  Setting up optimal appointment schedules requires the computation of an
inherently involved objective function, typically requiring distributional
knowledge of the clients' waiting times and the server's idle times (as a
function of the appointment times of the individual clients). A frequently used
idea is to approximate the clients' service times by their phase-type
counterpart, thus leading to explicit expressions for the waiting-time and
idle-time distributions. This method, however, requires the evaluation of the
matrix exponential of potentially large matrices, which already becomes
prohibitively slow from, say, 20 clients on. In this paper we remedy this issue
by recursively approximating the distributions involved relying on a
two-moments fit. More specifically, we approximate the sojourn time of each of
the clients by a low-dimensional phase-type, Weibull or Lognormal random
variable with the desired mean and variance. Our computational experiments show
that this elementary, yet highly accurate, technique facilitates the evaluation
of optimal appointment schedules even if the number of clients is large. The
three ways to approximate the sojourn-time distribution turn out to be roughly
equally accurate, except in certain specific regimes, where the low-dimensional
phase-type fit performs well across all instances considered. As this
low-dimensional phase-type fit is by far the fastest of the three alternatives,
it is the approximation that we recommend.
