A finite buffer, single part-type, single unreliable machine production system is considered. The part flow into the system is described through a fluid model. The problem is to determine a production control which minimizes an infinite horizon average back-log/surplus cost. In the infinite buffer capacity case, this problem is solved by a hedging point policy, and the hedging point can be analytically computed. In this paper we consider the finite capacity case, derive an equation to compute the optimal hedging point and solve it numerically. A comparison between the optimal hedging point in the finite buffer capacity case and the optimal hedging point in the infinite buffer capacity case is reported in the paper, together with the analysis of the effect of some meaningful parameters (like the buffer capacity and the demand loss cost) on the computed hedging point. Some numerical examples illustrate the results presented in the paper.