## 3 Performance Evaluation and Experimental Design
#### Analysis of Experimental Results  
To observe the ARL performance of PEWMA control schemes, 1000 data are generated from a Poisson mixture distribution with parameter vector \(\Phi=\begin{bmatrix}k=2\\ \theta=5,5.5\\ \pi=(0.90,0.10)\end{bmatrix}\). The ARL values are plotted against the parameter combinations of \(A=2.50\) to \(3.50\) and \(\lambda=0.1\) to \(0.75\) in Figure 2. The ARL is detected earlier with respect to large to small values of \(\lambda\). For \(\alpha_{1}=5\), \(\alpha_{2}=5.5\) and a small shift in mean as \(0.02\sigma\), ARL performances are illustrated for \(\lambda=0.1\), \(0.2\), \(0.3\), \(0.4\), \(0.5\), \(0.75\). The ARL is detected earlier with respect to larger to smaller values of \(\lambda\). However, for \(A>3.50\), the ARL performance of the PEWMA chart seems to become similar for different values of \(\lambda\).  
In Figure 3, for \(\alpha_{1}=5\), \(\alpha_{2}=7\) and a large shift in mean as \(0.76\sigma\), ARL performances are illustrated for \(\lambda=0.1\), \(0.2\), \(0.3\), \(0.4\), \(0.5\), \(0.75\). It is observed that out-of-control situations are quickly detected when the shift in the mean is large and \(\lambda\) is small.  
The performance of the ARL for a relatively large value such as \(0.76\sigma\) of the mean shift, for different values of \(\lambda\) and for the \(A\) values determined b  
Figure 3: ARLs for various values of \(A\) and a series of \(\lambda\) with \(\alpha_{1}=5\), \(\alpha_{2}=7\), shift in mean \(=0.76\sigma\).  
Figure 2: ARLs for various values of \(A\) and a series of \(\lambda\) with \(\alpha_{1}=5\), \(\alpha_{2}=5.5\), shift in mean \(=0.02\sigma\).  
in Figure 4. For given values of \(A\), ARL is detected earlier for small values of \(0.1<\lambda<0.3\), however for \(\lambda>0.3\) ARL performance declines generally for corresponding values of \(A\).  
A 3D graphic is presented in Figure 5 to provide another view of how the ARL performance is influenced by the parameters \(\lambda\), \(A\) and mean shifts. In Figure 5, the performance of ARL is given depending on \(\lambda\) and shift in mean for \(A=3.00\). It is seen that for larger shifts in mean greater than 0.10, \(\lambda\) does not affect the performance of ARL. However, the values of \(\lambda>0.5\) can be selected to detect out-of-control in the case of small shifts in the mean.  
Figure 4: ARLs for various values of \(\lambda\) for a series of \(A\) with \(\alpha_{1}=5\), \(\alpha_{2}=7\), shift in mean = 0.76\(\sigma\).  
Figure 5: ARLs for various values of \(\lambda\) and shifts in mean (multiple of \(\sigma\)) with \(A=3.00\) and \(\alpha_{1}=5\), \(\alpha_{2}=7\).  
The ARL performance of PEWMA chart is illustrated in Figure 6 where the parameters are set to \(\lambda=0.1\), \(\alpha_{1}=5\), \(\alpha_{2}=5.5\) with various shifts in mean and \(2.50<A<3.50\). The performance of ARL varies almost linearly for different levels of shifts in mean. However, for values of \(A>3.00\), the performance exhibits non-stationary behavior.  
Figure 7 demonstrates the changes in performance resulting from the shifts in mean where the parameters are set to \(\lambda=0.1\), \(\alpha_{1}=5\), \(\alpha_{2}=7\). In the case of larger shifts in mean, the same performance is observed for all values of \(A\) choosing \(\lambda=0.1\) when we compared the pure Poisson distribution that has no shifts in mean.