Objective: The aim of this study is to develop a novel framework for designing probability-limit-Based control chart to jointly monitor the time between events and event intensity while accounting for their dependence and the effects of observable covariates.

Research Methodology: In this study, First, the dependence between the time between events and event intensity is modeled using the Archimedean Clayton, Frank, and Gumbel copulas. Subsequently, the Cox proportional hazards model is employed to account for the effects of observable covariates on event occurrence times. Based on the marginal distributions of the variables and the dependence structure specified by the copula, a ratio-based monitoring statistic is defined and the corresponding upper control limit is determined. The performance of the proposed monitoring approaches is then evaluated and compared under various shift scenarios involving changes in the time between events, event intensity, and simultaneous shifts in both variables, across different levels of dependence and copula models, using the average run length (ARL) as the performance criterion.

Findings: The results show that incorporating the effects of observable covariates through the Cox proportional hazards model, particularly under simultaneous-shift scenarios, enhances the sensitivity of the proposed monitoring approach, resulting in lower ARL values. Furthermore, comparison of the copula functions demonstrates that the appropriate specification of the dependence structure plays a critical role in control chart performance, with its importance becoming particularly evident when simultaneous changes occur in the time between events and event intensity.

Originality/Research Value: The main novelty of this study lies in developing an integrated framework for monitoring event data that simultaneously accounts for the dependence between the time between events and event intensity and the effects of observable covariates within the control chart structure. In this framework, copula functions are employed to model the dependence structure, while the Cox proportional hazards model is used to incorporate the effects of observable covariates. This integration provides a more flexible and realistic framework for monitoring processes in which event timing and intensity are interdependent and influenced by observable factors.