Nonmonotonicity of phase transitions in a loss network with controls

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.

[1]  J. Doob.  Stochastic processes , 1953 .

[2]  Dah Ming Chiu,et al.  TRAM: A Tree-based Reliable Multicast Protocol , 1998 .

[3]  Jeff Kahn,et al.  On Phase Transition in the Hard-Core Model on ${\mathbb Z}^d$ , 2004, Combinatorics, Probability and Computing.

[4]  A. Mazel,et al.  Random surfaces with two-sided constraints: An application of the theory of dominant ground states , 1991 .

[5]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[6]  Stan Zachary,et al.  Markov random fields and Markov chains on trees , 1983 .

[7]  Olle Häggström.  A Monotonicity Result for Hard-core and Widom-Rowlinson Models on Certain $d$-dimensional Lattices , 2002 .

[8]  Frank Kelly,et al.  Stochastic Models of Computer Communication Systems , 1985 .

[9]  J. Laurie Snell,et al.  Markov Random Fields and Their Applications , 1980 .

[10]  Olle Häggström.  Ergodicity of the hard-core model onZ2 with parity-dependent activities , 1997 .

[11]  Peter Winkler,et al.  Hard constraints and the Bethe Lattice: adventures at the interface of combinatorics and statistical physics , 2002 .

[12]  Partha P. Mitra,et al.  Markov random field models of multicasting in tree networks , 2002, Advances in Applied Probability.

[13]  Paul Martin,et al.  POTTS MODELS AND RELATED PROBLEMS IN STATISTICAL MECHANICS , 1991 .

[14]  F. Spitzer.  Markov Random Fields on an Infinite Tree , 1975 .

[15]  J. Lehoczky,et al.  Insensitivity of blocking probabilities in a circuit-switching network , 1984 .

[16]  S. Zachary,et al.  Loss networks , 2009, 0903.0640.

[17]  M. Luby,et al.  Fast convergence of the Glauber dynamics for sampling independent sets , 1999, Random Struct. Algorithms.

[18]  E. Dynkin.  Sufficient Statistics and Extreme Points , 1978 .

[19]  Graham R. Brightwell,et al.  Graph Homomorphisms and Phase Transitions , 1999, J. Comb. Theory, Ser. B.

[20]  Christian Borgs,et al.  Phase diagram for the constrained integer partitioning problem , 2003, Random Struct. Algorithms.

[21]  F. Papangelou.  GIBBS MEASURES AND PHASE TRANSITIONS (de Gruyter Studies in Mathematics 9) , 1990 .

[22]  Stan Zachary,et al.  Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice , 1985 .

[23]  Olle Häggström,et al.  Nonmonotonic Behavior in Hard-Core and Widom–Rowlinson Models , 1999 .

[24]  R. Baxter.  Exactly solved models in statistical mechanics , 1982 .

[25]  J. Berg,et al.  Percolation and the hard-core lattice gas model , 1994 .

[26]  Robert L. Devaney,et al.  A First Course in Chaotic Dynamical Systems , 2020 .