In this article, a new analytical model is proposed for a limited-availability group serving a mixture of multiservice BPP (Binomial, Poisson, Pascal) traffic. The model assumes that the different traffic classes belonging to this traffic mixture have priorities that affect their ability to be served. The model includes for the first time the possibility of handling priority traffic through a limited availability group and assumes the possibility of handling priority BPP traffic. The proposed model has been subjected to a number of investigations in which a number of different BPP traffic classes and a number of different priority arrangements have been considered. In this article, the authors present exemplary results of the numerical experiments that illustrate the possible applications of this model to analyze links in a multiservice network. The presented computational results were also compared with the results of simulation experiments, which confirmed the satisfactory accuracy of the proposed model. This allows the model to be easily applied in practice for modeling, analysis, and dimensioning of modern multiservice networks, such as cellular or elastic optical networks.