Certain continuous production processes and early warning systems must be inspected in order to detect failures that would otherwise not be apparent. An optimal inspection policy is a sequence of inspection times which minimizes the average cost per inspection cycle relative to some cost model. Under both an existing cost model, and a new cost model proposed in this paper, the computation of such policies is difficult. Three sub-optimal inspection policies are proposed, each, like the optimal policy, having decreasing times between successive inspections when the time to system failure has increasing hazard rate. Numerical comparisons are given for both cost models in the case of Weibull failure times.