The Gauss hypergeometric distribution with the density proportional to xα−1 (1 − x)β−1 (1 + ξx)−γ, 0 < x < 1 arises in connection with the prior distribution of the parameter ρ (0 < ρ < 1) representing traffic intensity in a M/M/1 queue system. In this article, we define and study a multivariate generalization of this distribution and derive some of its properties like marginal densities, joint moments, and factorizations. A data application is given.