Reliability-based Modeling of System Performance E-Book

Abdelkhalak El Hami

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Sprache: Englisch
• Veröffentlichungsjahr: 2023

Seitenzahl: 256

Beschreibung

Probabilistic modeling of system performance, built primarily on the foundation of reliability theory, is referred to as reliability-based modeling. This book progressively builds up the functional complexity of system modeling. It starts by modeling the simplest system that can be designed, and then moves to the functional modeling of the multi-state system, step-by-step. Reliability-based Modeling of System Performance deals with modeling the functioning of systems by presenting its concepts, methods and mathematical tools. It is accessible to engineering students and academics who have already acquired basic knowledge of probability theory, applied statistics, differential calculus, integrals, combinatorial calculus and Boolean algebra.

Leseprobe

Reliability of Multiphysical Systems Set

coordinated by

Abdelkhalak El Hami

Volume 19

Reliability-based Modeling of System Performance

Abdelkhalak El Hami

Mohamed Eid

First published 2023 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 Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk

John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com

© ISTE Ltd 2023The rights of Abdelkhalak El Hami and Mohamed Eid to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s), contributor(s) or editor(s) and do not necessarily reflect the views of ISTE Group.

Library of Congress Control Number: 2023938839

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN 978-1-78630-835-1

Introduction

Probabilistic modeling of system performance is built primarily on a solid foundation of reliability theory. It can therefore be called “reliability-based modeling of system performance”. Although it has organic links with applied statistics, combinatorial calculations and Boolean algebra, reliability-based modeling is a distinct discipline that is conceptually different from the disciplines cited above. Currently, reliability-based modeling covers a very wide spectrum of applications, such as system reliability engineering, functional analysis, operational safety, risk analysis and management, risk management and maintenance optimization, system degradation and aging, system failure monitoring, diagnosis and prognosis, performance analysis, mass-production and quality control, structural reliability and software reliability.

The focus of the book is restricted to the reliability, availability, maintenance, and safety (RAMS) modeling of complex systems, excluding passive systems (structures) and software.

The book is accessible to engineering students and engineers who have already acquired the elementary knowledge of probability theory, applied statistics, differential calculus, integrals, combinatorial calculus and Boolean algebra.

Having specified above the areas contributing to the reliability-based modeling of system’s performance, the primary functional qualities concerned are the availability and the reliability of the system. These functional qualities are probabilistic in nature and statistically measurable. As soon as the concepts of availability and reliability are well understood, other derived qualities arise and complete the functional characterization of the system, such as the system functional space and the transition rates between the states defining the system functional space and the probabilities of sojourning in the states. Our recourse to mathematics does only serve to support the paradigm of reliability-based modeling of system’s performance practiced by system engineering.

Throughout the book, the complexity of systems modeling is gradually built up. We start by modeling the operation of the simplest system that engineers can conceive, which is the “binary-state system”. We will call it the elementary component. Progressively, we proceed towards the functional modeling of the multistate system. In this approach, the multistate system is built from independent and coherent elementary components. Independence and functional coherence of the constituents are well defined in the book. However, neither dependency nor functional incoherence of constituents has been addressed in the book.

Whether the system is binary or multistate, the definition of failure always remains of a functional nature. The physical roots of a failure may only appear at the lowest levels of the functional analysis where the elementary component failure appears. The lowest level of any functional analysis is conventionally decreed by the engineers and/or by the analysts. The functional boundaries separating between components and systems are also conventional. A system in a specific step of the analysis can become a component in the next step and vice versa. Gradually, this allows the modeling of the increasing functional complexity up to the system of systems. The book explores this possibility through the functional binarization of the multistate systems. The binarization of the multistate systems ensures the conservation of the fundamental mathematical models that describe the availability and the reliability of every functional entity through the full scale of complexity. Consequently, the availability and the reliability of every functional entity are governed by the same fundamental system of differential and integral equations, regardless of the degree of its complexity. The functional complexity is finally determined by the number of states defining the system, the functional space structure and the logical relations structuring the system functionalities. For low and medium complexity systems, the availability and the reliability can be determined by some analytical or numerical models. For higher complexities, analysts may call for the numerical simulation using the Monte Carlo simulation or similar methods. The book gives insights into these methods through simple didactic applications. The book is structured in eight chapters.

In Chapter 1, we insist on the identification of the system through its functionalities. As mentioned above, a system is a functional entity. As such, “availability” designates the aptitude of the system to supply its required functionalities on demand. It does not refer to its physical condition or being. A physically sound system may not be available because it is poorly cooled though its cooling unit is also sound. Indeed, it may not provide the required cooling capacity because the system operating temperature is beyond the safe thermal operating limits. Admittedly, the confusion between functionality and the physical system arises at the most elementary stage of the successive segmentation of functions and sub-functions. At the lowest stage of the segmentation, the elementary system would be arbitrarily called a component. The analyst arbitrarily sets the functional threshold below which the component would arise, in the most suitable way for his/her analysis. This threshold could obviously vary from one analysis to another. This elementary functional entity embodied in a component will be described in the simplest way that the systems engineer can imagine. It would exclusively be either “available” or “unavailable”. The component therefore operates in perpetual transitions between the availability state and the unavailability state, until the end of its operational life. The transitions are characterized by a failure rate and a repair rate. The elementary component has a binary operating pattern, which is characterized by its failure and repair rates. Thus, we introduce the concept of the elementary component. The mathematical modeling of the availability and the reliability of the elementary component is developed in detail in Chapter 1.

Chapter 2 deals with the mathematical modeling of the availability and the reliability of the multistate system using the elementary models developed in Chapter 1. The multistate system is built by the functional assembly of the elementary components under the conditions that the elementary components are independent and functionally coherent. The independence requires that the transitions of the elementary components do not mutually infer. The functional coherence requires that the transition of any of the elementary components to an unavailability state cannot increase the overall availability of the system and that the transition of any of the elementary components to an availability state cannot increase the overall unavailability of the system. That is to say, in other words, the functional degradation of an elementary component can only degrade the overall functional state of the system, and the functional upgrading of an elementary component can only upgrade the overall functional state of the system. Having established the principles of independence and functional coherence, we only need to describe the logical functional relationships that structure all the elementary components within a well-defined system. These logical functional relations operate in two elementary algebraic modes, which are the union (OR) and the intersection (AND), in the Boolean sense of the terms. Then, the functional assembly of the elementary components in a single system is carried out using the elementary Boolean operators: OR “∨” and AND “∧”1. To introduce the notion of multistate system, we start by modeling the simplest multistate system one may conceive. That is a multistate system built up by only two elementary components. Some other multistate systems of higher order will equally be modeled before establishing the generic demarch of multistate systems modeling. Finally, we also demonstrate that any multistate system can be reduced to a binary elementary component by dividing its global functional space into a (sub-) space of availability states and a (sub-) space of unavailability states. The binarization of a multistate system will be introduced with the help of two additional notions: the critical states and the critical transitions. The mathematical modeling of the availability and the reliability of the multistate systems and their binarization gives birth to the notions of the equivalent parameters, that is, the equivalent failure and equivalent repair rates.

In Chapter 3, we treat the modeling of the matrix-like system (n × l), with n elementary components in-series, which are repeated in-parallel l times. This system is found, quite often, in the electronic devices of detection and signal analysis systems, active and passive redundant systems and manufacturing processes. Quite often too, the n × l involved components are identical. Analysts may be interested in matrix systems for two different, but not decoupled, reasons. The first concerns the modeling of the availability and the reliability of the systems in the extension of Chapter 2 after a direct adaptation to the matricial structure. The second reason is rather for the modeling of the performance degradations of a system (matrix-like or not) if the operation of the system is rather classified in phases defined by thresholds which is determined by the number of the available or the unavailable elementary components. This is exactly the case of distributed systems such as networks which are structured in nodes and arcs. In these types of systems, analysts are more interested in modeling the performance of the network and its evolution from one performance phase to another, more than in the overall availability and reliability of the system. In other words, analysts are interested in transitions from and sojourns in different performance phases. In this approach, the performance states of the distributed system are not simply classified into availability space and unavailability space. The performance states are classified according to the number of available or unavailable elementary components.

Chapters 4 and 5 are devoted to the application of the concepts and notions already introduced above on simple systems in the sense of the structural simplicity of the system and the numerical simplicity of the elementary components’ parameters. The structural simplicity results from the small number of elementary components belonging to the system and the simplicity of their logical links. The numerical simplicity results from the following assumption: the elementary components are mutually independent, with time-independent failure and repair rates. This simplicity is purposely required as we assume that the training of the readers of the book should be done through active learning and hand-working on the proposed application cases.

In Chapter 4, we proceed for the systems with redundancy. Redundancy is a technical solution that generally improves system’s availability and reliability, but at a higher total cost of installation and maintenance. So, we are talking about systems with redundancy. In the literature, there are two complementary definitions of the system . Either it is a macro system made up of N systems that are functionally identical, of which the availability of at least n is sufficient to make the macro system available. Or, it is a macro system made up of N systems that are functionally identical, of which the unavailability of at least n systems is enough to make the macro system unavailable. In the first definition, n indicates the minimum availability condition of the macro system’s availability. In the second, n indicates the minimal unavailability condition of the macro system’s unavailability. The analyst should define which of the two stand points of view is the most appropriate to his/her analysis.

In Chapter 5, we develop some models for general systems. Assessing the reliability and the availability of general systems is among the most common activities in all the areas relating to systems engineering, such as design and concept validation, maintenance management and optimization, safety management, assessments of periodic testing strategy and operational safety (in normal and accident situations). Dynamic modeling of the availability and reliability of a system goes through several stages: modeling the availability and reliability of the elementary components, the construction of the functional space of the system, the identification of the states associated with the availability space and those associated with the unavailability space, the identification of the critical states, the determination of the equivalent transition rates and, finally, the determination of the availability/unavailability and the reliability/unreliability of the system. A didactic application case is illustrated. It is a simple system consisting of five elementary components.

In Chapter 6, we deal with the modeling of the sequential events. The modeling of sequential events occupies a specific place in the fields of accident analysis and system safety analysis. Often, analysts seek to identify the sequences of failures (systemic, human, procedural, etc.) that may produce serious accidents at the earliest stages of system design. They seek to eliminate or reduce the occurrence of dreaded sequences to interrupt the occurrence order of the events of the sequence and/or to mitigate their consequences. Analysts thus use several analysis tools, including the event tree analysis. Unlike the fault tree, the event tree considers the occurrence order of the events. In this chapter, we have developed two models fairly practiced in the modeling of sequential events: the dynamic model and the static model. Generally, the dynamic model is used more in accident analyses. On the other hand, the static model is widely used in system safety analyses. An academic case composed of four sequential events is treated. A comparative analysis between the models (dynamic and static), illustrated by numerical results, is also carried out.

In Chapter 7, numerical simulation by the Monte Carlo method is briefly presented and illustrated through some applications. We have emphasized on the notion of bijective analytical sampling, the generic approach (the game), the scoring and the statistical processing of the scored figures (standard deviation, variance, uncertainty, etc.) with applications in two simple cases. The first case describes the evolution over time of a system represented by its graph of states. The transitions between the states are numerically simulated to mainly determine the average sojourn time per state and the number of transitions to each state within a given time interval. The second case concerns a system described by its elementary components and its logical structure. Its reliability and its operating histogram are mainly determined from the numerical simulation of the operating histograms of its elementary components. We have not covered several important topics in the Monte Carlo simulation method to note the non-analytical sampling, the acceleration techniques and the propagation of uncertainties. However, the basic knowledge presented in the chapter allows the readers to independently complete their knowledge of these subjects according to their professional needs in real time.

Finally, mathematical modeling would not be of any engineering interest without acquiring some knowledge related to failure and repair data issued from the real system engineering world. Chapter 8 will therefore be dedicated to the acquisition of these data and their processing via either life-test campaigns or operational experience feedback analyses. The chapter presents the physical testing techniques used for system reliability estimation. These tests make it possible to obtain data relevant to the reliability of the designed or manufactured systems. When a new prototype includes innovations, data are often unavailable or even nonexistent. Subsequently, it is necessary for design optimization purposes to build a knowledge base using physical tests outcome. This makes it possible to estimate the reliability of the components as a function of the operational stresses to which they may be subjected.

Among the different tests used to estimate the reliability of industrial products, we underline the accelerated tests, the aggravated tests and the Bayesian tests.

Note

1

When it seems appropriate, we will indifferently use the symbols “∨” or “+” to designate the OR-operator and “∧” or “·” to designate the AND-operator throughout the book.

1Basic Notions

1.1. Introduction

We introduce in this chapter some basic notions that engineers use to perceive the operation of a system. Understanding these basic notions allows us to build mathematical models as close as possible to the real operation patterns of the system. These notions are of two different natures: logical and probabilistic.

Sections 1.2 and 1.3 briefly present these basic notions that the reader must acquire to be able to continue his/her navigation in the book. The basic notions are presented without demonstrations.

In section 1.4, we present the “functional” quality as being the only quality that the system can endorse, disregarding other measurable qualities of the system. Only functional qualities will intervene in the definition and, subsequently, in the modeling of the system operational patterns. Considering the system as a “functional entity” highlights the binary nature of the system operational patterns. This is the primary quality of the system. This functional binarity will be perfectly characterized by measurable quantities issued from the real world of systems as perceived by the engineer. These are precisely the system’s failure and repair rates.

In section 1.5, we develop mathematical models describing the evolution over time of the functional performance of the system because of a binary functional concept. These models give us the possibility of characterizing the operation of the system by a multitude of measurable secondary qualities. Each of these secondary qualities will be detailed in the following sections.

Section 1.6 will therefore be devoted to a secondary quality called “availability”, being an instantaneous quality by definition. It will be described by a first-order linear differential equation, generally inhomogeneous with variable coefficients.

Section 1.7 will be devoted to other secondary qualities called “reliability” and “maintainability”. Being integral qualities, they are described by homogeneous first-order linear differential equations, generally with variable coefficients.

Section 1.8