Mathematics and Statistics 9(5): 685-696, 2021 http://www.hrpub.org DOI: 10.13189/ms.2021.090508 Analytical Solutions of ARL for SAR(p) L Model on a Modified EWMA Chart Piyatida Phanthuna 1 , Yupaporn Areepong 2,* 1 Faculty of Science and Technology, Rajamangala University of Technology Phra Nakhon, 10800, Thailand 2 Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, 10800, Thailand Received June 7, 2021; Revised July 30, 2021; Accepted August 22, 2021 Cite This Paper in the following Citation Styles (a): [1] Piyatida Phanthuna, Yupaporn Areepong , "Analytical Solutions of ARL for SAR(p) L Model on a Modified EWMA Chart," Mathematics and Statistics, Vol. 9, No. 5, pp. 685 - 696, 2021. DOI: 10.13189/ms.2021.090508. (b): Piyatida Phanthuna, Yupaporn Areepong (2021). Analytical Solutions of ARL for SAR(p) L Model on a Modified EWMA Chart. Mathematics and Statistics, 9(5), 685 - 696. DOI: 10.13189/ms.2021.090508. Copyright©2021 by authors, all rights reserved. Authors agree that this article remains permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Abstract A modified exponentially weighted moving average (EWMA) scheme expanded from an EWMA chart is an instrument for immediate detection on a small shifted size. The objective of this research is to suggest the average run length (ARL) with the explicit formula on a modified EWMA control chart for observations of a seasonal autoregressive model of order p th (SAR(p) L ) with exponential residual. A numerical integral equation method is brought to approximate ARL for checking an accuracy of explicit formulas. The results of two methods show that their ARL solutions are close and the percentage of the absolute relative change (ARC) is obtained to less than 0.002. Furthermore, the modified EWMA chart with the SAR(p) L model is tested to shift detection when the parameters c and λ are changed. The ARL and the relative mean index (RMI) results are found to be better when c and λ are increased. In addition, the modified EWMA control chart is compared to performance with the EWMA scheme and such that their results encourage the modified EWMA chart for a small shift. Finally, this explicit formula can be applied to various real-world data. For example, two data about information and communication technology are used for the validation and the capability of our techniques. Keywords Explicit Formula, Seasonal Autoregressive, Average Run Length 1. Introduction A control chart is one of the instruments for Statistical Process Control (SPC). The ordinary control charts are well known such as a Shewhart control chart [1], an exponentially weighted moving average (EWMA) chart [2], a cumulative sum (CUSUM) scheme [3]. A Shewhart chart is the basis of a control chart that detected a large shift on 3-sigma control limits speedily. Next, EWMA and CUSUM control charts are developed to a small shift detection appropriately. For the EWMA control chart, many recent works of literature were found to this chart usage, for example, Li et al. [4] introduced the EWMA control chart to invent the transient patterns of motivation and potential of specifying seasonal faster motivation. Moreover, Nawaz et al. [5] presented the EWMA control chart by integrating multiscale principal component analysis for improving the monitoring efficiently and detecting an online multiscale mistake. Recently, Hu and Liu [6] detected the positive shifts of zero-inflated poisson models by using a weighted score test statistic on an upper-sided exponentially weighted moving average control chart. Meanwhile, the EWMA control chart is improved to a performance by many researchers. One of them is the modified EWMA control chart which was originally presented by Patel and Divecha [7] and developed by Khan et al. [8]. The modified EWMA statistic is expanded by adding a multiple of a previous shift term and a constant c for an abrupt detection of autocorrelated observations. The modified EWMA control chart was continuously studied in various literature [9-11]. For the ability comparison of control charts, one of the well-known measurements is the average run length (ARL) [12] presented by using numerous calculations such as a