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Heavy Traffic Analysis of Polling Systems in Tandem

Author

Listed:
  • Martin I. Reiman

    (Bell Laboratories, Lucent Technologies, Murray Hill, New Jersey 07974)

  • Lawrence M. Wein

    (Sloan School of Management, Massachusetts Institute of Technology, Cambridge, Massachusetts 02142-1347)

Abstract

We analyze the performance of a tandem queueing network populated by two customer types. The interarrival times of each type and the service times of each type at each station are independent random variables with general distributions, but the load on each station is assumed to be identical. A setup time is incurred when a server switches from one customer type to the other, and each server employs an exhaustive polling scheme. We conjecture that a time scale decomposition, which is known to occur at the first station under heavy traffic conditions, holds for the entire tandem system, and we employ heavy traffic approximations to compute the sojourn time distribution for a customer that arrives to find the network in a particular state. When setup times are zero (except perhaps at the first station) and additional “product-form” type assumptions are imposed, we find the steady-state sojourn time distribution for each customer type.

Suggested Citation

  • Martin I. Reiman & Lawrence M. Wein, 1999. "Heavy Traffic Analysis of Polling Systems in Tandem," Operations Research, INFORMS, vol. 47(4), pages 524-534, August.
    • Handle: RePEc:inm:oropre:v:47:y:1999:i:4:p:524-534
    DOI: 10.1287/opre.47.4.524
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    References listed on IDEAS

    as
    1. 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.
    2. Yamada, Keigo, 1986. "Multi-dimensional Bessel processes as heavy traffic limits of certain tandem queues," Stochastic Processes and their Applications, Elsevier, vol. 23(1), pages 35-56, October.
    3. Keigo Yamada, 1984. "Diffusion Approximations for Storage Processes with General Release Rules," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 459-470, August.
    4. Martin I. Reiman, 1984. "Open Queueing Networks in Heavy Traffic," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 441-458, August.
    5. Lawrence M. Wein, 1991. "Due-Date Setting and Priority Sequencing in a Multiclass M/G/1 Queue," Management Science, INFORMS, vol. 37(7), pages 834-850, July.
    6. William P. Peterson, 1991. "A Heavy Traffic Limit Theorem for Networks of Queues with Multiple Customer Types," Mathematics of Operations Research, INFORMS, vol. 16(1), pages 90-118, February.
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    Citations

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    Cited by:

    1. David M. Markowitz & Lawrence M. Wein, 2001. "Heavy Traffic Analysis of Dynamic Cyclic Policies: A Unified Treatment of the Single Machine Scheduling Problem," Operations Research, INFORMS, vol. 49(2), pages 246-270, April.
    2. Sem Borst & Onno Boxma, 2018. "Polling: past, present, and perspective," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 26(3), pages 335-369, October.
    3. Ravi Suman & Ananth Krishnamurthy, 2020. "Analysis of tandem polling queues with finite buffers," Annals of Operations Research, Springer, vol. 293(1), pages 343-369, October.
    4. S. Minkevičius & S. Steišūnas, 2006. "About the Sojourn Time Process in Multiphase Queueing Systems," Methodology and Computing in Applied Probability, Springer, vol. 8(2), pages 293-302, June.
    5. Gromoll, H. Christian & Terwilliger, Bryce & Zwart, Bert, 2020. "Heavy traffic limit for the workload plateau process in a tandem queue with identical service times," Stochastic Processes and their Applications, Elsevier, vol. 130(3), pages 1435-1460.
    6. Sigrún Andradóttir & Hayriye Ayhan & Douglas G. Down, 2001. "Server Assignment Policies for Maximizing the Steady-State Throughput of Finite Queueing Systems," Management Science, INFORMS, vol. 47(10), pages 1421-1439, October.
    7. Saulius Minkevičius & Vladimiras Dolgopolovas & Leonidas L. Sakalauskas, 2016. "A Law of the Iterated Logarithm for the Sojourn Time Process in Queues in Series," Methodology and Computing in Applied Probability, Springer, vol. 18(1), pages 37-57, March.
    8. 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.

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