Catastrophic failure and cumulative damage models involving two types of extended exponential distributions

The present study supposes a single unit and investigates cumulative damage and catastrophic failure models for the unit, in situations where the interarrival times between the shocks, and the magnitudes of the shocks, involve two different stochastic processes. In order to consider two essentially different stochastic processes, integer gamma and Weibull distributions are treated as distributions with two parameters and extensions of exponential distributions. With respect to the cumulative damage models, under the assumption that the interarrival times between shocks follow exponential distributions, the case in which the magnitudes of the shocks follow integer gamma distributions is analyzed. With respect to the catastrophic failure models, the respective cases in which the interarrival times between shocks follow integer gamma and Weibull distributions are discussed. Finally, the study provides some characteristic values for reliability in such models.