Abstract: In this study, the performance of a proposed CUSUM control chart was evaluated using the average run length (ARL) and average time to signal (ATS) First, the ARL was approximated using four numerical integral equation (NIE) methods based on the Gauss–Legendre quadrature, Midpoint, Trapezoidal, and Simpson’s rules to establish a unified computational framework for assessing CUSUM control chart behavior under diverse process conditions. Primary emphasis was placed on the ARL performance associated with rapid detections of process mean shifts, while ATS was used to characterize the temporal efficiency of signaling the out-of-control ARL state. Both ARL and ATS evaluations were conducted for the CUSUM control chart with a FISMAX(D, Q, r)s model, a long-memory process with exponential white noise pertinent to real-world processes. The numerical results reveal notable differences among the NIE-based methods. Simpson’s rule provided the smallest ARL and expected ARL values, followed by the Midpoint, Trapezoidal, and Gauss–Legendre methods. In contrast, when computational efficiency was considered, the Midpoint method consistently produced the lowest ATS and expected ATS values, with Gauss–Legendre, Trapezoidal, and Simpson’s rule requiring progressively more computational time. These findings highlight trade-offs between approximation accuracy and computational speed. Considering both detection speed and computational efficiency, the Midpoint method emerged as the most appropriate and practically effective option for the scenarios investigated in this study.

Keywords: Approximated ARL, numerical integral equation (NIE), average run length (ARL), average time to signal (ATS), long-memory FISMAX(D, Q, r)s process, exponential white noise

Pages: 140-156