Age replacement policies in two time scales

We develop and estimate optimal age replacement policies for devices whose age is measured in two time scales. For example, the age of a jet engine can be measured in the number of flight hours and the number of landings. Under a single-scale age replacement policy, a device is replaced at age τ or upon failure, whichever occurs first. We show that a natural generalization to two scales is to replace nonfailed devices when their usage path crosses the boundary of a two-dimensional region M, where M is a lower set with respect to the matrix partial order. For lifetimes measured in two scales, we consider devices that age along linear usage paths. We generalize the single-scale long-run average cost, estimate optimal two-scale policies, and give an example. We note that these policies are strongly consistent estimators of the true optimal policies under mild conditions, and study small-sample behavior using simulation.