## 1 Introduction  
Control charts are one of the most important Statistical Process Control (SPC) techniques used to diagnose and prevent problems in production and service processes such as production, maintenance, healthcare [1; 2; 3; 4; 5]. Control charts allow users to quickly identify assignable causes and implement necessary solutions. Various process control charts have been improved to effectively monitor process measurement data. Shewhart control charts are very common charts that are used in SPC. The scope of SPC is a two-phase application: Phase I (offline analysis), which is called the retrospective phase, and Phase II (online monitoring), which is called the prospective or monitoring phase. In the Phase I application, process data is analyzed, and the process is stabilized by detecting variability and taking actions, if any, for assignable causes. In this situation, where the process is under control, Phase I analysis is completed, and Phase II, where the process is monitored online, begins. While Shewhart control charts are effective during Phase I, they are less sensitive to small and medium-sized process shifts in Phase II. The cumulative sum (CUSUM) and Exponentially Weighted Moving Average (EWMA) control charts are superior alternatives to the Shewhart control chart for Phase II when process monitoring [6]. Although the performances of CUSUM and EWMA are closely aligned, EWMA stands out as the more user-friendly and straightforward option in terms of setup and usage. EWMA control chart was introduced by [PERSON] et al. [7] and is very effective in detecting small process shifts. There have been several theoretical studies of the Average Run Length (ARL) properties of the EWMA control charts. These studies provide ARL tables and graphs encompassing a variety of control chart parameter values. [PERSON] et al. [8] examined the ARL performance of the EWMA control charts for the case of nonnormal distributions. The EWMA control limits were revised for Poisson counts [9]. Poisson EWMA (PEWMA) control charts have a wide range of applications as they can be used to monitor processes in both production and service sectors. A good parameter estimation also positively affects the performance of the control charts; otherwise, there may be problems such as the inability to detect variability promptly and an increase in the false alarm rate. Since historical data in PEWMA also control statistics, the points on the chart are not independent of each other. Therefore, the distribution of Run Length (RL) values is also non-geometric. For this reason, the ARL performance calculations of PEWMA control charts are more difficult than other control charts. In several studies, the RL distributions and ARL performances are calculated by simulation, integral equations, and Markov Chain Approach [10; 11; 12; 13; 14; 15; 16; 17]. A new PEWMA chart is introduced by [PERSON] [18] improved based on square transformation of Poisson data. In considering PEWMA as ARL-biased, [PERSON] and [PERSON] [19] used the Markov chain to determine ARL accurately and compared some competing results of previous studies. Although the Poisson distribution is frequently used to describe count data, it assumes that variance is equal to the mean (equi dispersion), which could lead to incorrect inferences in the case of overdispersion [20]. One of the most common methods to cope with different levels of overdispersion is to use Poisson mixture distributions [21]. Several various control charts have been proposed for different types of Poisson distributions in ref. [22] and Poisson mixture distributions in ref. [21]. This study involves a comparative analysis of the ARL performance of the PEWMA control chart. This analysis encompasses diverse parameter combinations within the context of Poisson mixture distributions. Simulation methodology is applied to generate data and facilitate the charting process. The rest of the paper is organized as follows. In Section 2, we present the charting methodology.