The selection of an optimal checkpointing strategy has most often been considered in the transaction processing environment where systems are allowed unlimited repairs. In this environment an optimal strategy maximizes the time spent in the normal operating state and consequently the rate of transaction processing. This paper seeks a checkpoint strategy which maximizes the probability of critical-task completion on a system with limited repairs. These systems can undergo failure and repair only until a repair time exceeds a specified threshold, at which time the system is deemed to have failed completely. For such systems, a model is derived which yields the probability of completing the critical task when each checkpoint operation has fixed cost. The optimal number of checkpoints can increase as system reliability improves. The model is extended to include a constraint which enforces timely completion of the critical task.