  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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