This note considers a failure-prone manufacturing system producing one part-type in discrete mode. The problem is to find the optimal integrated production and backordering policy, in which control decisions include how much the production rate should take when the machine is up and whether an unmet demand should be backorder or rejected. The optimal policy is proved to be a threshold control and can be characterized by three threshold parameters for both discounted cost and long-run average cost cases. The structural relationships between three threshold parameters are established. It is found that the completely backordering policy may be optimal for the discounted cost but will never be optimal for the average cost. In addition, the explicit form of the stationary distribution under such threshold policy is derived, which is then be used to calculate interesting steady-state performance measures and seek the optimal threshold values. A numerical example illustrates the results.