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. For systems with a large population N and a commensurately rapid service rate, or infrequent arrivals, we obtain various asymptotic results. We analyse 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 behaviours. We also identify precisely when the finite population model can be approximated by the standard infinite population M/M/1 - PS queue.