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Observations from steady-state M/M/1 processes are often used by simulation-methodology researchers conducting Monte Carlo performance evaluations of output analysis, variance reduction, and optimization. The classical method for generating data from M/M/1 processes requires two random number streams. Motivated by the need for variance reduction in simulation experiments, we propose inverse-transformation algorithms for generating M/M/1 processes, including waiting times in queue (M/M/1-QT) and in system (M/M/1-ST), using a single random number stream. Moreover, we propose analytical results for computing the variance of the sample mean from M/M/1 processes. The proposed method for generating processes for the M/M/1-QT and the M/M/1-ST is superior (in terms of statistical effectiveness; i.e., variance reduction) to the corresponding classical methods using two random number streams. Moreover, the proposed generation method for M/M/1-QT is also superior to the classical method in terms of computational efficiency. Finally, although the proposed M/M/1-ST generation method is not as computationally efficient as the classical technique, it does have the same computational efficiency as the classical method if a parallel computing scheme is adopted with the Newton's search method.