TY - JOUR

T1 - Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and Processor-Sharing queues.

AU - Lambert, A.

AU - Simatos, F.

AU - Zwart, A.P.

PY - 2013

Y1 - 2013

N2 - 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 Lévy 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. © Institute of Mathematical Statistics, 2013.

AB - 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 Lévy 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. © Institute of Mathematical Statistics, 2013.

U2 - 10.1214/12-AAP904

DO - 10.1214/12-AAP904

M3 - Article

VL - 23

SP - 2357

EP - 2381

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

ER -