In ATM networks, a usage parameter control (UPC) strategy must regulate incoming traffic according to the characteristics of the sources declared at call set-up. Among various UPC schemes, Leaky Bucket is a conventionally method having been discussed extensively. This paper examines the characteristics of the multiplexer with a sufficient buffer in which cell arrivals are policed by Enhanced Leaky-Bucket (ELB) before entering the system. In addition to the factor of mean rate, peak rate and cell delay variation (CDV) are also considered for each ELB. We find out the worst output pattern from the ELB and derive the upper bound on average waiting time as a function of the ELB parameters.