Finite-pool queueing with heavy-tailed services

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

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
Volume54
Issue number3
DOIs
Publication statusPublished - 1 Sep 2017

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Queueing
Stable Process
Join
Queue
Customers
Single Server Queue
Busy Period
Scaling Limit
Exponential time
Queue Length
Population Size
Leverage
Brownian motion
Tail
Converge

Keywords

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

Cite this

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title = "Finite-pool queueing with heavy-tailed services",
abstract = "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, 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.",
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author = "G. Bet and {Van Der Hofstad}, R.W. and {van Leeuwaarden}, J.S.H.",
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Finite-pool queueing with heavy-tailed services. / Bet, G.; Van Der Hofstad, R.W.; van Leeuwaarden, J.S.H.

In: Journal of Applied Probability, Vol. 54, No. 3, 01.09.2017, p. 921-942.

Research output: Contribution to journalArticleAcademicpeer-review

TY - JOUR

T1 - Finite-pool queueing with heavy-tailed services

AU - Bet, G.

AU - Van Der Hofstad, R.W.

AU - van Leeuwaarden, J.S.H.

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