Multinary systems and reliability models from coherence to some kind of non-coherence

European Union

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• Publié par: EUROPEAN-UNION
• Nombre de lectures: 7
• Langue: English
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Commission of the European Communities
nuclear science and technology
MULTINARY SYSTEMS AND RELIABILITY MODELS
FROM COHERENCE TO SOME KIND
OF NON-COHERENCE Commission of the European Communities
nuclear science and technology
MULTINARY SYSTEMS AND RELIABILITY MODELS
FROM COHERENCE TO SOME KIND
OF NON-COHERENCE
N. MAZARS
Visiting scientist
from Toulouse University-France
Commission of the European Communities
Joint Research Centre
Ispra Establishment
1-21020 Ispra (Varese)
PARLE'"^ r '
Directorate-General Science, Research and Development _
Joint Research Centre " '
1986 l^ÉUaJJia2S^LL Published by the
COMMISSION OF THE EUROPEAN COMMUNITIES
Directorate-General
Telecommunications, Information Industries and Innovation
Bâtiment Jean Monnet
LUXEMBOURG
LEGAL NOTICE
Neither the Commission of the European Communities nor any person acting on behalf
of then is responsible for the use which might be made of the following
information
© ECSC — EEC — EAEC, Brussels-Luxembourg, 1986 1"TT
ABSTRACT
First restricted to models for binary systems, re­
liability theory is being generalized for multinary systems
of multinary components (i.e. which can assume a finite
number of performance levels ranging from the "complete
failure" up to the "perfect functioning"). After a general
viewpoint on reliability models for multinary systems,
coherence generalizations are examined. First studied in
terms of structure functions, the binary coherent systems
can be fully characterized in terms of their minimal path
(cut) sets as well as in terms of their life functions.
These three basic deterministic treatments are briefly
reviewed and fully generalized for the multinary case. The
(N+l)-level broad-sense coherent systems are first consi­
dered. Various fundamental notions such as minimal path
(cut) sets and relevance first are introduced in terms of
structure functions. The binary decompositions are studied
and used for characterizing the broad-sense coherence in
terms of sets; this leads to some fundamental relations use­
ful for multinary systems analysis. The binary-type cohe­
rence, the homogenous coherence and the various types of
strict-sense coherence are reviewed and fully characterized
in various ways. Life functions lead to some model useful
for reliability calculations whereas the results first
obtained in terms of (N+l) levels systems can be genera­
lized, in a straight way, for the whole multinary case.
Methods for determining, in a "exact" or "approximated"
way, reliability characteristics of multinary coherent sys­
tems are studied from both of the fundamental models of re­
liability, then possible. Futhermore, some kind of non-cohe­
rent multinary system is suggested. CONTENTS
Pages
Preamble
Chapter A: A General Viewpoint on Reliablity Models 1
for Multinary Systems
Chapter B: Three Basic Deterministic Treatments and Reliability
Models for Multinary Coherent Systems 11
Appendix A: On Some Positive Stochastic Dependences 137
Appendix B: Three Types of "Bounds"
for the Interval Reliability and Unreliability
of the binary Coherent Systems 140
Bibliography 144
* * VI
Chapter Aï A GENERAL VIEWPOINT ON RELIABILITY MODELS
FOR MULTTNARY SYSTEMS
I. FROM MÜLTINARY SYSTEMS TO RELIABILITY MODELS 1
1.1. Introduction 1
1.2. On Mili ti nary Systems 2
1.3. Same General Model of Reliability 3
II. RELIABILITY CHARACTERISTICS AND RELIABILITY METHODS 4
II. 1. Reliability Characteristics 4
1.1.1.y Functions
1.1.2. Availability Functions
1.1.3.y and Reliability Functions
II.2. Five Complexity Factors
for the Analytical Methods of Reliability 9 VTI
Chapter B: THREE BASIC DETERMINISTIC TREATMENTS AND RELIABILITY MODELS
FOR MULTINARY COHERENT SYSTEMS
I. FROM THE BINARY COHERENT SYSTEMS TO THE MULTTNARY ONES;
BACKGROUND, PURPOSE AND SUMMARY 11
1.1. Introduction
1.2. Multinary Semi-Coherent Systems, General Framework 12
1.3. The Three Fundamental Treatments of Binary Coherent Systems 16
1.3.1. Binary Coherent Systems in Terms of Boolean Functions
1.3.2. Binaryts in Terms of Sets
I.3.3. Binary Coherent Systems in Terms of Some Continuous
Real-Valued Functions
1.3.4. Binary Coherent Systems and Reliability Theory
T.4. A Retrospective Review of Multinary Coherent Systems 27
1.5. Purpose and Summary 31
II. A DETERMINISTIC CONCEPT: COHERENCE5
11.1. Two Types of Coherence in the Broad Sense 3
11.1.1. Definitions and Duality Relations
11.1.2. Deterministic Properties
11.2. Minimal Path Sets and Cut Sets 39
11.2.1. Definitions
11.2.2. Duality Relations and Fundamental Results
11.3. Binary Decompositions of the (N+1)-Level Coherent Systems 44
IT.3.1. Tn Terms of their Minimal Path Sets
IT.3.2. in Terms of theirl Cut Sets; Duality Relations VIII
11.4. Sane Set Characterizations of the (W-l)-Level Coherent Systems 52
11.4.1. From the Functional Formulation of the (N+1)-Level
Coherent Systems towards Their Set Characterizations
11.4.2. Characterizations of the (N+1)-Level Coherent Systems
in Terms of their Minimal Path Sets and Cut Sets
11.4.3. Dominance Relation between (N+l)-Level Coherent Systems
11.5. Binary-Type (N+1)-Level Coherent Systems 62
11.5.1. Definition and Various Characterizations of the Binary-Type
Coherent Systems
11.5.2. Homogenous (Broad-Sense) Coherent Systems:
11.6. Relevance 74
11.6.1. Two Types of Relevance
11.6.2. Exact Relevance and Binary-Type Coherent Systems
11.7. Six Relevance Conditions for the Coherence in the Strict Sense 79
11.7.1. Definitions and Comparisons
11.7.2. Some Deterministic Properties of the Strict-Sense
Coherent Systems
11.7.3. Binary-Type Strict-Sense Coherent Systems
III. THREE FÜRTHER DETERMINISTIC VIEWPOINTS
ON MULTINARY COHERENT SYSTEMS 91
111.1. Modules and Modular Décompositions
111.1.1. Definitions and Basic Relations
111.1.2. Modules and Relevance
111.2. Life Functions of the (N+l)-Level Coherent Systems 96
111.2.1. A Dynamical Viewpoint; Definitions
111.2.2. Some Characterizations of "Coherent Life Functions"
111.2.3. Life Functions of the Binary-Type Coherent Systems
111.2.4. Some Basic Notions in Terms of Life Functions
111.3. Multinary Coherent Systems 103
111.3.1. Multinary Broad-Sense Coherent Systems
111.3.2.y Strict-Sensets