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Estimating the reliability of complex systems using various bounds methods

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Abstract

The precise two-terminal reliability calculation becomes more difficult when the numeral of components of the complex system increases. The accuracy of approximation methods is often adequate for expansive coverage of practical applications, while the algorithms and computation time are typically simplified. As a result, the reliability bounds of two-terminal systems and estimation methods have been established. Our method for determining a complex system's reliability lower and upper bounds employs a set of minimal paths and cuts. This paper aims to present a modern assessment of reliability bounds for coherent binary systems and a comparison of various reliability bounds in terms of subjective, mathematical, and efficiency factors. We performed the suggested methods in Mathematica and approximated their interpretation with existing ones. The observed results illustrate that the proposed Linear and Quadratic bounds (LQb) constraint is superior to Esary-Proschan (EPb), Spross (Sb), and Edge-Packing (EDb) bounds in the lower bond, and the EDb bound is preferable to other methods above in the upper bond. This modification is attributed to sidestepping certain duplicative estimations that are part of the current methods. Given component test data, the new measure supplies close point bounds for the system reliability estimation. The Safety–Critical-System (SCS) uses an illustrative model to show the reliability designer when to implement certain constraints. The numerical results demonstrate that the proposed methods are computationally feasible, reasonably precise, and considerably speedier than the previous algorithm version. Extensive testing on real-world networks revealed that it is impossible to enumerate all minimal paths or cuts, allowing one to derive precise bounds.

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Abbreviations

Sb:

Spross bound

EPb:

Esary-proschan bound

EDb:

Edge-packing bound

LQb:

Linear and quadratic bound

SCS:

Safety–critical system

i. i. d.:

Independent and identically distributed

\({x}_{i}\) :

A system component

\(\overline{{x }_{i}}\) :

The complement of component \({x}_{i}\) such that \(\overline{{x }_{i}}=1-{x}_{i}\)

\(Pr({x}_{i})\) :

The Probability of component \({x}_{i}\) such that \(Pr\left({x}_{i}\right)={r}_{i}\)

\({r}_{i}\) :

The reliability of component \({x}_{i}\)

\({q}_{i}\) :

The unreliability of component \({x}_{i}\)

\({\varvec{X}}\) :

The vector of all component \({x}_{i}\) of the system

\({\varvec{P}}\) :

The vector of all minimal path \({P}_{i}\) of the system

\({\varvec{D}}{\varvec{P}}\) :

The vector of all disjoint minimal path \(D{P}_{i}\) of the system

\({\varvec{C}}\) :

The vector of all minimal cut \({C}_{i}\) of the system

\({\varvec{D}}{\varvec{C}}\) :

The vector of all disjoint minimal cut \(D{C}_{i}\) of the system

\(IM\) :

Incidence matrix of the system

\(CM\) :

Minimal cut matrix of the system

\({R}_{Exact}\) :

The exact reliability of the system

\({Q}_{Exact}\) :

The exact unreliability of the system

\({R}_{LB}\) :

The reliability lower bound of the system

\({R}_{UB}\) :

The reliability upper bound of the system

\({B}_{j,r}\) :

The Boolean product of \({x}_{i}\) and \(\overline{{x }_{i}}\) for some \({P}_{i}\) or \({C}_{i}\)

\({A}_{j,i}\) :

Sets {\({B}_{j,r}\)} of disjoint products derived from \({P}_{i}\) or \({C}_{i}\)

\({\alpha }_{i}\) :

Portion that are obtained successively from

\({A}_{j,i}\) :

For minimal path \({P}_{i}\)

\({\beta }_{j}\) :

Portion that are obtained successively from

\({A}_{j,i}\) :

For minimal cut \({C}_{j}\)

\(\mu \left(g\right)\) :

A specified lower bound for a spross bound method

\(\gamma \left(h\right)\) :

A specified upper bound for a spross bound method

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  1. Department of Mathematics, Directorate of Education Babylon, Babylon, 51005, Iraq

    Emad Kareem Mutar

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Appendix

Appendix

The following code allows the efficient computation of the IM ( in Eq. (3)) and CM ( in Eq. (4)) by using the MATHEMATICA software. Algorithms prepared by the author have been employed to find matrices in Eqs. (19) and (20).

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Mutar, E.K. Estimating the reliability of complex systems using various bounds methods. Int J Syst Assur Eng Manag 14, 2546–2558 (2023). https://doi.org/10.1007/s13198-023-02108-7

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