A new class of survival distribution for degradation processes subject to shocks

Mei-Ling Ting Lee,G A Whitmore

doi:10.1186/s40488-019-0095-1

Abstract

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. In this paper, we present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism. In our failure model, system strength follows a geometric degradation process. The degradation process itself is any Lévy process, that is, any stochastic process with stationary independent increments. The shock stream is a Fréchet stochastic process, a process derived from the Fréchet extreme-value distribution. Finally, the shock-degradation process is a convolution of the Fréchet shock process and any one of the candidate degradation processes. The system fails at the first occasion when a shock takes system strength across a threshold at zero. The paper presents results for Wiener diffusion processes and gamma processes as examples of Lévy degradation processes. The paper develops key statistical properties of the process model and its survival distribution, including several that are important for its practical application. As the failure mechanism is a first hitting time event, applications that require regression structures fall within the domain of threshold regression methodology.

Highlights

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
We present a family of stochastic processes, called shock-degradation processes, that describe this failure mechanism
We model the degradation process as a Lévy stochastic process and the shock stream by a Fréchet stochastic process

Summary

Introduction

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. We derive an expression for the survival function of our shock-degradation process. To obtain the survival function itself, which we denote by F(s), we take the expectation of P(S > s|C) over all possible degradation sample paths C as follows: F(s) = EC [P(S > s|C)] = EC {exp [−cQ(s)]} .

Results

Conclusion