System reliability analysis: A tutorial

Abstract

This paper deals with the logical formulation of a system for purposes of reliability analyses and both exact and approximate methods of calculating the system reliability. The first part deals with the logical concepts, and the second part with probability calculations. The logical formulation in Part 1 starts from first principles. The universal set U of system states is a “Boolean algebra”. The “power set” P is the set of subsets of U . The subset of system success states or “paths” is a “lattice” within U , also an element of P , represented by a Boolean polynomial. The term of the polynomials are monomials which give the indicators for the “sublattices” or “complete subsets” of U within the lattice of paths. The optimal representation of this lattice polynomial is in the minimalized form; the terms of the minimalized lattice of paths are called the “minimal paths”. Similarly, the subset of system failure states or “cuts” is a lattice polynomial whose terms are sublattices within the lattice of cuts; the terms of the minimalized lattice polynomial are the “minimal cuts”. Logically consistent systems (also known as “coherent structures”) have certain properties with respect to the partial ordering of paths and cuts. The concepts of Boolean logic, minimalization, etc., apply to systems that do not have the “consistency” property, as well as to those that do have it. The second part of this paper deals with ways of using a minimalized lattice polynomial to derive a numerical-valued probability function from which to calculate the system reliability: also computer software, error bounds and approximations. The reliability is the probability that the actual state of the system is an element of the lattice of paths. The exact probability is derived from the minimalized lattice polynomial of paths by the method of inclusion-exclusion, also known as Poincaré's Theorem. Dually the probability of failure, or system unreliability, is derived from the minimalized lattice of cuts. Modularization and/or inversion can be helpful in keeping the probability function reasonably small in size. Computer software is available both to generate the probability polynomial and to calculate either the reliability of unreliability. Approximations can be used based upon simplifying assumptions which delete low probability terms from the function. Conservative error bounds are obtainable for some models. The approximation techniques include serializing methods (“single-point failures” and “parts count”), very large system approximations for fault-tree applications and the Esary—Proschan bounds.