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.

Graphs models and algorithms for reliability assessment of coherent and non-coherent systems

2018 ◽  
Vol 232 (2) ◽  
pp. 201-215
Author(s):  
Nicolae Brînzei ◽  
Jean-François Aubry
Keyword(s):  
System Reliability ◽  
Block Diagram ◽  
Reliability Assessment ◽  
Order Relation ◽  
Hasse Diagram ◽  
Coherent Systems ◽  
Reliability Polynomial ◽  
Formal Definitions ◽  
Tree Models ◽  
System States

In this article, we propose new models and algorithms for the reliability assessment of systems relying on concepts of graphs theory. These developments exploit the order relation on the set of system components’ states which is graphically represented by the Hasse diagram. The monotony property of the reliability structure function of coherent systems allows us to obtain an ordered graph from the Hasse diagram. This ordered graph represents all the system states and it can be obtained from only the knowledge of the system tie-sets. First of all, this model gives a new way for the research of a minimal disjoint Boolean polynomial, and, second, it is able to directly find the system reliability without resorting to an intermediate Boolean polynomial. Browsing the paths from the minimal tie-sets to the maxima of the ordered graph and using a weight associated with each node, we are able to propose a new algorithm to directly obtain the reliability polynomial by the research of sub-graphs representing eligible monomials. This approach is then extended to non-coherent systems thanks to the introduction of a new concept of terminal tie-sets. These algorithms are applied to some case studies, for both coherent and non-coherent real systems, and the results compared with those computed using standard reliability block diagram or fault tree models validate the proposed approach. Formal definitions of used graphs and of developed algorithms are also given, making their software implementation easy and efficient.

A New GO Methodology Algorithm Based on BDD

2013 ◽  
Vol 791-793 ◽  
pp. 1134-1138
Author(s):  
Jian Fan ◽  
Yi Ren ◽  
Lin Lin Liu
Keyword(s):  
System Reliability ◽  
Binary Decision Diagram ◽  
Analysis Procedure ◽  
Logic Tree ◽  
Minimal Path ◽  
Binary Decision ◽  
Go Model ◽  
Go Methodology ◽  
Minimal Path Sets

To solve the problems on quick achieving reliability and the minimal path sets (MPS) of a system with GO Methodology, a Binary Decision Diagram (BDD) based new GO methodology algorithm is introduced. This technique can avoid the shared signals and combination explosion problems simultaneously. Detailed steps with a case study are presented to explain the analysis procedure of this technique, firstly, establish the GO model and transform it into a directed acyclic diagram (DAG), logic tree and BDD successively; then minimize the BDD according to the Without Rule; at last, this system reliability and MPS can be achieved. The results of the studied case verify the validity and effectiveness of this algorithm.

Multistate coherent systems

1978 ◽  
Vol 15 (4) ◽  
pp. 675-688 ◽  
Author(s):  
E. El-Neweihi ◽  
F. Proschan ◽  
J. Sethuraman
Keyword(s):  
Finite Number ◽  
System Reliability ◽  
System Performance ◽  
Basic Theory ◽  
The Other ◽  
Coherent System ◽  
Coherent Systems ◽  
Standard Notion ◽  
Complete Failure

The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.

Multistate coherent systems

1978 ◽  
Vol 15 (04) ◽  
pp. 675-688 ◽  
Author(s):  
E. El-Neweihi ◽  
F. Proschan ◽  
J. Sethuraman
Keyword(s):  
Finite Number ◽  
System Reliability ◽  
System Performance ◽  
Basic Theory ◽  
The Other ◽  
Coherent System ◽  
Coherent Systems ◽  
Standard Notion ◽  
Complete Failure

The vast majority of reliability analyses assume that components and system are in either of two states: functioning or failed. The present paper develops basic theory for the study of systems of components in which any of a finite number of states may occur, representing at one extreme perfect functioning and at the other extreme complete failure. We lay down axioms extending the standard notion of a coherent system to the new notion of a multistate coherent system. For such systems we obtain deterministic and probabilistic properties for system performance which are analogous to well-known results for coherent system reliability.

scholarly journals The number of failed components in a coherent working system when the lifetimes are discretely distributed

Metrika ◽  
2021 ◽  
Author(s):  
Krzysztof Jasiński
Keyword(s):  
Random Variables ◽  
Continuous Case ◽  
Coherent System ◽  
Coherent Systems ◽  
Discrete Random Variables ◽  
Independent And Identically Distributed

AbstractIn this paper, we study the number of failed components of a coherent system. We consider the case when the component lifetimes are discrete random variables that may be dependent and non-identically distributed. Firstly, we compute the probability that there are exactly i, $$i=0,\ldots ,n-k,$$ i = 0 , … , n - k , failures in a k-out-of-n system under the condition that it is operating at time t. Next, we extend this result to other coherent systems. In addition, we show that, in the most popular model of independent and identically distributed component lifetimes, the obtained probability corresponds to the respective one derived in the continuous case and existing in the literature.

Optimal Allocation of Components in k-out-of-R Parallel Modules Systems

1994 ◽  
Vol 8 (3) ◽  
pp. 435-441 ◽  
Author(s):  
Fan Chin Meng
Keyword(s):  
System Reliability ◽  
Optimal Allocation ◽  
Parallel Systems ◽  
Coherent Systems ◽  
Special Case

In this note using the notion of node criticality in Boland, Proschan, and Tong [2] and modular decompositions of coherent systems, we obtain algorithms and guidelines for allocating components in a k-out-of-R parallel modules system to maximize the system reliability. An illustrative example is given to compare a special case of our results with the previous result for series-parallel systems due to El-Neweihi, Proschan, and Sethuraman [5].

scholarly journals Improved bounds for the availability and unavailability in a fixed time interval for systems of maintained, interdependent components

1980 ◽  
Vol 12 (01) ◽  
pp. 200-221 ◽  
Author(s):  
B. Natvig
Keyword(s):  
Lower Bound ◽  
System Reliability ◽  
Nuclear Reactors ◽  
Random Variables ◽  
Fixed Time ◽  
Time Interval ◽  
Coherent System ◽  
Exact Results ◽  
Fixed Time Interval ◽  
Special Case

In this paper we arrive at a series of bounds for the availability and unavailability in the time interval I = [t A , t B ] ⊂ [0, ∞), for a coherent system of maintained, interdependent components. These generalize the minimal cut lower bound for the availability in [0, t] given in Esary and Proschan (1970) and also most bounds for the reliability at time t given in Bodin (1970) and Barlow and Proschan (1975). In the latter special case also some new improved bounds are given. The bounds arrived at are of great interest when trying to predict the performance process of the system. In particular, Lewis et al. (1978) have revealed the great need for adequate tools to treat the dependence between the random variables of interest when considering the safety of nuclear reactors. Satyanarayana and Prabhakar (1978) give a rapid algorithm for computing exact system reliability at time t. This can also be used in cases where some simpler assumptions on the dependence between the components are made. It seems, however, impossible to extend their approach to obtain exact results for the cases treated in the present paper.

On the equivalence of systems of different sizes, with applications to system comparisons

2016 ◽  
Vol 48 (2) ◽  
pp. 332-348 ◽  
Author(s):  
Bo H. Lindqvist ◽  
Francisco J. Samaniego ◽  
Arne B. Huseby
Keyword(s):  
Optimization Problem ◽  
Coherent System ◽  
Sufficient Condition ◽  
Equivalent System ◽  
Coherent Systems ◽  
Two Systems ◽  
Equivalent Systems ◽  
Engineered Systems ◽  
Signature Vector ◽  
Mixed Systems

Abstract The signature of a coherent system is a useful tool in the study and comparison of lifetimes of engineered systems. In order to compare two systems of different sizes with respect to their signatures, the smaller system needs to be represented by an equivalent system of the same size as the larger system. In the paper we show how to construct equivalent systems by adding irrelevant components to the smaller system. This leads to simpler proofs of some current key results, and throws new light on the interpretation of mixed systems. We also present a sufficient condition for equivalence of systems of different sizes when restricting to coherent systems. In cases where for a given system there is no equivalent system of smaller size, we characterize the class of lower-sized systems with a signature vector which stochastically dominates the signature of the larger system. This setup is applied to an optimization problem in reliability economics.

COMBINATORIAL APPROACH TO COMPUTING COMPONENT IMPORTANCE INDEXES IN COHERENT SYSTEMS

2011 ◽  
Vol 26 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Ilya B. Gertsbakh ◽  
Yoseph Shpungin
Keyword(s):  
System Reliability ◽  
Simple Formula ◽  
Analytic Formula ◽  
Importance Measure ◽  
Coherent Systems ◽  
Joint Reliability ◽  
Component Importance ◽  
Definition Of ◽  
Combinatorial Parameter ◽  
Combinatorial Representation

We consider binary coherent systems with independent binary components having equal failure probability q. The system DOWN probability is expressed via its signature's combinatorial analogue, the so-called D-spectrum. Using the definition of the Birnbaum importance measure (BIM), we introduce for each component a new combinatorial parameter, so-called BIM-spectrum, and develop a simple formula expressing component BIM via the component BIM-spectrum. Further extension of this approach allows obtaining a combinatorial representation for the joint reliability importance (JRI) of two components. To estimate component BIMs and JRIs, there is no need to know the analytic formula for system reliability. We demonstrate how our method works using the Monte Carlo approach. We present several examples of estimating component importance measures in a network when the DOWN state is defined as the loss of terminal connectivity.

scholarly journals On relative ageing of coherent systems with dependent identically distributed components

2020 ◽  
Vol 52 (1) ◽  
pp. 348-376
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Sufficient Conditions ◽  
System Level ◽  
Hazard Rates ◽  
Coherent System ◽  
Coherent Systems ◽  
Component Level ◽  
Distributed Components ◽  
Two Systems

AbstractRelative ageing describes how one system ages with respect to another. The ageing faster orders are used to compare the relative ageing of two systems. Here, we study ageing faster orders in the hazard and reversed hazard rates. We provide some sufficient conditions for one coherent system to dominate another with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

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