Stochastic networks with multiple stable points

Details Stochastic networks with multiple stable points Stochastic networks with multiple stable points

2006-01-12

arxiv.org/abs/math/0601296v4

Annals of Probability 2008, Vol. 36, No. 1, 255-278

Mathematics

Probability

Published in at http://dx.doi.org/10.1214/009117907000000105 the Annals of Probability (http://www.imstat.org/aop/) by the Ins

Scientific paper

10.1214/009117907000000105

This paper analyzes stochastic networks consisting of a set of finite capacity sites where different classes of individuals move according to some routing policy. The associated Markov jump processes are analyzed under a thermodynamic limit regime, that is, when the networks have some symmetry properties and when the number of nodes goes to infinity. An intriguing stability property is proved: under some conditions on the parameters, it is shown that, in the limit, several stable equilibrium points coexist for the empirical distribution. The key ingredient of the proof of this property is a dimension reduction achieved by the introduction of two energy functions and a convenient mapping of their local minima and saddle points. Networks with a unique equilibrium point are also presented.

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