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Route planning in uncertain and dynamic networks has recently emerged as an active and intense area of research, both due to industry needs and technological advances. This paper investigates methods to predict travel times along the arcs and estimate arrival times at the nodes of a stochastic and dynamic network in real time. It is shown that, under fairly mild conditions, the developed travel and arrival time estimators are unbiased and that the error variance of the arrival time estimator is bounded. Simulation results are used to demonstrate the efficiency of the proposed algorithm.