Scaling limits via excursion theory: interplay between Crump-Mode-Jagers branching processes and processor-sharing queues
The paper studies the convergence of the M/G/\(1\) processor-sharing, queue length process in heavy traffic regime, in the finite variance case. Section 2 contains general notations and states preliminary results. In Section 3, the authors explain the connections between Crump-Mode-Jagers processes, processor-sharing queues and Lévy processes. Section 4 is devoted to the proof of the main result of the paper, Theorem 4.1, which states the convergence of the processor-sharing queue length process toward a process that is defined through its excursion measure. The basic idea is to show that the convergence of suitably conditioned excursions implies the convergence of the full processes. Section 5 extends this result by explicitly identifying the limiting process as being a(nother) reflected Brownian motion with drift and by considering a general initial condition. Finally, Section 6 contains some comments about continuity properties of local time processes and possible implications of the state-space collapse property to branching processes.
scientific article; zbMATH DE number 6247413
scientific article