Propagation of Chaos and Poisson Hypothesis
We 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.
math.PR↗
arXiv subjects
Alexander Rybko
Publications and source records attributed to Alexander Rybko.
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.
math.PR↗
Absence of Breakdown of the Poisson Hypothesis I. Closed Networks at Low Load
We 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.
math-ph↗
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.
math-ph↗
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.
math-ph↗
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.
math-ph↗