## 3 Performance Evaluation and Experimental Design  
A dataset consisting of a mixture distribution is called a pure mixture distribution when it comes from the same distribution family, and it is referred to as a convolute mixture distribution when it comes from different distribution families. In evaluating the performance of the PEWMA control chart, we assumed that the data came from a Poisson mixture distribution within the same distribution family. We assumed that observations from this distribution are independent of each other and that the mixture proportions and distribution parameters remain constant over time.  
There are several measures available to assess the performance of a chart, with one commonly used metric being the ARL to evaluate how well the control chart responds to a signal that is out of control. The formulation of the ARL is given in Equation (26)  
\[ARL=\frac{\sum_{i=1}^{N}RL_{n}}{N}. \tag{26}\]  
Various approaches exist for estimating the run-length characteristics, including Monte Carlo simulation, Markov chain analysis, and integral equations [26]. In this study, simulation is employed to estimate the run-length characteristics of the PEWMA control chart due to its flexibility and efficiency. The steps for generating simulation results can be explained as follows:  
* Step 1: Set the simulation parameters, including the replication length (or the number of sample data) and the number of replications.
* Step 2: Select the parameters for the PEWMA control chart, including \(\lambda\), \(\sigma\), and \(A\).  
* Step 3: Define the Poisson mixture parameters (\(\alpha\) values) to represent desired shifts in the mean, based on \(\pi\) values ranging from 0 to 1.
* Step 4: Run simulation.
* Step 5: Calculate the PEWMA statistics \(z_{i}\) and determine the control limits UCL and LCL.
* Step 6: Check if \(z_{i}\) falls within the control limits. If \(z_{i}\leq\)LCL or \(z_{i}\geq\) UCL, record the sample index number as RL (Run Length).
* Step 7: Repeat Steps 4 and 6 for each replication.
* Step 8: Calculate the ARL by computing the average of all recorded RL values.
* Step 9: Return to Step 2 and run the simulation for different PEWMA control chart parameters.  
In order to detect shifts in the process, the design parameters of the EWMA chart are set for multiple of \(\sigma\) in terms of control limits (\(L\)) and various \(\lambda\) values. There have been several theoretical studies of the ARL properties of the control chart such as [10; 11; 17]. These studies provide ARL of tables and graphs for a range of values of \(\lambda\) and \(L\). In a PEWMA control chart, the ARL exhibits variability based on the sample mean and the control chart's parameters, namely \(\lambda\) and \(A\). Furthermore, alongside these parameters, the ARL performance of the system is influenced by the proportions \(\pi_{i}\) of the Poisson mixture distributions. Different combinations of these parameters are designed before conducting experiments to ascertain the desired ARL performance. Table 1 displays the results of 44 experiments conducted across four main samples. These samples are generated based on a Poisson mixture distribution with the given \(\alpha_{i}\) values, along with corresponding proportions \(\pi_{i}\) for each sample. In this study, each sample within the dataset consists of 1000 data points, generated using the software Arena version 14.00, and follows the characteristics of a Poisson mixture distribution. A simulation model was created to identify out-of-control points. This model utilizes the PEWMA control limits, as specified in Equations (10) and (11). The simulation is run for the combinations of \(A=2.50,2.55,\,2.60,\,2.65,\,2.70,\,2.75,\,2.80,\,2.85,\,2.90,\,2.95,\,3.00,\,3.05,\,3.10, \,3.15,\,3.20,\,3.25,\,3.30,\,3.35,\,3.40,\,3.45,\,3.50\) and \(\lambda=0.1,\,0.2,\,0.3,\,0.4,\,0.5,\,0.75\). So, the total number of simulation experiments is 5544. The replication number for