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We present and compare two ways of applying the regenerative method of simulation output analysis to simulate the steady-state behavior of irreducible discrete time Markov chains having a finite state space. The "standard" approach involves defining the regenerative cycles based on the times a particular "return" state is visited. The alternative approach involves using "splitting" to define the regenerative cycles. We present a way of selecting the splitting distribution that guarantees that the splitting cycles are shorter than the standard cycles, and that the variance estimates produced by the splitting approach are less variable than the variance estimates produced by the standard approach.