<p>Bivariate observations with identical values in each paired observation are called ties, which frequently occur in discrete bivariate data analysis. Especially, when dealing with bivariate ordinal data, with strong positive dependence, such tied observations are frequently observed due to the characteristic of ordinal data. In this paper, we develop several new general classes of discrete bivariate distributions for bivariate discrete data sets with a high positive dependence and high proportion of tied observations, focusing on the analysis of bivariate ordinal data. Specific families of discrete bivariate distributions are obtained and they are applied to real bivariate ordinal data to investigate the performance of these models. Due to the flexibility of the developed classes of distributions, the CUB model can be used as the underlying distributions to generate a new family of bivariate distribution for analyzing bivariate ordinal data. It is shown that the new family of bivariate distribution generated by using the CUB model performs much better than other compared models. Several computational results are provided as well to illustrate the characteristics of the developed model.</p>