# Calculation of the Dangerous Failure Rate of the Safety Function
## 2 CTMCs with Multiple Absorption States  
Let us consider a stochastic process that fulfils the Markov property (the probability of moving to the next state depends only on the present state and not on the previous states):  
\[Pr\{X(t)\leq x|\ X(t_{0})=x_{0},\ X(t_{1})=x_{1},\ \ldots,\ X(t_{n})=x_{n}\}=Pr \{X(t)\leq x|\ X(t_{n})=x_{n}\} \tag{1}\]  
where \(X(t)\) is the random variable, \(t\in T\) (\(T\) is the time range) is the time parameter and is valid, and \(0\leq t_{0}<t_{1}<\ldots<t_{n}<t\).  
If the value, which is acquired by \(X(t)\), is called state and if a set of states is countable, then the Markov process forms the Markov chain (MC). We distinguish two basic types of the MC:  
* discrete-time Markov chain;
* continuous-time Markov chain.  
For the homogenous CTMC the transition probability of the system from state \(i\) to state \(j\) can be calculated as the conditional probability that the system in time \(t=t+\Delta t\) goes to state \(j\) under the condition that the system in time \(t\) was in state \(i\), i.e.,  
\[p_{ij}(t+\Delta t)=Pr\{X(t+\Delta t)=j|\ X(t)=i\}. \tag{2}\]  
The CTMC is completely defined if the transition rate matrix (3) and the initial distribution (4) are defined. The transition rate matrix:  
\[\mathbb{Q}=\big{(}q_{ij}\big{)}\ \text{for}\ i,j\in\{1,\ \ldots,n\}, \tag{3}\]  
where \(q_{ij}\) is the transition rate from state \(i\) to state \(j\) and \(q_{ii}=-\sum_{j=1,j\
eq i}^{n}q_{ij}\) is the sojourn rate in state \(i\). If the CTMC is homogeneous, the transition rates are constant. The initial distribution:  
\[\overset{\rightarrow}{P_{0}}=P_{0}(\overset{\rightarrow}{t=0})=\{p_{1}(t=0 ),\ p_{2}(t=0),\ \ldots,\ p_{n}(t=0)\} \tag{4}\]  
where \(\overset{\rightarrow}{P_{0}}\) is the initial distribution in time \(t=0\), \(p_{i}(t=0)\) is the probability of state \(i\) in time \(t=0\).  
The CTMC distribution in time \(t\) can be calculated as a solution of the differential equations system (5) for the initial distribution (4)  
\[\frac{dP(t)}{dt}=P(t)\cdot\mathbb{Q}. \tag{5}\]  
Figure 2 shows a CTMC that contains \((n-k+1)\) transient states and \((k-1)\) absorption states. \(q_{ij}\) is the intensity of the transition from state \(i\) to state \(j\).  
The CTMC in Figure 2 can be described by a general matrix of transition rates (6):  
\[\mathbb{Q}=\left(\begin{array}{ccccc}-\sum\limits_{l=1,j\
eq 1}^{n}q_{1l}&q_{12}& \ldots&q_{1(k-1)}&\ldots&q_{1n}\\ q_{21}&-\sum\limits_{l=1,j\
eq 2}^{n}q_{2l}&\ldots&q_{2(k-1)}&\ldots&q_{2n}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ q_{(k-1)1}&q_{(k-1)2}&\ldots&-\sum\limits_{l=1,j\
eq(k-1)}^{n}q_{(k-1)l}& \ldots&q_{(k-1)n}\\ 0&0&\ldots&0&\ldots&0\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ 0&0&\ldots&0&\ldots&0\end{array}\right), \tag{6}\]  
where the state \(i=1,\ \ldots,\ (k-1)\) is from the set of transition states \(T\) and state \(j=k,\ \ldots,\ n\) is from the set of absorption states \(A\). Because the article solves the problem of calculating the dangerous failure rate of SF, the symbol \(\lambda\) will be used instead of the symbol \(q\) for the transition rates in the next part of the article.