Abstract
Symmetric switching functions (SSFs) play a prominent role in the reliability analysis of a binary k-out-of-n: G system, which is a dichotomous system that is successful if and only if at least k out of its n components are successful. The aim of this paper is to extend the utility of SSFs to the reliability analysis of a multi-state k-out-of-n: G system, which is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 (the lowest output level) and M (the highest output level)) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j. This paper is devoted to the analysis of non-repairable multi-state k-out-of-n: G systems with independent non-identical components. The paper utilizes algebraic techniques of multiple-valued logic (together with known properties of SSFs) to evaluate each of the multiple levels of the system output as an individual binary or propositional function of the system multi-valued inputs. 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 symbolic reliability analysis of a commodity-supply system (which serves as a standard gold example of a multi-state k-out-of-n: G system) is completed successfully herein, yielding results that have been checked symbolically, and also were shown to agree numerically with those obtained earlier.
Highlights
The reliability literature deals mainly with binary or dichotomous systems, in which both a system and its components have two states (i.e., either operational or non-operational)
The main contribution of the paper is to demonstrate that, to binary k-out-of-n systems, multi-state k-out-of-n systems can be conveniently analyzed with the aid of symmetric switching functions (SSFs)
Description of an Example Multi-State k-out-of-n System While a binary k-out-of-n: G system is a dichotomous system that is successful if and only if at least k out of its n components are successful, a multi-state k-out-of-n: G system is a multi-state system whose multi-valued success is greater than or equal to a certain value j (lying between 1 and M) whenever at least km components are in state m or above for all m such that 1 ≤ m ≤ j (Tian et al, 2008; Mo et al, 2015)
Summary
The reliability literature deals mainly with binary or dichotomous systems, in which both a system and its components have two states (i.e., either operational (successful) or non-operational (failed)). This analysis adapts several important concepts of switching algebra to multi-valued logic, including those of probability-ready expressions, Boolean quotients, and the Boole-Shannon expansion.
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