We propose nonstandard simulation estimators of expected time averages over finite intervals [0, t], seeking to enhance estimation efficiency. We make three key assumptions: (i) the underlying stochastic process has regenerative structure, (ii) the time average approaches a known limit as time t increases and (iii) time 0 is a regeneration time. To exploit those properties, we propose a residual-cycle estimator, based on data from the regenerative cycle in progress at time t, using only the data after time t. We prove that the residual-cycle estimator is unbiased and more efficient than the standard estimator for all sufficiently large t. Since the relative efficiency increases in t, the method is ideally suited to use when applying simulation to study the rate of convergence to the known limit. We also consider two other simulation techniques to be used with the residual-cycle estimator. The first involves overlapping cycles, paralleling the technique of overlapping batch means in steady-state estimation; multiple observations are taken from each replication, starting a new observation each time the initial regenerative state is revisited. The other technique is splitting, which involves independent replications of the terminal period after time t, for each simulation up to time t. We demonstrate that these alternative estimators provide efficiency improvement by conducting simulations of queueing models.