  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.
