In ATM networks, two main functions for achieving traffic control and congestion control are Call Admission Control (CAC) and Usage Parameter Control (UPC). Among various UPC schemes, Leaky Bucket is a popular one. In this paper, we study the characteristics of the system in which every traffic source is regulated by an enhanced leaky-bucket before entering the multiplexer at the edge node of the ATM network. In addition to the factor of mean cell rate, peak cell rate is also taken into consideration. Based on the criteria of average waiting time at the multiplexer, wee derive the performance bounds expressed as the functions of the LB parameters and numbers of connections.