具有平衡节点的开放Jackson网络的常返与暂态
Recurrence and transience of open Jackson networks with balanced nodes
- Centro de Matemática, University of Porto（波尔图大学数学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究开放Jackson网络中平衡节点的常返性，证明队列长度过程常返当且仅当负载为1的节点数不超过2，并讨论Lyapunov函数方法的应用。
AI中文摘要：
考虑一个具有泊松到达和指数服务的开放Jackson网络，其中没有节点过载，且恰好有$k_0$个节点的负载等于$1$。我们证明队列长度过程是常返的当且仅当$k_0\leq 2$。为此，我们证明该网络满足一个扇区条件，因此其容量与其对称化网络的容量相当；后者是一个可逆网络，其常返/暂态性可通过标准方法处理。我们还讨论了Lyapunov函数方法在此情形下的结果；特别地，我们证明每个平衡队列在任意大的时间都是空的，即使在暂态情形下也是如此。
英文摘要：
Consider an open Jackson network with Poisson arrivals and exponential services, in which no node is overloaded and exactly~$k_0$ nodes have load equal to~$1$. We prove that the queue-length process is recurrent if and only if $k_0\leq 2$. For that, we show that the network satisfies a sector condition, so its capacities are comparable to those of its symmetrisation; the latter is a reversible network whose recurrence/transience can be dealt with by standard means. We also discuss what the Lyapunov-function method gives in this setting; in particular, we show that every balanced queue is empty at arbitrarily large times, also in the transient case.
