Analysis of minimal path and cut vectors in multistate monotone systems and use it for detection of binary type multistate monotone systems

2020 ◽  
Vol 234 (5) ◽  
pp. 686-695
Author(s):  
Paweł Marcin Kozyra
Keyword(s):  
Coherent Systems ◽  
Minimal Path ◽  
Monotone Systems ◽  
Monotone System ◽  
Minimal Cut

Theorem specifying how all minimal cut (path) vectors to level j can be obtained from all minimal path (cut) vectors to level j in any multistate monotone system is proven. Characterizations of binary type multistate monotone systems and binary type multistate strongly coherent systems by minimal cut and path vectors and corresponding to them, binary type cut and path sets are also demonstrated.

ACTIVE REDUNDANCY ALLOCATION FOR COHERENT SYSTEMS WITH INDEPENDENT AND HETEROGENEOUS COMPONENTS

2018 ◽  
Vol 34 (1) ◽  
pp. 72-91 ◽  
Author(s):  
Rui Fang ◽  
Xiaohu Li
Keyword(s):  
Sufficient Conditions ◽  
Coherent Systems ◽  
Minimal Path ◽  
Redundancy Allocation ◽  
Numerical Examples ◽  
Minimal Cut Sets ◽  
Minimal Cut ◽  
Minimal Path Sets ◽  
Theoretical Results

This paper studies the allocation of active redundancies to coherent systems on the context that the base and redundancy components have mutual independent lifetimes. For systems with two symmetric components and systems with one component's minimal cut sets (minimal path sets) covering those of another, we derive sufficient conditions to compare the resultant system lifetimes. Some numerical examples are also presented to illustrate the theoretical results.

On Reliability Bounds via Conditional Inequalities

1998 ◽  
Vol 35 (01) ◽  
pp. 104-114
Author(s):  
M. Xie ◽  
C. D. Lai
Keyword(s):  
Computational Complexity ◽  
System Reliability ◽  
Minimal Path ◽  
Reliability Bounds ◽  
Minimal Cut ◽  
One Step

In this paper we study an approximation of system reliability using one-step conditioning. It is shown that, without greatly increasing the computational complexity, the conditional method may be used instead of the usual minimal cut and minimal path bounds to obtain more accurate approximations and bounds. We also study the conditions under which the approximations are bounds on the reliability. Some further extensions are also presented.

Computational Complexity of Coherent Systems and the Reliability Polynomial

1988 ◽  
Vol 2 (4) ◽  
pp. 461-469 ◽  
Author(s):  
R.E. Barlow ◽  
S. Iyer
Keyword(s):  
System Reliability ◽  
Graph Representation ◽  
Coherent System ◽  
Coherent Systems ◽  
Logic Tree ◽  
Network Graph ◽  
Minimal Path ◽  
Reliability Polynomial ◽  
First Finding ◽  
Pivoting Strategies

There are three general methods for system reliability evaluation, namely; (1) inclusion–exclusion, (2) sum of disjoint products, and (3) pivoting. Of these, only pivoting can be applied directly to a logic tree or network graph representation without first finding minimal path (or cut) sets. Domination theory provides the basis for selecting optimal pivoting strategies. Simple proofs of domination-theory results for coherent systems are given, based on the reliability polynomial. These results are related to the problem of finding efficient strategies for computing coherent system reliability. The original results for undirected networks are due to Satyanarayana and Chang [5] (cf. [1]). Many of the original set theoretic results are due to Huseby [3]. However, he does not use the reliability polynomial to prove his results.

scholarly journals Bounds for the Availabilities of Multistate Monotone Systems Based on Decomposition into Stochastically Independent Modules

2012 ◽  
Vol 44 (1) ◽  
pp. 292-308
Author(s):  
J. Gåsemyr
Keyword(s):  
Fixed Point ◽  
Complex Systems ◽  
Upper Bounds ◽  
Reliability Theory ◽  
Lower And Upper Bounds ◽  
Minimal Path ◽  
Monotone Systems ◽  
Modular Decomposition ◽  
The Relationship ◽  
Different Levels

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

scholarly journals Bounds for the Availabilities of Multistate Monotone Systems Based on Decomposition into Stochastically Independent Modules

2012 ◽  
Vol 44 (01) ◽  
pp. 292-308 ◽  
Author(s):  
J. Gåsemyr
Keyword(s):  
Fixed Point ◽  
Complex Systems ◽  
Upper Bounds ◽  
Reliability Theory ◽  
Lower And Upper Bounds ◽  
Minimal Path ◽  
Monotone Systems ◽  
Modular Decomposition ◽  
The Relationship ◽  
Different Levels

Multistate monotone systems are used to describe technological or biological systems when the system itself and its components can perform at different operationally meaningful levels. This generalizes the binary monotone systems used in standard reliability theory. In this paper we consider the availabilities and unavailabilities of the system in an interval, i.e. the probabilities that the system performs above or below the different levels throughout the whole interval. In complex systems it is often impossible to calculate these availabilities and unavailabilities exactly, but it is possible to construct lower and upper bounds based on the minimal path and cut vectors to the different levels. In this paper we consider systems which allow a modular decomposition. We analyse in depth the relationship between the minimal path and cut vectors for the system, the modules, and the organizing structure. We analyse the extent to which the availability bounds are improved by taking advantage of the modular decomposition. This problem was also treated in Butler (1982) and Funnemark and Natvig (1985), but the treatment was based on an inadequate analysis of the relationship between the different minimal path and cut vectors involved, and as a result was somewhat inaccurate. We also extend to interval bounds that have previously only been given for availabilities at a fixed point of time.

On Reliability Bounds via Conditional Inequalities

1998 ◽  
Vol 35 (1) ◽  
pp. 104-114 ◽  
Author(s):  
M. Xie ◽  
C. D. Lai
Keyword(s):  
Computational Complexity ◽  
System Reliability ◽  
Minimal Path ◽  
Reliability Bounds ◽  
Minimal Cut ◽  
One Step

In this paper we study an approximation of system reliability using one-step conditioning. It is shown that, without greatly increasing the computational complexity, the conditional method may be used instead of the usual minimal cut and minimal path bounds to obtain more accurate approximations and bounds. We also study the conditions under which the approximations are bounds on the reliability. Some further extensions are also presented.

Comparing Criticality of Nodes via Minimal Cut (Path) Sets for Coherent Systems

1994 ◽  
Vol 8 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Fan Chin Meng
Keyword(s):  
Importance Measure ◽  
Optimal Assignment ◽  
Coherent Systems ◽  
Minimal Cut ◽  
Importance Ranking

In 1989, Boland, Proschan, and Tong [2] introduced the notion of criticality ranking among nodes and developed a procedure for obtaining an optimal assignment of components in coherent systems. In this article we obtain characterizations of the criticality ranking in terms of minimal cut (path) sets for coherent systems. Furthermore, utilizing the characterizations, it is shown that the criticality ranking defined by Boland et al. [2] is consistent with the cut-importance ranking introduced by Butler in 1979 [4]. A relationship between the criticality ranking and the well-known and widely used Birnbaum reliability importance measure is also derived.

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