  The parallel execution of requests in a Cloud Computing platform, as for
Virtualized Network Functions, is modeled by an $M^{[X]}/M/1$ Processor-Sharing
(PS) system, where each request is seen as a batch of unit jobs. The
performance of such paralleled system can then be measured by the quantiles of
the batch sojourn time distribution. In this paper, we address the evaluation
of this distribution for the $M^{[X]}/M/1$-PS queue with batch arrivals and
geometrically distributed batch size. General results on the residual busy
period (after a tagged batch arrival time) and the number of unit jobs served
during this residual busy period are first derived. This enables us to provide
an approximation for the distribution tail of the batch sojourn time whose
accuracy is confirmed by simulation.
