  We consider the problem of scheduling appointments for a finite customer
population to a service facility with customer no-shows, to minimize the sum of
customer waiting time and server overtime costs. Since appointments need to be
scheduled ahead of time we refer to this problem as an optimization problem
rather than a dynamic control one. We study this optimization problem in fluid
and diffusion scales and identify asymptotically optimal schedules in both
scales. In fluid scale, we show that it is optimal to schedule appointments so
that the system is in critical load; thus heavy-traffic conditions are obtained
as a result of optimization rather than as an assumption. In diffusion scale,
we solve this optimization problem in the large horizon limit. Our explicit
stationary solution of the corresponding Brownian Optimization Problem
translates the customer-delay versus server-overtime tradeoff to a tradeoff
between the state of a reflected Brownian motion in the half-line and its local
time at zero. Motivated by work on competitive ratios, we also consider a
reference model in which an oracle provides the decision maker with the
complete randomness information. The difference between the values of the
scheduling problem for the two models, to which we refer as the stochasticity
gap (SG), quantifies the degree to which it is harder to design a schedule
under uncertainty than when the stochastic primitives (i.e., the no-shows and
service times) are known in advance. In the fluid scale, the SG converges to
zero, but in the diffusion scale it converges to a positive constant that we
compute.
