Batch sojourn time in the M[X]/M/1 processor sharing queue with geometric batch size

In this paper, we analyze the sojourn time of an entire batch in a M-[X]/M/1 processor sharing queue, where geometrically distributed batches arrive according to a Poisson process and individual jobs require exponentially distributed service times. By conditioning on the number of jobs in the queue and the number of jobs in a tagged batch, we establish recurrence relations for 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 series expansion can be computed by solving an infinite lower triangular linear system. Once this unknown function is determined, we determine the Laplace transform and the mean value of the sojourn time of a batch in the system.