# Evaluation of Machinery Readiness Using Semi-Markov Processes
## 2 Mathematical Modeling  
**Definition 1**. : _Let us consider a random process with a finite state space \(S=\{1,...,s\}\), \(s<\infty\). Let \(\left(\Omega,\ \mathcal{F},\ P\right)\) be a probabilistic space and \(\left\{X\left(t\right):\ t\in T\right\}\) a stochastic process defined for \(\left(\Omega,\ \mathcal{F},\ P\right)\), taking values from the finite or calculable set \(S\). Process \(\{X(t):\ t\in T\}\) is called a Markov process if for each \(i,\ j,\ i_{0},\ i_{1,\ldots,i_{n-1}\in S}\) and for each \(t_{0},\ t_{1}\,\ \ldots,\ t_{n},t_{n+1}\ \in T\) meeting the requirement \(t_{0}\ <\ t_{1}\ <\ t_{n}\ <\ t_{n+1}\) the dependency given below is met:_  
\[P(X(t_{n+1})=j\big{|}X(t_{n})=i,\ X(t_{n-1})=i_{n-1},\ \ldots,\ X(t_{0})=i_{0})=P (X(t_{n+1})=j\big{|}X(t_{n})=i), \tag{1}\]  
Assuming that \(t_{n}=u,\ t_{n+1}=\tau\), then the conditional probability:  
\[P(X(\tau)=j\big{|}X(u)=i)=p_{ij}(u,\ s), \tag{2}\]  
for \(i,j\in S\), where \(p_{ij}(u,s)\) denotes the probability of transition from state \(i\) at time \(u\), to state \(j\) at time \(s\). Assuming that \(t_{0},\ t_{1}\,\ \ldots,\ t_{n-1}\) denote time (instants) from the past, \(t_{n}\) denotes the present instant, and \(t_{n+1}\) the time in the future, the equation says that the future does not depend on the past when the present is known, thus the probability of the future state is independent of the past states, but only of the present state. This property is called [PERSON]'s property, and the stochastic process that satisfies it, a memoryless process. If instants of time are discrete, \(T=N_{0}=\{0,\ 1,\ 2,\ \ldots\}\), then we are dealing with [PERSON]'s chain, and when the process is realized in a continuous time \(T=R_{+}=[0,\infty)\), it is a continuous-time Markov process. For the stochastic process \(\{X(t):\ t>0\}\) taking values from the finite or countable set \(S\) with fixed and right-hand continuous phase trajectories in some sections, and for \(\tau_{0}=0\) which marks the start of the process and \(\tau_{1},\ \tau_{2},\ \ldots\) which denotes successive times of change of states, the random variable:  
\[T_{i}=\tau_{n+1}-\tau_{n}\big{|}X(\tau_{n})=i,\ i\ \in\ S, \tag{3}\]  
denotes the waiting time in the state \(i\) when a successor state is unknown. From the Chapman-Kolmogorov equation, it follows [21; 22] that the process dwelling times in the individual states constitute random variables with exponential distributions and with the parameter \(\lambda_{i}>0\):  
\[G_{i}(t)=P(T_{i}\leq t)=P(\tau_{n+1}-\tau_{n}\leq t\big{|}X(\tau_{n})=i)=1-e^{ -\lambda_{i}\ t},\ t\geq 0,\ i\in S, \tag{4}\]  
where \(G_{i}\) is a cumulative probability distribution of a random variable \(T_{i}\)[23] when a successor state is unknown. The generalization of Markov processes are semi-Markov processes, for which dwelling times in the individual states can have arbitrary distributions, concentrated in the set \([0,\infty]\). Based on [24; 25] it was assumed in this article to define the semi-Markov process with a finite set of states starting from Markov renewal process. In the probabilistic space \((\Omega,\mathcal{F},\mathrm{P})\) random variables are defined for each \(n\in\mathrm{N}\):  
\[\xi_{n}:\Omega\to S, \tag{5}\]  
\[\vartheta_{n}:\Omega\to R_{+} \tag{6}\]  
A two-dimensional sequence of random variables \(\{(\xi_{n},\vartheta_{n  
This definition shows that the Markov renewal process is a specific case of the two-dimensional Markov process. Transition probabilities of this process depend solely on the discrete value of the coordinate. The Markov renewal process \(\{(\xi_{n},\delta_{n}):n\in N\}\) is called homogeneous if the probabilities:  
\[P[\xi_{n+1}=j,\delta_{n+1}<t/\xi_{n}=i]=Q_{ij}(t), \tag{9}\]  
Do not depend on \(n\). From the above definition, it follows that for each pair \((i,j)\in\)\(SxS\) function \(Q_{ij}(t)\) is [24,25]:  
* non-decreasing,
* right-hand continuous,
* \(Q_{ij}(0)=0\),
* \(Q_{ij}(t)\leq 1\),
* \(\sum_{j\in S}\lim_{t\to\infty}Q_{ij}(t)=1\).