Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one-sided reflection.

中文翻译：

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布朗队列串联的吸引力

考虑串联的 n 个双无限和固定布朗队列的序列。假设进入第一个队列的到达过程是一个零均值遍历过程。我们证明，随着 n 趋于无穷大，从第 n 个队列的出发过程在分布上收敛于布朗运动。特别是这意味着布朗运动是布朗排队算子的有吸引力的不变测度。我们的证明利用了串联的布朗队列和最后通道布朗渗透模型之间的关系，在第二个设置中开发了一种耦合技术。结果也在布朗粒子在一侧反射下作用的相关上下文中得到解释。