This article presents an analytical model of a switching network that can be used in the nodes of a large number of modern multi-service networks. The model of a multi-service three-stage switching network proposed in this article has the advantage of providing the possibility to evaluate the point-to-point blocking probability for multicast connections. The information obtained in this way can then be used both at the stage of the estimation of the capacity of a new network and during the optimisation process of an existing network. In the calculations performed during the investigations, three scenarios for the fan out of multicast connections in the switching network are taken into consideration. The assumption is that the branching of a connection can occur in the first, second and the third stage of the switching network. The model proposed in the article is based on the concept of “effective” availability. The article discusses the methods for the determination of the effective availability parameter for successive connections that belong to a given multicast connection. The results of the analytical modelling are compared with the simulation data for three selected structures of multi-service switching networks to which different mixtures of multi-service BPP traffic are offered. The results of the calculations and the data provided by the simulation experiments validate and confirm high accuracy of the proposed model. This allows the model to be easily applied in practice to different environments to evaluate the capacities of the nodes in multi-service networks.