Managing service systems with an offline waiting option and customer abandonment

Many service providers offer customers the choice of either waiting in a line or going offline 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 make an upfront decision on how to allocate service capacity between the two lines. Then, during system operation, he must provide estimates of the waiting times for both lines to each arriving customer. The estimation of offline waiting times is complicated by the fact that some offline customers do not return for service at their appointed time. We show 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 in a line and offline collapses to one dimension, and we characterize the one-dimensional limit process as a reflected diffusion with linear drift. The analytic tractability of this one-dimensional limit process allows us to solve for the capacity allocation that minimizes average cost when there are costs associated with customer abandonments and queueing. We further show that in this limit regime, a simple scheme based on Little's Law to dynamically estimate in line and offline wait times is effective.