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Scaling limits via excursion theory: Interplay between Crump–Mode–Jagers branching processes and processor-sharing queues
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 L\'{e}vy processes, branching processes and queuing theory. These results yield the convergence of long excursions of the queue length processes, toward …