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Switching-Algebraic Analysis of System Reliability

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Advances in Reliability and System Engineering

Part of the book series: Management and Industrial Engineering ((MINEN))

Abstract

This chapter deals with the paradigm of handling system reliability analysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain. This paradigm is well-established within the academic circles of reliability theory and engineering, albeit virtually unknown outside these circles. The chapter lists and explains arguments in favor of this paradigm for systems described by verbal statements, fault trees, block diagrams, and network graphs. This is followed by a detailed exposition of the pertinent concept of the Real or Probability Transform of a switching (two-valued Boolean) function, and that of a Probability-Ready Expression (PRE). Some of the important rules used in generating a PRE are presented, occasionally along with succinct proofs. These include rules to achieve disjointness (orthogonality) of ORed formulas, and to preserve statistical independence, as much as possible, among ANDed formulas. Recursive relations in the Boolean domain are also discussed, with an application to the four versions of the AR algorithm for evaluating the reliability and unreliability of the k-out-of-n:G and the k-out-of-n:F systems. These four versions of the algorithm are explained in terms of signal flow graphs that are compact, regular, and acyclic, in addition to being isomorphic to the Reduced Ordered Binary Decision Diagram (ROBDD). An appendix explains some important properties of the concept of Boolean quotient, whose expectation in Boolean-based probability is the counterpart of conditional probability in event-based probability.

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Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, 21589, Saudi Arabia

    Ali Muhammad Rushdi

  2. Department of Electronics and Communications Engineering, Faculty of Engineering, Cairo University, Giza, 12613, Egypt

    Mahmoud Ali Rushdi

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  1. Ali Muhammad Rushdi
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Editors and Affiliations

  1. Dept of Mathematics, Clement Town, Graphic Era University Dept of Mathematics, Clement Town, Dehradun, India

    Mangey Ram

  2. Department of Mechanical Engineering, University of Aveiro Department of Mechanical Engineering, Aveiro, Portugal

    J. Paulo Davim

Appendix A: Boolean Quotient

Appendix A: Boolean Quotient

Let us define a literal to be a letter or its complement, where a letter is a constant or a variable. A Boolean term or product is a conjunction or ANDing of m literals in which no letter appears more than once. For m = 1, a term is a single literal and for m = 0, a term is the constant 1. Note that, according to this definition the constant 0 is not a term. Given a Boolean function and a term t, the Boolean quotient of with respect to t, denoted by (), is defined to be the function formed from by imposing the constraint {t = 1} explicitly [103], i.e.,

(19)

The Boolean quotient is also known as a ratio, a subfunction, or a restriction. Brown [103] lists and proves several useful properties of Boolean quotients, of which we reproduce the following ones:

(20)
(21)

{for n-variable functions and g and an m-variable term t with m ≤ n},

(22)
(23)
(24)
(25)

In this Appendix, we followed Brown [103] in denoting a Boolean quotient by an inclined slash . However, it is possible to denote it by a vertical bar to stress the equivalent meaning (borrowed from conditional probability) of conditioned by t or given t.

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Rushdi, A.M., Rushdi, M.A. (2017). Switching-Algebraic Analysis of System Reliability. In: Ram, M., Davim, J. (eds) Advances in Reliability and System Engineering. Management and Industrial Engineering. Springer, Cham. https://doi.org/10.1007/978-3-319-48875-2_6

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