## 4 RAM analysis of EPBM
### Availability Analysis
#### 4.1.2 Markov Method  
When using the Markov method, the system's behavior should be independent of memory. This means that the accident behavior of the system depends only on current data and is independent of its past behaviors. The assumptions for the Markov method are [25]:  
1. All system transition rates (i.e., failure and repair rates) are constant.
2. All occurrences are independent of each other.  
Figure 2: TBF and TTR of the machine for each subsystem.  
3. The probability of transition from one system state to another in the finite time interval dt is given by \(\lambda\) dt, where \(\lambda\) is the transition rate (e.g., system failure or repair rate) from one system state to another. iv. The probability of more than one transition occurrence in finite time interval dt from one system state to another is very small or negligible (e.g., (\(\lambda\) dt) (\(\lambda\) dt) \(\to\) 0). The first step is to develop the system transition diagram, including its subsystems, and the relation between those, as presented in Figure 3.  
From the transition diagram, a system transition matrix is formed, and the probability of each subsystem is calculated.  
\[\text{M}=\begin{bmatrix}X_{0}&\lambda_{C}&\lambda_{E}&\lambda_{M}\\ \mu_{C}&X_{C}&0&0\\ \mu_{E}&0&X_{E}&0\\ \mu_{M}&0&0&X_{M}\end{bmatrix} \tag{7}\]  
Since the total probability values of subsystems in a system should be equal to one, therefore:  
\[\text{X}_{0}\ =1-(\lambda_{\text{C}}+\lambda_{\text{E}}+\lambda_{\text{M}})=1- \sum\lambda_{i} \tag{8}\]  
\[\text{X}_{1}=1-\mu_{\text{C}} \tag{9}\]  
\[\text{X}_{2}=1-\mu_{\text{E}} \tag{10}\]  
\[\text{X}_{3}=1-\mu_{\text{M}} \tag{11}\]  
On the other hand, the probability matrix of the system can be presented as:  
\[\pi=\begin{bmatrix}P_{0}&P_{C}&P_{E}&P_{M}\end{bmatrix} \tag{12}\]  
Figure 3: Transition diagram of the EPBM and its subsystem.  
Regarding the fourth assumption of the Markov method:  
\[\pi\times M=\pi \tag{13}\]  
Or:  
\[\begin{bmatrix}P_{0}&P_{\mathrm{C}}&P_{E}&P_{M}\end{bmatrix}\begin{bmatrix}1- \sum\lambda_{i}&\lambda_{\mathrm{C}}&\lambda_{E}&\lambda_{M}\\ \mu_{\mathrm{C}}&1-\mu_{\mathrm{C}}&0&0\\ \mu_{E}&0&1-\mu_{E}&0\\ \mu_{M}&0&0&1-\mu_{M}\end{bmatrix}=\begin{bmatrix}P_{0}&P_{\mathrm{C}}&P_{E}&P _{M}\end{bmatrix} \tag{14}\]  
and also:  
\[P_{0}+P_{\mathrm{C}}+P_{E}+P_{M}=1 \tag{15}\]  
The probability values of each subsystem can then be calculated by equalizing the two equations above:  
\[P_{\mathrm{C}}=P_{0}\frac{\lambda_{\mathrm{C}}}{\mu_{\mathrm{C}}} \tag{16}\]  
\[P_{E}=P_{0}\frac{\lambda_{E}}{\mu_{E}} \tag{17}\]  
\[P_{M}=P_{0}\frac{\lambda_{M}}{\mu_{M}} \tag{18}\]  
\[P_{0}=\frac{1}{1+\sum\frac{\lambda_{i}}{\mu_{i}}} \tag{19}\]  
where, \(P_{0}\) is the availability of the system (AEPBM) and \(P_{C}\), \(P_{E}\) and \(P_{M}\) are the unavailability of cutterhead, electrical, and mechanical subsystems, respectively.  
The availability value of the system (AEPBM) in the network modeling is not equal to one in the Markov chain. By increasing the number of subsystems, the availability value calculated by the network modeling method will always be less than the Markov method.