Separability in Stochastic Binary Systems

A Stochastic Binary System (SBS) is a mathematical model of multi-component on-off systems subject to random failures. SBS models extend classical network reliability models (where the components subject to failure are nodes or links of a graph) and are able to represent more complex interactions between the states of the individual components and the operation of the system under study. The reliability evaluation of stochastic binary systems belongs to the class of NP-Hard computational problems. Furthermore, the number of states is exponential with respect to the size of the system (measured in the number of components). As a consequence, the representation of an SBS becomes a key element in order to develop exact and/or approximation methods for reliability evaluation. The contributions of this paper are three-fold. First, we present the concept of separable stochastic binary systems, showing key properties, such as an efficient representation and complexity in the reliability evaluation. Second, we fully characterize separable systems in two ways, using a geometrical interpretation and minimum-cost operational subsystems. Finally, we show the application of separable systems in network reliability models, specifically in the all-terminal reliability model, which has a wide spectrum of applications. Index Terms—Stochastic Binary System, Network Reliability, Computational Complexity, Chernoff Inequality

On the Reliability Estimation of Stochastic Binary System

A stochastic binary system is a multi-component 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. Stochastic binary systems (SBS) serve 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 non-decreasing structure. We explore the combinatorics of SBS, which provide building blocks for system reliability estimation, looking at minimal non-operational 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 (RVR) 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 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.Agencia Nacional de Investigación e InnovaciónMath-AMSU

GRASP/VND Optimization Algorithms for Hard Combinatorial Problems

Two hard combinatorial problems are addressed in this thesis. The first one is known as the ”Max CutClique”, a combinatorial problem introduced by P. Martins in 2012. Given a simple graph, the goal is to find a clique C such that the number of links shared between C and its complement C C is maximum. In a first contribution, a GRASP/VND methodology is proposed to tackle the problem. In a second one, the N P-Completeness of the problem is mathematically proved. Finally, a further generalization with weighted links is formally presented with a mathematical programming formulation, and the previous GRASP is adapted to the new problem. The second problem under study is a celebrated optimization problem coming from network reliability analysis. We assume a graph G with perfect nodes and imperfect links, that fail independently with identical probability ρ ∈ [0,1]. The reliability RG(ρ), is the probability that the resulting subgraph has some spanning tree. Given a number of nodes and links, p and q, the goal is to find the (p,q)-graph that has the maximum reliability RG(ρ), uniformly in the compact set ρ ∈ [0,1]. In a first contribution, we exploit properties shared by all uniformly most-reliable graphs such as maximum connectivity and maximum Kirchhoff number, in order to build a novel GRASP/VND methodology. Our proposal finds the globally optimum solution under small cases, and it returns novel candidates of uniformly most-reliable graphs, such as Kantor-Mobius and Heawood graphs. We also offer a literature review, ¨ and a mathematical proof that the bipartite graph K4,4 is uniformly most-reliable. Finally, an abstract mathematical model of Stochastic Binary Systems (SBS) is also studied. It is a further generalization of network reliability models, where failures are modelled by a general logical function. A geometrical approximation of a logical function is offered, as well as a novel method to find reliability bounds for general SBS. This bounding method combines an algebraic duality, Markov inequality and Hahn-Banach separation theorem between convex and compact sets

Analysis and optimization of highly reliable systems

In the field of network design, the survivability property enables the network to maintain a certain level of network connectivity and quality of service under failure conditions. In this thesis, survivability aspects of communication systems are studied. Aspects of reliability and vulnerability of network design are also addressed. The contributions are three-fold. First, a Hop Constrained node Survivable Network Design Problem (HCSNDP) with optional (Steiner) nodes is modelled. This kind of problems are N P-Hard. An exact integer linear model is built, focused on networks represented by graphs without rooted demands, considering costs in arcs and in Steiner nodes. In addition to the exact model, the calculation of lower and upper bounds to the optimal solution is included. Models were tested over several graphs and instances, in order to validate it in cases with known solution. An Approximation Algorithm is also developed in order to address a particular case of SNDP: the Two Node Survivable Star Problem (2NCSP) with optional nodes. This problem belongs to the class of N P-Hard computational problems too. Second, the research is focused on cascading failures and target/random attacks. The Graph Fragmentation Problem (GFP) is the result of a worst case analysis of a random attack. A fixed number of individuals for protection can be chosen, and a non-protected target node immediately destroys all reachable nodes. The goal is to minimize the expected number of destroyed nodes in the network. This problem belongs to the N P-Hard class. A mathematical programming formulation is introduced and exact resolution for small instances as well as lower and upper bounds to the optimal solution. In addition to exact methods, we address the GFP by several approaches: metaheuristics, approximation algorithms, polytime methods for specific instances and exact methods in exponential time. Finally, the concept of separability in stochastic binary systems is here introduced. Stochastic Binary Systems (SBS) represent a mathematical model of a multi-component on-off system subject to independent failures. The reliability evaluation of an SBS belongs to the N P-Hard class. Therefore, we fully characterize separable systems using Han-Banach separation theorem for convex sets. Using this new concept of separable systems and Markov inequality, reliability bounds are provided for arbitrary SBS

Confiabilidad diámetro acotada para el modelo hostil. Construcción de estrategias de selección de cortes minimales para el método de la reducción recursiva de la varianza

Los Sistemas Binarios Estocásticos (SBS por sus siglas en inglés) son modelos matemáticos que permiten generalizar la noción clásica de confiabilidad de redes de comunicación. Los SBS consisten en un conjunto finito de componentes sujetos a fallas aleatorias, y una función lógica que describe la operación o falla del sistema, para cada estado posible de sus componentes. Instancias particulares de estos problemas pueden ser modelados bajo funciones monótonas, clasificándolos como Sistemas Binarios Estocásticos Monótonos (SMBS), los cuales heredan el orden natural del producto cartesiano. En el marco de Internet e inspirado en aplicaciones sensibles a la latencia, el problema de Confiabilidad Diámetro Acotado analiza la probabilidad de que dos o más nodos que desean comunicarse estén conectados por caminos acotados, ante fallas en los elementos de la red. Este problema ha sido estudiado principalmente en estructuras de redes donde fallan únicamente aristas o únicamente fallan nodos, y no tanto en fallas conjuntas (o "fallas simultáneas" de nodos y aristas). El problema de Confiabilidad Diámetro Acotada para el modelo hostil es una instancia de los sistemas binarios estocásticos monótonos, y la complejidad de su evaluación se clasifica como NP-difícil. Las contribuciones de este documento son las siguientes. En primera instancia se implementa el método de Reducción Recursiva de la Varianza (RVR) para el cálculo de Confiabilidad Diámetro Acotada del modelo hostil. Se realiza un estudio de sensibilidad del método RVR ante distintas estrategias de selección de cortes minimales. Finalmente, se brinda un estudio comparativo entre el método Monte Carlo Crudo y RVR, observando el desempeño de los métodos elegidos.Stochastic Binary Systems (SBS) are mathematical models that allow extend the classical notion of reliability in communication networks. An SBS consists of a set of finite components subject to random failures, and a logical function that describes the operation or failure of the system, for each possible configuration. Particular instances of these problems are modeled under monotone functions, known as Stochastic Monotone Binary System (SMBS), which inherit the natural order of the cartesian product. Within the framework of the Internet and inspired by latency-sensitive applications, the Diameter Constrained Reliability is the probability that two or more target nodes are connected by bounded paths, in the resulting network after potential failures on its components. The scientific literature considers structural links or node failures. Here, we consider joint failures instead, what is known as the Hostile Network model. The Diameter Constrained Reliability problem for the hostile model is an instance of Stochastic Monotone Binary System, and its reliability evaluation belongs to the N P-Hard class. The contributions of this thesis can be summarized as follows. First, the Recursive Variance Reduction (RVR) method is implemented to estimate the Diameter Constrained Reliability in this novel hostile environment with joint node/link failures. Then, a sensibility analysis of RVR method is performed, following different cut-selection strategies. Finally, a performance analysis between Monte Carlo Crudo and RVR methods is provided