Many service providers offer customers the choice of either waiting in a line, or going offine and returning at a dynamically determined future time.
The best known example is the FASTPASS® system at Disneyland. To operate such a system, the service provider must first make an upfront decision on how to allocate service capacity between the two lines. Then, during system operation, he must dynamically provide estimates of the waiting times at both lines to each arriving customer. The estimation of offine waiting times is complicated by the fact that some offine customers do not return for service at their appointed time.
The paper Managing Service Systems with an Offine Waiting Option and Customer Abandonment shows that when demand is large and service is fast, for any fixed capacity allocation decision, the two-dimensional process tracking the number of customers waiting inline and offine collapses to one dimension, and characterize the one-dimensional limit process as a reflected diffusion with linear drift.
The analytic tractability of this one-dimensional limit process allows to solve for the capacity allocation that minimizes average cost, when there are costs associated with customer abandonments and queueing. It is further shown that in this limit regime, a simple scheme based on Little's law to dynamically estimate inline and offline wait times is effective.
Note: This paper was made possible with the help of an ExtendSim Research Grant for the final project of Vasiliki Kostami in obtaining his PhD in Operations Management from the University of Southern California.