On Busy Periods of the Critical GI/G/1 Queue and BRAVO

Yoni Nazarathy; Zbigniew Palmowski

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arXiv · 2203.15456

On Busy Periods of the Critical GI/G/1 Queue and BRAVO

Yoni Nazarathy·Zbigniew Palmowski

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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 BRAVO phenomenon (Balancing Reduces Asymptotic Variance of Outputs) 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 $2+\epsilon$ moments.

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Yoni Nazarathy, Zbigniew Palmowski. 2022-03-29. On Busy Periods of the Critical GI/G/1 Queue and BRAVO. https://arxiv.org/abs/2203.15456

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