  For the $M^{[X]}/M/1$ processor Sharing queue with batch arrivals, the
sojourn time $\Omega$ of a batch is investigated.
  We first show that the distribution of $\Omega$ can be generally obtained
from an infinite linear differential system. When further assuming that the
batch size has a geometric distribution with given parameter $q \in [0,1[$,
this differential system is further analyzed by means of an associated
bivariate generating function $(x,u,v) \mapsto E(x,u,v)$. Specifically,
denoting by $s \mapsto E^*(s,u,v)$ the one-sided Laplace transform of
$E(\cdot,u,v)$ and defining $$ \Phi(s,u,v) = P(s,u) \, (1-v) \, F^*(s,u,uv),
\quad 0 < \vert u \vert < 1, \, \vert v \vert < 1, $$ for some known polynomial
$P(s,u)$ and where $$ F^*(s,u,v) = \frac{E^*(s,u,v)-E^*(s,q,v)}{u-q}, $$ we
show that the function $\Phi$ verifies an inhomogeneous linear partial
differential equation (PDE) $$ \frac{\partial \Phi}{\partial u} - \left [
\frac{u - q}{P(s,u)} \right ] v(1-v) \, \frac{\partial \Phi}{\partial v} +
\ell(s,u,v) = 0 $$ for given $s$, where the last term $\ell(s,u,v)$ involves
both $E^*(s,q,v)$ and the first order derivative $\partial E^*(s,q,v)/\partial
v$ at the boundary point $u = q$. Solving this PDE for $\Phi$ via its
characteristic curves and with the required analyticity properties eventually
determines the one-sided Laplace transform $E^*$.
  By means of a Laplace inversion of this transform $E^*$, the distribution
function of the sojourn time $\Omega$ of a batch is then given in an integral
form. The tail behavior of the distribution of sojourn time $\Omega$ is finally
derived.
