Poisson Hypothesis for Information Networks II. Cases of Violations and Phase Transitions

We present examples of queuing networks that never come to equilibrium. That is achieved by constructing Non-linear Markov Processes, which are non-ergodic, and possess eternal transience property

Stationary States of the Generalized Jackson Networks

We consider Jackson Networks on general countable graphs and with arbitrary service times. We find natural sufficient conditions for existence and uniqueness of stationary distributions. They generalise these obtained earlier by Kelbert, Kontsevich and Rybko.Comment: 18 pages, minor change

Poisson Hypothesis for information networks (A study in non-linear Markov processes)

In this paper we prove the Poisson Hypothesis for the limiting behavior of the large queueing systems in some simple ("mean-field") cases. We show in particular that the corresponding dynamical systems, defined by the non-linear Markov processes, have a line of fixed points which are global attractors. To do this we derive the corresponding non-linear integral equation and we explore its self-averaging properties. Our derivation relies on a solution of a combinatorial problem of rode placements.Comment: 70 page

Spontaneous Resonances and the Coherent States of the Queuing Networks

We present an example of a highly connected closed network of servers, where the time correlations do not go to zero in the infinite volume limit. This phenomenon is similar to the continuous symmetry breaking at low temperatures in statistical mechanics. The role of the inverse temperature is played by the average load.Comment: 3 figures added, small correction

Phase transitions in the queuing networks and the violation of the Poisson hypothesis

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Absence of Breakdown of the Poisson Hypothesis I. Closed Networks at Low Load

International audienceWe prove that the general mean-field type networks at low load behave in accordance with the Poisson Hypothesis. That means that the network equilibrates in time independent of its size. This is a "high-temperature" counterpart of our earlier result, where we have shown that at high load the relaxation time can diverge with the size of the network ("low-temperature"). In other words, the phase transitions in the networks can happen at high load, but cannot take place at low load

Propagation of Chaos and Poisson Hypothesis

International audienceWe establish the Strong Poisson Hypothesis for symmetric closed networks. In particular, the asymptotic independence of the nodes as the size of the system tends to infinity is proved