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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<jats:inline-formula><jats:alternatives><jats:tex-math>$$2+\\epsilon $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mn>2<\/mml:mn><mml:mo>+<\/mml:mo><mml:mi>\u03f5<\/mml:mi><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>moments.<\/jats:p>","DOI":"10.1007\/s11134-022-09858-4","type":"journal-article","created":{"date-parts":[[2022,8,22]],"date-time":"2022-08-22T16:03:24Z","timestamp":1661184204000},"page":"219-225","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On busy periods of the critical GI\/G\/1 queue and BRAVO"],"prefix":"10.1007","volume":"102","author":[{"given":"Yoni","family":"Nazarathy","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9257-1115","authenticated-orcid":false,"given":"Zbigniew","family":"Palmowski","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,8,22]]},"reference":[{"issue":"1","key":"9858_CR1","doi-asserted-by":"publisher","first-page":"243","DOI":"10.1239\/aap\/1300198521","volume":"43","author":"A Al Hanbali","year":"2011","unstructured":"Al Hanbali, A., Mandjes, M., Nazarathy, Y., Whitt, W.: The asymptotic variance of departures in critically loaded queues. 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