The current literature continues to propose many extensions and refinements of shock models. The system is assumed to fail when the arriving shock exceeds some critical level ( Shanthikumar and Sumita 1983 Gut and Hüsler 1999). Later studies considered systems governed by bivariate stochastic processes consisting of one process that generated inter-arrival times for shocks and the other that generated the magnitudes of the shocks. The latter probability might increase or decrease as a function of the cumulative number of shocks. Early work considered shocks arriving according to a stochastic counting process, combined with a probability of surviving the incident shock ( Esary and Marshall 1973 Block and Savits 1978). Shock processes have been studied extensively to explain such diverse phenomena as device failures, insurance claims, earth quakes and business bankruptcies, to name but a few. As the failure mechanism is a first hitting time event, applications that require regression structures fall within the domain of threshold regression methodology. The paper develops key statistical properties of the process model and its survival distribution, including several that are important for its practical application.
The paper presents results for Wiener diffusion processes and gamma processes as examples of Lévy degradation processes. The system fails at the first occasion when a shock takes system strength across a threshold at zero. Finally, the shock-degradation process is a convolution of the Fréchet shock process and any one of the candidate degradation processes. The shock stream is a Fréchet stochastic process, a process derived from the Fréchet extreme-value distribution. The degradation process itself is any Lévy process, that is, any stochastic process with stationary independent increments. In our failure model, system strength follows a geometric degradation process.
In this paper, we present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism. Many systems experience gradual degradation while simultaneously being exposed to a stream of random shocks of varying magnitudes that eventually cause failure when a shock exceeds the residual strength of the system.
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