Alternative time scales for systems with random usage

In various reliability applications, there can be different time scales in which to measure time to failure, or assess the performance of objects. For instance, for automobiles the chronological age & the total number of driving hours, as measures of usage, are good candidates for the time scale. An impact of a random usage on some properties of time to failure distribution functions is studied. It is shown that a random usage can change the shape of the failure rate of an object compared with the shape of the failure rate for the deterministic usage. Specifically, the sharply increasing failure rate can turn into a decreasing one, which is, in fact, surprising. The random usage can also change the aging properties of distributions under consideration, and this should be taken into account in applications. Several simple examples illustrate the developed concept.