# Moscow Mathematical Journal

Volume 17, Issue 1, January–March 2017 pp. 145–160.

Poisson Hypothesis for Open Networks at Low Load

**Authors**:
A. Rybko (1), Senya Shlosman (2), and A. Vladimirov (1)

**Author institution:**(1) Inst. of the Information Transmission Problems, RAS, Moscow, Russia

(2) Skolkovo Institute of Science and Technology, Moscow, Russia,

Inst. of the Information Transmission Problems, RAS, Moscow, Russia

Aix Marseille Université, Université de Toulon, CNRS, CPT UMR 7332, 13288, Marseille, France

**Summary: **

We study large communication networks of the mean-field type. The input flows to the nodes of the network are supposed to be stationary and with low rate. We show that such a network is ergodic, i.e., it goes to the stationary state, which does not depend on the initial state of the network. This is in contrast with the high load regime, when the large time behavior of the network might depend on its initial state. Our technique is based on the coupling construction, which couples two Non-Linear Markov Processes.

2010 Math. Subj. Class. 60J27.

**Keywords:**Mean-field, non-linear Markov process, queuing theory.

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