Template:Background of weibull distribution

Background

Consider a life test of identical components. The components were placed in a test at age [math]\displaystyle{ T=0 }[/math] and were tested to failure, with their times-to-failure recorded. Further assume that the test covered the entire lifespan of the units, and different failure modes were observed over each region of life, namely early life (early failure mode), chance life (chance failure mode), and wear-out life (wear-out failure mode). Also, as items failed during the test, they were removed from the test, inspected and segregated into lots according to their failure mode. At the conclusion of the test, there will be [math]\displaystyle{ n }[/math] subpopulations of [math]\displaystyle{ {{N}_{1}},{{N}_{2}},{{N}_{3}},...,{{N}_{n}} }[/math] failed components. If the events of the test are now reconstructed, it may be theorized that at age [math]\displaystyle{ T=0 }[/math] there were actually [math]\displaystyle{ n }[/math] separate subpopulations in the test, each with a different times-to-failure distribution and failure mode, even though at [math]\displaystyle{ T=0 }[/math] the subpopulations were not physically distinguishable. The mixed Weibull methodology accomplishes this segregation based on the results of the life test.

If [math]\displaystyle{ N }[/math] identical components from a mixed population undertake a mission of [math]\displaystyle{ T }[/math] duration, starting the mission at age zero, then the number of components surviving this mission can be found from the following definition of reliability:

[math]\displaystyle{ {{R}_{1,2,...,n}}(T)=\frac{{{N}_{1,2,3,..,{{n}_{S}}}}(T)}{N} }[/math]

Then:

[math]\displaystyle{ \begin{align} {{N}_{1,2,...,{{n}_{S}}}}(T)= & N[{{R}_{1,2,...,n}}(T)] \\ \\ {{N}_{{{1}_{S}}}}(T)=& {{N}_{1}}{{R}_{1}}(T);{{N}_{{{2}_{S}}}}(T)={{N}_{2}}{{R}_{2}}(T) \\ {{N}_{{{3}_{S}}}}(T)=& {{N}_{3}}{{R}_{3}}(T);...;{{N}_{{{n}_{S}}}}={{N}_{n}}{{R}_{n}}(T) \end{align} }[/math]

The total number surviving by age [math]\displaystyle{ T }[/math] in the mixed population is the sum of the number surviving in all subpopulations or:

[math]\displaystyle{ {{N}_{1,2,...,{{n}_{S}}}}(T)={{N}_{{{1}_{S}}}}(T)+{{N}_{{{2}_{S}}}}(T)+{{N}_{{{3}_{S}}}}(T)+\cdots +{{N}_{{{n}_{S}}}}(T) }[/math]

Substituting into Eqn. (rel) yields:

[math]\displaystyle{ {{R}_{1,2,...,n}}(T)=\frac{1}{N}[{{N}_{1}}{{R}_{1}}(T)+{{N}_{2}}{{R}_{2}}(T)+{{N}_{3}}{{R}_{3}}(T)+\cdots +{{N}_{n}}{{R}_{n}}(T)] }[/math]

or:

[math]\displaystyle{ {{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T) }[/math]

This expression can also be derived by applying Bayes' theorem [20], which says that the reliability of a component drawn at random from a mixed population composed of [math]\displaystyle{ n }[/math] types of failure subpopulations is its reliability, [math]\displaystyle{ {{R}_{1}}(T) }[/math] , given that the component is from subpopulation 1, or [math]\displaystyle{ \tfrac{{{N}_{1}}}{N} }[/math] plus its reliability, [math]\displaystyle{ {{R}_{2}}(T) }[/math] , given that the component is from subpopulation 2, or [math]\displaystyle{ \tfrac{{{N}_{2}}}{N} }[/math] plus its reliability, [math]\displaystyle{ {{R}_{3}}(T) }[/math] , given that the component is from subpopulation 3, or [math]\displaystyle{ \tfrac{{{N}_{3}}}{N} }[/math] , and so on, plus its reliability, [math]\displaystyle{ {{R}_{n}}(T) }[/math] , given that the component is from subpopulation [math]\displaystyle{ n }[/math] , or [math]\displaystyle{ \tfrac{{{N}_{n}}}{N} }[/math] , and:

[math]\displaystyle{ \underset{i=1}{\overset{n}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}=1 }[/math]

This may be written mathematically as:

[math]\displaystyle{ {{R}_{1,2,...,n}}(T)=\frac{{{N}_{1}}}{N}{{R}_{1}}(T)+\frac{{{N}_{2}}}{N}{{R}_{2}}(T)+\frac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{R}_{n}}(T) }[/math]

Other functions of reliability engineering interest are found by applying the fundamentals to Eqn. (rel1).

For example, the probability density function can be found from:

[math]\displaystyle{ \begin{align} {{f}_{1,2,...,n}}(T)= & -\frac{d}{dT}[{{R}_{1,2,...,n}}(T)] \\ {{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}\left( -\frac{d}{dT}[{{R}_{1}}(T)] \right)+\frac{{{N}_{2}}}{N}\left( -\frac{d}{dT}[{{R}_{2}}(T)] \right) \\ & +\ \ \frac{{{N}_{3}}}{N}\left( -\frac{d}{dT}[{{R}_{3}}(T)] \right)+\cdots +\frac{{{N}_{n}}}{N}\left( -\frac{d}{dT}[{{R}_{n}}(T)] \right) \\ {{f}_{1,2,...,n}}(T)= & \frac{{{N}_{1}}}{N}{{f}_{1}}(T)+\frac{{{N}_{2}}}{N}{{f}_{2}}(T) \\ & +\ \ \frac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\frac{{{N}_{n}}}{N}{{f}_{n}}(T) \end{align} }[/math]

Also, the failure rate function of a population is given by:

[math]\displaystyle{ \begin{align} {{\lambda }_{1,2,...,n}}(T)= & \frac{{{f}_{1,2,...,n}}(T)}{{{R}_{1,2,...,n}}(T)}, \\ {{\lambda }_{1,2,...,n}}(T)= & \frac{\tfrac{{{N}_{1}}}{N}{{f}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{f}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{f}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{f}_{n}}(T)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\tfrac{{{N}_{3}}}{N}{{R}_{3}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)}. \end{align} }[/math]

The conditional reliability for a new mission of duration [math]\displaystyle{ t }[/math] , starting this mission at age [math]\displaystyle{ T }[/math] , or after having already operated a total of [math]\displaystyle{ T }[/math] hours, is given by:

[math]\displaystyle{ \begin{align} {{R}_{1,2,...,n}}(T,t)= & \frac{{{R}_{1,2,...,n}}(T+t)}{{{R}_{1,2,...,n}}(T)} \\ {{R}_{1,2,...,n}}(T,t)= & \frac{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T+t)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T+t)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T+t)}{\tfrac{{{N}_{1}}}{N}{{R}_{1}}(T)+\tfrac{{{N}_{2}}}{N}{{R}_{2}}(T)+\cdots +\tfrac{{{N}_{n}}}{N}{{R}_{n}}(T)} \end{align} }[/math]

The Mixed Weibull Equations

Depending on the number of subpopulations chosen, Weibull++ uses the following equations for the reliability and probability density functions:

[math]\displaystyle{ {{R}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}}{N}{{e}^{-{{\left( \tfrac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}}}}} }[/math]

and:

[math]\displaystyle{ {{f}_{1,...,S}}(T)=\underset{i=1}{\overset{S}{\mathop \sum }}\,\frac{{{N}_{i}}{{\beta }_{i}}}{N{{\eta }_{i}}}{{\left( \frac{T}{{{\eta }_{i}}} \right)}^{{{\beta }_{i}}-1}}{{e}^{-{{(\tfrac{T}{{{\eta }_{i}}})}^{{{\beta }_{i}}}}}} }[/math]

where [math]\displaystyle{ S=2 }[/math] , [math]\displaystyle{ S=3 }[/math] , and [math]\displaystyle{ S=4 }[/math] for 2, 3 and 4 subpopulations respectively. Weibull++ uses a non-linear regression method or direct maximum likelihood methods to estimate the parameters.