Multivariate modeling and analysis based on the multivariate normal distribution is well established and widely used. However, when the marginal distributions have only a positive support, such as time-to-event models, that are positively skewed, often the multivariate normal theory and resulting approximations fail. Accordingly, over the last fifty years, thousands of papers have been published suggesting many ways of generating families of positive support multivariate distributions, such as gamma, Weibull and exponential. As evidenced by recent literature, this quest is still rigorously pursued even today. In this work, we provide a large and flexible class of multivariate gamma distributions that contains both absolutely continuous and discontinuous distributions on the positive hypercube support. All of these models are applicable to the area of reliability and survival modeling. Also, since this theoretical research resulted from a real world application we also provide methods for parameter estimation for a variety of applications.