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Finite-pool queueing with heavy-tailed services

Part of: Operations research and management science Special processes Computer system organization

Published online by Cambridge University Press:  15 September 2017

Gianmarco Bet ,
Remco van der Hofstad  and
Johan S. H. van Leeuwaarden
Gianmarco Bet*
Affiliation:
Eindhoven University of Technology
Remco van der Hofstad*
Affiliation:
Eindhoven University of Technology
Johan S. H. van Leeuwaarden*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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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.

Keywords

Heavy-traffic approximationheavy-tailed distributionfunctional central limit theoremSkorokhod reflection map

MSC classification

Primary: 60K25: Queueing theory
Secondary: 90B22: Queues and service 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 54 , Issue 3 , September 2017 , pp. 921 - 942
Copyright
Copyright © Applied Probability Trust 2017 

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