## 3 The CTMC of the Braking System  
### System States Transition Process with Daily Maintenance  
The failing and daily maintenance processes of the braking system can be described by a multi-stage Markov chain. The system working time is \(\tau_{1}\) and the inspection and maintenance time is \(\tau_{2}\), then the life cycle of the system can be divided into intervals such as \([0,\tau_{1}]\), \([\tau_{1},\tau_{1}+\tau_{2}]\), \([\tau_{1}+\tau_{2},\,2\tau_{1}+\tau_{2}]\), etc. Each stage is similar to the finite pure death process (the maintenance stage can be considered as a reverse pure death process). Such processes can be described in terms of absorbing Markov chains. For absorbing Markov chains, the state probability will be in an absorbing state after some steps, regardless of which state they start from. In the model of this paper, both the failing process and the daily maintenance process can be described separately by absorbing Markov chains. The two Markov chains are interrelated, and the initial states of the latter stage are dependent on the final state of the former stage. The braking system starts with n brake motors normally intact, and the system is in a state in which only \(k\) brake motors are working properly before failing. Thus, the starting state of the Markov chain is the intact state \(n\), and the ending state is the pre-fault state \(k\). It is assumed that the system has a component failure rate of \(\lambda\) during the operating phase and a component repair rate of \(\mu\) during the preventive inspection and maintenance phase. Both the failing process and the daily maintenance process follow an exponential distribution for \(t\). For such a K-out-of-N: G system, the total number of states can be expressed as \(I=n-k+1\), which includes \(n-k+1\) operating states and one shutdown state. Therefore, the system state probability can be expressed as \(\overrightarrow{P_{t}}=[P_{t}(1),P_{t}(2),\cdots,P_{t}(I)]\). The initial state probability of the system \(\overrightarrow{P_{0}}\) can be expressed as follows:  
\[\overrightarrow{P_{0}}=[1,0,0,\cdots,0] \tag{1}\]  
As the braking system is in state \(i\), there are \(n-i+1\) brake motors operating normally. When one of the brake motors fails, the number of motors in normal operation is left at \(n-i\), and the system will transfer to state \(i+1\). This transfer process happens one by one. Thus, when the system is in the working phase, its state transfer rate is:  
\[c_{i\to i+1}=(n-k+2-i)\lambda \tag{2}\]  
Therefore, the Markov state transfer rate \(C=\left(c_{(i,j)}\right)_{(I\times I)}\) of the working phase can be expressed as:  
\[c_{i,j}=\left\{ \begin{array}{ll}-(n-k+2)\lambda&j=i<n-k+2\\ (n-k+2)\lambda&j=i+1<n-k+1\\ 0&others\end{array} \right. \tag{3}\]  
When one of the motors is repaired, the system will transfer from state \(i\) to state \(i-1\). As with the failure process, the transfer process happens one by one. Thus, when the system is under daily maintenance, the state transfer process is:  
\[m_{i\to i-1}=i\mu \tag{4}\]  
Therefore, the daily maintenance phase Markov state transfer matrix \(M=\left(m_{(i,j)}\right)_{(I\times I)}\) can be expressed as:  
\[m_{i,j}=\left\{ \begin{array}{ll}(n-k+2)\mu&i=j<n-k+2\\ -(n-k+2)\mu&i=j+1<n-k+1\\ 0&others\end{array} \right. \tag{5}\]When the system is in daily maintenance, the instantaneous state probabilities of the system at different moments can be expressed as \(\overrightarrow{P_{t}}\) which is obtained from the above state transfer matrix. The state probability of each phase depends on the state transfer matrix of that phase, and the state probability at the beginning of each phase is the state probability at the end of the previous phase. Therefore, when \(t\in[0,\tau_{1}]\), the system state probability can be expressed as:  
\[\overrightarrow{P_{t}}=\overrightarrow{P_{0}}\bullet exp(C\bullet t) \tag{6}\]  
When \(t\in[\tau_{1},\tau_{1}+\tau_{2}]\), the system enters the daily maintenance phase, and the start state of this phase is the end state of the previous working phase.