Steady-State Simulation with Replication-Dependent Initial Transients: Analysis and Examples

The replicated batch means (RBM) method for steady-state simulation output analysis generalizes both the independent replications (IR) and batch means (BM) methods. We analyze the performance of RBM in situations where the underlying stochastic process possesses an additive initial transient. Our analysis differs from prior work in that the initial transient is stochastic, and hence the sample paths of the transient process may be replication dependent, and possibly also correlated across replications. We provide asymptotic expressions for the mean and variance of the RBM estimators of the steady-state mean and variance parameter of the stochastic process being simulated. We then use our results to study the performance of RBM as a function of the number of replications, initialization method for the replications, and decay rate of the associated initialization bias. Our results provide guidance on when IR, BM, or a combination thereof is the best choice, and also on effective choices of initial states for the replications.