## 4 Materials and Methods  
### Markov Chains  
Below, we assume the following: \(R\)--the set of all real numbers; \(N\)--a set of natural numbers including zero; \((\Omega,F,P)\)--probabilistic space.  
**Definition 1**.: _A stochastic process \(\left\{X_{t}\right\}_{t\in T}\) is a family of random variables \(X_{t}:\Omega\to S\) defined on the probabilistic space \((\Omega,F,P)\) for any \(t\in T\subset\mathbb{R}\) and \(S\) is the set of realizations._  
**Definition 2**.: _A stochastic process \(\left\{X_{t}\right\}_{t\in T}\) is a Markov process [1; 35] if for every \(n\), for any finite subset \(\left\{t_{1},t_{2},\ldots,t_{n}\right\}\), where \(t_{1}<t_{2}<,\ldots,<t_{n}\), \(t_{i}\in T\), and for any states \(\left\{x_{1},x_{2},\ldots,x_{n}\right\}\), \(x_{i}\in S\), \(1\leq i\leq n\) the property_  
\[P\big{(}X_{t_{n}}<x_{n}\big{|}X_{t_{n-1}}=x_{n-1},X_{t_{n-2}}=x_{n-2},\ldots,X _{t_{1}}=x_{1}\big{)}=P\big{(}X_{t_{n}}<x_{n}\big{|}X_{t_{n-1}}=x_{n-1}\big{)} \tag{1}\]  
_is satisfied._  
**Definition 3**.: _A stochastic process \(\left\{X_{t}\right\}_{t\in T}\) is called a homogeneous process [35] if for any \(s,t\in T\), \(s<t\ x\,,y\in S\) the property_  
\[P(X_{t}<y|X_{s}=x)=P(X_{t-s}<y|X_{0}=x) \tag{2}\]  
_is satisfied._  
Figure 3: Tires stored in the case study company. Source: Photo taken during the research.  
Figure 2: General overview of the process.  
The condition (1.1) is called the Markov property. In the paper, we assume that the set of moments in which we observe the realizations of the stochastic process is a subset of the set of natural numbers, \(T\subset N\), while the set of realizations of random variables \(X_{t}\), \(t\in N\), is a finite set \(S=\left\{S_{1},S_{2},\ldots,S_{m}\right\}\), \(m\in N\). The set \(S\) is called the set of states of the stochastic process. A Markov process with discrete time is called a Markov chain.  
**Definition 4**.: _A sequence of random variables \(\left\{X_{t}\right\}_{t\in N}\) with the values in the set of states \(S\) is called a Markov chain if, and only if, for any \(n\in N\) and \(x_{t}\in S\), \(t=0,1,\ldots,n\), the property [10; 39]_  
\[P(X_{n}=x_{n}|X_{0}=x_{0},X_{1}=x_{1},X_{2}=x_{2},\ldots,X_{n-1}=x_{n-1})=P(X_ {n}=x_{n}|X_{n-1}=x_{n-1}) \tag{3}\]  
_is satisfied._  
For a homogeneous Markov chain \(\left\{X_{t}\right\}_{t\in N}\), the distribution of transition probabilities between the states is determined by the transition probability matrix  
\[P=\left[p_{ij}\right]_{1\leq i,j\leq m}, \tag{4}\]  
where \(p_{ij}\geq 0\) for \(1\leq i,j\leq m\) and \(\sum_{j=1}^{m}p_{ij}=1\) for \(1\leq i\leq m\), and \(p_{ij}\stackrel{{ def}}{{=}}p_{S_{i}S_{j}}=P\big{(}X_{n+1}=S_{j} \big{|}X_{n}=S_{i}\big{)}\) is the probability of transition from state \(S_{i}\) at time \(n\) to state \(S_{j}\) at time \(n+1\) for any \(n\in N\).  
For a homogeneous Markov chain, the probability matrix for \(k\) steps is  
\[P^{(k)}=\left[p_{ij}^{(k)}\right]_{1\leq i,j\leq m}, \tag{5}\]  
where \(p_{ij}^{(k)}=P\big{(}X_{n+k}=S_{j}\big{|}X_{n}=S_{i}\big{)}\) is the probability of transition from state \(S_{i}\) at time \(n\) to state \(S_{j}\) at time \(n+k\) for any \(n\in N\), and  
\[P^{(k)}=P^{(k-l)}P^{(l)} \tag{6}\]  
for \(1\leq l<k\). Equation (6) is called the Chapman-Kolmogorov equation [1; 39; 40].  
Let \(\pi=(\pi_{1},\pi_{2},\ldots,\pi_{m})\), \(\pi_{j}=P\big{(}X_{0}=S_{j}\big{)}\) for \(S_{j}\in S\) be the initial distribution for a homogeneous Markov chain \(\left\{X_{t}\right\}_{t\in N}\) with a one-step transition matrix equal to \(P\) and \(\sum_{j=1}^{m}\pi_{j}=1\).  
**Definition 5**.: _The initial distribution \(\pi=(\pi_{1},\pi_{2},\ldots,\pi_{m})\) of a homogeneous Markov chain \(\left\{X_{t}\right\}_{t\in N}\) with transition matrix \(P\) is stationary if the condition_  
\[\pi=\pi P \tag{7}\]  
_is satisfied._  
A homogeneous Markov chain with a stationary initial distribution is called a stationary Markov chain [1; 39; 40].