Service system with dependent service and patience times

Motivated by recent empirical evidence, we consider a large service system in which the patience time of each customer depends on his service requirement. Our goal is to study the impact of such dependence on key performance measures, such as expected waiting times and average queue length, as well as on optimal capacity decisions. Since the dependence structure renders exact analysis intractable, we employ a stationary fluid approximation that is based on the entire joint distribution of the service and patience times. Our results show that even moderate dependence has significant impacts on system performance, so considering the patience and service times to be independent when they are in fact dependent is futile. We further demonstrate that Pearson's correlation coefficient, which is commonly used to measure and rank dependence, is an insufficient statistic, and that the entire joint distribution is required for comparative statics. Thus, we propose a novel framework, incorporating the fluid model with bivariate dependence orders and copulas, to study the impacts of the aforementioned dependence. We then demonstrate how that framework can be applied to facilitate revenue optimization when staffing and abandonment costs are incurred. Finally, the effectiveness of the fluid-based approximations and optimal-staffing prescriptions is demonstrated via simulations.