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Universal, delay-limited simulation of an unknown information source of a certain parametric family (e.g., the family of memoryless sources or Markov sources of a given order), given a training sequence from that source and a stream of independent and uniformly distributed bits, is considered. The goal of universal simulation is that the probability law of the generated sequence be identical to that of the training sequence, with minimum mutual information between the random processes generating both sequences. In the delay-limited setting, the simulation algorithm generates a random sequence sequentially, by delivering one symbol for each training symbol that is made available after a given initial delay, whereas the random bits are assumed to be available on demand. In this paper, the optimal universal delay-limited simulation scheme is characterized for broad parametric families, and the mutual information achieved by the proposed scheme is analyzed. The results are extended to a setting of variable delay.