When Service Times Depend on Customers’ Delays: A Relationship Between Two Models of Dependence

As empirically observed in restaurants, call centers, and intensive care units, service times needed by customers are often related to the delay they experience in queue. Two forms of dependence mechanisms in service systems with customer abandonment immediately come to mind: First, the service requirement of a customer may evolve while waiting in queue, in which case the service time of each customer is endogenously determined by the system’s dynamics. Second, customers may arrive ( exogenously ) to the system with a service and patience time that are stochastically dependent, so that the service-time distribution of the customers that end up in service is different than that of the entire customer population. We refer to the former type of dependence as endogenous and to the latter as exogenous . Because either dependence mechanism can have significant impacts on a system’s performance, it should be identified and taken into consideration for performance-evaluation and decision-making purposes. However, identifying the source of dependence from observed data is hard because both the service times and patience times are censored due to customer abandonment. Further, even if the dependence is known to be exogenous, there remains the difficult problem of fitting a joint service-patience times distribution to the censored data. We address these two problems and provide a solution to the corresponding statistical challenges by proving that both problems can be avoided. We show that, for any exogenous dependence, there exists a corresponding endogenous dependence, such that the queuing dynamics under either dependence have the same law. We also prove that there exist endogenous dependencies for which no equivalent exogenous dependence exists. Therefore, the endogenous dependence can be considered as a generalization of the exogenous dependence. As a result, if dependence is observed in data, one can always consider the system as having an endogenous dependence, regardless of the true underlying dependence mechanism. Because estimating the structure of an endogenous dependence is substantially easier than estimating a joint service-patience distribution from censored data, our approach facilitates statistical estimations considerably.