Asynchronous transfer mode ( ATM ) is a promising multiplexing and switching technique for implementing an integrated access as well as transport network and has been adopted by CCITT as a basis for the future broadband integrated services digital network ( BISDN ). The ATM technique allows digital communication of any type to share common transmission links and switching devices on a statistical multiplexing basis. Information is transmitted in the form of constant length cells. In an ATM network, the major parameters to cause ATM network performance deterioration are the cell loss and the cell delay at the buffer queue in the ATM multiplexer. Therefore, the performance parameters of an ATM multiplexer are specifically focused on the cell loss probability, the cell delay, and the distribution of queueing length at buffer m this study. The performance of an ATM multiplexer is studied, whose input consists of the superposition of homogeneous bursty ( ON/OFF ) sources, 1e , all the superposed sources are characterized by the bursty sources of the same parameter values. The cell loss probability and the distribution of queuing length at buffer under different offered load and buffer size conditions are evaluated. An ATM multiplexer with three priority classes is simulated using the priority assignment control method of [15]. Under the priority assignment period P and the priority assignment ratio WD in this method have been defined, the relationship between the traffic balance of classes and buffer size of each is studied. The cell loss probability and delay time of each class ( same sources and different sources between classes ) are evaluated. The results are useful to design a economic and effective ATM multiplexer.