A control problem of a production system involving identical items is presented here. The requests for the items arrive at random, and operational costs are due to unsold items or non-supplied demand. The production rate of an item can be modified during production, and eventually the production can be interrupted, to meet the corresponding demand. This forms a Markov decision process in which one has to choose the production rate to minimize the expected value of a discounted cost. We rely on the structure of the model to provide a characterization of the optimal control, including the general qualitative behavior of solutions. Under some conditions on the data, we show that the optimal policy possesses a simple form. The model and the analysis also apply to a capacity expansion problem, and an extension to the production problem involving two types of items is pursued.