  We consider a finite population processor-sharing (PS) queue, with Markovian
arrivals and an exponential server. Such a queue can model an interactive
computer system consisting of a bank of terminals in series with a central
processing unit (CPU). For systems with a large population $N$ and a
commensurately rapid service rate, or infrequent arrivals, we obtain various
asymptotic results. We analyze the conditional sojourn time distribution of a
tagged customer, conditioned on the number $n$ of others in the system at the
tagged customer's arrival instant, and also the unconditional distribution. The
asymptotics are obtained by a combination of singular perturbation methods and
spectral methods. We consider several space/time scales and parameter ranges,
which lead to different asymptotic behaviors. We also identify precisely when
the finite population model can be approximated by the standard infinite
population $M/M/1$-PS queue.
