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A two-queue polling model with priority on one queue and heavy-tailed On/Off sources: a heavy-traffic limit

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  • Rosario Delgado

    (Universitat Autònoma de Barcelona)

Abstract

We consider a single-server polling system consisting of two queues of fluid with arrival process generated by a big number of heavy-tailed On/Off sources, and application in road traffic and communication systems. Class-j fluid is assigned to queue j, $$j=1,2$$ j = 1 , 2 . Server 2 visits both queues to process or let pass the corresponding fluid class. If there is class-2 fluid in the system, it is processed by server 2 until the queue is empty, and only then server 2 visits queue 1, revisiting queue 2 and restarting the cycle as soon as new class-2 fluid arrives, with zero switchover times. Server 1 is an “extra” server which continuously processes class-1 fluid (if there is any). During the visits of server 2 to queue 1, class-1 fluid is simultaneously processed by both servers (possibly at different speeds). We prove a heavy-traffic limit theorem for a suitable workload process associated with this model. Our limit process is a two-dimensional reflected fractional Brownian motion living in a convex polyhedron. A key ingredient in the proof is a version of the Invariance Principle of Semimartingale reflecting Brownian motions which, in turn, is also proved.

Suggested Citation

  • Rosario Delgado, 2016. "A two-queue polling model with priority on one queue and heavy-tailed On/Off sources: a heavy-traffic limit," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 57-85, June.
    • Handle: RePEc:spr:queues:v:83:y:2016:i:1:d:10.1007_s11134-016-9479-9
    DOI: 10.1007/s11134-016-9479-9
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    References listed on IDEAS

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    1. Delgado, Rosario, 2007. "A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic," Stochastic Processes and their Applications, Elsevier, vol. 117(2), pages 188-201, February.
    2. E. G. Coffman & A. A. Puhalskii & M. I. Reiman, 1998. "Polling Systems in Heavy Traffic: A Bessel Process Limit," Mathematics of Operations Research, INFORMS, vol. 23(2), pages 257-304, May.
    3. Otis B. Jennings, 2008. "Heavy-Traffic Limits of Queueing Networks with Polling Stations: Brownian Motion in a Wedge," Mathematics of Operations Research, INFORMS, vol. 33(1), pages 12-35, February.
    4. E. Morozov & B. Steyaert, 2013. "Stability analysis of a two-station cascade queueing network," Annals of Operations Research, Springer, vol. 202(1), pages 135-160, January.
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