# Calculation of the Dangerous Failure Rate of the Safety Function
## 4 Rate of Dangerous Failure of a Safety Function for Dual Architecture with Two Absorption States
### Dangerous Failure Rate of the SF for Dual Architecture with Two Absorption States  
Based on Equations (13) and (14), for the CTMC in Figure 6 deduce that  
\[\lambda_{D}(t)=\frac{\frac{dp_{D}(t)}{dt}}{1-(p_{D}(t)+p_{S}(t))}=\frac{-2 \lambda(\lambda c-\delta)\cdot e^{-2\lambda t}-2\delta\cdot\lambda\cdot e^{-(1+ \epsilon)\lambda t}+2c\lambda^{2}\cdot e^{-(\lambda+\delta)t}}{-(\lambda c- \delta)\cdot e^{-2\lambda t}+2\lambda c\cdot e^{-(\lambda+\delta)t}-2\delta \cdot e^{-(1+c)\lambda t}}, \tag{24}\]  
\[\lambda_{S}(t)=\frac{dp_{S}(t)}{dt}=\frac{2c\lambda\delta\cdot e^{-(\lambda+ \delta)t}-2\delta c\lambda\cdot e^{-(1+\epsilon)\lambda t}}{-(\lambda c- \delta)\cdot e^{-2\lambda t}+2\lambda c\cdot e^{-(\lambda+\delta)t}-2\delta \cdot e^{-(1+c)\lambda t}}. \tag{25}\]  
By merging states 1, 2, and 3 into state X (Figure 6), the CTMC from Figure 6, can be replaced by CTMC that corresponds to the CTMC in Figure 4. CTMC with merged states is shown in Figure 7.  
For the probability of state X, it is valid that:  
\[p_{X}(t)=p_{1}(t)+p_{2}(t)+p_{3}(t)=\frac{-(\lambda c-\delta)\cdot e^{-2 \lambda t}+2\lambda c\cdot e^{-(\lambda+\delta)t}-2\cdot\delta\cdot e^{-(1+ \epsilon)\lambda t}}{(\lambda c-\delta)}, \tag{26}\]  
\[\frac{dp_{X}(t)}{dt}=\frac{2\lambda\cdot(\lambda c-\delta)\cdot e^{-2\lambda t }-2\lambda c\cdot(\lambda+\delta)e^{-(\lambda+\delta)t}+2\delta\cdot\lambda \cdot(1+c)\cdot e^{-(1+c)\lambda t}}{(\lambda c-\delta)} \tag{27}\]  
If the applied substitution is correct, then it must be true that:  
\[\frac{dp_{X}(t)}{dt}=-(\lambda_{D}(t)+\lambda_{S}(t))\cdot p_{X}(t) \tag{28}\]  
After inserting Equations (24)-(26) to Equation (28), we get:  
\[\begin{array}{l}-(\lambda_{D}(t)+\lambda_{S}(t))\cdot p_{X}(t)\\ =-\left(\frac{-2\lambda(\lambda c-\delta)\cdot e^{-2\lambda t}-2\delta\cdot \lambda\cdot e^{-(1+\epsilon)\lambda t}+2c\lambda^{2}\cdot e^{-(\lambda+ \delta)t}}{-(\lambda c-\delta)\cdot e^{-2\lambda t}+2\lambda c\cdot e^{-( \lambda+\delta)t}-2\delta\cdot e^{-(1+c)\lambda t}}\right.\\ \left.+\frac{2c\lambda\delta\cdot e^{-(\lambda+\delta)t}-2\delta(\lambda\cdot e ^{-(1+\epsilon)\lambda t}}{-(\lambda c-\delta)\cdot e^{-2\lambda t}+2\lambda c \cdot e^{-(1+\epsilon)\lambda t}-2\delta\cdot e^{-(1+\epsilon)\lambda t}} \right)\cdot\frac{-(\lambda c-\delta)\cdot e^{-2\lambda t}+2\lambda c\cdot e ^{-(\lambda+\delta)t}-2\cdot\delta\cdot e^{-(1+\epsilon)\lambda t}}{(\lambda c -\delta)}\\ =\frac{2\lambda(\lambda c-\delta)\cdot e^{-2\lambda t}+2\delta\cdot\lambda(1+c) \cdot e^{-(1+\epsilon)\lambda t}-2c\lambda(\lambda+\delta)\cdot e^{-(\lambda +\delta)t}}{(\lambda c-\delta)}=\frac{dp_{X}(t)}{dt}\end{array} \tag{29}\]  
Which confirms the correctness of this reasoning.  
By merging states S and D to state Y in the CTMC in Figure 7, a CTMC with two states is formed (Figure 8).  
If it is true that  
\[p_{Y}(t)=p_{D}(t)+p_{S}(t), \tag{30}\]  
then the rate of the transition from state X to state Y can be expressed by:  
Figure 7: Merging transient states of the model.  
\[\lambda_{Y}(t)=\frac{\frac{d(p_{D}(t)+p_{S}(t))}{dt}}{1-(p_{D}(t)+p_{S}(t))}= \frac{\frac{dp_{D}(t)}{dt}}{1-(p_{D}(t)+p_{S}(t))}+\frac{\frac{dp_{S}(t)}{dt}}{1 -(p_{D}(t)+p_{S}(t))}=\lambda_{D}(t)+\lambda_{S}(t) \tag{31}\]