An alternative approach to heavy-traffic limits for finite-pool queues

Abstract

We consider a model for transitory queues in which only a finite number of customers can join. The queue thus operates over a finite time horizon. In this system, also known as the \(\Delta _{(i)}/G/1\) queue, the customers decide independently when to join the queue by sampling their arrival time from a common distribution. We prove that, when the queue satisfies a certain heavy-traffic condition and under the additional assumption that the second moment of the service time is finite, the rescaled queue length process converges to a reflected Brownian motion with parabolic drift. Our result holds for general arrival times, thus improving on an earlier result Bet et al. (Math Oper Res 2019, https://doi.org/10.1287/moor.2018.0947) which assumes exponential arrival times.

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Acknowledgements

The author is very grateful to Debankur Mukherjee and Jori Selen for suggesting many improvements to the manuscript and for the numerous helpful discussions.

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Correspondence to G. Bet.

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Bet, G. An alternative approach to heavy-traffic limits for finite-pool queues. Queueing Syst 95, 121–144 (2020). https://doi.org/10.1007/s11134-020-09653-z

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Keywords

  • Queueing theory
  • Transitory queueing systems
  • Heavy-traffic approximations
  • Continuous-mapping approach

Mathematics Subject Classification

  • Primary 60K25
  • 90B22
  • Secondary 68M20