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

Poisson Hypothesis for Information Networks (A study in non-linear Markov processes) I. Domain of Validity

In this paper we study the Poisson Hypothesis, which is a device to analyze approximately the behavior of large queueing networks. We prove it in some simple limiting cases. We show in particular that the corresponding dynamical system, defined by the non-linear Markov process, has a line of fixed points which are global attractors. To do this we derive the corresponding non-linear equation and we explore its self-averaging properties. We also argue that in cases of havy-tail service times the PH can be violated.

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.

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.

关于非可逆马尔可夫跳跃过程的泊松方程

On the Poisson equation for nonreversible Markov jump processes

本文利用矩阵森林定理和平均首达时间，为非可逆马尔可夫跳跃过程的泊松方程解建立界，并应用于估计弛豫到稳态非平衡过程中时间累积差异。

Scaling hypothesis for the Euclidean bipartite matching problem II. Correlation functions

We analyze the random Euclidean bipartite matching problem on the hypertorus in $d$ dimensions with quadratic cost and we derive the two--point correlation function for the optimal matching, using a proper ansatz introduced by Caracciolo et al. to evaluate the average optimal matching cost. We consider both the grid--Poisson matching problem and the Poisson--Poisson matching problem. We also show that the correlation function is strictly related to the Green's function of the Laplace operator on the hypertorus.

Spreading maps (polymorphisms), symmetries of Poisson processes and matching summation

The matrix of a permutation is a partial case of Markov transition matrices. In the same way, a measure preserving bijection of a space A with finite measure is a partial case of Markov transition operators. A Markov transition operator also can be considered as a map (polymorphism) A to A, which spreads points of A into measures on A. In this paper, we discuss R-polymorphisms and $\vee$-polymorphisms, who are analogues of the Markov transition operators for the groups of bijections A to A leaving the measure quasiinvariant; two types of the polymorphisms correspond to the cases, when A has fi…