Modelling process durations with gamma mixtures for right-censored data: Applications in customer clustering, pattern recognition, drift detection, and rationalisation

Customer modelling, particularly concerning length of stay or process duration, is vital for identifying customer patterns and optimising business processes. Recent advancements in computing and database technologies have revolutionised statistics and business process analytics by producing heterogeneous data that reflects diverse customer behaviours. Different models should be employed for distinct customer categories, culminating in an overall mixture model. Furthermore, some customers may remain “alive” at the conclusion of the observation period, meaning their journeys are incomplete, resulting in right-censored (RC) duration data. This combination of heterogeneous and right-censored data introduces complexity to process duration modelling and analysis. This paper presents a general approach to modelling process duration data using a gamma mixture model, where each gamma distribution represents a specific customer pattern. The model is adapted to account for RC data by modifying the likelihood function during model fitting. The paper explores three key application scenarios: (1) offline pattern clustering, which categorises customers who have completed their journeys; (2) online pattern tracking, which monitors and predicts customer behaviours in real-time; and (3) concept drift detection and rationalisation, which identifies shifts in customer patterns and explains their underlying causes. The proposed method has been validated using synthetically generated data and real-world data from a hospital billing process. In all instances, the fitted models effectively represented the data and demonstrated strong performance across the three application scenarios.