Reliability Polynomial Research Articles

The Reliability of a Class of Two-Layer Networks with Unreliable Edges

It is well known that networks are dynamic graphs, and the topology of a network can be described by a graph. Thus, the reliability of a network under edge failure is defined as the probability that its corresponding topological graph remains connected under the condition that the edges fail with independent probabilities. In this paper, the reliability of a class of two-layer networks is considered, where each layer is a complete graph and the edges joining different layers are one-to-one correspondingly connected. The edge failure probability is uniform in the same layer and distinct in different layers and between layers. The recursive formula for the reliability (reliability polynomial) of the two-layer network is obtained, and the corresponding algorithm is also given. Furthermore, the reliability of several networks is computed by Python, which verifies the correctness of the algorithm.

Reliability polynomial chaos metamodel for the dynamic behaviour of reinforced concrete bridges

The approximation of complex engineering problems and mathematical regressions serves as the authentic inspiration behind the artificial intelligence metamodeling methods. Among these methods, polynomial chaos expansion, along with artificial neural networks, has emerged at the forefront and become the most practical technique. Previous studies have highlighted their robust capabilities in solving complex problems and their wide utilization across numerous applications, particularly in structural analysis, optimization design problems, and predictive models of uncertainty outcomes. The aim of this article is to present a methodology that introduces their implementation of for structural engineering, primarily focusing on reinforced concrete bridges. The proposed approach consists of demonstrating the applicability of the polynomial chaos to evaluate the dynamic behavior of two-span reinforced concrete bridges through a predictive model of natural vibration properties for eigenvalues modal analysis. Subsequently, response spectral method is conducted according to the Moroccan guide for bridge seismic design and the prescription of the EUROCODE 8 within the context of reliability assessment using Monte Carlo simulation. The efficacy of the proposed approach is illustrated by a comparison between the predicted vibration properties and the resulting values obtained through finite element modal analysis and artificial neural networks. The polynomial chaos process is based on a collected dataset of multiple reinforced concrete bridges sourced from technical studies offices and the Regional Administration of the East, affiliated with the Moroccan Ministry of Equipment and Water. Finally, this work contributes to the field by enhancing predictive modeling and reliability evaluation for bridge engineering using artificial intelligence metamodels.

QRP: quantum reliability polynomial for stochastic reliability analysis of reversible circuits

QRP: quantum reliability polynomial for stochastic reliability analysis of reversible circuits

Estimating the reliability bounds of communication system by using sum of disjoint product method

In this paper, an algorithm is present that uses the sum of disjoint products (SDP) to calculate the reliability bounds of a complex system. The algorithm requires a set of all minimal paths and cuts to compute the bounds of the reliability polynomial. Furthermore, using Boolean algebra to construct a sum-of-products expression. This research aims to create the lower and upper bounds of reliability for a complex communication system (Double-bridge). By minimizing the number of disjointed (mutually exclusive) variables, the method can reduce the computer time required to estimate the probability of the principal events. The communication system is also used to establish the usefulness of the provided algorithm, which is better than the currently used SDP method for big data sets and more specific to understand and implement. Furthermore, simulations and numerical examples have shown that the new technique typically improves Spross bounds, particularly for highly reliable systems.

New approximations for network reliability

New approximations for network reliability

Improvement of Network Reliability by Hybridization of the Penalty Technique Based on Metaheuristic Algorithms

In this study, a network created by converting the Petri net into a network for shutdown system, by calculatinga network’s nonlinear reliability polynomial by splitting the network into two, each having a source node and anterminal node. This the procedure converts the system into two parallel-series block diagrams that are connected ina series. The metaheuristic algorithm using the honey badger algorithm (HBA) and Dwarf Mongoose OptimizationAlgorithm (DMO) have been used to improve the problem for the shutdown network hybridizing these algorithmsemploying the penalty method, we will obtained the (PFMHBA, PFMDMO) algorithms. Thereafter, we comparedthe results of the use of the hybridization technique with the results of the process that does not use this technique. Theobjective of this comparison was to ascertain whether the use of hybridization makes the results increasing reliable,reducing cost, and shortening the duration of implementation.

The Roots of the Reliability Polynomials of Circular Consecutive-k-out-of-n:F Systems

The zeros of the reliability polynomials of circular consecutive-k-out-of-n:F systems are studied. We prove that, for any fixed k≥2, the set of the roots of all the reliability polynomials (for all n≥k) is unbounded in the complex plane. In the particular case k=2, we show that all the nonzero roots are real, distinct numbers and find the closure of the set of roots. For every n≥k, the expressions of the minimum root and the maximum root are given, both for circular as well as for linear systems.

Reliability polynomials of consecutive‐k‐out‐of‐n:Fsystems have unbounded roots

AbstractThis article studies the roots of the reliability polynomials of linear consecutive‐k‐out‐of‐n:Fsystems. We prove that these roots are unbounded in the complex plane, for any fixed . In the particular case , we show that the reliability polynomials have only real roots and highlight the closure of these roots by establishing their explicit formulas. We also point out that in this case, for any fixedn, the nonzero roots of the reliability polynomial are distinct numbers.

On the Roots of a Family of Polynomials

The aim of this paper is to give a characterization of the set of roots of a special family of polynomials. This family is relevant in reliability theory since it contains the reliability polynomials of the networks created by series-parallel compositions. We prove that the set of roots is bounded, being contained in the two disks of the radius equal to the golden ratio, centered at 0 and at 1. We study the closure of the set of roots and prove that it includes two disks centered at 0 and 1 of a radius slightly greater than 1, as well as the sinusoidal spirals centered at 0 and at 1, respectively. The expression of some limit points is also provided.

More reliable graphs are not always stronger

AbstractA graph is stronger than a graph if has at least as many connected spanning subgraphs of size as for any positive integer . Counting the number of connected spanning subgraphs of fixed size allows us to compute the reliability of a graph. Formally, the reliability polynomial of a graph is the probability that the graph is connected when each edge is deleted independently with the same fixed probability. A graph is uniformly more reliable than if its reliability polynomial is greater than or equal to the reliability polynomial of for all probabilities. As a direct consequence of the definition, a sufficient condition for to be uniformly more reliable than is for to be stronger than . In this paper, we show that the sufficient condition is not necessary by providing an example of two infinite families of graphs, and , such that is uniformly more reliable than but is not stronger than .

Signature-Based Analysis of the Weighted-r-within-Consecutive-k-out-of-n: F Systems

In the present paper, we deliver a reliability study of the weighted-r-consecutive-k-out-of-n: F reliability systems consisting of independent and identically distributed components. The signature vector of the structures is arithmetically determined by the aid of an appropriate Monte-Carlo simulation scheme. Several conclusions for the corresponding reliability polynomials are also drawn. An extensive numerical investigation is carried out to evaluate the performance of the weighted-r-consecutive-k-out-of-n: F reliability systems. Some stochastic signature-based comparisons among several members of the underlying family of reliability structures are also discussed.

Approximating the Level Curves on Pascal’s Surface

It is well-known that in general the algorithms for determining the reliability polynomial associated to a two-terminal network are computationally demanding, and even just bounding the coefficients can be taxing. Obviously, reliability polynomials can be expressed in Bernstein form, hence all the coefficients of such polynomials are fractions of the binomial coefficients. That is why we have very recently envisaged using an extension of the classical discrete Pascal’s triangle (which comprises all the binomial coefficients) to a continuous version/surface. The fact that this continuous Pascal’s surface has real values in between the binomial coefficients makes it appealing as being a mathematical concept encompassing all the coefficients of all the reliability polynomials (which are integers, as resulting from counting processes) and more. This means that, the coefficients of any reliability polynomial can be represented as discrete steps (on level curves of integer values) on Pascal’s surface. The equation of this surface was formulated by means of the gamma function, for which quite a few approximation formulas are known. Therefore, we have started by reviewing many of those results, and have used a selection of those approximations for the level curves problem on Pascal’s surface. Towards the end, we present fresh simulations supporting the claim that some of these could be quite useful, as being both (reasonably) easy to calculate as well as fairly accurate.

A directional ghost-cell immersed boundary method for low Mach number reacting flows with interphase heat and mass transfer

This paper presents a directional ghost-cell immersed boundary method for low Mach number reacting flows with general boundary conditions, extending the approach described in Chi et al. (2020) [17]. The method employs locally directional schemes for ghost value reconstruction and utilization along each discretization direction. In this manner, the boundary condition can be naturally imposed on the boundary intersection point along each coordinate direction, allowing an easy and straightforward implementation of complex boundary conditions. Using Taylor series approximation of fluid points near the immersed boundary, the boundary variable and its gradient governed by arbitrary boundary condition can be implicitly involved, leading to a reliable polynomial extrapolation for the ghost values. In this way, Dirichlet, Neumann and Robin boundary conditions are implemented for general variables with formally second-order accuracy. For reacting gas-solid flow with surface reactions, the reaction-related coefficients in Robin boundary condition lead to a complex implementation in conventional IBM techniques, while the present directional framework leads to a straightforward algorithm while preserving accuracy. The proposed method has been checked by a series of test cases with different boundary conditions, including basic flow, heat and species transport, Stefan problem, and finally two practical applications involving heat and mass transfer. The local accuracy of all boundary conditions exhibits nearly second-order convergence, as expected. While only a single solid object is considered in this first work, the same method can be simply extended to multiple objects of arbitrary shape, leading to fully resolved simulation of reactive particle-laden flows.

On a Surface Associated with Pascal’s Triangle

An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.

Efficient methods for computing the reliability polynomials of graphs and complex networks

Efficient methods for computing the reliability polynomials of graphs and complex networks

Entropy and enumeration of spanning connected unicyclic subgraphs in self-similar network

The number of spanning connected unicyclic subgraphs is a critical quantity to characterize the reliability of networks. In 1993, Colbourn in [Colbourn (1993)] proposed a problem about reliability polynomial: Can the number of spanning connected unicyclic subgraphs of a graph be computed efficiently? Up to now, there is no research about this problem. In this paper, we mainly study the entropy and the enumeration of spanning connected unicyclic subgraphs in self-similar networks. We propose a linear algorithm for computing the number of spanning connected unicyclic subgraphs in self-similar network (x, y)-flower. By the algorithm, the exact number of spanning connected unicyclic subgraphs is determined in (x, y)-flower networks. In addition, we define the entropy of spanning connected unicyclic subgraphs and obtain the entropies of spanning connected unicyclic subgraphs in (x, y)-flower networks.

Generalized Convexity Properties and Shape-Based Approximation in Networks Reliability

Some properties of generalized convexity for sets and functions are identified in case of the reliability polynomials of two dual minimal networks. A method of approximating the reliability polynomials of two dual minimal network is developed based on their mutual complementarity properties. The approximating objects are from the class of quadratic spline functions, constructed based on both interpolation conditions and shape knowledge. It is proved that the approximant objects preserve both the high-order convexity and some extremum properties of the exact reliability polynomials. It leads to pointing out the area of the network where the maximum number of paths is achieved. Numerical examples and simulations show the performance of the algorithm, both in terms of low complexity, small error and shape preserving. Possibilities of increasing the accuracy of approximation are discussed.

Plotting in a Formally Verified Way

An invaluable feature of computer algebra systems is their ability to plot the graph of functions. Unfortunately, when one is trying to design a library of mathematical functions, this feature often falls short, producing incorrect and potentially misleading plots, due to accuracy issues inherent to this use case. This paper investigates what it means for a plot to be correct and how to formally verify this property. The Coq proof assistant is then turned into a tool for plotting function graphs using reliable polynomial approximations. This feature is provided as part of the CoqInterval library.

On the reliability estimation of stochastic binary systems

AbstractA stochastic binary system (SBS) is a multicomponent on‐off system subject to random independent failures on its components. After potential failures, the state of the subsystem is ruled by a logical function (called structure function) that determines whether the system is operational or not. A SBS serves as a natural generalization of network reliability analysis, where the goal is to find the probability of correct operation of the system (in terms of connectivity, network diameter, or different measures of success). A particular subclass of interest is stochastic monotone binary systems (SMBS), which are characterized by nondecreasing structure. We explore the combinatorics of SBS, which provide building blocks for system reliability estimation, looking at minimal nonoperational subsystems, called mincuts. One key concept to understand the underlying combinatorics of SBS is duality. As methods for exact evaluation take exponential time, we discuss the use of Monte Carlo algorithms. In particular, we discuss the F‐Monte Carlo method for estimating the reliability polynomial for homogeneous SBS, the recursive variance reduction for SMBS, which builds upon the efficient determination of mincuts, and three additional methods that combine in different ways the well‐known techniques of permutation Monte Carlo and splitting. These last three methods are based on a stochastic process called the creation process, a temporal evolution of the SBS which is static by definition. All the methods are compared using different topologies, showing large efficiency gains over the basic Monte Carlo scheme.

Network reliability in hamiltonian graphs

The reliability polynomial of a graph gives the probability that a graph remains operational when all its edges could fail independently with a certain fixed probability. In general, the problem of finding uniformly most reliable graphs inside a family of graphs, that is, one graph whose reliability is at least as large as any other graph inside the family, is very difficult. In this paper, we study this problem in the family of graphs containing a hamiltonian cycle.