We are interested in estimating, through simulation, the probability of entering a rare failurestate before a regeneration state. Since this probability is typically small, we apply importancesampling. The method that we use is based on nding the most likely paths to failure. We presentan algorithm that is guaranteed to produce an estimator that meets the conditions presented inL'Ecuyer and Tun (2011) for vanishing relative error. We furthermore demonstrate how theprocedure that is used to obtain the change of measure can be executed a second time to achieveeven further variance reduction, using ideas from Juneja (2007), and also apply this techniqueto the method of failure biasing, with which we compare our results.