Scaling limits via excursion theory: interplay between Crump-Mode-Jagers branching processes and processor-sharing queues
type: Item
instance of: scholarly article
title: Scaling limits via excursion theory: interplay between Crump-Mode-Jagers branching processes and processor-sharing queues
zbMATH Open document ID: 1285.60034
DOI: 10.1214/12-AAP904
author: Florian Simatos
author: Amaury Lambert
author: Bert Zwart
published in: The Annals of Applied Probability
publication date: 2014-01-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1102.5620
review text: The paper studies the convergence of the M/G/\(1\) processor-sharing, queue length process in heavy traffic regime, in the finite variance case. Section 2 contains general notations and states preliminary results. In Section 3, the authors explain the connections between Crump-Mode-Jagers processes, processor-sharing queues and Lévy processes. Section 4 is devoted to the proof of the main result of the paper, Theorem 4.1, which states the convergence of the processor-sharing queue length process toward a process that is defined through its excursion measure. The basic idea is to show that the convergence of suitably conditioned excursions implies the convergence of the full processes. Section 5 extends this result by explicitly identifying the limiting process as being a(nother) reflected Brownian motion with drift and by considering a general initial condition. Finally, Section 6 contains some comments about continuity properties of local time processes and possible implications of the state-space collapse property to branching processes.
reviewed by: Oleg K. Zakusilo
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J80
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60K25
zbMATH DE Number: 6247413
zbMATH Keywords: scaling limit
zbMATH Keywords: excursion theory
zbMATH Keywords: processor-sharing queue
zbMATH Keywords: local time process of Lévy processes
zbMATH Keywords: Crump-Mode-Jagers branching processes
MaRDI profile type: Publication
arXiv ID: 1102.5620
cites work: Q4888858
cites work: Q4269108
cites work: Q4265385
cites work: Q3928760
cites work: On the character of convergence to Brownian local time. II
cites work: State space collapse with application to heavy traffic limits for multiclass queueing networks
cites work: Proof(s) of the Lamperti representation of continuous-state branching processes
cites work: On strong invariance for local time of partial sums
cites work: Q3721531
cites work: On the transition from a Markov chain to a continuous time process
cites work: On a relationship between processor-sharing queues and Crump–Mode–Jagers branching processes
cites work: Diffusion approximation for a processor sharing queue in heavy traffic.
cites work: The fluid limit of a heavily loaded processor sharing queue
cites work: Continuity of a class of random time trnsformations
cites work: Q4778955
cites work: Q2774021
cites work: Q3855869
cites work: Q5808656
cites work: An embedding of compensated compound Poisson processes with applications to local times
cites work: Q3906873
cites work: Random Walks and A Sojourn Density Process of Brownian Motion
cites work: The contour of splitting trees is a Lévy process
cites work: Continuous state branching processes
cites work: The Limit of a Sequence of Branching Processes
cites work: A LIFO queue in heavy traffic
cites work: Q3099670
cites work: Weak invariance principles for local time
cites work: Sojourn times of diffusion processes
cites work: Q4411306
cites work: A key limit theorem for critical branching processes
cites work: Some Useful Functions for Functional Limit Theorems
cites work: Diffusion approximations for open multiclass queueing networks: Sufficient conditions involving state space collapse
OpenAlex ID: W2043196557
description: scientific article; zbMATH DE number 6247413