In this paper a new analytical model of a switching node carrying a mixture of different traffic streams, including multicast streams, has been proposed. It is assumed that each of outgoing directions of the node could be modelled by a limited-availability group composed of links of various capacities. Hence, the model enables calculations of blocking probabilities for calls of all traffic classes for any structure of the groups forming the outgoing directions. The problem of blocking probability evaluation in the switching node with multicast connections is solved in the paper by the fixed point methodology. The proposed method is based on an approximation of the service process occurring in outgoing directions by appropriately constructed one-dimensional Markov chain. The results of analytical calculations have been compared with the simulation results of a switching node with three outgoing directions carrying multicast and unicast traffic streams.