Time and magnitude monitoring based on the renewal reward process

Time between event (TBE) charts are SPC tools for monitoring the occurrence of unwanted events, such as the appearance of a defective item or a failure of a piece of equipment. In some cases, a magnitude, indicating the severity of the event, is also measured. Time and magnitude charts, which are based on the assumption that the stochastic process underlying the occurrence of events is the marked Poisson process, are the preferred option. However, these charts are not suitable to deal with damage events caused by repeatedly occurring shocks or stress conditions. To bridge this gap, we introduce a new control chart based on the assumption of a renewal process with rewards, where the reward represents magnitude, and a magnitude-over-threshold condition represents the occurrence of an event. In particular, we consider two cases for magnitude: (i) magnitude is cumulative over time and (ii) magnitude is non-cumulative or independent over time. We use known results in renewal theory to provide expressions of the probability distributions needed to compute the control limits and perform a simulation analysis of the control chart performance.