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;
- \(1\leq i,j\leq N\ \ 0\leq p_{\text{ij}}\leq 1\ ,i,j\in S\)
- \(1\leq i,j\leq N\ \sum p_{\text{ij}}=1\ ,i,j\in S\)
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.