This paper presents the Optimistic Atomic Broadcast algorithm (OPT-ABcast) which exploits the spontaneous total order message reception 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 message reception property does not hold, processes terminate the current stage and start a new one by solving a Consensus problem leading to the delivery of some messages. We evaluate the efficiency of the OPT-ABcast algorithms using the notion of deliver latency.