Markov random field models of multicasting in tree networks

In this paper, we analyse a model of a regular tree loss network that supports two types of calls: unicast calls that require unit capacity on a single link, and multicast calls that require unit capacity on every link emanating from a node. We study the behaviour of the distribution of calls in the core of a large network that has uniform unicast and multicast arrival rates. At sufficiently high multicast call arrival rates the network exhibits a .phase transition., leading to unfairness due to spatial variation in the multicast blocking probabilities. We study the dependence of the phase transition on unicast arrival rates, the coordination number of the network, and the parity of the capacity of edges in the network. Numerical results suggest that the nature of phase transitions is qualitatively different when there are odd and even capacities on the links. These phenomena are seen to persist even with the introduction of nonuniform arrival rates and multihop multicast calls into the network. Finally, we also show the inadequacy of approximations such as the Erlang fixed-point approximations when multicasting is present.