Random Walk Analysis in a Reliability System under Constant Degradation and Random Shocks

Ryan T White,Jewgeni H Dshalalow

doi:10.3390/axioms10030199

Abstract

In this paper, we study a reliability system subject to occasional random shocks hitting an underlying device in accordance with a general marked point process with position dependent marking. In addition, the system ages according to a linear path that eventually fails even without any external shocks that accelerate the total failure. The approach for obtaining the distribution of the failure time falls into the area of random walk analysis. The results obtained are in closed form. A special case of a marked Poisson process with exponentially distributed marks is discussed that supports our claim of analytical tractability. The example is further confirmed by simulation. We also provide a classification of the literature pertaining to various reliability systems with degradation and shocks.

Highlights

We apply and embellish the theory of fluctuations to arrive at analytically closed formulas and establish the main functional for the joint probability distribution of the first passage time and the position of walker or escape location associated with the failure time and the extent of the overall damage, respectively
If the times between shocks δk are exponentially distributed with parameter λ and the impacts of the shocks Xk’s are exponentially distributed with parameter μ, the joint Laplace– Stieltjes transform (LST) Φν(α, β, θ, θ) of the deterioration upon the shock before the failure, the deterioration upon failure, the time of the shock before the failure, and the failure time satisfies
We studied a reliability system with linear degradation and external shocks that further accelerate its deterioration and lead to an inevitable failure when the overall degradation crosses or exceeds an M, its sustainability threshold

Summary

Introduction

A threshold is set so that if underlying system’s operational capacity crosses that threshold when going downhill, the system fails, and a key target is the prediction of the failure time

Methods

Results

Conclusion