A Shock-Based Model for the Reliability of Three-State Networks

Abstract

This paper is a study on the reliability of a three-state network under shocks in which each shock may cause the failure of more than one component at each time instant. The network is assumed to have n binary components and three states: up, partial performance, and down. The components are subject to failure due to the occurrence of shocks appearing based on a counting process, and some of the components may fail as a result of each shock. To give a model for the reliability of the network, a new variant of the notion of two-dimensional signature is introduced, which is called two-dimensional t-signature. Based on this new notion, some mixture representations are given for the joint reliability function of the entrance times into a partial performance state T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and a down state T. Several stochastic orderings and dependence properties regarding T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and T are provided. The results are also explored for the special case when the shocks appear according to a nonhomogeneous pure birth process under different conditions.