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

Yoni Nazarathy () and Zbigniew Palmowski ()
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Yoni Nazarathy: The University of Queensland
Zbigniew Palmowski: Wrocław University of Science and Technology

Queueing Systems: Theory and Applications, 2022, vol. 102, issue 1, No 10, 219-225

Abstract: 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.

Keywords: Busy period; BRAVO; Single server queue; Asymptotics; 60J27; 60K25 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s11134-022-09858-4

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