# Evaluation of Machinery Readiness Using Semi-Markov Processes
## 4 Case Study
### Estimation of the Semi-Markov Model Parameters
\begin{array}{l}\pi_{2}\text{:}0.029+\pi_{3}\text{:}0.043+\pi_{4} \text{:}0.003+\pi_{5}\text{:}0.676+\pi_{6}\text{:}0.982=\pi_{1}\\ \pi_{1}\text{:}0.141+\pi_{5}\text{:}0.004=\pi_{2}\\ \pi_{1}\text{:}0.143+\pi_{2}\text{:}0.117+\pi_{5}\text{:}0.001+\pi_{6}\text{: }0.018=\pi_{3}\\ \pi_{1}\text{:}0.716+\pi_{2}\text{:}0.854+\pi_{3}\text{:}0.957=\pi_{4}\\ \pi_{4}\text{:}0.997=\pi_{5}\\ \pi_{5}\text{:}0.319=\pi_{6}\\ \pi_{1}+\pi_{2}+\pi_{3}+\pi_{4}+\pi_{5}+\pi_{6}=1\end{array} \right.\text{.} \tag{26}\]  
The solution are the stationary probabilities presented in Table 5.  
The analysis of the stationary distribution showed (Table 5) that the limit highest transitions probabilities concern states (S\({}_{1}\), S\({}_{4}\), S\({}_{5}\)) related to standard activities resulting from the production process technology (all over 27%). Indications of undesirable conditions such as failure (S\({}_{2}\)) or downtime (S\({}_{3}\), S\({}_{6}\)) range from 4% to 8%, which is a good result. The calculated limit probabilities relate to the frequency of observations in the sample and do not take into account the duration of individual states, therefore, the limit distribution of the semi-Markov process represents more significant diagnostics. It can be determined using the stationary distribution \begin{table}
\begin{tabular}{c c c c c c} \hline  & **S\({}_{1}\)** & **S\({}_{2}\)** & **S\({}_{3}\)** & **S\({}_{4}\)** & **S\({}_{5}\)** & **S\({}_{6}\)** \\ \hline \(p_{ij}\) & 0.276 & 0.04 & 0.046 & 0.276 & 0.275 & 0.088 \\ \hline \end{tabular}
\end{table} 
Table 5: Stationary probabilities of the Markov chain. of the Markov chain and the expected duration of the process states [24; 25]. Then the limit probabilities of semi-Markov process are expressed by the formula:  
\[P_{j}=\underset{t\rightarrow\infty}{\lim}P(t)=\frac{\pi_{j}E\Big{(}\mathbb{T}_{j }\Big{)}}{\sum_{j\in\mathbb{S}}\pi_{j}E\Big{(}\mathbb{T}_{j}\Big{)}}. \tag{27}\]  
The solution requires to calculate the following forms from the sample of average conditional durations of the process states:  
\[\mathbb{T}=\Big{[}\overline{\mathbb{T}}_{ij}\Big{]},\quad i,j=1,\,2,\,\ldots, \,6. \tag{28}\]  
that are presented in Table 6.  
Based on the transitions probabilities matrix \(P=\Big{[}p_{ij}\Big{]}\) (Table 5) and the matrix of average conditional durations of the states of the process \(\mathbb{T}=\Big{[}T_{ij}\Big{]}\) of random variables \(\overline{\mathbb{T}}_{ij}\) (Table 6), dependencies describing average unconditional durations of the process states were determined \(\overline{\mathbb{T}}_{j}\) according to the formula:  
\[\overline{\mathbb{T}}_{j}=\sum\
olimits_{i=1}^{6}p_{ij}\,\overline{\mathbb{T }}_{ij}. \tag{29}\]  
For this purpose, the following equation system was solved:  
\[\left\{ \begin{array}{l}\overline{\mathbb{T}}_{1}=p_{12}\cdot T_{12}+p_{13} \cdot T_{13}+p_{14}\cdot T_{14}\\ \overline{\mathbb{T}}_{2}=p_{21}\cdot T_{21}+p_{23}\cdot T_{23}+p_{24}\cdot T _{24}\\ \overline{\mathbb{T}}_{3}=p_{31}\cdot T_{31}+p_{34}\cdot T_{34}\\ \overline{\mathbb{T}}_{4}=p_{41}\cdot T_{41}+p_{45}\cdot T_{45}\\ \overline{\mathbb{T}}_{5}=p_{51}\cdot T_{51}+p_{52}\cdot T_{52}+p_{53}\cdot T_{ 53}+p_{56}\cdot T_{56}\\ \overline{\mathbb{T}}_{6}=p_{61}\cdot T_{61}+p_{63}\cdot T_{63}\end{array}  \right.. \tag{30}\]  
The obtained expected values of unconditional dwelling \(\overline{\mathbb{T}}_{i}\) times of the process X(t) in the individual operational states are presented in Table 7.  
The calculated random variables \(\mathbb{T}_{i}\) have finite, positive expected values. This allows to calculate, based on theorem (27), the limit probabilities \(P_{j}\) which are presented in Table 8.