Generalized δδ-shock model via runs

According to the δδ-shock model, the system fails when the time between two consecutive shocks falls below a fixed threshold δδ. This model has a potential application in various fields such as inventory, insurance and system reliability. In this paper, we study run-related generalization of this model such that the system fails when kk consecutive interarrival times are less than a threshold δδ. The survival function and the mean value of the failure time of the system are explicitly derived for exponentially distributed interarrival times. We also propose a new combined shock model which considers both the magnitudes of successive shocks and the interarrival times.