A Rank‐Based CUSUM Monitoring Scheme for Time Between Events and Amplitude Data

ABSTRACT Monitoring time between events and amplitude (TBEA) data is critical in many applications, particularly in manufacturing, reliability engineering, and service operations. Yet, most existing TBEA control charts focus on a single quality characteristic and are typically developed under restrictive distributional assumptions. In practice, each event is characterized by two independent variables: the waiting time since the previous event and the magnitude (amplitude) of the event itself, whose underlying distributions are often unknown. This paper proposes a novel rank‐based one‐sided cumulative sum (CUSUM) control chart for univariate TBEA monitoring, designed to jointly monitor shifts in both the interarrival time and the amplitude. The run‐length properties of the proposed nonparametric CUSUM‐TBEA chart are derived via a Markov chain approximation, enabling computation of in‐control (IC) and out‐of‐control (OOC) performance measures using closed‐form expressions. Monte Carlo simulations confirm the accuracy of the Markov chain results and provide guidance on the required simulation size for reliable performance estimation. Under identical IC conditions, the proposed CUSUM–TBEA chart generally exhibits faster detection and reduced run‐length variability compared to the classical exponentially weighted moving average (EWMA) chart, particularly for moderate to large shifts. The effects of Phase I sample size, unknown shift magnitudes, and correlation are also investigated. The results reveal robust performance across a wide range of monitoring scenarios. An illustrative example involving wildfire occurrence data further highlights the practical applicability of the proposed method. Overall, the results show that the proposed chart is a powerful distribution‐free and a practical tool for monitoring event‐driven processes with concurrent changes in event timing and magnitude.