# Evaluation of Machinery Readiness Using Semi-Markov Processes
## 1 Introduction  
### Background Introduction to the Study  
Ensuring desired availability of all machine tools in a production line is an important issue [1; 2]. It stands for their ability to obtain and maintain the functional state necessary to produce the required performance [3; 4; 5]. The technical readiness of machines is an important element of the company diagnostics and should be estimated, as its evaluation helps shape the capacity of a production line. High machine tools reliability translates into no unnecessary downtime and, consequently, greater process efficiency. Machine tools must be technically sound, adequately controlled and supplied with necessary materials, energy and information [6]. The availability of a machine tool is determined using a probability theory-based reliability model. In probability theory, the state of an object is defined as the result of one and only one event in a sequence of trials of finite or computable set of elementary events excluding each other in pairs [7]. This makes it possible to use the tools of probability calculus and mathematical statistics to analyze technical systems. When machine tools are in operation they stochastically transit from one state to another. As a result, transition probabilities are associated withall machine tools in a production line. Therefore, Markov chain and its derivatives are often used to set a model of reliability. Some of the relevant articles where Markov chain-based reliability models are used to study the availability of machines tools in a production line are described below.  
The use of Markov processes and their generalization--semi-Markov processes--are popular. Their use is dictated by the multi-states condition of the technical objects and the assumption that the assessment of individual functional states the object is in, is a better measure than the readiness of the object as a whole. However, the use of these models is subject to restrictions. First of all, it is necessary to fulfil the Markov property which states that the probability of a future state is independent of the past states, and depends only on the present state. Identifying a model without meeting this assumption may lead to false conclusions, which is suggested by many authors [8; 9]. They point out that ignoring the Markov property examination will results in incorrect analysis results, e.g., [PERSON] et al. [10], [PERSON] et al. [8], or [PERSON] et al. [11]. Therefore, it is necessary to examine the randomness of sequences of subsequent operational states, as it is done by [PERSON] et al. [12] or [PERSON] and [PERSON] [13].  
In addition, [PERSON]'s models require meeting the assumption that the unconditional process dwell times in the individual states and the conditional durations of an individual state, are random variables of exponential distribution, provided that the next one is one of the remaining states [14; 15]. Many authors point out that proper matching of distributions affects the reliability of results [13; 16]. They use [PERSON]'s model for exponential distributions [17; 18], and the semi-Markov model for the remaining ones, e.g., Weibull [19] or Gamma [20]. Adoption of only the assumption on the form of distribution without a statistical survey of the collected sample may lead to wrong conclusions.