An exposition of system reliability analysis with an ecological perspective
TL;DR: The paper treats the problem of survival reliability which is the probability of successful migration of a specific species from a critical habitat patch to destination habitat patches via heterogeneous imperfect corridors and contributes methods for computing a new measure of reliability that arises when paths to destination habitats patches share common corridors.
About: This article is published in Ecological Indicators.The article was published on 2016-04-01. It has received 23 citations till now. The article focuses on the topics: Reliability theory & Critical habitat.
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01 Jan 2017
TL;DR: This chapter deals with the paradigm of handling system reliabilityAnalysis in the Boolean domain as a supplement to (rather than a replacement to) analysis in the probability domain, and explains some important properties of the concept of Boolean quotient.
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.
31 citations
TL;DR: The paper demonstrates that the prime implicants of the system threshold function are its Minimal Winning Coalitions (MWC), and stresses the utility of threshold Boolean functions in the understanding, study, analysis, and design of weighted voting systems irrespective of size.
Abstract: Weighted voting systems play a crucial role in the investigation and modeling of manyengineering structures and political and socio-economic phenomena. There is an urgentneed to describe these systems in a simplified powerful mathematical way that can begeneralized to systems of any size. An elegant description of voting systems is presentedin terms of threshold Boolean functions. This description benefits considerably fromthe wealth of information about these functions, and of the potpourri of algebraic andmap techniques for handling them. The paper demonstrates that the prime implicantsof the system threshold function are its Minimal Winning Coalitions (MWC). Thepaper discusses the Boolean derivative (Boolean difference) of the system thresholdfunction with respect to each of its member components. The prime implicants of thisBoolean difference can be used to deduce the winning coalitions (WC) in which thepertinent member cannot be dispensed with. Each of the minterms of this Booleandifference is a winning coalition in which this member plays a pivotal role. However,the coalition ceases to be winning if the member defects from it. Hence, the numberof these minterms is identified as the Banzhaf index of voting power. The conceptsintroduced are illustrated with detailed demonstrative examples that also exhibit someof the known paradoxes of voting- system theory. Finally, the paper stresses the utilityof threshold Boolean functions in the understanding, study, analysis, and design ofweighted voting systems irrespective of size.
19 citations
18 Mar 2019
TL;DR: The analysis of a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components yields a Multi-Valued Karnaugh Map (MVKM), which serves as an explicit function of the multi-valued inputs of the system.
Abstract: 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 (the lowest non-zero 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 a commodity-supply system that serves as a standard gold example of a non-repairable multi-state k-out-of-n: G system with independent non-identical components. We express each instance of the multi-state system output as an explicit function of the multi-valued inputs of the system. The ultimate outcome of our analysis is a Multi-Valued Karnaugh Map (MVKM), which serves as a natural, unique, and complete representation of the multi-state system. To construct this MVKM, we use “binary” entities to relate each of the instances of the output to the multi-valued inputs. These binary entities are represented via an eight-variable Conventional Karnaugh Map (CKM) that is adapted to a map representing four variables that are four-valued each. Despite the relatively large size of the maps used, they are still very convenient, thanks to their regular structure. No attempt was made to draw loops on the maps or to seek minimal formulas. The maps just served as handy tools for combinatorial representation and for collectively implementing the operations of ANDing, ORing, and complementation. The MVKM obtained serves as a means for symbolic analysis yielding results that agree numerically with those obtained earlier. The map is a useful tool for visualizing many system properties, and is a valuable resource for computing a plethora of Importance Measures for the components of the system.
15 citations
Additional excerpts
... The map offers a convenient pictorial mechanism for decomposing its output function into various sub-functions, thereby constructing a multi-valued expansion tree or decision diagram for this function [1-4, 19-23, 41, 65-71]....
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01 Jan 2016
TL;DR: It is demonstrated here that in many cases this analysis is still possible via elementary faulttree manipulations that use the concept of a Boolean quotient to effectively implement Bayes’ Theorem in the Boolean domain.
Abstract: Fault trees are top-down formal deductive analytic tools with diverse applications in many fields such as reliability, safety and security. Forward fault tree analysis (FTA) can be termed a priori analysis since it predicts the top-event probability in terms of basic-event probabilities. This paper offers a tutorial exposition and a detailed comparison of two kinds of backward or a posteriori FTA that are implemented in the probability domain and in the Boolean domain, respectively. For the probability-domain a posteriori FTA, it is assumed that the top event probability is known. For example, when the top event is presumed to have occurred, then it has a probability of one. The analysis proceeds recursively in the probability domain to assess the probabilities of lower events under certain realistic assumptions such as mutual exclusiveness or statistical independence of the input events for a specific gate, and with the utilization of educated guesses on certain ratios of probabilities of such events. This paper offers a detailed mathematical procedure for implementing this a posteriori FTA that makes the most of the concept of duality. The procedure is demonstrated via a detailed illustrative example. The paper also considers the a posteriori FTA in the Boolean domain. Such an analysis is available in the literature in terms of the very powerful tool of Bayesian Networks (BNs). We demonstrate here that in many cases this analysis is still possible via elementary faulttree manipulations that use the concept of a Boolean quotient to effectively implement Bayes’ Theorem in the Boolean domain. Again, a demonstrative example is given to illustrate the Boolean a posteriori FTA, explain its details, and show that the power of BNs is not really warranted in simple cases. A detailed comparison between the two kinds of a posteriori FTA is also given to identify their similarities and differences.
10 citations
27 Apr 2021
TL;DR: A simple method for handling the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, and a novel method for combining the MS-PRE and MS-IE concepts together are discussed.
Abstract: This paper deals with an emergent variant of the classical problem of computing the probability of the union of n events, or equivalently the expectation of the disjunction (ORing) of n indicator variables for these events, i.e., the probability of this disjunction being equal to one. The variant considered herein deals with multi-valued variables, in which the required probability stands for the reliability of a multi-state delivery network (MSDN), whose binary system success is a two-valued function expressed in terms of multi-valued component successes. The paper discusses a simple method for handling the afore-mentioned problem in terms of a standard example MSDN, whose success is known in minimal form as the disjunction of prime implicants or minimal paths of the pertinent network. This method utilizes the multi-state inclusion-exclusion (MS-IE) principle associated with a multi-state generalization of the idempotency property of the ANDing operation. The method discussed is illustrated with a detailed symbolic example of a real-case study, and it produces a more precise version of the same numerical value that was obtained earlier. The example demonstrates the notorious shortcomings and the extreme inefficiency that the MS-IE Original Research Article Rushdi and Amashah; AJRCOS, 8(1): 21-45, 2021; Article no.AJRCOS.67438 22 method suffers, but, on the positive side, it reveals the way to alternative methods, in which such a shortcoming is (partially) mitigated. A prominent and well known example of these methods is the construction of a multi-state probability-ready expression (MS-PRE). Another candidate method would be to apply the MS-IE principle to the union of fewer (factored or composite) paths that is converted (at minimal cost) to PRE form. A third candidate method, employed herein, is a novel method for combining the MS-PRE and MS-IE concepts together. It confines the use of MS-PRE to ‘shellable’ disjointing of ORed terms, and then applies MS-IE to the resulting partially orthogonalized disjunctive form. This new method makes the most of both MS-PRE and MS-IE, and bypasses the troubles caused by either of them. The method is illustrated successfully in terms of the same real-case problem used with the conventional MS-IE.
9 citations
Cites background or methods from "An exposition of system reliability..."
...With this interpretation, an application of the IE principle results in the following expression of reliability [32-36]....
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...If neither of the two terms and in the sum (A ∨ B) subsumes the other ( A ∨ B ≠ A and A ∨ B ≠ B ) and the two terms are not already disjoint (A ∧ B ≠ 0 ), then can be disjointed with by using the formula [24,36,43,48-57]...
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...For this case, the disjointing formula (8) simplifies to the Reflection law [36]....
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...Moreover, it involves so many subtractions that make it highly sensitive to round-off errors in the ultra-reliable regime [36,40-42]....
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References
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11,456 citations
TL;DR: In this paper, the authors present a data structure for representing Boolean functions and an associated set of manipulation algorithms, which have time complexity proportional to the sizes of the graphs being operated on, and hence are quite efficient as long as the graphs do not grow too large.
Abstract: In this paper we present a new data structure for representing Boolean functions and an associated set of manipulation algorithms. Functions are represented by directed, acyclic graphs in a manner similar to the representations introduced by Lee [1] and Akers [2], but with further restrictions on the ordering of decision variables in the graph. Although a function requires, in the worst case, a graph of size exponential in the number of arguments, many of the functions encountered in typical applications have a more reasonable representation. Our algorithms have time complexity proportional to the sizes of the graphs being operated on, and hence are quite efficient as long as the graphs do not grow too large. We present experimental results from applying these algorithms to problems in logic design verification that demonstrate the practicality of our approach.
9,021 citations
TL;DR: Probability and Statistics with Reliability, Queuing and Computer Science Applications, Second Edition, offers a comprehensive introduction to probabiliby, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics.
Abstract: Probability and Statistics with Reliability, Queuing and Computer Science Applications, Second Edition, offers a comprehensive introduction to probabiliby, stochastic processes, and statistics for students of computer science, electrical and computer engineering, and applied mathematics. Its wealth of practical examples and up-to-date information makes it an excellent resource for practitioners as well.
2,738 citations