Scaling limits via excursion theory: Interplay between Crump-Mode-Jagers branching processes and Processor-Sharing queues
Abstract
We investigate scaling limits of two non Markovian processes: Crump-Mode-Jagers (CMJ) branching processes and the queue length process of Processor-Sharing (PS) queues. We start from two observations: (1) the Lamperti transformation maps CMJ processes to busy cycles of the PS queue length process; (2) a CMJ process is the local time process of a Lévy process killed at 0, called contour process. As the rescaled contour processes converge weakly to Brownian motion with negative drift, we show that the rescaled CMJ processes conditioned on their extinction time converge to the conditioned excursion measure of the Feller diffusion. Tightness relies on a simple queueing argument on the departure process of symmetric queues. Thanks to the Lamperti transformation, this shows that busy cycles of the PS queue converge to the positive, drifted Brownian excursion (with suitable conditionings). We show that this characterizes the convergence of the whole process toward reflected Brownian motion. This original approach makes it possible to strengthen previous results and to shed light on the infinite variance case. Finally, we discuss potential implications of the state space collapse property, well-known in the queueing literature, to branching processes.