This paper treats the interworking of LAN-based networks like TCP over the ATM protocol stack in an ATM multicast session. Multicast connection will cause CIP since multicast group members form a connection tree by some tree methods and share the connected tree. The paper solve the CIP problem through a block-by-block transmission using ABT/IT method. ABT/IT RM cell is modified and block scheduling algorithm considering the traffic types is applied to each ATM switch using the enhanced RM cell. Block scheduling algorithm will avoid the indiscriminate discard of UBR traffic when congestion occurs and it can provide an efficient and fair service. The paper builds a block scheduler system and suggests the block scheduling algorithm for a multicast session in an ATM switch. UBR traffics arriving at the switch trough each VC is classified by the traffic type and stored at class buffer and thereafter indisciminately transmitted. When block scheduling algorithm is applied it will improve the UBR traffic performance such as end-to-end delay cell block loss ration etc. This paper evaluated the performance of block scheduling algorithm through the simulation using the C language and data structure.