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Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines

Author

Listed:
  • H. Thorsdottir

    (CWI)

  • I. M. Verloop

    (CNRS, IRIT
    Universitè de Toulouse, INP)

Abstract

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue-length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse.

Suggested Citation

  • H. Thorsdottir & I. M. Verloop, 2016. "Markov-modulated M/G/1-type queue in heavy traffic and its application to time-sharing disciplines," Queueing Systems: Theory and Applications, Springer, vol. 83(1), pages 29-55, June.
    • Handle: RePEc:spr:queues:v:83:y:2016:i:1:d:10.1007_s11134-016-9477-y
    DOI: 10.1007/s11134-016-9477-y
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    References listed on IDEAS

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    1. I. M. Verloop & U. Ayesta & R. Núñez-Queija, 2011. "Heavy-Traffic Analysis of a Multiple-Phase Network with Discriminatory Processor Sharing," Operations Research, INFORMS, vol. 59(3), pages 648-660, June.
    2. Gennadi Falin & Anatoli Falin, 1999. "Heavy traffic analysis of M/G/1 type queueing systems with Markov-modulated arrivals," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 7(2), pages 279-291, December.
    3. Kiran M. Rege & Bhaskar Sengupta, 1996. "Queue-Length Distribution for the Discriminatory Processor-Sharing Queue," Operations Research, INFORMS, vol. 44(4), pages 653-657, August.
    4. Peter Purdue, 1974. "The M / M /1 Queue in a Markovian Environment," Operations Research, INFORMS, vol. 22(3), pages 562-569, June.
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    Cited by:

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