  Consider a set of jobs with independent random service times to be scheduled
on a single machine. The jobs can be surgeries in an operating room, patients'
appointments in outpatient clinics, etc. The challenge is to determine the
optimal sequence and appointment times of jobs to minimize some function of the
server idle time and service start-time delay. We introduce a generalized
objective function of delay and idle time, and consider $l_1$-type and
$l_2$-type cost functions as special cases of interest. Determining an
index-based policy for the optimal sequence in which to schedule jobs has been
an open problem for many years. For example, it was conjectured that `least
variance first' (LVF) policy is optimal for the $l_1$-type objective. This is
known to be true for the case of two jobs with specific distributions. A key
result in this paper is that the optimal sequencing problem is non-indexable,
i.e., neither the variance, nor any other such index can be used to determine
the optimal sequence in which to schedule jobs for $l_1$ and $l_2$-type
objectives. We then show that given a sequence in which to schedule the jobs,
sample average approximation yields a solution which is statistically
consistent.
