The traffic modeling is a key element in analyzing and simulating communications network. Most of the previously studied approaches to the connection admission control (CAC) are based on an assumed arrival process model. However, there are some limits to apply the assumed model to the real environment. In this paper, we study a connection admission control (CAC) method without assuming arrival processes in an ATM multiplexer when homogeneous traffics are multiplexed. To obtain the equivalent bandwidth (EB) without modeling arrival processes, the concept of equivalent bandwidth convergence (EBC) is proposed and implemented by using the upper bound of the first derivative of cell loss rate (CLR) for the load. With the EBC concept, one can avoid modeling of the arrival process and implement the CAC simply and effectively. We show by numerical examples the convergence process of the average CLR and the EB for voice and video traffics, and the robustness of the EBC concept.