SCALING LIMITS VIA EXCURSION THEORY: INTERPLAY BETWEEN CRUMP-MODE-JAGERS BRANCHING PROCESSES AND PROCESSOR-SHARING QUEUES

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 Levy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the que...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:The Annals of applied probability 2013-12, Vol.23 (6), p.2357-2381
Hauptverfasser: Lambert, Amaury, Simatos, Florian, Zwart, Bert
Format: Artikel
Sprache:eng
Schlagworte:
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung: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 Levy 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.
ISSN:1050-5164
2168-8737
DOI:10.1214/12-AAP904