In this paper, discrete-time versions of a model by Akella and Kumar [1] are presented for the production of a single commodity on a machine subject to random failures and breakdowns. The successive up and down times of the machines form an alternating renewal sequence, with arbitrary distributions, while the demands are i.i.d. over slots. The infinite-horizon discounted cost problem is investigated with a cost-per-stage which is a convex function of the inventory/backlog level. Under the assumptions made, the convexity of the value function is established by elementary arguments, and as in [1], the form of the optimal production strategy is shown to be characterized by a critical inventory level function. The results are specialized to the cases where the up time distribution is geometric and the demand is constant over time. Easy extensions are briefly discussed.