When Service Times Depend on Customers' Delays : A Solution to Two Empirical Challenges

Service times of customers often depend on the delay they experience in queue, as was recently demonstrated empirically in restaurants, call centers and intensive care units. 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 (and do not abandon the queue) 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. Since 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 the patience times are censored due to the 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 in the current paper, and provide a solution to the corresponding statistical challenges by proving that both problems can be avoided. In particular, we show that, for any exogenous dependence, there exists a corresponding endogenous dependence, such that the queueing dynamics under either dependence are equivalent (both have the same law). As a result, if dependence is observed in data, one can consider the system as having an endogenous dependence, regardless of the true underlying dependence mechanism. Since estimating the structure of an endogenous dependence is substantially easier than estimating a joint service-patience distribution from censored data, this approach facilitates statistical estimations considerably, and is recommended even when the dependence is thought or is known to be exogenous