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


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, which assumes exponential arrival times.

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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).

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

Mathematics Subject Classification

  • Primary 60K25
  • 90B22
  • Secondary 68M20