Systems Dependability Assessment : Modeling With Graphs And Finite State Automata

FOCUS RISK MANAGEMENT AND DEPENDABILITY SERIES Systems Dependability Assessment Modeling with Graphs and Finite State Automata Jean-François Aubry Nicolae Brinzei Systems Dependability Assessment FOCUS SERIES Series Editor Jean-François Aubry Systems Dependability Assessment Modeling with Graphs and Finite State Automata Jean-François Aubry Nicolae Brînzei First published 2015 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2015 The rights of Jean-François Aubry and Nicolae Brînzei to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2014956809 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISSN 2051-2481 (Print) ISSN 2051-249X (Online) ISBN 978-1-84821-765-2 Contents P REFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix I NTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . xiii PART 1. P REDICTED R ELIABILITY OF S TATIC S YSTEMS ; A G RAPH -T HEORY BASED A PPROACH . . . . . . . . . . . . 1 C HAPTER 1. S TATIC AND T IME I NVARIANT S YSTEMS B OOLEAN R EPRESENTATION . . . . . . . . . . . . . . 3 WITH 1.1. Notations . . . . . . . . . . . . . . . . . . . . . 1.2. Order relation on U . . . . . . . . . . . . . . . 1.3. Structure of a system . . . . . . . . . . . . . . . 1.3.1. State diagram of a system . . . . . . . . . . 1.3.2. Monotony of an SF, coherence of a system 1.4. Cut-set and tie-set of a system . . . . . . . . . . 1.4.1. Tie-set . . . . . . . . . . . . . . . . . . . . . 1.4.2. Cut-set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C HAPTER 2. R ELIABILITY OF A C OHERENT S YSTEM 2.1. Demonstrating example . . . . . . . 2.2. The reliability block diagram (RBD) 2.3. The fault tree (FT) . . . . . . . . . . 2.4. The event tree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 4 6 6 7 9 9 10 . . 13 . . . . 15 18 21 26 . . . . vi Systems Dependability Assessment 2.5. The structure function as a minimal union of disjoint monomials . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1. Ordered graph of a monotone structure function . . 2.5.2. Maxima and minima of the ordered graph . . . . . . 2.5.3. Ordered subgraphs of the structure function . . . . . 2.5.4. Introductory example . . . . . . . . . . . . . . . . . 2.5.5. Construction of the minimal Boo