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We considerheavy-tailed<\/jats:italic>service-time distributions with tails decaying asx<\/jats:italic>-\u03b1<\/jats:sup>, \u03b1 \u2208 (1, 2). We consider the asymptotic regime in which the population size grows to \u221e and establish that the scaled queue-length process converges to an \u03b1-stable process with a negative quadratic drift. We leverage this asymptotic result to characterize the head start that is needed to create a long period of uninterrupted activity (a busy period). The heavy-tailed service times should be contrasted with the case of light-tailed service times, for which a similar scaling limit arises (Betet al.<\/jats:italic>(2015)), but then with a Brownian motion instead of an \u03b1-stable process.<\/jats:p>","DOI":"10.1017\/jpr.2017.42","type":"journal-article","created":{"date-parts":[[2017,9,15]],"date-time":"2017-09-15T09:18:55Z","timestamp":1505467135000},"page":"921-942","source":"Crossref","is-referenced-by-count":2,"title":["Finite-pool queueing with heavy-tailed services"],"prefix":"10.1017","volume":"54","author":[{"given":"Gianmarco","family":"Bet","sequence":"first","affiliation":[]},{"given":"Remco","family":"van der Hofstad","sequence":"additional","affiliation":[]},{"given":"Johan S. 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