This paper studies the diffusion limit for a network of infinite-server queues operating under Markov modulation (meaning that the system's parameters depend on an autonomously evolving background process). In previous papers on (primarily single-node) queues with Markov modulation, two variants were distinguished: one in which the server speed is modulated, and one in which the service requirement is modulated (i.e., depends on the state of the background process upon arrival). The setup of the present paper, however, is more general, as we allow both the server speed and the service requirement to depend on the background process. For this model we derive a Functional Central Limit Theorem: we show that, after accelerating the arrival processes and the background process, a centered and normalized version of the network population vector converges to a multivariate Ornstein-Uhlenbeck process. The proof of this result relies on expressing the queueing process in terms of Poisson processes with a random time change, an application of the Martingale Central Limit Theorem, and continuous-mapping arguments.

中文翻译：

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马尔可夫调制无限服务器队列网络的扩散限制

本文研究了在马尔可夫调制下运行的无限服务器队列网络的扩散限制（这意味着系统的参数取决于自主演化的后台进程）。在之前关于使用马尔可夫调制的（主要是单节点）队列的论文中，区分了两种变体：一种是调制服务器速度，另一种是调制服务需求（即，取决于后台进程的状态）到达）。然而，本文的设置更通用，因为我们允许服务器速度和服务要求取决于后台进程。对于这个模型，我们推导出一个函数中心极限定理：我们表明，在加速到达过程和背景过程之后，网络人口向量的中心化和归一化版本收敛到多元 Ornstein-Uhlenbeck 过程。这个结果的证明依赖于用随机时间变化的泊松过程、鞅中心极限定理的应用和连续映射参数来表达排队过程。