Generalized Shock Model Based On The Frequency of Shocks: A Simple Approach

TL;DR

This paper introduces a generalized shock model where system failure occurs after the accumulation of a certain number of shocks within a threshold interval, providing explicit failure time distributions and approximations.

Contribution

It extends the traditional delta-shock model by considering multiple shocks and derives explicit failure time distributions and normal approximations.

Findings

Explicit distribution of failure time derived

Normal approximation to failure time proposed

Model generalizes existing shock failure models

Abstract

In a $\delta-$shock model, a system subject to randomly occurring shocks, the system fails when the time between two successive shocks lies below a threshold $\delta$. In this note, we study the generalization of this model where such $\delta-$shocks are accumulated and the system fails on the occurrence of $k^{th}$ such a $\delta-$shock. The probability distribution of the system failure time and the statistical characteristics are explicitly obtained. Normal approximation to the failure time distribution is proposed.