Finite-pool queueing with heavy-tailed services

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We consider the Δ(i)/G/1 queue, in which a total of n customers join a single-server queue for service. Customers join the queue independently after exponential times. We consider heavy-tailed service-time distributions with tails decaying as x, α ⊂ (1, 2). We consider the asymptotic regime in which the population size grows to ∞ and establish that the scaled queue-length process converges to an α-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 (Bet et al. (2015)), but then with a Brownian motion instead of an α-stable process.

Original languageEnglish
Pages (from-to)921-942
Number of pages22
JournalJournal of Applied Probability
Issue number3
Publication statusPublished - 1 Sep 2017


  • functional central limit theorem
  • heavy-tailed distribution
  • Heavy-traffic approximation
  • Skorokhod reflection map

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