Template:Ipl weibull

IPL-Weibull

The IPL-Weibull model can be derived by setting [math]\displaystyle{ \eta =L(V) }[/math] , yielding the following IPL-Weibull [math]\displaystyle{ pdf\ \ : }[/math]

[math]\displaystyle{ f(t,V)=\beta K{{V}^{n}}{{\left( K{{V}^{n}}t \right)}^{\beta -1}}{{e}^{-{{\left( K{{V}^{n}}t \right)}^{\beta }}}} }[/math]

This is a three parameter model. Therefore it is more flexible but it also requires more laborious techniques for parameter estimation. The IPL-Weibull model yields the IPL-exponential model for [math]\displaystyle{ \beta =1. }[/math]

IPL-Weibull Statistical Properties Summary

Mean or MTTF

The mean, [math]\displaystyle{ \overline{T} }[/math] (also called [math]\displaystyle{ MTTF }[/math] ), of the IPL-Weibull model is given by:

[math]\displaystyle{ \overline{T}=\frac{1}{K{{V}^{n}}}\cdot \Gamma \left( \frac{1}{\beta }+1 \right) }[/math]

where [math]\displaystyle{ \Gamma \left( \tfrac{1}{\beta }+1 \right) }[/math] is the gamma function evaluated at the value of [math]\displaystyle{ \left( \tfrac{1}{\beta }+1 \right) }[/math] .

Median

The median, [math]\displaystyle{ \breve{T}, }[/math] of the IPL-Weibull model is given by:

[math]\displaystyle{ \breve{T}=\frac{1}{K{{V}^{n}}}{{\left( \ln 2 \right)}^{\tfrac{1}{\beta }}} }[/math]

Mode

The mode, [math]\displaystyle{ \tilde{T}, }[/math] of the IPL-Weibull model is given by:

[math]\displaystyle{ \tilde{T}=\frac{1}{K{{V}^{n}}}{{\left( 1-\frac{1}{\beta } \right)}^{\tfrac{1}{\beta }}} }[/math]

Standard Deviation

The standard deviation, [math]\displaystyle{ {{\sigma }_{T}}, }[/math] of the IPL-Weibull model is given by:

[math]\displaystyle{ {{\sigma }_{T}}=\frac{1}{K{{V}^{n}}}\cdot \sqrt{\Gamma \left( \frac{2}{\beta }+1 \right)-{{\left( \Gamma \left( \frac{1}{\beta }+1 \right) \right)}^{2}}} }[/math]

IPL-Weibull Reliability Function

The IPL-Weibull reliability function is given by:

[math]\displaystyle{ R(T,V)={{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}} }[/math]

Conditional Reliability Function

The IPL-Weibull conditional reliability function at a specified stress level is given by:

[math]\displaystyle{ R(T,t,V)=\frac{R(T+t,V)}{R(T,V)}=\frac{{{e}^{-{{\left[ K{{V}^{n}}\left( T+t \right) \right]}^{\beta }}}}}{{{e}^{-{{\left( K{{V}^{n}}T \right)}^{\beta }}}}} }[/math]

or:

[math]\displaystyle{ R(T,t,V)={{e}^{-\left[ {{\left( K{{V}^{n}}\left( T+t \right) \right)}^{\beta }}-{{\left( K{{V}^{n}}T \right)}^{\beta }} \right]}} }[/math]

Reliable Life

For the IPL-Weibull model, the reliable life, .. , of a unit for a specified reliability and starting the mission at age zero is given by:

[math]\displaystyle{ {{T}_{R}}=\frac{1}{K{{V}^{n}}}{{\left\{ -\ln \left[ R\left( {{T}_{R}},V \right) \right] \right\}}^{\tfrac{1}{\beta }}} }[/math]

IPL-Weibull Failure Rate Function

The IPL-Weibull failure rate function, [math]\displaystyle{ \lambda (T) }[/math] , is given by:

[math]\displaystyle{ \lambda \left( T,V \right)=\frac{f\left( T,V \right)}{R\left( T,V \right)}=\beta K{{V}^{n}}{{\left( K{{V}^{n}}T \right)}^{\beta -1}} }[/math]

Parameter Estimation

Maximum Likelihood Estimation Method

Substituting the inverse power law relationship into the Weibull log-likelihood function yields:

[math]\displaystyle{ \begin{align} \Lambda = \underset{i=1}{\overset{{{F}_{e}}}{\mathop \sum }}\,{{N}_{i}}\ln \left[ \beta KV_{i}^{n}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta -1}}{{e}^{-{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}}} \right] \\ -\underset{i=1}{\overset{S}{\mathop \sum }}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \\ +\overset{FI}{\mathop{\underset{i=1}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\ln [R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }] \end{align} }[/math]

where:

[math]\displaystyle{ R_{Li}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Li}^{\prime \prime } \right)}^{\beta }}}} }[/math]

[math]\displaystyle{ R_{Ri}^{\prime \prime }={{e}^{-{{\left( KV_{i}^{n}T_{Ri}^{\prime \prime } \right)}^{\beta }}}} }[/math]

and:

• [math]\displaystyle{ {{F}_{e}} }[/math] is the number of groups of exact times-to-failure data points.
• [math]\displaystyle{ {{N}_{i}} }[/math] is the number of times-to-failure data points in the [math]\displaystyle{ {{i}^{th}} }[/math] time-to-failure data group.
• [math]\displaystyle{ \beta }[/math] is the Weibull shape parameter (unknown, the first of three parameters to be estimated).
• .. is the IPL parameter (unknown, the second of three parameters to be estimated).
• [math]\displaystyle{ n }[/math] is the second IPL parameter (unknown, the third of three parameters to be estimated).
• [math]\displaystyle{ {{V}_{i}} }[/math] is the stress level of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ {{T}_{i}} }[/math] is the exact failure time of the [math]\displaystyle{ {{i}^{th}} }[/math] group.
• [math]\displaystyle{ S }[/math] is the number of groups of suspension data points.
• [math]\displaystyle{ N_{i}^{\prime } }[/math] is the number of suspensions in the .. group of suspension data points.
• [math]\displaystyle{ T_{i}^{\prime } }[/math] is the running time of the [math]\displaystyle{ {{i}^{th}} }[/math] suspension data group.
• [math]\displaystyle{ FI }[/math] is the number of interval data groups.
• [math]\displaystyle{ N_{i}^{\prime \prime } }[/math] is the number of intervals in the [math]\displaystyle{ {{i}^{th}} }[/math] group of data intervals.
• [math]\displaystyle{ T_{Li}^{\prime \prime } }[/math] is the beginning of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.
• [math]\displaystyle{ T_{Ri}^{\prime \prime } }[/math] is the ending of the [math]\displaystyle{ {{i}^{th}} }[/math] interval.

The solution (parameter estimates) will be found by solving for [math]\displaystyle{ \beta , }[/math] [math]\displaystyle{ K, }[/math] [math]\displaystyle{ n }[/math] so that [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial \beta }=0, }[/math] [math]\displaystyle{ \tfrac{\partial \Lambda }{\partial K}=0 }[/math] and .. , where:

[math]\displaystyle{ \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( KV_{i}^{n}{{T}_{i}} \right) \\ -\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }}\ln \left( KV_{i}^{n}{{T}_{i}} \right) \\ -\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( \,KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }}\ln \left( KV_{i}^{n}T_{i}^{\prime } \right) \\ \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{\left( KV_{i}^{n} \right)}^{\beta }}\left[ R_{Li}^{\prime \prime }T_{Li}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Li}^{\prime \prime }) \right)-R_{Ri}^{\prime \prime }T_{Ri}^{\prime \prime \beta }\left( \ln (KV_{i}^{n}T_{Ri}^{\prime \prime }) \right) \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \\ \frac{\partial \Lambda }{\partial K}= & \frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}-\frac{\beta }{K}\underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}{{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\ -\frac{\beta }{K}\underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }{{\left( KV_{i}^{n}T_{i}^{\prime } \right)}^{\beta }} \\ \overset{{}}{\mathop{-\beta \underset{i=1}{\mathop{\overset{FI}{\mathop{\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,}}\,N_{i}^{\prime \prime }\frac{{{K}^{\beta -1}}V_{i}^{n\beta }\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \end{align} }[/math] [math]\displaystyle{ \begin{align}\frac{\partial\Lambda }{\partial n}=\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}) \\ -\beta \underset{i=1}{\overset{{{F}_{e}}}{\mathop{\sum }}}\,{{N}_{i}}\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\ -\beta \underset{i=1}{\overset{S}{\mathop{\sum }}}\,N_{i}^{\prime }\ln ({{V}_{i}}){{\left( KV_{i}^{n}{{T}_{i}} \right)}^{\beta }} \\ \overset{FI}{\mathop{\underset{i=1}{\mathop{-\underset{}{\overset{}{\mathop \sum }}\,}}\,}}\,N_{i}^{\prime \prime }\frac{n{{K}^{\beta }}V_{i}^{\beta (n-1)}\left[ T_{Li}^{\prime \prime \beta }R_{Li}^{\prime \prime }-T_{Ri}^{\prime \prime \beta }R_{Ri}^{\prime \prime } \right]}{R_{Li}^{\prime \prime }-R_{Ri}^{\prime \prime }} \end{align} }[/math]