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We consider the problem of scheduling appointments for a finite customerpopulation to a service facility with customer no-shows, to minimize the sum ofcustomer waiting time and server overtime costs. Since appointments need to bescheduled ahead of time we refer to this problem as an optimization problemrather than a dynamic control one. We study this optimization problem in fluidand diffusion scales and identify asymptotically optimal schedules in bothscales. In fluid scale, we show that it is optimal to schedule appointments sothat the system is in critical load; thus heavy-traffic conditions are obtainedas a result of optimization rather than as an assumption. In diffusion scale,we solve this optimization problem in the large horizon limit. Our explicitstationary solution of the corresponding Brownian Optimization Problemtranslates the customer-delay versus server-overtime tradeoff to a tradeoffbetween the state of a reflected Brownian motion in the half-line and its localtime at zero. Motivated by work on competitive ratios, we also consider areference model in which an oracle provides the decision maker with thecomplete randomness information. The difference between the values of thescheduling problem for the two models, to which we refer as the stochasticitygap (SG), quantifies the degree to which it is harder to design a scheduleunder uncertainty than when the stochastic primitives (i.e., the no-shows andservice times) are known in advance. In the fluid scale, the SG converges tozero, but in the diffusion scale it converges to a positive constant that wecompute.

Asymptotically optimal appointment schedules with customer no-shows