A MGF based approximation to cumulative exposure models

Online monitoring data contains various measurements of the activity of the system. The amounts of works are also measured in various ways. When we model the reliability of a system, the intensity or the risk of failure events, we need to choose a time scale. Though there should be genuine time scales for each failure phenomenon, the field data including online monitoring data may not be able to provide evidence for them. There are many uncontrollable factors in the field. Many variables are monotone increasing and highly correlated with each other within a system. Yet they also represent the differences among systems. This article tries to build a bridge between two useful approaches, alternative time scale ( Kordonsky and Gertsbakh (1997), Duchesne and Lawless (2002)) and cumulative exposure model ( Hong and Meeker (2013)), by assuming the stationarity of the increments of these measurements within a system.