A new model of a queueing system with two buffers, both finite in one case and only finite in other case were considered. The study of the features like blocking, splitting, feedback, balking and reneging can be done. In present work the study of splitting and up to some extent the study of blocking is done. The analytical solution of the tandem queue with splitting feature is in itself novel. In addition an algorithm is also studied for the stationary process of each case to determine the distribution of the queue length and other queue parameters as consequences. Although, solutions were not explicitly displayed in the closed forms, it can be accessed through the algorithm. The numerical examples demonstrate the behavior of the model and some imbalances that occurs.