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Although new high-bandwidth network technologies are being introduced and widely deployed, asynchronous transfer mode (ATM) is still considered one of the most important network technologies currently in use. A number of ATM switch architectures have been proposed in the literature. However, industry has shown that is better to use the well-known shared-medium technique in the design of these ATM switches. In this paper, four variations of a new distributed scheme are proposed for the arbitration of a shared bus of an ATM switch. These schemes are based on learning automata. By taking advantage of the bursty nature of ATM traffic, the new arbitration scheme demonstrates superb performance compared to the time division multiple access (TDMA) scheme.