Template:Actual failure intensity function

Actual Failure Intensity Function
Consider a test plan consisting of $$i$$  phases. Taking into account the fix delay within each phase, we expect the actual failure intensity to be different (i.e. shifted) from the nominal failure intensity. This is because fixes are not incorporated instantaneously, thus growth is realized at a later time compared to the nominal case. Specifically, the actual failure intensity will be estimated as follows:

Test Phase 1 For the first phase of a test plan, the actual idealized curve failure intensity, $${{r}_{AI}}(t)$$, is $$:$$


 * $${{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}} \right)t \right]}^{(\beta -1)}}\text{ for }0{{L}_{1}}+{{t}_{0}}$$. The actual idealized curve initialization time for Phase 1, $$T_{0}^{AIC},$$  is calculated from:


 * $${{r}_{AI}}(T_{0}^{AIC})={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}$$

Where $${{r}_{AI}}(T_{0}^{AIC})={{r}_{I}}.$$ Therefore, using Eqn. (lambdaqualsominal):


 * $${{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ \left( \frac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)T_{0}^{AIC} \right]}^{(\beta -1)}}={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta t_{0}^{(\beta -1)}$$

Solving Eqn. (ActIhase1) for $$T_{0}^{AIC}$$  we get:


 * $$T_{0}^{AIC}=\frac{\left( \tfrac{{{T}_{1}}-{{L}_{1}}}{{{T}_{1}}} \right)}$$

Test Phase $$i$$ For any test phase $$i$$, the actual idealized curve failure intensity is given by:


 * $${{r}_{AI}}(t)={{\lambda }_{A}}+(1-d){{\lambda }_{B}}+d\lambda \beta {{\left[ {{T}_{i-1}}-{{L}_{i-1}}+\left( \frac{{{T}_{i}}-{{L}_{i}}-{{T}_{i-1}}+{{L}_{i-1}}}{{{T}_{i}}-{{T}_{i-1}}} \right)(t-{{T}_{i-1}}) \right]}^{(\beta -1)}}$$

where $${{T}_{i-1}}\le t\le {{T}_{i}}$$  and  $${{T}_{i}}$$  is the test time of each corresponding test phase. The actual idealized curve MTBF is:


 * $${{M}_{AI}}=\frac{1}{{{r}_{AI}}(t)}$$