Congestion is one of the most important economic concepts, which has been widely investigated in many data envelopment analysis (DEA) studies. The conventional congestion DEA approaches do not consider the internal structures of the multi-stage production processes (or DMUs (decision-making units)) and treat them as “black-boxes”, i.e., their inputs enter and their outputs exit, without consideration of their internal structures. Hence, the conventional black-box congestion approaches are incapable of providing specific information concerning the sources of inefficiency of the internal structures, which is more important from the managerial and economic viewpoints. This is the underlying problem of these approaches. Therefore, in this study, to tackle this problem, we first introduce a new concept called “generalized congestion” in a network DEA framework. The concept of generalized congestion states that a decrease in some inputs of a DMU along with or without an increase in some other inputs cause an increase in some outputs of the DMU and/or a decrease in some otherof its outputs. Second, we propose a DEA congestion approach with considering the internal structures of the two-stage production processes such that it allows us to look into the two-stage DMUs and to recognize organizational congestion and, generalized congestion status and their component congestion and generalized congestion status. Finally, in order to demonstrate the applicability of the proposed approach in the real world, we provide a numerical example and an empirical application to Chinese leading universities and make a comparison among our proposed congestion approach and known congestion approaches.