A Shock Model Based Approachto Network Reliability

Abstract

We consider a network consisting of n components (links or nodes) and assume that the network has two states, up and down. We further suppose that the network is subject to shocks that appear according to a counting process and that each shock may lead to the component failures. Under some assumptions on the shock occurrences, we present a new variant of the notion of signature which we call it t-signature. Then t-signature based mixture representations for the reliability function of the network are obtained. Several stochastic properties of the network lifetime are investigated. In particular, under the assumption that the number of failures at each shock follows a binomial distribution and the process of shocks is non-homogeneous Poisson process, explicit form of the network reliability is derived and its aging properties are explored. Several examples are also provided.Networks include a wide variety of real-life systems in communication, industry, software engineering, etc. A network is defined to be a collection of nodes (vertices) and links (edges) in which some particular nodes are called terminals. For instance, nodes can be considered as road intersections, telecommunications switches, servers, and computers; and examples of links can be telecommunication fiber, railways, copper cable, wireless channels, etc.According to the existing literature, a network can be modeled by the triplet N = (V, E, T ), in which V shows the node set, where we assume |V | = m, E stands for link set, with |E| = n, and T ⊆ V is a set of all terminals. When all terminals of the network are connected to each other, the network is called T −connected. We assume that the components (links or nodes) of a network are subject to failure, where the failure of the components may occur according to a stochastic mechanism. A link failure means that the link is obliterated and a node failure means that all links incident to that node are erased. Assuming that the network has two states up, and down, the failure of the components may result in the change of the state of the network.In reliability engineering literature, several approaches are proposed to assess the reliability of a network. An approach, to study the reliability of a network with n components, is based on the assumption that the components of the network have statistically independent and identically distributed (i.i.d.) lifetimes X 1 , X 2 , . . . , X n , and the network has a lifetime T which is a function of X 1 , . . . , X n . An important concept in this approach is the notion of signature that is presented in the following definition; see [17] and [9].Definition 1. Assume that π = (e i 1 , e i 2 , . . . , e in ) is a permutation of the network components numbers. Suppose that all components in this permutation are up. We move along the permutation, from left to right, and turn the state of each component from up to down state. Under the assumption that all permutations are equally likely, the signature vector of the network is defined as s = (s 1 , ..., s n ) wherewh...

Maintenance Overtime Policy with Cumulative Damage

“…So that, replacing working number with shock number, we can consider replacement policies for cumulative damages models. There were also many studies in cumulative damage models (Stallmeyer and Walker, 1968;Bogdanoff and Kozin, 1985;Nakagawa, 2007), and network reliability models under the assumption that the components are subject to shock was introduced (Zarezadeh et al, 2016). We propose a maintenance policy of an operating unit which extends the overtime policy to a cumulative damage model: It would be reasonable for the unit to make appropriate policies with scheduled time or number of shocks to maintain or replace it.…”

Signature-Based Information Measures of Multi-State Networks

“…Note that in representation (1.2), s i,j = 0 for i ≥ j. For results on signature-based properties of the lifetime of networks and coherent systems with two or more states, we refer the reader to [1,2,7,11,14,15,24]. It should be mentioned that the probability matrix with elements defined as (1.1) is equal to the bivariate signature matrix (joint signature) corresponding to two systems with shared components given in [12] (see also [22]).…”

An Algorithm for computing the t-signature of two-state networks

“…It is more realistic to consider the case where more than one link may fail at each time. Motivated by this, Zarezadeh et al [13] studied the reliability of two-state networks under the aforementioned assumption.…”

“…see Lemma 1 of [13]. Let the discrete random variable M denote the minimum number of links whose failures cause to fail the network in a way of links failure order.…”