  We consider the $M/M/1$-PS queue with processor sharing. We study the
conditional sojourn time distribution of an arriving customer, conditioned on
the number of other customers present. A new formula is obtained for the
conditional sojourn time distribution, using a discrete Green's function. This
is shown to be equivalent to some classic results of Pollaczeck and Vaulot from
1946. Then various asymptotic limits are studied, including large time and/or
large number of customers present, and heavy traffic, where the arrival rate is
only slightly less than the service rate.
