## The Annals of Applied Probability

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

#### 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 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.

#### Article information

Source
Ann. Appl. Probab., Volume 23, Number 6 (2013), 2357-2381.

Dates
First available in Project Euclid: 22 October 2013

https://projecteuclid.org/euclid.aoap/1382447691

Digital Object Identifier
doi:10.1214/12-AAP904

Mathematical Reviews number (MathSciNet)
MR3127938

Zentralblatt MATH identifier
1285.60034

#### Citation

Lambert, Amaury; Simatos, Florian; Zwart, Bert. Scaling limits via excursion theory: Interplay between Crump–Mode–Jagers branching processes and processor-sharing queues. Ann. Appl. Probab. 23 (2013), no. 6, 2357--2381. doi:10.1214/12-AAP904. https://projecteuclid.org/euclid.aoap/1382447691

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