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, $${{\lambda }_{I}},$$  until the initialization time,  $${{t}_{0}}$$ :


 * $${{r}_{NI}}({{t}_{0}})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}$$


 * Therefore:


 * $${{\lambda }_{I}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}$$


 * Then:


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

Using Eqn. (lambda initial) to substitute $${{\lambda }_{I}}$$  we have:


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


 * Then:


 * $${{t}_{0}}={{\left( \frac{{{\lambda }_{B}}}{\lambda \cdot \beta } \right)}^{\tfrac{1}{\beta -1}}}$$

The initialization time, $${{t}_{0}},$$  allows for growth to start after a Type B failure mode has occurred.