Reliability Characterization of Binary-Imaged Multi-State Coherent Threshold Systems

Abstract

A notable reliability model is the binary threshold system (also called the weighted-k-out-of-n system), which is a dichotomous system that is successful if and only if the weighted sum of its component successes exceeds or equals a particular threshold. The aim of this paper is to extend the utility of this model to the reliability analysis of a homogeneous binary-imaged multi-state coherent threshold system of (m+1) states, which is a non-repairable system with independent non-identical components. The paper characterizes such a system via switching-algebraic expressions of either system success or system failure at each non-zero level. These expressions are given either (a) as minimal sum-of-products formulas, or (b) as probability–ready expressions, which can be immediately converted, on a one-to-one basis, into probabilities or expected values. The various algebraic characterizations can be supplemented by a multitude of map representations, including a single multi-value Karnaugh map (MVKM) (giving a superfluous representation of the system structure function S), (m+1) maps of binary entries and multi-valued inputs representing the binary instances of S, or m maps, again of binary entries and multi-valued inputs, but now representing the success/failure at every non-zero level of the system. We demonstrate how to reduce these latter maps to conventional Karnaugh maps (CKMs) of much smaller sizes. Various characterizations are inter-related, and also related to pertinent concepts such as shellability of threshold systems, and also to characterizations via minimal upper vectors or via maximal lower vectors.

A pseudo Karnaugh mapping approach for datasets imbalance

“…Other contributions like variable-entered Karnaugh map has been proposed in Banzhaf voting problems [6] and multi-state system analysis [7][8][9] also proposed a numbered Karnaugh Map to assist their researches.…”

تحليل رمزي مستند للجبر البولاني لمعولية النظم المتسقة متعددة الحالات ذات المركبات غير المتجانسة

“… The system is a coherent one enjoying the properties of causality, monotonicity, and component relevancy [1,2,4,[31][32][33][34] .…”

“…Component State Vectors and  A vector is larger than another vector (denoted ) if every element of is at least as large as the corresponding element of , and at least one element of is larger than the corresponding element of . 1 is a function of 1 only) [34] . For a binary-imaged system, elements of the set of MUVs are vectors of or 0 components only, and elements of the set of MLVs are vectors of or components only [34] .…”

“…The literature abounds with standard research techniques for the reliability analysis of MSSs [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34] . Most of these standard techniques rely on the utilization of discrete non-binary functions [35][36][37] or multiple-valued logic [38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] .…”

“…The formula of each of these levels is then written as a probability-ready expression, thereby allowing its immediate conversion, on a one-to-one basis, into a probability or expected value. The analysis will be seen to be particularly simple when the multi-state system is binary-imaged, i.e., when its success at each specific level is dependent only on component successes at the same level [8,9,16,[30][31][32][33][34] . The paper strives to provide a pedagogically-oriented treatment that establishes a clear and insightful interrelationship between binary modeling and MSS modeling by stressing that multi-valued concepts are natural and simple extensions of two-valued ones.…”