Template:Initialization time rga

Initialization Time

Reliability growth can only begin after a Type B failure mode occurs, which cannot be at a time equal to zero. Therefore, there is a need for an initialization time, different than zero, to be defined. The nominal idealized growth curve failure intensity is initially set equal to the initial failure intensity, [math]\displaystyle{ {{\lambda }_{I}}, }[/math] until the initialization time, [math]\displaystyle{ {{t}_{0}} }[/math] :

[math]\displaystyle{ {{r}_{NI}}({{t}_{0}})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)} }[/math]

Therefore:

[math]\displaystyle{ {{\lambda }_{I}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)} }[/math]

Then:

[math]\displaystyle{ {{t}_{0}}={{\left[ \frac{{{\lambda }_{I}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\lambda \beta } \right]}^{\tfrac{1}{\beta -1}}} }[/math]

Using Eqn. (lambda initial) to substitute [math]\displaystyle{ {{\lambda }_{I}} }[/math] we have:

[math]\displaystyle{ {{t}_{0}}={{\left[ \frac{{{\lambda }_{A}}+{{\lambda }_{B}}-{{\lambda }_{A}}-(1-d){{\lambda }_{B}}}{d\cdot \lambda \cdot \beta } \right]}^{\tfrac{1}{\beta -1}}} }[/math]

Then:

[math]\displaystyle{ {{t}_{0}}={{\left( \frac{{{\lambda }_{B}}}{\lambda \cdot \beta } \right)}^{\tfrac{1}{\beta -1}}} }[/math]

The initialization time, [math]\displaystyle{ {{t}_{0}}, }[/math] allows for growth to start after a Type B failure mode has occurred.