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In this paper we consider stochastic scheduling models where all relevant data (like processing times, release dates, due dates, etc.) are independent random variables, exponentially distributed. We are interested in the computational complexity of determining optimal policies for these stochastic scheduling models. We give a number of examples of models in which the optimal policies can be determined by polynomial time algorithms while the deterministic counterparts of these models are NP-complete. We also give some examples of stochastic scheduling models for which there exists no polynomial time algorithm if P is not equal NP. (Author)