  In this paper, we analyze the sojourn of an entire batch in a processor
sharing $M^{[X]}/M/1$ processor queue, where geometrically distributed batches
arrive according to a Poisson process and jobs require exponential service
times. By conditioning on the number of jobs in the systems and the number of
jobs in a tagged batch, we establish recurrence relations between conditional
sojourn times, which subsequently allow us to derive a partial differential
equation for an associated bivariate generating function. This equation
involves an unknown generating function, whose coefficients can be computed by
solving an infinite lower triangular linear system. Once this unknown function
is determined, we compute the Laplace transform and the mean value of the
sojourn time of a batch in the system.
