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This paper addresses statistical issues that arise when discrete event simulations are run on parallel processing computers. First, the statistical properties of estimates obtained by parallel distributed simulation and parallel independent replications are compared. This comparison shows that, when estimating steady state quantities, the run length, the strength of the initial transient and the asymptotic variance must be taken into account in addition to the parallel processing speed-up and the number of processors in order to determine which method is statistically more efficient. Second, the statistical properties of estimators of transient quantities obtained by the method of parallel independent replications are considered. The analysis shows that strongly consistent estimates are not obtained in finite expected time as the number of processors increases unless the computational time to complete a single replication is bounded.