  We consider the processor sharing $M/M/1$-PS queue which also models balking.
A customer that arrives and sees $n$ others in the system "balks" (i.e.,
decides not to enter) with probability $1-b_n$. If $b_n$ is inversely
proportional to $n+1$, we obtain explicit expressions for a tagged customer's
sojourn time distribution. We consider both the conditional distribution,
conditioned on the number of other customers present when the tagged customer
arrives, as well as the unconditional distribution. We then evaluate the
results in various asymptotic limits. These include large time (tail behavior)
and/or large $n$, lightly loaded systems where the arrival rate $\lambda\to 0$,
and heavily loaded systems where $\lambda\to\infty$. We find that the
asymptotic structure for the problem with balking is much different from the
standard $M/M/1$-PS queue. We also discuss a perturbation method for deriving
the asymptotics, which should apply to more general balking functions.
