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On busy periods of the critical GI/G/1 queue and BRAVO

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  • Yoni Nazarathy

    (The University of Queensland)

  • Zbigniew Palmowski

    (Wrocław University of Science and Technology)

Abstract

We study critical GI/G/1 queues under finite second-moment assumptions. We show that the busy-period distribution is regularly varying with index half. We also review previously known M/G/1/ and M/M/1 derivations, yielding exact asymptotics as well as a similar derivation for GI/M/1. The busy-period asymptotics determine the growth rate of moments of the renewal process counting busy cycles. We further use this to demonstrate a Balancing Reduces Asymptotic Variance of Outputs (BRAVO) phenomenon for the work-output process (namely the busy time). This yields new insight on the BRAVO effect. A second contribution of the paper is in settling previous conjectured results about GI/G/1 and GI/G/s BRAVO. Previously, infinite buffer BRAVO was generally only settled under fourth-moment assumptions together with an assumption about the tail of the busy period. In the current paper, we strengthen the previous results by reducing to assumptions to existence of $$2+\epsilon $$ 2 + ϵ moments.

Suggested Citation

  • Yoni Nazarathy & Zbigniew Palmowski, 2022. "On busy periods of the critical GI/G/1 queue and BRAVO," Queueing Systems: Theory and Applications, Springer, vol. 102(1), pages 219-225, October.
    • Handle: RePEc:spr:queues:v:102:y:2022:i:1:d:10.1007_s11134-022-09858-4
    DOI: 10.1007/s11134-022-09858-4
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    References listed on IDEAS

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    1. S. Foss & A. Sapozhnikov, 2004. "On the Existence of Moments for the Busy Period in a Single-Server Queue," Mathematics of Operations Research, INFORMS, vol. 29(3), pages 592-601, August.
    2. Robert, Christian Y. & Segers, Johan, 2008. "Tails of random sums of a heavy-tailed number of light-tailed terms," Insurance: Mathematics and Economics, Elsevier, vol. 43(1), pages 85-92, August.
    3. Baltrunas, A. & Daley, D. J. & Klüppelberg, C., 2004. "Tail behaviour of the busy period of a GI/GI/1 queue with subexponential service times," Stochastic Processes and their Applications, Elsevier, vol. 111(2), pages 237-258, June.
    4. A. P. Zwart, 2001. "Tail Asymptotics for the Busy Period in the GI/G/1 Queue," Mathematics of Operations Research, INFORMS, vol. 26(3), pages 485-493, August.
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