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This paper presents the Optimistic Atomic Broadcast algorithm (OPT-ABcast) which exploits the spontaneous total-order property experienced in local-area networks in order to allow fast delivery of messages. The OPT-ABcast algorithm is based on a sequence of stages, and messages can be delivered during a stage or at the end of a stage. During a stage, processes deliver messages fast. Whenever the spontaneous total-order property does not hold, processes terminate the current stage and start a new one by solving a Consensus problem which may lead to the delivery of some messages. We evaluate the efficiency of the OPT-ABcast algorithm using the notion of delivery latency.