Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and processor-sharing queues

TL;DR

This paper investigates the heavy traffic limits of the M/G/1 processor-sharing queue by combining excursion theory, Lévy processes, and branching processes, demonstrating convergence to reflected Brownian motion and exploring implications for Crump-Mode-Jagers processes.

Contribution

It introduces a novel approach linking excursion theory with queue and branching process convergence, providing new invariance principles and insights into the state space collapse phenomenon.

Findings

Convergence of queue length excursions to reflected Brownian motion with drift.

Establishment of invariance principles for Crump-Mode-Jagers processes.

Discussion of the implications of state space collapse in branching processes.

Abstract

We study the convergence of the $M/G/1$ processor-sharing, queue length process in the heavy traffic regime, in the finite variance case. To do so, we combine results pertaining to L\'{e}vy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward excursions obtained from those of some reflected Brownian motion with drift, after taking the image of their local time process by the Lamperti transformation. We also show, via excursion theoretic arguments, that this entails the convergence of the entire processes to some (other) reflected Brownian motion with drift. Along the way, we prove various invariance principles for homogeneous, binary Crump-Mode-Jagers processes. In the last section we discuss potential implications of the state space collapse property, well known in the queuing literature, to branching processes.