Signature-based block diagram analysis. Calculating complex-structure signatures

Abstract

Aim. The signature theory allows comparing the structural dependability of complex systems consisting of identical components using a numeric vector referred to as signature that is determined using a certain algorithm. The signature describes a system’s structure and is associated with the order statistics of the components’ times to failure. The paper aims to familiarize the Russian readers with the theory, as well as to build simple algorithms that would allow finding the signature of an arbitrary VooL system, including a series parallel system using the known signatures of its subsystems.Methods. Calculations and theorem proving primarily involve the use of combinatorics with various element selection patterns. Additionally, classical methods of the probability theory and mathematical theory of dependability are used.Conclusions. The paper introduces such concept of the mathematical dependability theory as technical system signature. The primary purpose of this characteristic consists in comparing the structural dependability of several, in particular, two systems. The algorithm is sufficiently simple and involves comparing two finite numeric arrays according to a certain rule. Structural dependability is understood as the dependability of connection circuits of components that are identical in terms of dependability. That significantly reduces the capabilities of the signature theory, as practically all technical systems consist of components with various dependability. However, with a certain conservatism, it can be considered that all of a system’s components have a dependability identical to that of the least dependable one. The degree of conservatism will be determined by the difference in the component dependability that will be a sufficiently low value for highly dependable components. Additionally, we should note the purely mathematical beauty of the signature theory that has recently been gaining significant momentum in foreign scientific publications. It must be noted that according to the classical laws of combinatorics, as the number n of system component elements grows, the construction of a technical system signature speeds up significantly. Therefore, it becomes necessary to develop sufficiently simple algorithms for calculating the signature of a random system. As part of the presented work, analytical approaches to signature acquisition were developed for both simple cases when a single component is, in series or parallely, added to a subsystem, and more complex situations when one or more subsystems are added to another subsystem. A number of cases of signature construction were considered using both the classical method, and the one suggested in this paper. An example was analysed of system dependability comparison using their respective signatures. The structural dependability of systems with various numbers of components was compared. The paper proves a number of theorems that allow calculating the signature of a random structure diagram consisting of components that are identical in terms of dependability.