An Equilibrium Analysis of a Multiclass Queue with Endogenous Abandonments in Heavy Traffic

This paper studies a multiclass queueing system with endogenous abandonments where congestion affects customers' abandonment behavior and vice versa. Our model captures this interaction by developing two closely related models: an abandonment model and a queueing model. In the abandonment model, customers take the virtual waiting time distribution as given. Class k customers receive a reward r_k from service and incur a cost c_k per period of waiting. Customers are forward looking and make wait or abandon decisions dynamically to maximize their expected discounted utilities. The queueing model takes the customers' abandonment time distribution as an input and studies the resulting virtual waiting time distribution. Because the multiclass queueing system is not amenable to exact analysis, we resort to an approximate analysis in the conventional heavy traffic limit (under the hazard rate scaling). Leveraging the so-called state-space collapse property, we provide a characterization of the system performance. Combining the results for the two models, we show that there exists a unique equilibrium in which the customers' abandonment time and the virtual waiting time for the various classes are consistent in the two models. Lastly, we provide a computational scheme to calculate the equilibrium numerically and apply that using data from an Israeli bank call center