Rate monotonic algorithm has recently been proposed to be used along with traffic shapers to provide quality of service guarantee in ATM networks , . Admission criterion and a fast admission control procedure were developed assuming that cell arrivals are periodic. For bursty traffic, this assumption is conservative and may significantly reduce the efficiency of bandwidth utilization. In this paper, we present an efficient admission criterion for bursty sources. We assume that the traffic generated by each source is regulated with a leaky bucket regulator. Our admission criterion takes into consideration of traffic characteristics. Since the arrival pattern changes in every hop, every network node is required to know how many hops a connection has traversed. We derive the admission criterion for a network (such as the Internet) which allows different users to transmit packets of different lengths. Numerical results show that, compared with the conservative criterion which assumes periodic traffic sources, system utilization could be largely improved.