Nonmonotonicity of phase transitions in a loss network with controls

Details Nonmonotonicity of phase transitions in a loss network with controls Nonmonotonicity of phase transitions in a loss network with controls

2006-10-10

arxiv.org/abs/math/0610321v1

Annals of Applied Probability 2006, Vol. 16, No. 3, 1528-1562

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051606000000303 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051606000000303

We consider a symmetric tree loss network that supports single-link (unicast) and multi-link (multicast) calls to nearest neighbors and has capacity $C$ on each link. The network operates a control so that the number of multicast calls centered at any node cannot exceed $C_V$ and the number of unicast calls at a link cannot exceed $C_E$, where $C_E$, $C_V\leq C$. We show that uniqueness of Gibbs measures on the infinite tree is equivalent to the convergence of certain recursions of a related map. For the case $C_V=1$ and $C_E=C$, we precisely characterize the phase transition surface and show that the phase transition is always nonmonotone in the arrival rate of the multicast calls. This model is an example of a system with hard constraints that has weights attached to both the edges and nodes of the network and can be viewed as a generalization of the hard core model that arises in statistical mechanics and combinatorics. Some of the results obtained also hold for more general models than just the loss network. The proofs rely on a combination of techniques from probability theory and dynamical systems.

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