This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.
Suppose that a system has three states up, partial performance and down. We assume that for a random time T 1 the system is in state up, then it moves to state partial performance for time T 2 and then the system fails and goes to state down. We also denote the lifetime of the system by T , which is clearly T = T 1 + T 2 . In this paper, several stochastic comparisons are made between T , T 1 and T 2 and their reliability properties are also investigated. We prove, among other results, that different concepts of dependence between the elements of the signatures (which are structural properties of the system) are preserved by the lifetimes of the states of the system (which are aging properties of the system). Various illustrative examples are provided.
This article develops information optimal models for the joint distribution based on partial information about the survival function or hazard gradient in terms of inequalities. In the class of all distributions that satisfy the partial information, the optimal model is characterized by well-known information criteria. General results relate these information criteria with the upper orthant and the hazard gradient orderings. Applications include information characterizations of the bivariate Farlie-Gumbel-Morgenstern, bivariate Gumbel, and bivariate generalized Gumbel, for which no other information characterization are available. The generalized bivariate Gumbel model is obtained from partial information about the survival function and hazard gradient in terms of marginal hazard rates. Other examples include dynamic information characterizations of the bivariate Lomax and generalized bivariate Gumbel models having marginals that are transformations of exponential such as Pareto, Weibull, and extreme value. Mixtures of bivariate Gumbel and generalized Gumbel are obtained from partial information given in terms of mixtures of the marginal hazard rates.
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...
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