The Annals of Applied Probability

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

Amaury Lambert, Florian Simatos, and Bert Zwart

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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.

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

First available in Project Euclid: 22 October 2013

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Zentralblatt MATH identifier

Primary: 60F17: Functional limit theorems; invariance principles
Secondary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J55: Local time and additive functionals 60K25: Queueing theory [See also 68M20, 90B22]

Scaling limit excursion theory processor-sharing queue local time process of Lévy processes Crump–Mode–Jagers branching processes


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.

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