This paper describes two different approaches to dealing with the initial transient problem. In the first approach, the length of the "warm-up period" is determined by obtaining analytical estimates on the rate of convergence to stationarity. Specifically, we obtain an upper bound on the "second eigenvalue" of the transition matrix of a Markov chain, thereby providing one with a theoretical device that potentially can give estimates of the desired form. The second approach is data-driven, and

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... volves using observed data from the simulation to determine an estimate of the "warm-up period". For the method we study, we are able to use a coupling argument to establish a number of important theoretical properties of the algorithm.