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Batch Arrival Processor-Sharing with Application to Multi-Level Processor-Sharing Scheduling

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Abstract

We analyze a Processor-Sharing queue with Batch arrivals. Our analysis is based on the integral equation derived by Kleinrock, Muntz and Rodemich. Using the contraction mapping principle, we demonstrate the existence and uniqueness of a solution to the integral equation. Then we provide asymptotical analysis as well as tight bounds for the expected response time conditioned on the service time. In particular, the asymptotics for large service times depends only on the first moment of the service time distribution and on the first two moments of the batch size distribution. That is, similarly to the Processor-Sharing queue with single arrivals, in the Processor-Sharing queue with batch arrivals the expected conditional response time is finite even when the service time distribution has infinite second moment. Finally, we show how the present results can be applied to the Multi-Level Processor-Sharing scheduling.

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References

  1. S. Aalto, U. Ayesta, and E. Nyberg-Oksanen, Two-level processor sharing scheduling disciplines: Mean delay analysis, in Proceedings of ACM SIGMETRICS, 2004, pp. 97–105.

  2. K.E. Avrachenkov, U. Ayesta, P. Brown, and E. Nyberg, Differentiation between short and long TCP flows: Predictability of the response time, in Proceedings of INFOCOM, 2004.

  3. F. Baccelli and P. Bremaud, Elements of queuing theory: Palm martingale calculus and stochastic recurrences, Springer, 2003.

  4. N. Bansal, Analysis of the M/G/1 processor-sharing queue with bulk arrivals, Operations Research Letters 31(5) 2003.

  5. C. Bisdikian, A note on the conservation law for queues with batch arrivals, IEEE Transactions on Communications 41(6) (1993) 832–835.

    Article  Google Scholar 

  6. K. Claffy, G. Miller, and K. Thompson, The nature of the beast. recent traffic measurements from an Internet backbone, in INET, July 1998.

  7. R. Cooper, Introduction to Queueing Theory, North-Holland, 1981.

  8. H. Feng and V. Misra, Asymptotic bounds for MX/G/1 processor sharing queues, Tech. Rep. CUCS-006-04, Columbia University, July 2003.

  9. L. Guo and I. Matta, Differentiated control of web traffic: A numerical analysis, in Proceedings of SPIE ITCOM: Scalability and Traffic Control in IP Networks, 2002.

  10. L. Guo and I. Matta, Scheduling flows with unknown sizes: Approximate analysis, Tech. Rep. BU-CS-2002-009, Boston University, March 2002.

  11. D.P. Heyman, T.V. Lakshman, and A.L. Neidhardt, A new method for analysing feedback-based protocols with applications to engineering web traffic over the Internet, in Proceedings ACM SIGMETRICS, 1997, pp. 24–38.

  12. L. Kleinrock, Queueing Systems, vol. 2 (John Wiley and Sons, 1976).

  13. L. Kleinrock and R.R. Muntz, Multilevel processor-sharing queueing models for time-shared models, in Proceedings of ITC6, Aug. 1970, pp. 341/1–341/8.

  14. L. Kleinrock, R.R. Muntz, and E. Rodemich, The processor sharing queueing model for time-shared systems with bulk arrivals, Networks Journal 1(1) (1971) 1–13.

    Google Scholar 

  15. L. Massoulie and J. Roberts, Bandwidth sharing and admission control for elastic traffic, Telecommunication Systems 15(1) (2000) 185–201.

    Article  Google Scholar 

  16. T.M. O’Donovan, Distribution of attained service and residual service in general queueing systems, Operations Research 22 (1974) 570–575.

    Google Scholar 

  17. I. Rai, G. Urvoy-Keller, and E. Biersack, Analysis of LAS scheduling for job size distributions with high variance, in Proceedings of ACM SIGMETRICS, 2003, pp. 218–228.

  18. K.M. Rege and B. Sengupta, The M/G/1 processor-sharing queue with bulk arrivals, in Proceedings of Modelling and Evaluation of ATM Networks 1993, pp. 417–432.

  19. R. Righter and J.G. Shanthikumar, Scheduling multiclass single server queueing systems to stochastically maximize the number of successful departures, Prob. Eng. Inf. Sci. 3 (1989) 323–334.

    Google Scholar 

  20. L.E. Schrage, The queue M/G/1 with feedback to lower priority queues, Management Science 13 (1967) 466–471.

    Google Scholar 

  21. S.F. Yashkov, Processor-sharing queues: Some progress in analysis, Queueing Systems 2 (1987) 1–17.

    Article  Google Scholar 

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Authors and Affiliations

  1. INRIA Sophia Antipolis, 2004 rte. des lucioles, 06902, Sophia Antipolis Cedex, France

    K. Avrachenkov

  2. CWI, Kruislaan 413, P.O. Box 94079, 1090GB, Amsterdam, The Netherlands

    U. Ayesta

  3. France Telecom R&D, 905, rue A. Einstein, 06921, Sophia Antipolis Cedex, France

    P. Brown

Authors
  1. K. Avrachenkov
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  2. U. Ayesta
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  3. P. Brown
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Correspondence to U. Ayesta.

Additional information

The work of Urtzi Ayesta was carried out while he was a PhD student at INRIA Sophia Antipolis and France Telecom R & D.

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Avrachenkov, K., Ayesta, U. & Brown, P. Batch Arrival Processor-Sharing with Application to Multi-Level Processor-Sharing Scheduling. Queueing Syst 50, 459–480 (2005). https://doi.org/10.1007/s11134-005-1666-z

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  • DOI: https://doi.org/10.1007/s11134-005-1666-z

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