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Heavy traffic limit for the workload plateau process in a tandem queue with identical service times

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  • Gromoll, H. Christian
  • Terwilliger, Bryce
  • Zwart, Bert

Abstract

We consider a two-node tandem queueing network in which the upstream queue has renewal arrivals with generally distributed service times, and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. The reuse of service times creates strong dependence at the second queue, making its workload difficult to analyze. To investigate the evolution of workload in the second queue, we introduce and study a process M, called the plateau process, which encodes most of the information in the workload process. We focus on the case of infinite-variance service times and show that under appropriate scaling, workload in the first queue converges, and although the workload in the second queue does not converge, the plateau process does converge to a limit M∗ that is a certain function of two independent Lévy processes. Using excursion theory, we derive some useful properties of M∗ and compare a time changed version of it to a limit process derived in previous work.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:3:p:1435-1460
    DOI: 10.1016/j.spa.2019.05.007
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    References listed on IDEAS

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    1. 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.
    2. Ward Whitt, 1980. "Some Useful Functions for Functional Limit Theorems," Mathematics of Operations Research, INFORMS, vol. 5(1), pages 67-85, February.
    3. H. Christian Gromoll & Bryce Terwilliger & Bert Zwart, 2018. "Heavy traffic limit for a tandem queue with identical service times," Queueing Systems: Theory and Applications, Springer, vol. 89(3), pages 213-241, August.
    4. Boxma, O. J., 1978. "On the longest service time in a busy period of the M[+45 degree rule]G[+45 degree rule]1 queue," Stochastic Processes and their Applications, Elsevier, vol. 8(1), pages 93-100, November.
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