We are interested in E [N], the mean time until the most recent k values of a sequence of independent and identically distributed random variables exceeds a specified constant. Using recent results, we present a simulation procedure for determining E [N]. These results are also used to obtain upper and lower bounds for E [N]. These bounds, however, are in terms of a quantity ω that is not easily calculated. A recursive procedure for evaluating ω when the data distribution is Bernoulli is given. Efficient simulation procedures for estimating ω in the cases of normal and exponential population distributions are also presented, as is a Markov chain monte carlo procedure when the distribution is general.