# Evaluation of Machinery Readiness Using Semi-Markov Processes
## 4 Case Study
### Calculations According to the Markov Model  
Markov processes concern exponential distributions, the most popular ones in reliability theory [11; 35]. They are described by two parameters, which fully define them. The first of these is the already calculated probability matrix of interstate transitions \(p_{ij}\) (Table 4). The second, important parameter is the function describing the transitions of objects between states, called the process transition intensity \(\lambda_{ij}(t)\), which characterizes the rate of changes in the probability of transition \(p_{ij}(t)\)[36]. \[\lambda_{ij}(t)=\lim_{\Delta t\to 0}\frac{1}{\Delta t}p_{ij}(t,\ t+\Delta t)\ \text{for}\ i,\ j=0,1,2,\ldots i\
eq j, \tag{32}\]  
For homogeneous Markov processes, the transition intensity is constant and equal to the inverse of the expected duration \(t_{ij}\) of the state \(S_{i}\) before \(S_{j}\)[37]:  
\[\lambda_{ij}(t)=\frac{1}{\text{E}\Big{(}t_{ij}\Big{)}}, \tag{33}\]  
where:  
\(\lambda_{ij}(t)\)--intensity of transitions from the state \(i\) to state \(j\),  
\(E(t_{ij})\)--expected duration value \(t_{ij}\). The intensities \(\lambda_{ii}\leq\ 0\) for \(i=j\) are defined as a complement to the sum of transition intensity from state S\({}_{i}\) for \(i\ \
eq\ j\) to 0:  
\[\lambda_{ii}+\Sigma_{j}\,\lambda_{ii}=0 \tag{34}\]  
thus:  
\[\lambda_{ii}=-\Sigma_{j}\,\lambda_{ii}. \tag{35}\]  
The modules \(|\lambda_{ii}|=-\lambda_{ii}\) are called the exit intensities from the state \(S_{i}\). \begin{table}
\begin{tabular}{l l l l} \hline \hline \(P_{1}\) & 0.6579 & \(P_{1}\) [\%] & 65.79 \\ \(P_{2}\) & 0.0396 & \(P_{2}\) [\%] & 3.96 \\ \(P_{3}\) & 0.0740 & \(P_{3}\) [\%] & 7.4 \\ \(P_{4}\) & 0.1667 & \(P_{4}\) [\%] & 16.67 \\ \(P_{5}\) & 0.0279 & \(P_{5}\) [\%] & 2.79 \\ \(P_{6}\) & 0.0337 & \(P_{6}\) [\%] & 3.37 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Values of limit probabilities \(P_{j}\) of the 6-state model. Calculated according to the above formulas (33)-(35), the element \(\lambda_{ij}\) of the matrix \(\Lambda\) of transition intensity is shown in Table 9.  
Then, using the relationship (36), ergodic probabilities \(p_{j}\) were calculated for the Markov model in continuous time. \[\prod^{T}\star\Lambda=0, \tag{36}\]  
where:  
* \(\prod^{T}=\left[p_{j}\right]^{T}=\left[p_{1};p_{ns}\right]\)--transposed vector of limit probabilities \(p_{j}\),
* \(|\Lambda|\)--transition intensity matrix:
* \(\sum_{j}p_{j}=1\)--the normalization condition. This way, for the process studied, we obtain the following matrix Equation (37):  
\[\left[ \begin{array}{c}p_{1}\\ p_{2}\\ p_{3}\\ p_{4}\\ p_{5}\\ p_{6}\end{array} \right]^{T}\left[ \begin{array}{cccccc}-\lambda_{11}&\lambda_ {12}&\lambda_{13}&\lambda_{14}&0&0\\ \lambda_{21}&-\lambda_{22}&\lambda_{23}&\lambda_{24}&0&0\\ \lambda_{31}&0&-\lambda_{33}&\lambda_{34}&0&0\\ \lambda_{41}&0&0&-\lambda_{44}&p_{45}&0\\ \lambda_{51}&\lambda_{52}&\lambda_{53}&0&-\lambda_{55}&\lambda_{56}\\ \lambda_{61}&0&\lambda_{63}&0&0&-\lambda_{66}\end{array} \right]=\left[ \begin{array}{c}0\\ 0\\ 0\\ 0\\ 0\\ 0\\ 0\end{array} \right], \tag{37}\]  
Taking into account the normalization condition: \(\sum_{j=1}^{6}p_{j}=1\), we get the limit probabilities \(p_{j}\) of the system's dwelling time in the states S\({}_{1}\)-S\({}_{6}\), which are shown in Table 10.  
The results obtained deviate from the values determined for the semi-Markov process, disturbingly revealing that the system studied tends primarily to remain in the downtime state (S\({}_{3}\)). The state in which the production takes place (S\({}_{1}\)) comes only second and takes the value lower by over 35% in relation to calculations made according to the semi-Markov process. A comparison of the other results is presented in Table 11. The highest difference concerns state S\({}_{3}\), and is over 534%.