In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

A cascading failure is a failure in a system of interconnected parts in which the failure of a part can trigger the failure of successive parts. Although an initial and introductory approach for probabilistic modeling and analysis of the cascading failures was suggested in the literature, any general framework and fundamental results have yet to be reported. In this paper, applying the point process approach, we suggest a general framework for modeling and analysis of the cascading failures. Furthermore, a new concept of 'information-based residual lifetime' will be defined and discussed.

This paper is concerned with a nonstationary Markovian chain of cascading damage that constitutes an iterated version of a classical damage model. The main problem under study is to determine the exact distribution of the total outcome of this process when the cascade of damages finally stops. Two different applications are discussed, namely the final size for a wide class of SIR (susceptible → infective → removed) epidemic models and the total number of failures for a system of components in reliability. The starting point of our analysis is the recent work of Lefèvre (2007) on a first-crossing problem for the cumulated partial sums of independent parametric distributions, possibly nonstationary but stable by convolution. A key mathematical tool is provided by a nonstandard family of remarkable polynomials, called the generalised Abel–Gontcharoff polynomials. Somewhat surprisingly, the approach followed will allow us to relax some model assumptions usually made in epidemic theory and reliability. To close, approximation by a branching process is also investigated to a certain extent.

What is a ‘cascading failure’? Whilst most people have an intuitive idea of what is meant by cascading, no definition exists within the framework of reliability theory. The aim of this paper is to explore the field of component dependence within a network structure, and to construct a definition of cascading failure.