Adaptive Behavior of Impatient Customers in Tele-Queues: Theory and Empirical Support

We address the modeling and analysis of abandonments from a queue that is invisible to its occupants. Such queues arise in remote service systems, notably the Internet and telephone call centers; hence, we refer to them as tele-queues. A basic premise of this paper is that customers adapt their patience (modeled by an abandonment-time distribution) to their service expectations, in particular to their anticipated waiting time. We present empirical support for that hypothesis, and propose an M/M/m-based model that incorporates adaptive customer behavior. In our model, customer patience depends on the mean waiting time in the queue. We characterize the resulting system equilibrium (namely, the operating point in steady state), and establish its existence and uniqueness when changes in customer patience are bounded by the corresponding changes in their anticipated waiting time. The feasibility of multiple system equilibria is illustrated when this condition is violated. Finally, a dynamic learning model is proposed where customer expectations regarding their waiting time are formed through accumulated experience. We demonstrate, via simulation, convergence to the theoretically anticipated equilibrium, while addressing certain issues related to censored-sampling that arise because of abandonments.