# Optimal Design of One-Sided Exponential Adaptive EWMA Scheme Based on Median Run Length
## 3 RL Properties of the One-Sided Exponential AEWMA Schemes  
### A Markov Chain Method to RL Properties  
The performance of a monitoring scheme is typically evaluated by RL measures, such as ARL, MRL, and standard deviation of RL (SDRL), etc. In general, evaluating the RL properties of EWMA-type schemes can be accomplished through three primary methods: integral equations [25], which provide exact solutions but can be mathematically complex; Markov chain methods [28; 29; 30], which offer a flexible and computationally efficient approach; and Monte Carlo simulations [31; 32], which are useful for obtaining empirical results and handling more complex scenarios. In this paper, a Markov chain model is developed to assess the RL and MRL performance of the proposed one-sided exponential AEWMA schemes. Without loss of generality, the in-control region is divided into \(N\) subintervals, labeled as \(i=1,2,\cdots,N\). Each state of the Markov chain is defined by these subintervals, and the value of the corresponding one-sided exponential AEWMA statistic is approximated using the midpoint of each subinterval. The scheme statistic \(A_{t}^{+}(A_{t}^{-})\) is considered to be in transient-state \(i\) if it falls within the \(i\)-th subinterval. If \(A_{t}^{+}(A_{t}^{-})\) falls outside the control limits, it is considered to be in the absorbing-state, indicating that the process is out-of-control. Based on the above state space partition, a \((N+1)\times(N+1)\) one-step transition probability matrix \(\mathbf{P}\) can be constructed as follows:  
\[\mathbf{P}=\left[ \begin{array}{cc}\mathbf{Q}&(\mathbf{I}-\mathbf{Q})\mathbf{ 1}\\ \mathbf{0}^{\intercal}&1\end{array} \right], \tag{11}\]  
where \(\mathbf{I}\) is an identity matrix, and \(\mathbf{1}\) is a column vector of ones. The transient transition probability matrix \(\mathbf{Q}=[p_{i,j}]_{N\times N}\) contains the probabilities \(p_{i,j}\) of transitioning from state \(i\) to state \(j\). For the upper-sided exponential AEWMA scheme, the in-control interval \([B^{+},UCL^{+})\) is divided into \(N\) subintervals, each subinterval \([H_{i}^{+}-\Delta^{+},H_{i}^{+}+\Delta^{+})\), \(i=1,2,\cdots,N\) with a width of \(2\Delta^{+}=\frac{UCL^{+}-B^{+}}{N}\), where \(H_{i}^{+}\) represents the center value of the \(i\)th subinterval:  
\[H_{i}^{+}=B^{+}+(2i-1)\Delta^{+},\quad i=1,2,\cdots,N. \tag{12}\]  
The transient transition probability matrix \(\mathbf{Q}=[p_{i,j}]_{N\times N}\) is given by:  
\[p_{i,j}=\left\{ \begin{array}{ll}\Pr(A_{t}^{+}<H_{j}^{+}+\Delta^{+}|A_{t-1}^ {+}=H_{i}^{+}),&j=1,\\ \\ \Pr(H_{j}^{+}-\Delta^{+}\leq A_{t}^{+}<H_{j}^{+}+\Delta^{+}|A_{t-1}^{+}=H_{i}^ {+}),&j=2,\ldots,N.\end{array} \right. \tag{13}\]  
Substituting Equations (3) and (7) into Equation (13) yields  
\[p_{i,j}=\left\{ \begin{array}{ll}\Pr\Bigl{(}\phi(cS_{t}-H_{i}^{+})<H_{j}^{+}- H_{i}^{+}+\Delta^{+}\Bigr{)},&j=1,\\ \\ \Pr\Bigl{(}H_{j}^{+}-H_{i}^{+}-\Delta^{+}\leq\phi(cS_{t}-H_{i}^{+})<H_{j}^{+}- H_{i}^{+}+\Delta^{+}\Bigr{)},&j=2,\ldots,N.\end{array} \right. \tag{14}\]From Equation (8), it is known that \(\phi(e)\) is invertible, and its inverse function is invertible \(\phi^{-1}(z)\) given by:  
\[\phi^{-1}(z)=\left\{ \begin{array}{ll}z-(1-\lambda)k&\text{if }z<-\lambda k\\ z/\lambda&\text{if }|z|\leq\lambda k\\ z+(1-\lambda)k&\text{if }z>\lambda k\end{array} \right., \tag{15}\]  
Substituting Equation (15) into Equation (14) yields:  
\[p_{i,j}=\left\{ \begin{array}{ll}\Pr\biggl{(}S_{t}^{+}<\frac{\phi^{-1}(H_{j}^ {+}-H_{i}^{+}+\Delta^{+})+H_{i}^{+}}{c}\biggr{)},&j=1,\\ \Pr\biggl{(}\frac{\phi^{-1}(H_{j}^{+}-H_{i}^{+}-\Delta^{+})+H_{i}^{+}}{c}\leq S _{t}^{+}<\frac{\phi^{-1}(H_{j}^{+}-H_{i}^{+}+\Delta^{+})+H_{i}^{+}}{c}\biggr{)},&j=2,\ldots,N.\end{array} \right. \tag{16}\]  
Since \(S_{t}^{+}\sim\exp(1)\), its c.d.f. is given by Equation (4), and we have:  
\[p_{i,j}=\left\{