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 $$\Delta _{(i)}/G/1$$ Δ ( 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.

中文翻译：

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有限池队列大流量限制的替代方法

我们考虑一个临时队列模型，其中只有有限数量的客户可以加入。因此，队列在有限的时间范围内运行。在这个系统中，也称为 $$\Delta _{(i)}/G/1$$ Δ ( i ) / G / 1 队列，客户通过从一个队列中采样他们的到达时间来独立决定何时加入队列。共同分布。我们证明，当队列满足一定的大流量条件并且在服务时间的二阶矩是有限的附加假设下，重新缩放的队列长度过程收敛到具有抛物线漂移的反射布朗运动。我们的结果适用于一般到达时间，从而改进了 Bet 等人的早期结果。（Math Oper Res 2019，https://doi.org/10.1287/moor.2018.0947）假设指数到达时间。