2-parameter Weibull distribution is given by:

::<math>f(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{\eta } \right)}^{\beta }}}}\,\!</math>

The scale parameter (or characteristic life) of the Weibull distribution is <math>\eta \,\!</math>.

The Arrhenius-Weibull model ''pdf'' can then be obtained by setting <math>\eta =L(V)\,\!</math>:

::<math>\eta =L(V)=C\cdot {{e}^{\tfrac{B}{V}}}\,\!</math>

and substituting for <math>\eta \,\!</math> in the 2-parameter Weibull distribution equation:

::<math>f(t,V)=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}\,\!</math>

An illustration of the ''pdf'' for different stresses is shown in the next figure. As expected, the ''pdf'' at lower stress levels is more stretched to the right, with a higher scale parameter, while its shape remains the same (the shape parameter is approximately 3). This behavior is observed when the parameter <math>B\,\!</math> of the Arrhenius model is positive.

[[Image:ALTA6.6.png|center|400px|Behavior of the probability density function at different stresses and with the parameters held constant.]]

The advantage of using the Weibull distribution as the life distribution lies in its flexibility to assume different shapes. The Weibull distribution is presented in greater detail in [[The Weibull Distribution]].

==Arrhenius-Weibull Statistical Properties Summary==
====Mean or MTTF====
The mean, <math>\overline{T}\,\!</math> (also called <math>MTTF\,\!</math> by some authors), of the Arrhenius-Weibull relationship is given by:

::<math>\overline{T}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right)\,\!</math>

where <math>\Gamma \left( \tfrac{1}{\beta }+1 \right)\,\!</math> is the gamma function evaluated at the value of <math>\left( \tfrac{1}{\beta }+1 \right)\,\!</math>.

====Median====
The median, <math>\breve{T},\,\!</math>
for the Arrhenius-Weibull model is given by:

::<math>\breve{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}}\,\!</math>

====Mode====
The mode, <math>\tilde{T},\,\!</math>
for the Arrhenius-Weibull model is given by:

::<math>\tilde{T}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}}\,\!</math>

====Standard Deviation====
The standard deviation, <math>{{\sigma }_{T}},\,\!</math> for the Arrhenius-Weibull model is given by:

::<math>{{\sigma }_{T}}=C\cdot {{e}^{\tfrac{B}{V}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}}\,\!</math>

====Arrhenius-Weibull Reliability Function====
The Arrhenius-Weibull reliability function is given by:

::<math>R(T,V)={{e}^{-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}}}\,\!</math>

If the parameter <math>B\,\!</math> is positive, then the reliability increases as stress decreases.

[[Image:ALTA6.7.png|center|500px|Behavior of the reliability function at different stress and constant parameter values.]]

The behavior of the reliability function of the Weibull distribution for different values of <math>\beta \,\!</math> was illustrated [[Distributions used in Accelerated Testing#The Weibull Distribution|here]]. In the case of the Arrhenius-Weibull model, however, the reliability is a function of stress also. A 3D plot such as the ones shown in the next figure is now needed to illustrate the effects of both the stress and <math>\beta .\,\!</math>

[[Image:ALTA6.8.png|center|800px|Reliability function for <math>\Beta<1 \,\!</math>, <math>\Beta=1 \,\!</math>, and <math>\Beta>1 \,\!</math>.]]

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

::<math>R((t|T),V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left( \tfrac{T+t}{\eta } \right)}^{\beta }}}}}{{{e}^{-{{\left( \tfrac{T}{\eta } \right)}^{\beta }}}}}\,\!</math>

or:

::<math>R((t|T),V)={{e}^{-\left[ {{\left( \tfrac{T+t}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }}-{{\left( \tfrac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta }} \right]}}\,\!</math>

====Reliable Life====
For the Arrhenius-Weibull relationship, the reliable life, <math>{{t}_{R}}\,\!</math>, of a unit for a specified reliability and starting the mission at age zero is given by:

::<math>{{t}_{R}}=C\cdot {{e}^{\tfrac{B}{V}}}{{\left\{ -\ln \left[ R\left( {{t}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}}\,\!</math>

This is the life for which the unit will function successfully with a reliability of <math>R({{t}_{R}})\,\!</math>. If <math>R({{t}_{R}})=0.50\,\!</math> then <math>{{t}_{R}}=\breve{T}\,\!</math>, the median life, or the life by which half of the units will survive.

====Arrhenius-Weibull Failure Rate Function====
The Arrhenius-Weibull failure rate function, <math>\lambda (T)\,\!</math>, is given by:

::<math>\lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\frac{\beta }{C\cdot {{e}^{\tfrac{B}{V}}}}{{\left( \frac{T}{C\cdot {{e}^{\tfrac{B}{V}}}} \right)}^{\beta -1}}\,\!</math>

[[Image:ALTA6.9.png|center|800px|Failure rate function for <math>\Beta<1 \,\!</math>, <math>\Beta=1 \,\!</math>, and <math>\Beta>1 \,\!</math>.]]

==Parameter Estimation==
====Maximum Likelihood Estimation Method====
The Arrhenius-Weibull log-likelihood function is as follows:

::<math>\begin{align}
& \Lambda = & \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{\beta }{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta -1}}{{e}^{-{{\left( \tfrac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}} \right] \ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \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}\,\!</math>

where:

::<math>R_{Li}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}\,\!</math>

::<math>R_{Ri}^{\prime \prime }={{e}^{-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}}}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure data points in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>\beta \,\!</math> is the Weibull shape parameter (unknown, the first of three parameters to be estimated).

*<math>B\,\!</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

*<math>C\,\!</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

*<math>{{V}_{i}}\,\!</math> is the stress level of the <math>{{i}^{th}}\,\!</math> group.

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

The solution (parameter estimates) will be found by solving for <math>\widehat{\beta },\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial \beta }=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
\frac{\partial \Lambda }{\partial \beta }=\ & \frac{1}{\beta }\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right) \\
& \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Li}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Li}^{\prime \prime }-{{\left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}\ln \left( \tfrac{T_{Ri}^{\prime \prime }}{C{{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)R_{Ri}^{\prime \prime }}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}+\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\frac{1}{{{V}_{i}}}{{\left( \frac{{{T}_{i}}}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }}+\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}{{\left( \frac{T_{i}^{\prime }}{\widehat{C}{{e}^{\tfrac{\widehat{B}}{{{V}_{i}}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{{{V}_{i}}}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= -\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}+\frac{\beta }{C}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( \frac{{{T}_{i}}}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }}+\frac{\beta }{C}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \frac{T_{i}^{\prime }}{C\cdot {{e}^{\tfrac{B}{{{V}_{i}}}}}} \right)}^{\beta }} +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\beta }{C}\frac{{{(T_{Li}^{\prime \prime })}^{\beta }}R_{Li}^{\prime \prime }-{{(T_{Ri}^{\prime \prime })}^{\beta }}R_{Ri}^{\prime \prime }}{{{\left( C{{e}^{\tfrac{B}{{{V}_{i}}}}} \right)}^{\beta }}\left( R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime } \right)}
\end{align}\,\!</math>

==Arrhenius-Weibull example==
{{:Arrhenius_Example}}

=Arrhenius-Lognormal=

The ''pdf'' of the lognormal distribution is given by:

::<math>f(T)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\bar{{{T}'}}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}\,\!</math>

where:

::<math>\begin{align}
{T}'=\ln(T)
\end{align}\,\!</math>

and:

*<math>T=\,\!</math> times-to-failure.

*<math>{T}'=\,\!</math> mean of the natural logarithms of the times-to-failure.

*<math>{{\sigma }_{{{T}'}}}=\,\!</math> standard deviation of the natural logarithms of the times-to-failure.

The median of the lognormal distribution is given by:

::<math>\breve{T}={{e}^{{{\overline{T}}^{\prime }}}}\,\!</math>

The Arrhenius-lognormal model ''pdf'' can be obtained first by setting <math>\breve{T}=L(V)\,\!</math>. Therefore:

::<math>\breve{T}=L(V)=C{{e}^{\tfrac{B}{V}}}\,\!</math>

or:

::<math>{{e}^{{{\overline{T}}^{\prime }}}}=C{{e}^{\tfrac{B}{V}}}\,\!</math>

Thus:

::<math>{{\overline{T}}^{\prime }}=\ln (C)+\frac{B}{V}\,\!</math>

Substituting the above equation into the lognormal ''pdf'' yields the Arrhenius-lognormal model ''pdf'' or:

::<math>f(T,V)=\frac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}\,\!</math>

Note that in the Arrhenius-lognormal ''pdf'', it was assumed that the standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}},\,\!</math> is independent of stress. This assumption implies that the shape of the distribution does not change with stress ( <math>{{\sigma }_{{{T}'}}}\,\!</math> is the shape parameter of the lognormal distribution).

==Arrhenius-Lognormal Statistical Properties Summary==
====The Mean====
*The mean life of the Arrhenius-lognormal model (mean of the times-to-failure), <math>\bar{T}\,\!</math>, is given by:

::<math>\begin{align}
& \bar{T}= & {{e}^{\bar{{T}'}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}} =\ & {{e}^{\ln (C)+\tfrac{B}{V}+\tfrac{1}{2}\sigma _{{{T}'}}^{2}}}
\end{align}\,\!</math>

*The mean of the natural logarithms of the times-to-failure, <math>{{\bar{T}}^{^{\prime }}}\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{{\sigma }_{T}}\,\!</math> is given by:

::<math>{{\bar{T}}^{\prime }}=\ln \left( {\bar{T}} \right)-\frac{1}{2}\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)\,\!</math>

====The Standard Deviation====
*The standard deviation of the Arrhenius-lognormal model (standard deviation of the times-to-failure), <math>{{\sigma }_{T}}\,\!</math>, is given by:

::<math>\begin{align}
& {{\sigma }_{T}}= & \sqrt{\left( {{e}^{2\bar{{T}'}+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)} =\ & \sqrt{\left( {{e}^{2\left( \ln (C)+\tfrac{B}{V} \right)+\sigma _{{{T}'}}^{2}}} \right)\left( {{e}^{\sigma _{{{T}'}}^{2}}}-1 \right)}
\end{align}\,\!</math>

*The standard deviation of the natural logarithms of the times-to-failure, <math>{{\sigma }_{{{T}'}}}\,\!</math>, in terms of <math>\bar{T}\,\!</math> and <math>{{\sigma }_{T}}\,\!</math> is given by:

::<math>{{\sigma }_{{{T}'}}}=\sqrt{\ln \left( \frac{\sigma _{T}^{2}}{{{{\bar{T}}}^{2}}}+1 \right)}\,\!</math>

====The Mode====
*The mode of the Arrhenius-lognormal model is given by:

::<math>\begin{align}
& \tilde{T}=\ {{e}^{{{\overline{T}}^{\prime }}-\sigma _{{{T}'}}^{2}}} =\ {{e}^{\ln (C)+\tfrac{B}{V}-\sigma _{{{T}'}}^{2}}}
\end{align}\,\!</math>

====Arrhenius-Lognormal Reliability Function====
The reliability for a mission of time <math>T\,\!</math>, starting at age 0, for the Arrhenius-lognormal model is determined by:

::<math>R(T,V)=\int_{T}^{\infty }f(t,V)dt\,\!</math>

or:

::<math>R(T,V)=\int_{{{T}^{^{\prime }}}}^{\infty }\frac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt\,\!</math>

There is no closed form solution for the lognormal reliability function. Solutions can be obtained via the use of standard normal tables. Since the application automatically solves for the reliability, we will not discuss manual solution methods.

====Reliable Life====
For the Arrhenius-lognormal model, the reliable life, or the mission duration for a desired reliability goal, <math>{{t}_{R}},\,\!</math> is estimated by first solving the reliability equation with respect to time, as follows:

::<math>{{T}'_{R}}= ln(C)+\frac{B}{V}+z \cdot {{\sigma}_{{T}'}}\,\!</math>

where:

::<math>z={{\Phi }^{-1}}\left[ F\left( T_{R}^{\prime },V \right) \right]\,\!</math>

and:

::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}',V)}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>

Since <math>{T}'=\ln (T)\,\!</math> the reliable life, <math>{{t}_{R}},\,\!</math> is given by:

::<math>{{t}_{R}}={{e}^{T_{R}^{\prime }}}\,\!</math>

====Arrhenius-Lognormal Failure Rate====
The Arrhenius-lognormal failure rate is given by:

::<math>\lambda (T,V)=\frac{f(T,V)}{R(T,V)}=\frac{\tfrac{1}{T\text{ }{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}}{\int_{{{T}'}}^{\infty }\tfrac{1}{{{\sigma }_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{{T}'-\ln (C)-\tfrac{B}{V}}{{{\sigma }_{{{T}'}}}} \right)}^{2}}}}dt}\,\!</math>

==Parameter Estimation==
====Maximum Likelihood Estimation Method====

The lognormal log-likelihood function for the Arrhenius-lognormal model is as follows:

::<math>\begin{align}
& \ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \frac{1}{{{\sigma }_{{{T}'}}}{{T}_{i}}}\phi \left( \frac{\ln \left( {{T}_{i}} \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] \text{ }+\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\ln \left[ 1-\Phi \left( \frac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right) \right] +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime })]
\end{align}\,\!</math>

where:

::<math>z_{Li}^{\prime \prime }=\frac{\ln T_{Li}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}\,\!</math>

::<math>z_{Ri}^{\prime \prime }=\frac{\ln T_{Ri}^{\prime \prime }-\ln C-\tfrac{B}{{{V}_{i}}}}{\sigma _{T}^{\prime }}\,\!</math>

and:

*<math>{{F}_{e}}\,\!</math> is the number of groups of exact times-to-failure data points.

*<math>{{N}_{i}}\,\!</math> is the number of times-to-failure data points in the <math>{{i}^{th}}\,\!</math> time-to-failure data group.

*<math>{{\sigma}_{{T}'}}\,\!</math> is the standard deviation of the natural logarithm of the times-to-failure (unknown, the first of three parameters to be estimated).

*<math>B\,\!</math> is the Arrhenius parameter (unknown, the second of three parameters to be estimated).

*<math>C\,\!</math> is the second Arrhenius parameter (unknown, the third of three parameters to be estimated).

*<math>{{V}_{i}}\,\!</math> is the stress level of the <math>{{i}^{th}}\,\!</math> group.

*<math>{{T}_{i}}\,\!</math> is the exact failure time of the <math>{{i}^{th}}\,\!</math> group.

*<math>S\,\!</math> is the number of groups of suspension data points.

*<math>N_{i}^{\prime }\,\!</math> is the number of suspensions in the <math>{{i}^{th}}\,\!</math> group of suspension data points.

*<math>T_{i}^{\prime }\,\!</math> is the running time of the <math>{{i}^{th}}\,\!</math> suspension data group.

*<math>FI\,\!</math> is the number of interval data groups.

*<math>N_{i}^{\prime \prime }\,\!</math> is the number of intervals in the <math>{{i}^{th}}\,\!</math> group of data intervals.

*<math>T_{Li}^{\prime \prime }\,\!</math> is the beginning of the <math>{{i}^{th}}\,\!</math> interval.

*<math>T_{Ri}^{\prime \prime }\,\!</math> is the ending of the <math>{{i}^{th}}\,\!</math> interval.

The solution (parameter estimates) will be found by solving for <math>{{\widehat{\sigma }}_{{{T}'}}},\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math> so that <math>\tfrac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}=0,\,\!</math> <math>\tfrac{\partial \Lambda }{\partial B}=0\,\!</math> and <math>\tfrac{\partial \Lambda }{\partial C}=0\,\!</math>, where:

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial B}= \frac{1}{\sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\frac{1}{{{V}_{i}}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{1}{{{V}_{i}}}\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }{{V}_{i}}(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}\,\!</math>

::<math>\begin{align}
& \frac{\partial \Lambda }{\partial C}= \frac{1}{C\cdot \sigma _{{{T}'}}^{2}}\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}(\ln ({{T}_{i}})-\ln (C)-\frac{B}{{{V}_{i}}}) +\frac{1}{C\cdot {{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{\varphi (z_{Ri}^{\prime \prime })-\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }C(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))} \\
& & \\
& \frac{\partial \Lambda }{\partial {{\sigma }_{{{T}'}}}}= \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\left( \frac{{{\left( \ln ({{T}_{i}})-\ln (C)-\tfrac{B}{{{V}_{i}}} \right)}^{2}}}{\sigma _{{{T}'}}^{3}}-\frac{1}{{{\sigma }_{{{T}'}}}} \right) +\frac{1}{{{\sigma }_{{{T}'}}}}\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }\frac{\left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)\phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)}{1-\Phi \left( \tfrac{\ln \left( T_{i}^{\prime } \right)-\ln (C)-\tfrac{B}{{{V}_{i}}}}{{{\sigma }_{{{T}'}}}} \right)} \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{z_{Ri}^{\prime \prime }\varphi (z_{Ri}^{\prime \prime })-z_{Li}^{\prime \prime }\varphi (z_{Li}^{\prime \prime })}{\sigma _{T}^{\prime }(\Phi (z_{Ri}^{\prime \prime })-\Phi (z_{Li}^{\prime \prime }))}
\end{align}\,\!</math>

and:

::<math>\phi \left( x \right)=\frac{1}{\sqrt{2\pi }}\cdot {{e}^{-\tfrac{1}{2}{{\left( x \right)}^{2}}}}\,\!</math>

::<math>\Phi (x)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{x}{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt\,\!</math>

=Arrhenius Confidence Bounds=
==Approximate Confidence Bounds for the Arrhenius-Exponential==
There are different methods for computing confidence bounds. ALTA utilizes confidence bounds that are based on the asymptotic theory for maximum likelihood estimates, most commonly referred to as the Fisher matrix bounds.
 
====Confidence Bounds on the Mean Life====

The Arrhenius-exponential distribution is given by setting <math>m=L(V)\,\!</math> in the exponential ''pdf'' equation. The upper <math>({{m}_{U}})\,\!</math> and lower <math>({{m}_{L}})\,\!</math> bounds on the mean life are then estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

where <math>{{K}_{\alpha }}\,\!</math> is defined by:

::<math>\alpha =\frac{1}{\sqrt{2\pi }}\int_{{{K}_{\alpha }}}^{\infty }{{e}^{-\tfrac{{{t}^{2}}}{2}}}dt=1-\Phi ({{K}_{\alpha }})\,\!</math>

If <math>\delta \,\!</math> is the confidence level (i.e., 95%=0.95), then <math>\alpha =\tfrac{1-\delta }{2}\,\!</math> for the two-sided bounds, and <math>\alpha =1-\delta \,\!</math> for the one-sided bounds. The variance of <math>\widehat{m}\,\!</math> is given by:

::<math>\begin{align}
& Var(\widehat{m})= & {{\left( \frac{\partial m}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial m}{\partial B} \right)}^{2}}Var(\widehat{B}) +2\left( \frac{\partial m}{\partial C} \right)\left( \frac{\partial m}{\partial B} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>Var(\widehat{m})={{e}^{\tfrac{2\widehat{B}}{V}}}\left[ Var(\widehat{C})+\frac{{{\widehat{C}}^{2}}}{{{V}^{2}}}Var(\widehat{B})+\frac{2\widehat{C}}{V}Cov(\widehat{B},\widehat{C}) \right]\,\!</math>

The variances and covariance of <math>B\,\!</math> and <math>C\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B}\,\!</math>, <math>\widehat{C})\,\!</math> as follows:

::<math>\left[ \begin{matrix}
Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Confidence Bounds on Reliability====
The bounds on reliability for any given time, <math>T\,\!</math>, are estimated by:

::<math>\begin{align}
& {{R}_{U}}(T)= & {{e}^{-\tfrac{T}{{{m}_{U}}}}} \\
& {{R}_{L}}(T)= & {{e}^{-\tfrac{T}{{{m}_{L}}}}}
\end{align}\,\!</math>

where <math>{{m}_{U}}\,\!</math> and <math>{{m}_{L}}\,\!</math> are estimated estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds on time (ML estimate of time) for a given reliability are estimated by first solving the reliability function with respect to time:

::<math>\widehat{T}=-\widehat{m}\cdot \ln (R)\,\!</math>

The corresponding confidence bounds are then estimated from:

::<math>\begin{align}
& {{T}_{U}}= -{{m}_{U}}\cdot \ln (R) \\
& {{T}_{L}}= -{{m}_{L}}\cdot \ln (R)
\end{align}\,\!</math>

where <math>{{m}_{U}}\,\!</math> and <math>{{m}_{L}}\,\!</math> are estimated estimated by:

::<math>\begin{align}
& {{m}_{U}}= \widehat{m}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}} \\
& {{m}_{L}}= \widehat{m}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{m})}}{\widehat{m}}}}
\end{align}\,\!</math>

==Approximate Confidence Bounds for the Arrhenius-Weibull==
====Bounds on the Parameters====
From the asymptotically normal property of the maximum likelihood estimators, and since <math>\widehat{\beta },\,\!</math> and <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln (\widehat{\beta }),\,\!</math> and <math>\ln (\widehat{C})\,\!</math> can then be treated as normally distributed. After performing this transformation, the bounds on the parameters can be estimated from:

::<math>\begin{align}
& {{\beta }_{U}}= \widehat{\beta }\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}} \\
& {{\beta }_{L}}= \widehat{\beta }\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{\beta })}}{\widehat{\beta }}}}
\end{align}\,\!</math>

also:

::<math>\begin{align}
& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})} \\
& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}
\end{align}\,\!</math>

and:

::<math>\begin{align}
& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}} \\
& {{C}_{L}}= \widehat{C}\cdot {{e}^{-\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}
\end{align}\,\!</math>

The variances and covariances of <math>\beta ,\,\!</math> <math>B,\,\!</math> and <math>C\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{\beta },\,\!</math> <math>\widehat{B},\,\!</math> <math>\widehat{C})\,\!</math>, as follows:

::<math>\left[ \begin{matrix}
Var(\widehat{\beta }) & Cov(\widehat{\beta },\widehat{B}) & Cov(\widehat{\beta },\widehat{C}) \\
Cov(\widehat{B},\widehat{\beta }) & Var(\widehat{B}) & Cov(\widehat{B},\widehat{C}) \\
Cov(\widehat{C},\widehat{\beta }) & Cov(\widehat{C},\widehat{B}) & Var(\widehat{C}) \\
\end{matrix} \right]={{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\beta }^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial \beta \partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial \beta } & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Confidence Bounds on Reliability====
The reliability function for the Arrhenius-Weibull model (ML estimate) is given by:

::<math>\widehat{R}(T,V)={{e}^{-{{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}}}}\,\!</math>

or:

::<math>\widehat{R}(T)={{e}^{-{{e}^{\ln \left[ {{\left( \tfrac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]}}}}\,\!</math>

Setting:

::<math>\widehat{u}=\ln \left[ {{\left( \frac{T}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \right]\,\!</math>

or:

::<math>\widehat{u}=\widehat{\beta }\left[ \ln (T)-\ln (\widehat{C})-\frac{\widehat{B}}{V} \right]\,\!</math>

The reliability function now becomes:

::<math>\widehat{R}(T,V)={{e}^{-{{e}^{\widehat{u}}}}}\,\!</math>

The next step is to find the upper and lower bounds on <math>\widehat{u}\ \ :\,\!</math>

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

where:

::<math>\begin{align}
& Var(\widehat{u})= {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{u})= {{\left( \frac{\widehat{u}}{\widehat{\beta }} \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\widehat{\beta }}{V} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\widehat{\beta }}{\widehat{C}} \right)}^{2}}Var(\widehat{C}) -\frac{2\widehat{u}}{V}Cov(\widehat{\beta },\widehat{B})-\frac{2\widehat{u}}{\widehat{C}}Cov(\widehat{\beta },\widehat{C})+\frac{2{{\widehat{\beta }}^{2}}}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

The upper and lower bounds on reliability are:

::<math>\begin{align}
& {{R}_{U}}(T,V)= {{e}^{-{{e}^{\left( {{u}_{L}} \right)}}}} \\
& {{R}_{L}}(T,V)= {{e}^{-{{e}^{\left( {{u}_{U}} \right)}}}}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds on time for a given reliability are estimated by first solving the reliability function with respect to time:

::<math>\begin{align}
\ln (R)&= -{{\left( \frac{\widehat{T}}{\widehat{C}\cdot {{e}^{\tfrac{\widehat{B}}{V}}}} \right)}^{\widehat{\beta }}} \\
\ln (-\ln (R))&= \widehat{\beta }\left( \ln \widehat{T}-\ln \widehat{C}-\frac{\widehat{B}}{V} \right)
\end{align}\,\!</math>

or:

::<math>\widehat{u}=\frac{1}{\widehat{\beta }}\ln (-\ln (R))+\ln \widehat{C}+\frac{\widehat{B}}{V}\,\!</math>

where <math>\widehat{u}=\ln \widehat{T}\,\!</math>.

The upper and lower bounds on <math>u\,\!</math> are estimated from:

::<math>{{u}_{U}}=\widehat{u}+{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

::<math>{{u}_{L}}=\widehat{u}-{{K}_{\alpha }}\sqrt{Var(\widehat{u})}\,\!</math>

where:

::<math>\begin{align}
& Var(\widehat{u})= & {{\left( \frac{\partial \widehat{u}}{\partial \beta } \right)}^{2}}Var(\widehat{\beta })+{{\left( \frac{\partial \widehat{u}}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial \widehat{u}}{\partial C} \right)}^{2}}Var(\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial B} \right)Cov(\widehat{\beta },\widehat{B})+2\left( \frac{\partial \widehat{u}}{\partial \beta } \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{\beta },\widehat{C}) +2\left( \frac{\partial \widehat{u}}{\partial B} \right)\left( \frac{\partial \widehat{u}}{\partial C} \right)Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{u})= & \frac{1}{{{\widehat{\beta }}^{4}}}{{\left[ \ln (-\ln (R)) \right]}^{2}}Var(\widehat{\beta })+\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{\widehat{C}}^{2}}}Var(\widehat{C})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}V}Cov(\widehat{\beta },\widehat{B})-\frac{2\ln (-\ln (R))}{{{\widehat{\beta }}^{2}}\widehat{C}}Cov(\widehat{\beta },\widehat{C}) +\frac{2}{V\widehat{C}}Cov(\widehat{B},\widehat{C})
\end{align}\,\!</math>

The upper and lower bounds on time can then found by:

::<math>\begin{align}
& {{T}_{U}}= & {{e}^{{{u}_{U}}}} \\
& {{T}_{L}}= & {{e}^{{{u}_{L}}}}
\end{align}\,\!</math>

==Approximate Confidence Bounds for the Arrhenius-Lognormal==
====Bounds on the Parameters====
The lower and upper bounds on <math>B\,\!</math> are estimated from:

::<math>\begin{align}
& {{B}_{U}}= \widehat{B}+{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Upper bound)} \\
& {{B}_{L}}= \widehat{B}-{{K}_{\alpha }}\sqrt{Var(\widehat{B})}\text{ (Lower bound)}
\end{align}\,\!</math>

Since the standard deviation, <math>{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>, and the parameter <math>\widehat{C}\,\!</math> are positive parameters, <math>\ln ({{\widehat{\sigma }}_{{{T}'}}})\,\!</math> and <math>\ln (\widehat{C})\,\!</math> are treated as normally distributed. The bounds are estimated from:

::<math>\begin{align}
& {{C}_{U}}= \widehat{C}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}\text{ (Upper bound)} \\
& {{C}_{L}}= \frac{\widehat{C}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var(\widehat{C})}}{\widehat{C}}}}}\text{ (Lower bound)}
\end{align}\,\!</math>

and:

::<math>\begin{align}
& {{\sigma }_{U}}= {{\widehat{\sigma }}_{{{T}'}}}\cdot {{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}\text{ (Upper bound)} \\
& {{\sigma }_{L}}= \frac{{{\widehat{\sigma }}_{{{T}'}}}}{{{e}^{\tfrac{{{K}_{\alpha }}\sqrt{Var({{\widehat{\sigma }}_{{{T}'}}})}}{{{\widehat{\sigma }}_{{{T}'}}}}}}}\text{ (Lower bound)}
\end{align}\,\!</math>

The variances and covariances of <math>B,\,\!</math> <math>C,\,\!</math> and <math>{{\sigma }_{{{T}'}}}\,\!</math> are estimated from the local Fisher matrix (evaluated at <math>\widehat{B},\,\!</math> <math>\widehat{C}\,\!</math>, <math>{{\widehat{\sigma }}_{{{T}'}}}),\,\!</math> as follows:

::<math>\left[ \begin{matrix}
Var\left( {{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) & Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{B} \right) & Var\left( \widehat{B} \right) & Cov\left( \widehat{B},\widehat{C} \right) \\
Cov\left( {{\widehat{\sigma }}_{{{T}'}}},\widehat{C} \right) & Cov\left( \widehat{C},\widehat{B} \right) & Var\left( \widehat{C} \right) \\
\end{matrix} \right]= {{\left[ \begin{matrix}
-\tfrac{{{\partial }^{2}}\Lambda }{\partial \sigma _{{{T}'}}^{2}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{\sigma }_{{{T}'}}}\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{B}^{2}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial B\partial C} \\
-\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial {{\sigma }_{{{T}'}}}} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial C\partial B} & -\tfrac{{{\partial }^{2}}\Lambda }{\partial {{C}^{2}}} \\
\end{matrix} \right]}^{-1}}\,\!</math>

====Bounds on Reliability====
The reliability of the lognormal distribution is:

::<math>R({T}',V;B,C,{{\sigma }_{{{T}'}}})=\int_{{{T}'}}^{\infty }\frac{1}{{{\widehat{\sigma }}_{{{T}'}}}\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{\left( \tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}} \right)}^{2}}}}dt\,\!</math>

Let <math>\widehat{z}(t,V;B,C,{{\sigma }_{T}})=\tfrac{t-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}},\,\!</math> then <math>\frac{d \widehat{z}}{dt}=\frac{1}{{\widehat{\sigma }}_{{{T}'}}}\,\!</math>.

For <math>t={T}'\,\!</math>, <math>\widehat{z}=\tfrac{{T}'-\ln (\widehat{C})-\tfrac{\widehat{B}}{V}}{{{\widehat{\sigma }}_{{{T}'}}}}\,\!</math>, and for <math>t=\infty ,\,\!</math> <math>\widehat{z}=\infty .\,\!</math> The above equation then becomes:

::<math>R(\widehat{z})=\int_{\widehat{z}({T}')}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\,\!</math>

The bounds on <math>z\,\!</math> are estimated from:

::<math>\begin{align}
& {{z}_{U}}= & \widehat{z}+{{K}_{\alpha }}\sqrt{Var(\widehat{z})} \\
& {{z}_{L}}= & \widehat{z}-{{K}_{\alpha }}\sqrt{Var(\widehat{z})}
\end{align}\,\!</math>

where:

::<math>\begin{align}
Var(\widehat{z})=& \left( \frac{\partial \widehat{z}}{\partial B} \right)_{\widehat{B}}^{2}Var(\widehat{B})+\left( \frac{\partial \widehat{z}}{\partial C} \right)_{\widehat{C}}^{2}Var(\widehat{C})+\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)_{{{\widehat{\sigma }}_{{{T}'}}}}^{2}Var({{\widehat{\sigma }}_{T}}) +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}Cov\left( \widehat{B},\widehat{C} \right) \\
& +2{{\left( \frac{\partial \widehat{z}}{\partial B} \right)}_{\widehat{B}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{B},{{\widehat{\sigma }}_{T}} \right) +2{{\left( \frac{\partial \widehat{z}}{\partial C} \right)}_{\widehat{C}}}{{\left( \frac{\partial \widehat{z}}{\partial {{\sigma }_{{{T}'}}}} \right)}_{{{\widehat{\sigma }}_{{{T}'}}}}}Cov\left( \widehat{C},{{\widehat{\sigma }}_{T}} \right)
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var(\widehat{z})= & \frac{1}{\widehat{\sigma }_{{{T}'}}^{2}}[\frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +\frac{2}{C\cdot V}Cov\left( \widehat{B},\widehat{C} \right)+\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)+\frac{2\widehat{z}}{C}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right)]
\end{align}\,\!</math>

The upper and lower bounds on reliability are:

::<math>\begin{align}
& {{R}_{U}}= & \int_{{{z}_{L}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Upper bound)} \\
& {{R}_{L}}= & \int_{{{z}_{U}}}^{\infty }\frac{1}{\sqrt{2\pi }}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\text{ (Lower bound)}
\end{align}\,\!</math>

====Confidence Bounds on Time====
The bounds around time, for a given lognormal percentile (unreliability), are estimated by first solving the reliability equation with respect to time, as follows:

::<math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=\ln (\widehat{C})+\frac{\widehat{B}}{V}+z\cdot {{\widehat{\sigma }}_{{{T}'}}}\,\!</math>

where:

::<math>\begin{align}
{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}})=&\ \ln (T) \\
z= & \ {{\Phi }^{-1}}\left[ F({T}') \right]
\end{align}\,\!</math>

and:

::<math>\Phi (z)=\frac{1}{\sqrt{2\pi }}\int_{-\infty }^{z({T}')}{{e}^{-\tfrac{1}{2}{{z}^{2}}}}dz\,\!</math>

The next step is to calculate the variance of <math>{T}'(V;\widehat{B},\widehat{C},{{\widehat{\sigma }}_{{{T}'}}}):\,\!</math>

::<math>\begin{align}
Var({T}')= & {{\left( \frac{\partial {T}'}{\partial B} \right)}^{2}}Var(\widehat{B})+{{\left( \frac{\partial {T}'}{\partial C} \right)}^{2}}Var(\widehat{C})+{{\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial C} \right)Cov\left( \widehat{B},\widehat{C} \right) \\
& +2\left( \frac{\partial {T}'}{\partial B} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) +2\left( \frac{\partial {T}'}{\partial C} \right)\left( \frac{\partial {T}'}{\partial {{\sigma }_{{{T}'}}}} \right)Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}\,\!</math>

or:

::<math>\begin{align}
& Var({T}')= \frac{1}{{{V}^{2}}}Var(\widehat{B})+\frac{1}{{{C}^{2}}}Var(\widehat{C})+{{\widehat{z}}^{2}}Var({{\widehat{\sigma }}_{{{T}'}}}) +\frac{2}{B\cdot C}Cov\left( \widehat{B},\widehat{C} \right) +\frac{2\widehat{z}}{V}Cov\left( \widehat{B},{{\widehat{\sigma }}_{{{T}'}}} \right) +\frac{2\widehat{z}}{C}Cov\left( \widehat{C},{{\widehat{\sigma }}_{{{T}'}}} \right)
\end{align}\,\!</math>

The upper and lower bounds are then found by:

::<math>\begin{align}
& T_{U}^{\prime }= & \ln {{T}_{U}}={T}'+{{K}_{\alpha }}\sqrt{Var({T}')} \\
& T_{L}^{\prime }= & \ln {{T}_{L}}={T}'-{{K}_{\alpha }}\sqrt{Var({T}')}
\end{align}\,\!</math>

Solving for <math>{{T}_{U}}\,\!</math> and <math>{{T}_{L}}\,\!</math> yields:

::<math>\begin{align}
& {{T}_{U}}= & {{e}^{T_{U}^{\prime }}}\text{ (Upper bound)} \\
& {{T}_{L}}= & {{e}^{T_{L}^{\prime }}}\text{ (Lower bound)}
\end{align}\,\!</math>

Proportional Hazards Model

2021-10-29T16:26:54Z

Cruz Daniel: changed the listed 'unknowns to solve for' to not include eta, since eta has been renamed a_0, which we already list. Suggestion came from Sayyed Vazirizade at University of Arizona

<noinclude>{{Navigation box}}
''This article also appears in the [[Multivariable_Relationships:_General_Log-Linear_and_Proportional_Hazards|Accelerated Life Testing Data Analysis Reference]] book.'' </noinclude>

Introduced by D. R. Cox, the Proportional Hazards (PH) model was developed in order to estimate the effects of different covariates influencing the times-to-failure of a system.
The model has been widely used in the biomedical field, as discussed in Leemis [[Appendix_E:_References|[22]]], and recently there has been an increasing interest in its application in reliability engineering. In its original form, the model is non-parametric, (i.e., no assumptions are made about the nature or shape of the underlying failure distribution). In this reference, the original non-parametric formulation as well as a parametric form of the model will be considered utilizing a Weibull life distribution. In ALTA, the proportional hazards model is included in its parametric form and can be used to analyze data with up to eight variables. The GLL-Weibull and GLL-exponential models are actually special cases of the proportional hazards model. However, when using the proportional hazards in ALTA, no transformation on the covariates (or stresses) can be performed.

==Non-Parametric Model Formulation==
According to the PH model, the failure rate of a system is affected not only by its operation time, but also by the covariates under which it operates. For example, a unit may have been tested under a combination of different accelerated stresses such as humidity, temperature, voltage, etc. It is clear then that such factors affect the failure rate of a unit.

The instantaneous failure rate (or hazard rate) of a unit is given by:

::<math>\lambda (t)=\frac{f(t)}{R(t)}\,\!</math>

where:

*<math>f(t)\,\!</math> is the probability density function.

*<math>R(t)\,\!</math> is the reliability function.

Note that for the case of the failure rate of a unit being dependent not only on time but also on other covariates, the above equation must be modified in order to be a function of time and of the covariates.
The proportional hazards model assumes that the failure rate (hazard rate) of a unit is the product of:

*an arbitrary and unspecified baseline failure rate, <math>{{\lambda }_{0}}(t),\,\!</math> which is a function of time only.

*a positive function <math>g(x,\underline{A})\,\!</math>, independent of time, which incorporates the effects of a number of covariates such as humidity, temperature, pressure, voltage, etc.

The failure rate of a unit is then given by:

::<math>\lambda (t,\underline{X})={{\lambda }_{0}}(t)\cdot g(\underline{X},\underline{A})\,\!</math>

where:

*<math>\underline{X}\,\!</math> is a row vector consisting of the covariates:

::<math>\underline{X}=({{x}_{1}},{{x}_{2}},...,{{x}_{m}})\,\!</math>

*<math>\underline{A}\,\!</math> is a column vector consisting of the unknown parameters (also called regression parameters) of the model:

::<math>\underline{A}={{({{a}_{1}},{{a}_{2}},...{{a}_{m}})}^{T}}\,\!</math>

:where:

::<math>\quad \quad m\,\!</math> = number of stress related variates (time-independent).

It can be assumed that the form of <math>g(\underline{X},\underline{A})\,\!</math> is known and <math>{{\lambda }_{0}}(t)\,\!</math> is unspecified. Different forms of <math>g(\underline{X},\underline{A})\,\!</math> can be used.

However, the exponential form is mostly used due to its simplicity and is given by:

::<math>g(\underline{X},\underline{A})={{e}^{{{\underline{A}}^{T}}{{\underline{X}}^{T}}}}={{e}^{\mathop{\sum}_{j=1}^{m}{{a}_{j}}{{x}_{j}}}}\,\!</math>

The failure rate can then be written as:

::<math>\lambda (t,\underline{X})={{\lambda }_{0}}(t)\cdot {{e}^{\mathop{\sum}_{j=1}^{m}{{a}_{j}}{{x}_{j}}}}\,\!</math>

==Parametric Model Formulation==
A parametric form of the proportional hazards model can be obtained by assuming an underlying distribution. In ALTA, the Weibull and exponential distributions are available. In this section we will consider the Weibull distribution to formulate the parametric proportional hazards model. In other words, it is assumed that the baseline failure rate is parametric and given by the Weibull distribution. In this case, the baseline failure rate is given by:

::<math>{{\lambda }_{0}}(t)=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}\,\!</math>

The PH failure rate then becomes:

::<math>\lambda (t,\underline{X})=\frac{\beta }{\eta }{{\left( \frac{t}{\eta } \right)}^{\beta -1}}\cdot {{e}^{\mathop{\sum}_{j=1}^{m}{{a}_{j}}{{x}_{j}}}}\,\!</math>

It is often more convenient to define an additional covariate, <math>{{x}_{0}} = 1\,\!</math>, in order to allow the Weibull scale parameter raised to the beta (shape parameter) to be included in the vector of regression coefficients. The PH failure rate can then be written as:

::<math>\lambda (t,\underline{X})=\beta \cdot {{t}^{\beta -1}}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}\,\!</math>

The PH reliability function is given by:

::<math>\begin{align}
R(t,\underline{X})=\ {{e}^{-\int_{0}^{t}\lambda (u)du}} =\ {{e}^{-\int_{0}^{t}\lambda (u,\underline{X})du}} =\ {{e}^{-{{t}^{\beta }}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}}}}
\end{align}\,\!</math>

The ''pdf'' can be obtained by taking the partial derivative of the reliability function with respect to time. The PH ''pdf'' is:

::<math>\begin{align}
f(t,\underline{X})= & \lambda (t,\underline{X})\cdot R(t,\underline{X}) =\ \beta \cdot {{t}^{\beta -1}}{{e}^{\left[ \mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}-{{t}^{\beta }}\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{j}}}} \right]}}
\end{align}\,\!</math>

The total number of unknowns to solve for in this model is <math>m+2\,\!</math> (i.e., <math>\beta ,{{a}_{0}},{{a}_{1}},...{{a}_{m}}\,\!</math>).

The maximum likelihood estimation method can be used to determine these parameters. The log-likelihood function for this case is given by:

::<math>\begin{align}
\ln (L)= & \Lambda =\underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left( \beta \cdot T_{i}^{\beta -1}{{e}^{-T_{i}^{\beta }\cdot {{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}}}{{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}} \right) -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( T_{i}^{\prime } \right)}^{\beta }}{{e}^{\mathop{\sum}_{j=0}^{m}{{a}_{j}}{{x}_{i,j}}}}+\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>

where:

::<math>\begin{align}
& R_{Li}^{\prime \prime }= & {{e}^{-T_{Li}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}} \\
& R_{Ri}^{\prime \prime }= & {{e}^{-T_{Ri}^{\prime \prime \beta }{{e}^{\underset{j=0}{\mathop{\overset{n}{\mathop{\mathop{\sum}_{}^{}}}\,}}\,{{\alpha }_{j}}{{x}_{j}}}}}}
\end{align}\,\!</math>

Solving for the parameters that maximize the log-likelihood function will yield the parameters for the PH-Weibull model. Note that for <math>\beta =1 \,\!</math>, the log-likelihood function becomes the log-likelihood function for the PH-exponential model, which is similar to the original form of the proportional hazards model proposed by Cox and Oakes [[Appendix_E:_References|[39]]].

Note that the likelihood function of the GLL model is very similar to the likelihood function for the proportional hazards-Weibull model. In particular, the shape parameter of the Weibull distribution can be included in the regression coefficients as follows:

::<math>{{a}_{i,PH}}=-\beta \cdot {{a}_{i,GLL}}\,\!</math>

where:

*<math>{{a}_{i,PH}}\,\!</math> are the parameters of the PH model.

*<math>{{a}_{i,GLL}}\,\!</math> are the parameters of the general log-linear model.

In this case, the likelihood functions are identical. Therefore, if no transformation on the covariates is performed, the parameter values that maximize the likelihood function of the GLL model also maximize the likelihood function for the proportional hazards-Weibull (PHW) model. Note that for <math>\beta = 1\,\!</math> (exponential life distribution), the two likelihood functions are identical, and <math>{{a}_{i,PH}}=-{{a}_{i,GLL}}.\,\!</math>

<noinclude>=Indicator Variables=
Another advantage of the multivariable relationships used in ALTA is that they allow for simultaneous analysis of continuous and categorical variables. Categorical variables are variables that take on discrete values such as the lot designation for products from different manufacturing lots. In this example, lot is a categorical variable, and it can be expressed in terms of indicator variables. Indicator variables only take a value of 1 or 0. For example, consider a sample of test units. A number of these units were obtained from Lot 1, others from Lot 2, and the rest from Lot 3. These three lots can be represented with the use of indicator variables, as follows:

*Define two indicator variables, <math>{{X}_{1}}\,\!</math> and <math>{{X}_{2}}.\,\!</math>

*For the units from Lot 1, <math>{{X}_{1}}=1,\,\!</math> and <math>{{X}_{2}}=0.\,\!</math>

*For the units from Lot 2, <math>{{X}_{1}}=0,\,\!</math> and <math>{{X}_{2}}=1.\,\!</math>

*For the units from Lot 3, <math>{{X}_{1}}=0,\,\!</math> and <math>{{X}_{2}}=0.\,\!</math>

Assume that an accelerated test was performed with these units, and temperature was the accelerated stress. In this case, the [[General_Log-Linear_Relationship|GLL relationship]] can be used to analyze the data. From this relationship we get:

::<math>L(\underline{X})={{e}^{{{\alpha }_{0}}+{{\alpha }_{1}}{{X}_{1}}+{{\alpha }_{2}}{{X}_{2}}+{{\alpha }_{3}}{{X}_{3}}}}\,\!</math>

where:

*<math>{{X}_{1}}\,\!</math> and <math>{{X}_{2}}\,\!</math> are the indicator variables, as defined above.

*<math>{{X}_{3}}=\tfrac{1}{T},\,\!</math> where <math>T\,\!</math> is the temperature.

The data can now be entered in ALTA and, with the assumption of an underlying life distribution and using MLE, the parameters of this model can be obtained.</noinclude>

Voting OR Gate Example

2021-08-16T17:47:40Z

Cruz Daniel: /* Voting OR Gate */

<noinclude>{{Banner BlockSim Examples}}
''This example also appears in the [[Fault_Tree_Diagrams_and_System_Analysis|System Analysis Reference book]]''. </noinclude>
===Voting OR Gate===

[[Image:I10.7.png|center|100px|link=|]]

In a Voting OR gate, the output event occurs if <math>k\,\!</math> or more of the input events occur. In system reliability terms, this implies that if any ''k''-out-of-''n'' components fail (input) then the system will fail (output).

The equivalent RBD construct is a node and is similar to a ''k''-out-of-''n'' parallel configuration with a distinct difference, as discussed next. To illustrate this difference, consider a fault tree diagram with a 2-out-of-4 Voting OR gate, as shown in the following figure.

[[Image:3.png|center|300px|Illustration of a 2-out-or-4 Voting OR gate.|link=]]

In this diagram, the system will fail if any two of the blocks below fail. Equivalently, this can be represented by the RBD shown in the next figure using a 3-out-of-4 node.

[[Image:4.png|center|300px|Equivalent representation of the 2-out-of-4 Voting OR gate.|link=]]

In this configuration, the system will not fail if three out of four components are operating, but will fail if more than one fails. In other words, the fault tree considers ''k''-out-of-''n'' failures for the system failure while the RBD considers ''k''-out-of-''n'' successes for system success.

====Note: Complexity of k-out-of-n configurations====
Note that for large values of <math>n</math> and intermediate values of <math>k</math> it is possible to create a configuration that appears simple but is actually extremely mathematically complex. This arises from the equation describing the number of combinations of <math>k</math> choices out of <math>n</math> units:

::<math>\binom{n}{k} = \frac{n!}{k! (n-k)!}</math>

With <math>k = 1, n = 50</math> this expression evaluates to 50, meaning there are 50 unique failure combinations. With <math> k = 49, n = 50</math> this number is again 50. Further, with <math>k = 2, n = 50</math> this number increases to 1225. But with <math>k = 25, n = 50</math> there are 126,410,606,437,752 unique combinations that must all be evaluated to provide answers. Attempting to calculate results or cut sets from such a fault tree analytically therefore requires an inordinate amount of calculation time. As such, similar combinations of <math>k</math> and <math>n</math> should be avoided, or solved with simulation if possible.

====Increasing the Flexibility====
Classical Voting OR gates have no properties and cannot fail or be repaired (i.e., they cannot be an event themselves). In BlockSim, Voting OR gates behave like nodes in an RBD; thus, they can also fail and be repaired just like any other event. By default, when a Voting OR gate is inserted into a fault tree diagram within BlockSim, the gate is set so that it cannot fail (classical definition). However, this property can be modified to allow for additional flexibility.

====Example====
Consider a system with three components, ''A'', ''B'' and ''C''. The system fails if any two components fail. Draw the fault tree and reliability block diagram for the system. The next two figures show both the FTD and RBD representations.

<center>[[Image:I10.8.png|center|250px|link=|]]

[[Image:I10.9.png|center|250px|link=|]]</center>

The reliability equation for either configuration is:

::<math>{{R}_{System}}=-2\cdot {{R}_{A}}\cdot {{R}_{B}}\cdot {{R}_{C}}+{{R}_{A}}\cdot {{R}_{B}}+{{R}_{A}}\cdot {{R}_{C}}+{{R}_{B}}\cdot {{R}_{C}}\,\!</math>

Equation above assumes a classical Voting OR gate (i.e., the voting gate itself cannot fail). If the gate can fail then the equation is modified as follows:

::<math>{{R}_{System}}={{R}_{Voting}}\left( -2\cdot {{R}_{A}}\cdot {{R}_{B}}\cdot {{R}_{C}}+{{R}_{A}}\cdot {{R}_{B}}+{{R}_{A}}\cdot {{R}_{C}}+{{R}_{B}}\cdot {{R}_{C}} \right)\,\!</math>

Note that while both the gate and the node are 2-out-of-3, they represent different circumstances. The Voting OR gate in the fault tree indicates that if two components fail then the system will fail; while the node in the reliability block diagram indicates that if at least two components succeed then the system will succeed.

Reliability Phase Diagrams (RPDs)

2021-04-30T14:21:35Z

Cruz Daniel:

{{Template:bsbook|10}}
The term ''phase diagram'' is used in many disciplines with different meanings. In physical chemistry, mineralogy and materials science, a phase diagram is a type of graph used to show the equilibrium conditions among the thermodynamically-distinct phases. In mathematics and physics, a phase diagram is used as a synonym for a phase space. In reliability engineering, we introduce the term phase diagram, or more specifically Reliability Phase Diagram or RPD, as an extension of the reliability block diagram (RBD) approach to graphically describe the sequence of different operational and/or maintenance phases experienced by a system. Whereas a reliability block diagram (RBD) is used to analyze the reliability of a system with a fixed configuration, a phase diagram can be used to represent/analyze a system whose reliability configuration and/or other properties change over time. In other words, during a mission the system may undergo changes in its reliability configuration (RBD), available resources or the failure, maintenance and/or throughput properties of its individual components. Examples of this include: 

::#Systems whose components exhibit different failure distributions depending on changes in the stress on the system.
::#Systems or processes requiring different equipment to function over a cycle, such as start-up, normal production, shut-down, scheduled maintenance, etc.
::#Systems whose RBD configuration changes at different times, such as the RBD of the engine configuration on a four-engine aircraft during taxi, take-off, cruising, and landing.
::#Systems with different types of machinery operating during day and night shifts and with different amounts of throughput during each shift.

To analyze such systems, each stage during the mission can be represented by a phase whose properties are inherited from an RBD corresponding to that phase's reliability configuration, along with any associated resources of the system during that time. A phase diagram is then a series of such phases drawn (connected) in a sequence signifying a chronological order.

To better illustrate this, consider the four-engine aircraft mentioned previously. Assume that when a critical failure (system failure) occurs during taxiing, the airplane does not take off and is sent for maintenance instead. However, when a critical failure occurs during take-off, cruising, landing, the system is assumed to be lost. Furthermore, assume that the taxi phase requires only one engine, the take-off phase requires all four engines, the cruising phase requires any three of the four engines and the landing phase requires any two of the four engines. To model this, each one of these cases would require a different k-out-of-n redundancy on the engines and thus a different RBD. Creating an RBD for each phase is trivial. However, what you need is a way to transition from one RBD to the next, in a specified sequence, while maintaining all the past history of each component during the transition.

In other words, a new engine would transition to the take-off phase with an age equal to the time it was used during taxi, or an engine that failed while in flight would remain failed in the next phase (i.e., landing). To model this, a block for each phase would be used in the phase diagram, and each phase block would be linked to the appropriate RBD. This is illustrated in the figure below.

[[Image:11.1.png|center|500px|Phase diagram illustrating the mission of a four-engine aircraft|link=]]

In this figure, the taxiing, take-off, cruising and landing blocks represent the operational phases and the final block is a maintenance phase. Each of the operational phases in this diagram has two paths leading from it: a success path and a failure path. This graphically illustrates the consequences in each case. For instance, if the first taxiing phase is successful, the airplane will proceed to the take-off phase; if it is unsuccessful, the airplane will be sent for maintenance. The failure paths for the take-off, cruising and landing phases point to stop blocks, which indicate that the simulation of the mission ends. For the final taxiing phase, both the success and failure paths lead to the maintenance path; the node block allows you to model this type of shared outcome.

The execution of a phase diagram from its first phase to its last phase is referred to as one cycle. For example, for the airplane mentioned above, a single mission from the initial taxiing through the post-flight maintenance is one cycle. If the simulation end time exceeds the total duration of one cycle of a phase diagram, the simulation continues and the phase diagram is executed multiple times until the simulation end time is reached. Execution of a phase diagram multiple times during a simulation is referred to as cycling. During cycling, the age of components accumulated in the last phase of the previous cycle is carried over to the first phase of the next cycle. In summary, cycling is used to model the continuous operation of a system involving repetition of the same phases in the same sequence (e.g., an airplane making multiple flights).

The sections that follow describe the types of phase blocks and other blocks that can be used in a phase diagram, as well as how those blocks can be connected. A more detailed example is then presented, followed by a discussion of the rules and assumptions that apply in phase diagram simulation. Finally, throughput in phase diagrams is discussed.

= Phase Blocks =

=== Operational Phases ===

In RPDs, two types of phases are used: operational phases and maintenance phases. An operational phase is used to represent any stage of the system's mission that is not exclusively dedicated to the execution of maintenance tasks. Operational phases are always defined by (linked to) an RBD. Each operational phase has a fixed, predefined time duration. 

==== Operational Phase Properties ====

'''Diagram'''
The diagram property is used to associate an RBD with a phase. You can select and associate any existing simulation RBD with a phase. Note that common components across different RBDs are identified by name. In other words, a component with the exact same name in two RBDs is assumed to be the same component working in two different phases.

'''Phase Duration'''
The duration of an operational phase is fixed and needs to be specified. However, this duration may be affected by the choice of path you choose followed by this phase. If a failure has not occurred by the end of the specified phase duration, the simulation will proceed along the success path leading from the phase block. If a failure occurs, the simulation will proceed along the failure path leading from the phase block.

'''Phase Duty Cycle'''
This property allows you to specify a common duty cycle value for the entire RBD that the phase represents, thereby modeling situations where the actual usage of the RBD during system operation is not identical to the usage for which you have data (either from testing or from the field). This can include situations where the item:

*Does not operate continuously (e.g., a DVD drive that was tested in continuous operation, but in actual use within a computer accumulates only 18 minutes of usage for every hour the computer operates).
*Is subjected to loads that are greater than or less than the rated loads (e.g., a motor that is rated to operate at 1,000 rpm but is being used at 800 rpm).
*Is affected by changes in environmental stress (e.g., a laptop computer that is typically used indoors at room temperature, but is being used outdoors in tropical conditions).

In these cases, continuous operation at the rated load is considered to be a duty cycle of 1. Any other level of usage is expressed as a percentage of the rated load value or operating time. For example, consider the DVD drive mentioned above; its duty cycle value would be 18 min / 60 min = 0.3. A duty cycle value higher than 1 indicates a load in excess of the rated value.

If a duty cycle is specified for the phase and there are also duty cycles specified for blocks within the RBD, their effects are compounded. For instance, consider the aircraft example given earlier. During the take-off phase, the subsystems experience 1.5 times the normal stress, so you would use a phase duty cycle value of 1.5. We also know that the landing gear is not used continuously during take-off. Assume that the landing gear is actually in use only 30% of the time during take-off. Each landing gear block in the RBD, then, would have a duty cycle value of 0.3. For each block, the effects of the phase duty cycle and the block duty cycle are compounded, yielding an effective duty cycle value of 1.5 x 0.3 = 0.45. 

=== Maintenance Phases ===

A maintenance phase represents the portion of a system's mission time where the system is down and maintenance actions are performed on some or all of its components. For representation ease a maintenance phase is defined by (linked to) a maintenance template. This template can be thought of as a list, or a collection, of the specific components (blocks) that are designated to undergo inspection, repair or replacement actions during the maintenance phase, along with their maintenance priority order. In other words, if blocks A, B and C are to undergo maintenance during a specific phase, they are placed in a maintenance template in a priority sequence. Depending on the resources available, the actions are prioritized as resources permit. That is, if three repair crews were available along with three spare parts, actions on A, B and C would be carried out simultaneously. However, if only one crew was available, the actions would be carried out based on the priority order defined in the template. Given that all aspects of maintenance can be probabilistically defined, the duration of a maintenance phase, unlike an operational phase, is not fixed and the phase lasts as long as it takes to complete all actions specified in the phase. To illustrate this, consider a race car that competes in two races, and even though corrective repair actions can be done during each race as needed, the race car then undergoes a major overhaul (i.e., series of maintenance actions). For this example assume the major sub-systems of the car undergoing these maintenance tasks are the engine, the transmission, the suspension system and the tires. The operation of the race car can then be represented as a phase diagram consisting of two operational phases, representing the two races, and one maintenance phase representing the maintenance activities. The figure below shows such a phase diagram along with the maintenance template. 
[[Image:11_2.png|center|300px|Phase diagram illustrating the three-phase mission of the race car along with the maintenance template|link=]] 

==== Maintenance Phase Properties ==== 

'''Maintenance Template'''

This specifies the maintenance template to be used in the currently selected maintenance phase.

'''Interval Maintenance Threshold'''

The Interval Maintenance Threshold property provides the ability to add some flexibility to the timing of scheduled maintenance tasks. In other words, tasks based on system or item age intervals (fixed or dynamic) will be performed if the start of the maintenance phase is within (1-X)% of the scheduled time for the action. It is used to specify an age interval when a maintenance task will be performed. This helps in optimizing the resources allocated to repair the system during a maintenance phase by performing preventive maintenance actions or inspections when the system is already down in a maintenance phase. For example, a preventive maintenance action is scheduled for a car (e.g., an oil change, tire rotation, etc.) every 60,000 miles, but a system downing failure of an unrelated component occurs at 55,000 miles. Here the system age threshold will determine whether the preventive maintenance will be performed earlier than scheduled. If the Interval Maintenance Threshold is 0.9, the preventive maintenance will be performed since the failure occurred after the system accumulated 91.67% of the time to the scheduled maintenance or is within 8.33%, (60,000-55,000)/60,000= 8.33%, of the system age at which the preventive maintenance was originally scheduled. If the system age threshold was 0.95, the preventive maintenance will not be performed at 55,000 miles, since the system failure did not occur within 5% of the system age at which the preventive maintenance was originally scheduled (1-0.95=0.05 or 5%).
{{:Interval Maintenance Threshold Example}}

{{:Example Using Success Failure Paths in Phase Diagrams}}

= Node Blocks and Stop Blocks =

Starting with Version 8, the two possible outcomes of an operational phase block are modeled using success and failure paths. Where previously a failure outcome was defined as part of the operational phase block's properties, it is now graphically represented within the diagram. Node blocks and stop blocks are provided to allow you to build configurations that are both accurate and readable.

=== Node Blocks ===

The purpose of a node block is simply to enable configurations that would otherwise not be possible due to limitations on connecting blocks. For example, consider an instance where maintenance is scheduled to be performed after the operational phase has completed successfully, and if a failure occurs during simulation, that maintenance will take place upon failure. In this case, the operational phase block's success and failure outcomes are identical. Success paths and failure paths cannot be identical in phase diagrams, however, so you would model this configuration in one of two ways:

*If the operational phase stops upon failure of the block and the simulation moves to the next phase along the success path, you would use a node block to model this configuration, as shown next.

[[Image:Phase node block.png|center|450px|link=]]

*If the operational phase continues for the specified duration despite failure and the simulation then moves to the next phase along the success path, you would simply not create a failure path.
:*If there is only one path, the success path observed for a phase, then on system failure at this phase, the "continue simulation" rule of BlockSim 7 applies. Under "continue simulation," when a system failure occurs, repairs begin as per the repair policy selected and the time to restore the system is part of the operational phase's time. In other words, the repairs continue in the operational phase until the system is up again. If the repairs are not completed before the phase ends, the repairs continue into the next phase. Thus, under this rule the duration of an operational phase is not affected by a system failure. As an example of this rule, consider a production line operating in two phases: a day shift and a night shift. A failure occurs in the day shift that renders the production line non-operational. Repair of the production line begins immediately and continues beyond the day shift. The production line is back up after midnight. In this case, the repair of the production line exhausts all of the duration of the day shift phase from the time of the failure to the end of the phase. Some part of the night shift phase is also exhausted.

Node blocks can have unlimited incoming connections and a single outgoing connection.

=== Stop Blocks ===

Stop blocks indicate that the simulation of the mission ends. A new simulation may then begin, if applicable. This is useful in situations where maintenance is not possible upon failure.

[[Image:Phase stop block.png|center|450px|link=]]

Stop blocks can have unlimited incoming connections. No outgoing connections can be defined for stop blocks.

When a path leads to a stop node, it is the same as the option "Start New Simulation" in BlockSim 7, which would halt the simulation and effectively means the end of the mission if the system fails. Specifically, if a failure path leads to a stop node the execution of the current operational phase and all phases that follow the current phase is halted, and the mission aborted. The stop node can be used to model a system whose failure cannot be repaired and the mission has to be aborted if a failure occurs. A good example of this would be the aircraft case discussed previously. A catastrophic failure during cruising would end the mission.

= Subdiagram Phase Blocks =

Subdiagram phase blocks represent other phase diagrams within the project. Using subdiagram phase blocks allows you to incorporate phase diagrams as phases within other phase diagrams. This allows you to break down extremely complex configurations into smaller diagrams, increasing understandability and ease of use and avoiding unnecessary repetition of elements.

Subdiagram phase blocks can have unlimited incoming connections and up to two outgoing connections, which may include one success path and one failure path. The success path and the failure path must be different; if both success and failure of the block actually lead to the same outcome, you can use a node block to model this configuration.
{{:Example Using Subdiagram Phase Blocks}}

=Example: Aircraft Phases with Forced Landing=
{{:BlockSim Example: Aircraft Phases with Forced Landing}}

= Simulation Rules and Assumptions =

=== When Transferring Interrupted Maintenance Tasks Across Phases ===

Maintenance tasks in progress during one operational phase can be interrupted if that phase ends before the repair is completed. For example, a crew delay or spare parts order may extend the duration of a repair beyond the duration of the phase. As described next, the software handles these interruptions differently, based on the stage in which the repair was interrupted and whether or not the failed block is present in the next contiguous phase.

::1. If a phase ends during the repair of a failed block and the block is present in the next contiguous phase:
:::a) If the same task is present in both phases, then the task will continue as-is in the next phase. This is considered an uninterrupted event, and counts as a single unique event at both the block and the system level.
:::b) If the interrupted task is not used in the next phase, then the task is cancelled and new tasks are applied as needed. In this case, all crew calls are cancelled and spare parts are restocked.
::::1) If the repair has started or the crew is delayed (crew logistic delay), the call will be assumed accepted and the component will be charged for it. If the crew was occupied with another component’s repair, the call will be assumed rejected and hence not charged to the component.
::::2) If the call for spare parts incurred emergency charges, those are charged to the block; otherwise, there are no other charges to the block.

::2. If a phase ends during the repair of a failed block and the block is not present in the next contiguous phase, then the task is cancelled and new tasks are applied as needed. All crew calls are cancelled and spare parts are restocked:
:::a) If the repair has started or the crew is delayed (crew logistic delay), the call will be assumed accepted and the component will be charged for it. If the crew was occupied with another component’s repair, the call will be assumed rejected and hence not charged to the component.
:::b) If the call for spare parts incurred emergency charges, those are charged to the block; otherwise, there are no other charges to the block.
:::c) Discontinuous events are counted as two distinct events at both the block and the system level.
:::d) When the system fails in a phase that has a failure path leading to a stop block, the system will remain down for the remainder of the simulation. From that point on, the blocks that are down are assumed unavailable and the blocks that are up are assumed operational for availability calculations.

=== For Stop Blocks ===

When a system failure occurs in a phase where the failure path points to a stop block, the simulation is aborted. Once this failure occurs, the following assumptions apply to the results: 

:• Components that are under repair or maintenance remain down and unavailable for the rest of the simulation.
:• Components that are operating remain up for the rest of the simulation.

=== For a Maintenance Phase ===

A system is considered down and unavailable during the execution of a maintenance phase and remains down until all components have been repaired or maintained according to the properties specified for the maintenance phase. A maintenance phase is executed when the simulation reaches the phase while progressing through the phase diagram, either following a success path or a failure path. The following assumptions apply to both cases.

::1. When a component enters a maintenance phase in a down state, the following rules apply:
:::a) If a task is in progress for this component, the event will transfer to the maintenance phase provided that the same task is present in the maintenance phase. The rules for interrupted tasks apply as noted above.
:::b) If the component is failed but no corrective maintenance is in progress (either because the component was non-repairable in the phase where it failed or because it had a task scheduled to be executed upon inspection and was waiting for an inspection), a repair is initiated according to the corrective maintenance properties specified for the component in the maintenance phase.
:::c) Failed components are fixed in the order in which they failed.
::2. When a component enters a maintenance phase in an operating state, the following rules apply:
:::a) Maintenance will be scheduled as follows:
::::1) Tasks based on intervals or upon start of a maintenance phase
::::2) Tasks based on events in a maintenance group, where the triggering event applies to a block
::::3) Tasks based on system down
::::4) Tasks based on events in a maintenance group, where the triggering event applies to a subdiagram
:::Within these categories, order is determined according to the priorities specified in the maintenance template (i.e., the higher the task is on the list, the higher the priority).

:::b) An inspection or preventive task may be initiated, if applicable, with inspections taking precedence over preventive tasks. Inspections and/or preventive tasks are initiated if one of the following applies:
::::1) Upon certain events:
:::::a) The task is set to be performed when a maintenance phase starts.
:::::b) The policy is set to be performed based on events in a maintenance group and one of those events occurs within the one of the specified maintenance groups. Note that such a triggered maintenance does not follow the priorities specified in the maintenance template, but is sent to the end of the queue for repair.
:::::c) The task is set to be performed whenever the system is down.
::::2) At certain intervals:
:::::a) The task is set to be performed at a fixed time interval, based on either item age or calendar time, and the maintenance falls within the maintenance threshold specified in the maintenance phase.

:::If the inspection task is not set to bring either the item or the system down, the inspection will still be considered a downing inspection.

Finally, if a block enters a maintenance phase in a failed state:

::1. If the block does not have a corrective task in the maintenance phase but does have an on condition task, the preventive portion of the on condition task is triggered immediately in order to restore the block.
::2. A maintenance phase will not end until all components are restored. Therefore, if any failed block does not have a task that restores it, the maintenance phase will not end.

= Phase Throughput =

Phase throughput is the maximum number of items that a system can process during a particular phase. It is defined at the phase level as a phase property in an operational phase. For a detailed discussion of throughput at the block level see [[Additional_Analyses|Throughput Analysis]]. Phase throughput can be thought of as the initial throughput that enters the system. For example, imagine a textile factory that receives different quantities of raw materials during different seasons. These seasons could be treated as different phases. In this case a phase may be seen as sending a certain quantity of units to the first component of the system (the textile factory in this case). Depending on the capacity and availability of the factory, these units may be all processed or a backlog may accumulate.

Alternatively, phase throughput can be used as a constraint to the throughput of the system. An example would be the start up period in a processing plant. When the plant stops operating, the equipment requires a warm up period before reaching its maximum production capacity. In this case the phase throughput may be used to limit the capacity of the first component which in turn would limit the throughput of the rest of the system. Note that there is no phase-related backlog for this example. In BlockSim this can be modeled by checking the '''Ignore backlog''' option in the Block Properties window for the first component.

Phase throughput can be one of the following:
:*'''Unrestricted throughput''' specifies that the amount of output the system can process during the phase will be equal to the amount that the linked diagram can process during the specified phase duration.
:*'''Constant throughput''' allows you to specify an amount of throughput per unit time that is the same throughout the entire simulation.
:*'''Variable throughput''' allows you to specify an amount of throughput per unit time that varies over the course of the simulation. For example, the flow of oil from a well may drop over time as the oil reserves are depleted so the amount of throughput per unit time will decrease as the simulation time increases.

== Constant Throughput Example ==
{{:Constant Throughput Example}}

== Variable (Time-Varying) Throughput ==

Time-varying throughput can be specified at the phase level through the Variable Throughput property of an operational phase. Variable throughput permits modeling of scenarios where the throughput changes over time. For variable throughput, three general models are available in BlockSim. Each of these models has two parameters ''a'' and ''b'' which are specified by the user. These models are discussed below: 

:1. Linear model:

::<math>\begin{align}
y=ax+b
\end{align}\,\!</math>

:This model describes the change in throughput <math>y\,\!</math> as a linear function of time <math>x\,\!</math>. Throughput processed between any two points of time <math>{{x}}_{{1}}\,\!</math> and <math>{{x}}_{{2}}\,\!</math> is obtained by integration of the linear function as:

::<math>\begin{align}
\text{Linearly varying throughput}= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}ydx \\
= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}(ax+b)dx \\
= & \frac{a}{2}(x_{2}^{2}-x_{1}^{2})+b({{x}_{2}}-{{x}_{1}})
\end{align}\,\!</math>

:2. Exponential model:

::<math>\begin{align}
y={be}^{ax}
\end{align}\,\!</math>

:This model describes the change in throughput <math>y\,\!</math> as an exponential function of time <math>x\,\!</math>. Throughput processed in a period of time between any two points <math>{{x}}_{{1}}\,\!</math> and <math>{{x}}_{{2}}\,\!</math> is obtained as:

:<math>\begin{align}
\text{Exponentially varying throughput}= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}ydx \\
= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}b{{e}^{ax}}dx \\
= & \text{ }\frac{b}{a}({{e}^{a{{x}_{2}}}}-{{e}^{a{{x}_{1}}}})
\end{align}\,\!</math>
:3. Power model:

::<math>\begin{align}
y={bx}^{a}
\end{align}\,\!</math>

:This model describes the change in throughput <math>y\,\!</math> as a power function of time <math>x\,\!</math>. Throughput processed between two points of time <math>{{x}}_{{1}}\,\!</math> and <math>{{x}}_{{2}}\,\!</math> is obtained as:

::<math>\begin{align}
\text{Power varying throughput}= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}ydx \\
= & \mathop{}_{{{x}_{1}}}^{{{x}_{2}}}b{{x}^{a}}dx \\
= & \frac{b}{a+1}(x_{2}^{a+1}-x_{1}^{a+1})
\end{align}\,\!</math>

All of the above models also have a user defined maximum throughput capacity value. Once this maximum throughput capacity value is reached, the throughput per unit time becomes constant and equal in value to the maximum throughput capacity specified by the user. In this situation, the variable throughput model would then act as a constant throughput model. The above models may at first glance seem limited, when in fact they do provide ample modeling flexibility. This flexibility is achieved by using these functions as building blocks for more complex functions. As an example, a step model can be easily created by using multiple phases, each with a constant throughput. A ramp model would use phases with linearly increasing functions in conjunction with constant phases, and so forth.

==Variable Throughput Example==
{{:Variable Throughput Example}}

BlockSim Example: Default ON unless SCT Overridden

2021-03-26T18:19:45Z

Cruz Daniel: /* Example 1 */

<noinclude>{{Banner BlockSim Examples}}
</noinclude>
The examples illustrate, for BlockSim's [[State Change Triggers]], how selecting the '''Default ON unless SCT overridden''' option in the '''State upon repair''' field affects simulation. Two cases are presented.

=== Example 1 ===
This example is used to illustrate the following state change trigger options:

#State Upon Repair: Default ON unless SCT overridden 
#Activate a block if any item from these associated maintenance group(s) goes down 
#Deactivate a block if any item from these associated maintenance group(s) is restored

'''BlockSim Solution'''

Consider the system shown in the figure below:

[[Image:Primary and Standby.png|center|500px]]

*Block P is the primary device. It belongs to maintenance group P.
*Block S is the standby device. It has state change triggers. The initial state is OFF. If Block P goes down, then activate this block; if Block P is restored, then deactivate this block. The State Upon Repair is "Default ON unless SCT Overridden."
*Both Block P and Block S have a Weibull distribution with Beta = 1.5 and Eta = 100 for reliability and repair action.
*Both Block P and Block S are as good as new after repair.

'''Block Up/Down Plot'''

The system event log is shown in the figure below and explained next.

#At 123 hours, Block P fails and activates Block S.
#At 186 hours, Block S fails and is restored at 208 hours. According to the block's settings, it is ON upon repair.
#At 223 hours, Block S fails again.
#At 301 hours, Block P is restored, and put a request to deactivate Block S. However, Block S is down for repair at this point. The request overwrites the default state upon repair setting of Block S. Thus when Block S is done with repair at 385 hours, it is OFF.
#At 439 hours, Block P fails and activates Block S.
#At 523 hours, Block P is restored and deactivates Block S.
#At 694 hours, Block P fails and activates Block S.
#At 702 hours, Block S fails and the repair is finished at 775 hours. According to the block's settings, it is ON upon repair.
#At 788 hours, Block P is restored and deactivates Block S.
#At 845 hours, Block P fails and activates Block S.

[[Image:Block up down plot for primary and standby example.png|center|600px]]

=== Example 2 ===

This example illustrates the following state change trigger options:

#State Upon Repair: Default ON unless SCT overridden
#Deactivate a block if any item from these associated maintenance group(s) goes down
#Activate a block if any item from these associated maintenance group(s) is restored

'''BlockSim Solution'''

Consider the system is shown in the figure below.

*Block A fails every 350 hours and the duration for the repair action is 100 hours. It belongs to maintenance group A.
*Block B cannot fail. It has state change triggers. The initial state is ON, and the state upon trigger is "Default On unless SCT overridden." If any item from maintenance group A goes down, this block is deactivated; if any item from maintenance group A is restored, this block is activated.
*Block C fails every 300 hours and the duration for the repair action is 100 hours. It is has state change triggers. The initial state is ON, and the state upon trigger is "Default On unless SCT overridden." If any item from maintenance group A goes down, this block is deactivated; if any item from maintenance group A is restored, this block is activated.
*Block D fails every 330 hours and the duration for repair action is 150 hours . It has state change triggers. The initial state is ON, and the state upon trigger is "Default On unless SCT overridden." If any item from maintenance group A goes down, this block is deactivated; if any item from maintenance group A is restored, this block is activated.
*Blocks A, C and D all are as good as new after repair.

[[Image:One way dependence.png|center|500px]]

'''Block Up/Down Plot'''

The system event log, for a duration of 1,200 hours, is shown in the figure below and explained next

#At 300 hours and at 330 hours, Blocks C and D fail respectively. 
#At 350 hours, Block A fails. This brings the system down, turns Block B OFF and also puts a request to turn Block C OFF. However, Block C is down for repair at this time, thus after the repair on Block C is finished, the default setting of Block C (default ON unless overridden) will be overridden and it will stay OFF upon repair.
#At 450 hours, Block A is restored, which activate Blocks B and C. Block D is still down for repair at this point, so nothing happens to Block D.
#At 480 hours, Block D is restored. According to its settings, Block D is ON upon repair.
#At 750 hours, Block C fails.
#At 800 hours, Block A fails, and deactivates Blocks B and D. It also makes a request to turn Block C OFF. However, Block C is down for repair at this point, so the state upon repair of Block C is overridden. Thus when repair on Block C is finished at 850 hours, it stays OFF.
#At 900 hours, Block A is restored, and activates Blocks B, C and D.
#From 910 hours to 1060 hours, Block D fails and is repaired. There is no trigger in this period, so it is ON upon repair.

[[Image:Block up down plot for one way dependence example.png|center|600px]]

Degradation Data Analysis

2021-03-23T17:31:00Z

Cruz Daniel: /* Using Extrapolated Intervals */

{{template:LDABOOK|21|Degradation Data Analysis}}
Given that products are more frequently being designed with higher reliability and developed in a shorter amount of time, it is often not possible to test new designs to failure under normal operating conditions. In some cases, it is possible to infer the reliability behavior of unfailed test samples with only the accumulated test time information and assumptions about the distribution. However, this generally leads to a great deal of uncertainty in the results. Another option in this situation is the use of degradation analysis. Degradation analysis involves the measurement of performance data that can be directly related to the presumed failure of the product in question. Many failure mechanisms can be directly linked to the degradation of part of the product, and degradation analysis allows the analyst to extrapolate to an assumed failure time based on the measurements of degradation over time.

In some cases, it is possible to directly measure the degradation of a physical characteristic over time, as with the wear of brake pads, the propagation of crack size, or the degradation of a performance characteristic over time such as the voltage of a battery or the luminous flux of an LED bulb. These cases belong to the Non-Destructive Degradation Analysis category. In other cases, direct measurement of degradation might not be possible without invasive or destructive measurement techniques that would directly affect the subsequent performance of the product; therefore, only one degradation measurement is possible. Examples are the measurement of corrosion in a chemical container or the strength measurement of an adhesive bond. These cases belong to the Destructive Degradation Analysis Category. In either case, however, it is necessary to be able to define a level of degradation or performance at which a failure is said to have occurred.

With this failure level defined, it is a relatively simple matter to use basic mathematical models to extrapolate the measurements over time to the point where the failure is said to occur. Once these have been determined, it is merely a matter of analyzing the extrapolated failure times in the same manner as conventional time-to-failure data.

==Non-Destructive Degradation Analysis==
The Non-Destructive Degradation Analysis applies to cases where multiple degradation measurements over time can be obtained for each sample in the test. Given a defined level of failure (or the degradation level that would constitute a failure), basic mathematical models are used to extrapolate the degradation measurements over time of each sample to the point in time where the failure will occur. Once these extrapolated failure times are obtained, it is merely a matter of analyzing the extrapolated failure times in the same manner as conventional time-to-failure data. As with conventional life data analysis, the amount of certainty in the results is directly related to the number of samples being tested. The following figure combines the steps of the analysis by showing the extrapolation of the degradation measurements to a failure time and the subsequent distribution analysis of these failure times.

[[Image:Non-Destructive Degradation.png|center|500px|]]

===Non-Destructive Degradation Models===
Once the degradation information has been recorded, the next task is to extrapolate the measurements to the defined failure level in order to estimate the failure time. Weibull++ allows the user to perform such extrapolation using a linear, exponential, power or logarithmic model. These models have the following forms:

:* Linear: <math>y=a\cdot x+b \!</math>
:* Exponential: <math>y=b\cdot {{e}^{a\cdot x}}\!</math>
:* Power: <math>y=b\cdot {{x}^{a}} \!</math>
:* Logarithmic: <math>y=a\cdot ln(x)+b \!</math>
:* Gompertz: <math>y=a\cdot {{b}^{{{c}^{x}}}} \!</math>
:* Lloyd-Lipow: <math>y=a-\frac{b}{x} \!</math>

where <math>y\,\!</math> represents the performance, <math>x\,\!</math> represents time, and <math>a,\,\!</math> <math>b\,\!</math> and <math>c\,\!</math> are model parameters to be solved for.

Once the model parameters <math>{{a}_{i}}\,\!</math>, <math>{{b}_{i}}\,\!</math> and <math>{{c}_{i}}\,\!</math> are estimated for each sample <math>i\,\!</math>, a time <math>{{x}_{i}}\,\!</math> can be extrapolated, which corresponds to the defined level of failure <math>y\,\!</math>. The computed <math>{{x}_{i}}\,\!</math> values can now be used as our times-to-failure for subsequent life data analysis. As with any sort of extrapolation, one must be careful not to extrapolate too far beyond the actual range of data in order to avoid large inaccuracies (modeling errors).

====Example====
'''Crack Propagation Example (Point Estimation)'''

Five turbine blades are tested for crack propagation. The test units are cyclically stressed and inspected every 100,000 cycles for crack length. Failure is defined as a crack of length 30mm or greater. The following table shows the test results for the five units at each cycle:

::<math>\begin{matrix}
Cycles (x1000) & Unit A (mm)& Unit B (mm) & Unit C (mm) & Unit D (mm)& Unit E (mm) \\
100 & 15 & 10 & 17 & 12 & 10 \\
200 & 20& 15 & 25 & 16 & 15 \\
300 & 22 & 20 &26 & 17 & 20 \\
400 & 26 &25 & 27 & 20 & 26 \\
500 & 29 & 30 & 33 &26 & 33 \\
\end{matrix}\,\!</math>

Use the exponential degradation model to extrapolate the times-to-failure data.

'''Solution'''

The first step is to solve the equation <math>y=b\cdot {{e}^{a\cdot x}}\,\!</math> for <math>a\,\!</math> and <math>b\,\!</math> for each of the test units. Using regression analysis, the values for each of the test units are:

::<math>\begin{matrix}
{} & a & b \\
Unit A & 0.00158 & 13.596 \\
Unit B & 0.00271 & 8.272 \\
Unit C & 0.00140 & 16.435 \\
Unit D & 0.00177 & 10.361 \\
Unit E & 0.00294 & 7.931 \\
\end{matrix}\,\!</math>

Substituting the values into the underlying exponential model, solve for <math>x\,\!</math> or:

::<math>x=\frac{\text{ln}(y)-\text{ln}(b)}{a}\,\!</math>

Using the values of <math>a\,\!</math> and <math>b\,\!</math>, with <math>y=30\,\!</math>, the resulting time at which the crack length reaches 30mm can then found for each sample:

::<math>\begin{matrix}
{} & Cycles-to-Failure \\
Unit A & \text{500,622} \\
Unit B & \text{475,739} \\
Unit C & \text{428,739} \\
Unit D & \text{600,810} \\
Unit E & \text{452,832} \\
\end{matrix}\,\!</math>

These times-to-failure can now be analyzed using traditional life data analysis to obtain metrics such as the probability of failure, B10 life, mean life, etc. This analysis can be automatically performed in the Weibull++ degradation analysis folio.

<div class="noprint">
{{Examples Box|Weibull++ Examples|More degradation analysis examples are available! See also:
{{Examples Both|http://www.reliasoft.com/Weibull/examples/rc4/index.htm|Degradation Analysis|http://www.reliasoft.tv/weibull/appexamples/weibull_app_ex_4|Watch the video...}}<nowiki/>
}}
</div>

===Using Extrapolated Intervals===
The parameters in a degradation model are estimated using available degradation data. If the data is large, the uncertainty of the estimated parameters will be small; otherwise, the uncertainty will be large. Since the failure time for a test unit is predicted based on the estimated model, we sometimes would like to see how the parameter uncertainty affects the failure time prediction. Let’s use the exponential model as an example. Assume that the critical degradation value is <math>{{y}_{crit}}\,\!</math>. The predicted failure time will be:

::<math> \hat{x}=\frac{\ln ({{y}_{crit}})-\ln (\hat{b})}{{\hat{a}}}\,\!</math>

The variance of the predicted failure time will be:

::<math>Var(\hat{x})={{\left( \frac{\partial x}{\partial a} \right)}^{2}}Var(\hat{a})+{{\left( \frac{\partial x}{\partial b} \right)}^{2}}Var(\hat{b})+2\left( \frac{\partial x}{\partial a} \right)\left( \frac{\partial x}{\partial b} \right)Cov(\hat{a},\hat{b})\,\!</math>

The variance and covariance of the model parameters are calculated from using Least Squares Estimation. The details of the calculation are not given here.

The 2-sided upper and lower bounds for the predicted failure time, with a confidence level of <math>1-\alpha \,\!</math> are:

::<math>{{x}_{U}}=\hat{x}+{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>
::<math>{{x}_{L}}=\hat{x}-{{K}_{1-\alpha /2}}\sqrt{Var(\hat{x})}\,\!</math>

In Weibull++, the confidence level used is 90%.

====Example====
'''Crack Propagation Example (Extrapolated Intervals)'''

Using the same data set from the previous example, predict the interval failure times for the turbine blades.

'''Solution'''

In the Weibull++ degradation analysis folio, select the '''Use extrapolated intervals''' check box, as shown next.

[[Image:Check Use Extrapolated Intervals.png|center|250px| ]]

Use the exponential degradation model for the degradation analysis, and the Weibull distribution parameters with MLE for the life data analysis. The following report shows the estimated degradation model parameters.

[[Image:Model Parameters and Stds.png|center|500px| ]]

The following report shows the extrapolated failure time intervals.

[[Image:Extraploated Failure Time Intervals.png|center|500px| ]]

==Destructive Degradation Analysis==
The Destructive Degradation Analysis applies to cases where the sample has to be destroyed in order to obtain a degradation measurement. As a result, degradation measurements for multiple samples are required at different points in time. The analysis performed is very similar to Accelerated Life Testing Analysis (ALTA). In this case, the "stress" used in ALTA becomes time, while the random variable becomes the degradation measurement instead of the time-to-failure. Given a defined level of failure (or the degradation level that would constitute a failure), the probability that the degradation measurement will be beyond that level at a given time can be obtained. The following plot shows the relationship between the distribution of the degradation measurement and time. The red shaded area of the last two pdfs represents the probability that the degradation measurement will be less than the critical degradation level at the corresponding times.

[[Image:Destructive_Degradation_1.1.png|center|500px| ]]

===Destructive Degradation Models===
The first step of destructive degradation analysis involves using a statistical distribution to represent the variability of a degradation measurement at a given time. The following distributions can be used:

:*Weibull
:*Exponential
:*Normal
:*Lognormal
:*Gumbel

Similar to accelerated life testing analysis, the assumption is that the location or log-location parameter of the degradation measurement distribution will change with time while the shape parameter will remain constant. For each distribution:

:* Weibull: <math>\ln (\eta )\,\!</math> is set as a function of time while <math>\beta \,\!</math> remains constant.
:* Exponential: <math>\ln (MTTF)\,\!</math> is set as a function of time.
:* Normal: <math>\mu \,\!</math> is set as a function of time while <math>\sigma \,\!</math> remains constant.
:* Lognormal: <math>{\mu }'\,\!</math> is set as a function of time while <math>{\sigma }'\,\!</math> remains constant.
:* Gumbel: <math>\mu \,\!</math> is set as a function of time while <math>\sigma \,\!</math> remains constant.

Finally, given the selected distribution, a degradation model is used to represent the change of the location (or log-location) parameter with time. The following degradation models can be used:

:* Linear: <math>\mu (t)=b+a\times t \,\!</math>
:* Exponential: <math>\mu (t)=b\times {{e}^{a\times t}}\,\!</math>
:* Power: <math>\mu (t)=b\times {{t}^{a}}\,\!</math>
:* Logarithm: <math>\mu (t)=a\times \ln (t)+b\,\!</math>
:* Lloyd-Lipow: <math>\mu (t)=a-\frac{b}{t}\,\!</math>

The distribution and degradation models parameters are then calculated using Maximum Likelihood Estimation (MLE).

For example, if a normal distribution is used to represent the degradation measurement and a linear degradation model is assumed, then the standard deviation, <math>\sigma \,\!</math>, will be assumed constant with time, and the mean, <math>\mu \,\!</math>, will be:

:: <math>\mu (t)=b+a\times t\,\!</math>

The CDF of the degradation measurement <math>x(t)\,\!</math> is:

:: <math>\Pr (x(t)<X)=\Phi \left( \frac{X-\mu (t)}{\sigma } \right)\,\!</math>

Given the CDF, the parameters <math>\sigma \,\!</math>, <math>b\,\!</math> and <math>a\,\!</math> are estimated using Maximum Likelihood Estimation.

Assuming that the requirement is that the measurement needs to be greater than a critical degradation value <math>{{D}_{crit}}\,\!</math> for the product to fail, the probability of failure at time <math>t\,\!</math> will be:

:: <math>F(t)=\Pr (x(t)>{{D}_{crit}})=1-\Phi \left( \frac{{{D}_{crit}}-\mu (t)}{\sigma } \right)\,\!</math>

It should be noted that the failure threshold could be specified as a degradation measurement less than (for the case of decreasing degradation) or greater than (for the case of increasing degradation) a critical degradation value.

The relationship between the distribution of degradation measurement and the distribution of failure time is illustrated in the following plot.

[[Image:Destructive_Degradation_2.png|center|500px| ]]

====Example====
A company has been collecting degradation data over a period of 4 years with the purpose of calculating reliability after 5 years. With the degradation measurement decreasing with time, failure is defined as a measurement of 150 or below. In order to obtain such measurements, the unit has to be destroyed and therefore removed from the population.

The following table gives the degradation measurements over 4 years.

::<math>\begin{matrix}
Year 1 & Year 2 & Year 3 & Year 4 & \\
437 & 412 & 246 & 125 \\
446 & 420 & 324 & 208 \\
497 & 451 & 330 & 229 \\
503 & 454 & 426 & 242 \\
705 & 554 & 499 & 273 \\
737 & 580 & 546 & 297 \\
748 & 608 & 554 & 311 \\
788 & 610 & 559 & 318 \\
818 & 727 & 625 & 393 \\
860 & 825 & & 403 \\
875 & 925 & & 470 \\
934 & & & \\
1124 & & & \\
1250 & & & \\
1350 & & & \\
\end{matrix}\,\!</math>

Do the following:
# Estimate parameters using the Linear degradation model and the 2-Parameter Weibull distribution
# Plot the degradation curve vs. time
# Calculate the reliability at 5 years.

'''Solution'''

Enter the data into a destructive degradation folio in Weibull++. Select '''Linear''' under the Degradation Model and '''2P-Weibull''' under the Measurement Distribution. Set the critical degradation to '''150'''. Click '''Calculate'''.

:1. The estimated parameters are <math>\hat{\beta }=3.618336\,\!</math>, <math>\hat{a}=-0.331695\,\!</math> and <math>\hat{b}=7.154643\,\!</math>.
:2. The following plot shows the degradation curve vs. time

[[Image:Destructive_Degradation_Example_Plot1.png|center|550px| ]]

:3. Using the QCP, the reliability at 5 years, or in other words the probability that the degradation measurement will be less than 150 at 5 years, is:

[[Image:Destructive_Degradation_Example_QCP.png|center|450px| ]]