Markov chain approach

Designing variable sampling schemes are based on Markov chain concepts and transition states in a process. This approach expresses that for predicting the action of a system in the future, it is only sufficient to consider the current state of the system. That is, the current state of the process is essential, and the previous states does not have any effect on future states. Therefore, if Xi is a random variable defined in the probability region, then:
\(P\left[X_{n+1}=j\middle|X_{1}=i_{1},\ X_{2}=i_{2},\ldots,X_{n}=i_{n}\right]=P\left[X_{n+1}=j\middle|X_{n}=i_{n}\right]\)(1)
So, the probability of transition (\(p_{\text{ij}}\)) in the Markov chain is:
\(P\left[X_{n+1}=j\middle|X_{n}=i\right]=P\left[X_{1}=j\middle|X_{0}=i\right]=p_{\text{ij}}\)(2)
Where;
Therefore, the Markov chain for sampling schemes can define by a transfer probability matrix (P), where the element is in row i and column j. The transition probability matrix with n state is expressed as follows:
\(\text{\ \ \ \ State~{}}\par \begin{matrix}\ \ 1&\ \ 2&\ \ \cdots&\text{~{}~{}~{}j}&\ \cdots&\text{~{}~{}~{}n}\\ \end{matrix}\)
\begin{equation} P=\begin{matrix}1\\ 2\\ \vdots\\ i\\ \vdots\\ n\\ \end{matrix}\text{~{}~{}}\begin{bmatrix}p_{11}&p_{12}&\cdots&p_{1j}&\cdots&p_{n1}\\ p_{21}&p_{22}&\cdots&p_{2j}&\cdots&p_{n2}\\ \vdots&\vdots&\ddots&\vdots&\ddots&\vdots\\ p_{i1}&p_{i2}&\cdots&p_{\text{ij}}&\cdots&p_{\text{in}}\\ \vdots&\vdots&\ddots&\vdots&\ \ddots&\vdots\\ p_{n1}&p_{n2}&\cdots&p_{\text{nj}}&\cdots&p_{\text{nn}}\\ \end{bmatrix}\nonumber \\ \end{equation}
In designing sampling schemes based on Markov chain concepts at each sampling stage according to the process state (in-control or out-of-control) and other states, the process has several transition states and one absorbing state. The method used in this paper to design sampling schemes is based on the researches by Costa [16], Faraz and Moghadam [17], Faraz et al. [18] and Shojaie-Navokh et al. [15], which can refer to them. For example, according to Shojaie-Navokh et al. [15], diagram of different process states in the VSSI sampling scheme, which has five transition states and one absorbing state, can be explained in Fig. 1.