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Production lines with limited storage capacities can be modeled as cyclic queueing networks with finite buffers and general service times. A new technique, called perturbation analysis of discrete event dynamic systems, is applied to these queueing networks. An estimate of the gradient of the system throughput is obtained by perturbation analysis based on only one sample trajectory of the system. We show that the estimate is strongly consistent. Using this perturbation analysis estimate of gradient, we can apply the Robbins-Monro stochastic procedure in optimizing the system throughput. Compared to the conventional Kiefer-Wolfowitz optimization procedure, this approach saves a large amount of computation. For a real system, the gradient estimate can be obtained without changing any parameters in the system. The results also hold for systems with general routing but in which no server can block more than one other server simultaneously.