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To simulate a continuous-time, discrete-state Markov chain with known transition probabilities, we can generate exponentially-distributed waiting time according to current total transition rate, and then select one transition according to their relative probabilities.

However, now I hope to simulate Markov chains alongside with a couple of differential equations, and I hope to simulate the entire system in a fixed time step. The Markov chains are actually several discrete-step one-direction random walks, and different inputs are sent to the differential equations when different random walks reach the thresholds. What is the best method to convert the exponentially distributed time step to a fixed time step?

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