This paper addresses the bias characteristics of estimators produced from Monte Carlo simulations. If the computer time allocated to the simulation is t, then N(t), the number of replications complete by time t, is a renewal process. The simulation implications of a known exact expression for the expected value of a sample mean based on N(t) replications are explored and a similar exact expression for a sample mean based on N(t)+1 replications is derived. Bias expansions for a sample mean based on N(t) or N(t)+1 replications are obtained. The bias in the sample mean based on N(t) replications is at most of order o(1/t). Under suitable moment conditions, the bias decreases at a much faster rate than o(1/t); on the other hand, the estimator based on N(t)+1 replications has bias of order 1/t. The exact expressions also lead to simple and totally unbiased estimators. Using Taylor series, the bias expansions of a general function of means based on N(t) or N(t)+1 replications are determined. The leading terms in these expansions are of order 1/t, although the coefficients are different. Based on these expansions, a Tin-style adjusted estimator is proposed to reduce the bias. These expansions are specialized to the case of ratio estimation in regenerative simulation. Due to a cancellation effect, the ratio estimator based on N(t)+1 cycles is biased only to order o(1/t) providing confirmation and reinterpretation of a result of M. Meketon and P. Heidelberger.
