**5. Conclusions**

The study achieved two important goals. The first of them was a presentation of the method of evaluating the readiness of a selected element of a machine tools in the production system. The analysis according to the Markov chain allowed to determine the probabilities of interstate transitions, which reflect the frequency of the occurrence of individual states. The highest values were achieved for relations *S*6*–S*4, *S*3*–S*4, *S*4*–S*5. They suggest a high incidence of unsuitability states—*S*<sup>3</sup> and *S*6—and the need to determine their causes and reduce their occurrence.

The limit values of transition probability were also calculated. The analysis of stationary distribution showed that the greatest indications concern states related to activities resulting from production process technology (*S*1, *S*4, *S*5), which is a good result.

However, a complete evaluation is only ensured by an analysis according to the semi-Markov process, taking into account the average dwell times of an object in the individual operating states. The calculated probability limits, examining the behavior of the object for *t* → ∞, were the highest for state *S*1—operation (over 65%) and state *S*4—service (almost 17%). The remaining limit values were found to be satisfactorily low, which means that the operation of the machine should be considered as proper. The calculated technical readiness rate of 85% should also be viewed as positive.

Such an analysis not only provides information on the assessment of the current and expected functioning of the machine, but also reveals areas where modifications can be made in order to increase the level of availability and, as a result, ensure more efficient execution of production orders.

Another goal was to compare the results according to the assumptions made, concerning the forms of distribution of the examined variables. In the literature this analysis is often omitted and it is assumed that the examined variable has an exponential distribution. This allows to use Markov's processes, whose parameter estimation is simpler and is described in more detail in publications. Such an assumption—as the study has shown—may lead to different results and effectively to form an incorrect assessment of the process/system studied. The intention of the authors was to indicate that omitting an important stage of statistical analysis of the collected data and assuming a priori the form of distributions does not guarantee the correctness of the obtained analyses.

In the presented study, the differences in the values of the calculated limit probabilities are large, reaching even over 530%. The overall evaluation of system readiness indicates a value lower by 46% in the case of the Markov process analysis.

However, the problem is not only the value of the calculated probabilities, but also the main aim of the system. According to the semi-Markov process, the system tends primarily to occupy state *S*<sup>1</sup> (operation) which is a satisfactory result, emphasizing the proper implementation of tasks. The results according to the Markov process show that the system tends to occupy mainly state *S*3—downtime, which indicates mismanagement and system inactivity.

The goals set by the authors have been achieved, but it should be stressed out that the results obtained concern only one selected machine. As part of further research, it is worth considering a comprehensive analysis of the entire production system using the method indicated in the article. It will provide complete information on its readiness, determine the level of impact of individual elements (machines), and identify areas for improvement.

**Author Contributions:** Conceptualization, A.B. and L.G.; Formal analysis, A.B. and A.S.; Methodology, A.B.; ´ Resources, M.G.; Writing—original draft, A.B. and A.S.; Writing—review & editing, A.B. and M.G. All authors ´ have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
