On the sojourn time of a batch in the $M^{[X]}/M/1$ Processor Sharing Queue

TL;DR

This paper models and analyzes the total time a batch spends in an M^{[X]}/M/1 processor sharing queue, deriving equations to compute the batch sojourn time distribution and mean.

Contribution

It introduces a novel approach using recurrence relations and PDEs to analyze batch sojourn times in a complex queueing system.

Findings

Derived a PDE for the generating function of batch sojourn times

Computed the Laplace transform of batch sojourn time

Calculated the mean batch sojourn time

Abstract

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.