Inspecting a Vital Component Needed upon Emergency

We consider a stochastically failing component that will be needed at a random future time when an emergency occurs. If the component is not operational at that time, the system incurs a large penalty, which we want to avoid through inspections and replacements. We propose a model and solution algorithm for finding an inspection policy that minimizes the infinite horizon discounted expected penalty, replacement, and inspection costs. We also discuss structural properties of the solution, as well as insights based on numerical results.