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

TL;DR

This paper analyzes the asymptotic behavior of busy periods in critical GI/G/1 queues, demonstrates a BRAVO phenomenon reducing variance in the work-output process, and improves moment assumptions needed for these results.

Contribution

It provides new asymptotic results for busy periods under weaker moment assumptions and establishes the BRAVO effect for GI/G/1 queues with minimal conditions.

Findings

Busy period distribution is regularly varying with index half.

BRAVO phenomenon reduces asymptotic variance of busy-time.

Results hold under existence of 2+epsilon moments, weakening previous assumptions.

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.