In this paper, a new multivariate counting process model (called Multivariate Poisson Generalized Gamma Process) is developed and its main properties are studied. Some basic stochastic properties of the number of events in the new multivariate counting process are initially derived. It is shown that this new multivariate counting process model includes the multivariate generalized Pólya process as a special case. The dependence structure of the multivariate counting process model is discussed. Some results on multivariate stochastic comparisons are also obtained.