Separability in Stochastic Binary Systems

Abstract

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.