Stochastic Models in Reliability and Maintenance

Transcription

1 Stochastic Models in Reliability and Maintenance Springer-Verlag Berlin Heidelberg GmbH

2 Shunji Osaki Editor Stochastic Models in Reliability and Maintenance With 46 Figures and 11 Tables " Springer

3 Professor Shunji Osaki Nanzan University Department of Information & Telecommunication Engineering Faculty of Mathematical Sciences and lnformation Engineering 27 Seirei-cho, Seto-shi Aichi, , Japan ISBN ISBN (ebook) DOI / Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Stochastic Models in Reliability and Maintenance: Il Tables / Ed.: Shunji Osaki. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Tokyo: Springer, 2002 This work is subject to copyright. Ali rights are reserved. whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm ar in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 2002 Originally published by Springer-Verlag Berlin Heide\berg in 2002 Softcover reprint of the hardcover Ist edition 2002 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: Erich Kirchner, Heidelberg

4 Preface Our daily lives can be maintained by the high-technology systems. Computer systems are typical examples of such systems. We can enjoy our modern lives by using many computer systems. Much more importantly, we have to maintain such systems without failure, but cannot predict when such systems will fail and how to fix such systems without delay. A stochastic process is a set of outcomes of a random experiment indexed by time, and is one of the key tools needed to analyze the future behavior quantitatively. Reliability and maintainability technologies are of great interest and importance to the maintenance of such systems. Many mathematical models have been and will be proposed to describe reliability and maintainability systems by using the stochastic processes. The theme of this book is "Stochastic Models in Reliability and Maintainability." This book consists of 12 chapters on the theme above from the different viewpoints of stochastic modeling. Chapter 1 is devoted to "Renewal Processes," under which classical renewal theory is surveyed and computational methods are described. Chapter 2 discusses "Stochastic Orders," and in it some definitions and concepts on stochastic orders are described and aging properties can be characterized by stochastic orders. Chapter 3 is devoted to "Classical Maintenance Models," under which the so-called age, block and other replacement models are surveyed. Chapter 4 discusses "Modeling Plant Maintenance," describing how maintenance practice can be carried out for plant maintenance. Chapter 5 is devoted to "Imperfect Preventive Maintenance," reviewing some imperfect maintenance models. Chapter 6 discusses "Generalized Renewal Processes," and generalized renewal processes and general repair models are described. Chapter 7 is devoted to "Two-Unit Redundant Models," under which several two-unit redundant models are surveyed and their optimization policies are analyzed. Chapter 8 focuses on "Markovian Deteriorating Systems," from which optimal inspection and/or replacement policies are derived. Chapter 9 is devoted to "Semi-Markov Reliability Models," where transient analyses of semi-markov reliability models are surveyed and numeric al solution techniques are described. Chapter 10 discusses "Software Reliability Models," reviewing typical software reliability models. Chapter 11 is devoted to "Data Communication Systems," showing how data communication systems can be formulated by means of reliability models, and

5 VI Preface optimal policies are discussed. Chapter 12 discusses "Monte-Carlo Methods": here some Monte-Carlo methods are surveyed in reliability models. This book will be of interest and importance to such users of "Stochastic Models in Reliability and Maintenance" as operations research analysts, probabilists, statisticians, applied mathematicians, engineers, and graduate students who are interested in reliability and maintenance. References leading further into the subject matter are cited in the end of each chapter. 1 believe that au the chapters of this book will introduce the readers to the major up-to-date theory and practice in stochastic models in reliability and maintenance. 1 would like to express my sincere appreciation to all the contributors to this book. 1 am also indebted to Professor Tadashi Dohi, Dr. Takashi Satow, and Dr. Hiroyuki Okamura, Hiroshima University, Japan, for their patient support in complet ion of the editing. Finally, 1 would like to express my sincere appreciation to Dr. Werner A. Miiller, Springer-Verlag, Heidelberg, Germany, for his friendly and patient help. Shunji Osaki

6 Contents 1. Renewal Processes and Their Computational Aspects T. Dohi, N. Kaio, S. Osaki 1.1 Introduction Basic Renewal Theory Continuous renewal theory Discrete renewal theory Some Useful Properties of the Renewal Function Specific examples Asymptotic properties Analytical Approximation Methods Phase renewal processes Gamma approximations Methods based on equilibrium distribution Bounds Numerical Methods Laplace inversion technique Cubic spline algorithm Discritization algorithm Approximation by rational functions Concluding Remarks Stochastic Orders in Reliability Theory M.Ohnishi 2.1 Introduction Definitions and Basic Properties Stochastic orders generated from univariate functions Conditional stochastic orders Bivariate characterization of stochastic orders Applications in Reliability Theory Notions of aging Useful stochastic inequalities in reliability theory Stochastic comparisons of system reliabilities Redundancy improvement Stochastic comparisons of maintenance policies... 53

7 VIII Contents Replacements upon failures Age replacement Block replacement Minimal repair Minimal repair with block replacement Stochastic comparison of different maintenance policies A TP2 Functions Classical Maintellance Models N. Kaio, T. Dohi, S. Osaki 3.1 Introduction Block Replacement Age Replacement Order Replacement Inspection Strategies Conclusions A Review of Delay Time Analysis for Modelling Plant Maintenance A. Christer 4.1 Introduction Maintenance Practice The Delay Time Concept Basic Delay Time Maintenance Model: Complex Plant Basic Maintenance Model: Component Tracking Relaxation of Assumptions Non-perfect Inspection Non-steady-state Condition Non-homogeneous Defect Arrival Rate> Condition-dependent Cost and Downtime for Repair Case Experience Using Subjective Data: Case Experience Revision of Subjectively Estimated Delay Time Distribution Correction for Sampling Bias Subjective Estimation of the Delay Time Distribution Directly Objective Estimation of Delay Time Parameters Case Experience Using Objective Data: HPP of Defect Arrival Discussion of Further Developments in Delay Time Modelling Conclusions Imperfect Preventive Maintenance Models T. Nakagawa 5.1 Introduction Sequential Imperfect Preventive Maintenance Introduction

8 Contents IX Model A - age Model B - failure rate Numerical examples Shock Model with Imperfect Preventive Maintenance Introduction Model and expected cost Optimal policies Conclusions Generalized Renewal Processes and General Repair Models 145 M. Kijima 6.1 Background and Motivation Generalized Renewal Processes g-renewal Processes in Discrete Time Monotonicity and Asymptotic Properties of the g-renewal Density..., On the g-renewal Function A General Repair Model Two-Unit Redundant Models..., 165 T. Nakagawa 7.1 Introduction Two-Unit Standby System Model and assumptions First-passage time distributions Expected numbers of visits to state..., Transition probabilities Preventive Maintenance of Two-Unit Systems Model and analysis Optimum preventive maintenance policies Replacement of a two-unit parallel system Other Two-Unit Systems Two-unit parallel system Two-unit priority standby system Two-unit standby system with imperfect switchover Other models Optimal Maintenance Problems for Markovian Deteriorating Systems H. Kawai, J. Koyanagi, M. Ohnishi 8.1 A Basic Optimal Replacement Problem for a Discrete Time Markovian Deteriorating System Some conditions on transition probabilities and cost structure Formulation by Markovian decision process (MDP).., 194

9 X Contents Optimality of controllimit rule An Optimal Inspection and Replacement Problem Transition probability Formulation by semi-markov decision process (SMDP) Structure of optimal inspection and replacement policy An Optimal Inspection and Replacement Policy with Incomplete Information Some notations and conditions Formulation by partially observable Markov decis ion process (POMDP) Some properties of TP2 order Some properties of optimal function Structure of optimal inspection and replacement policy A Continuous Time Markovian Deteriorating System A continuous time Markovian deteriorat ing system Transition probability Formulation by semi-markov decision process Structure of optimal policy An Optimal Maintenance Problem for a Queueing System Model description Formulation by semi-markov decision process Properties of value function Structure of optimal policy Transient Analysis of Semi-Markov Reliability Models - A Tutorial Review with Emphasis on Discrete-Parameter Approaches A. Csenki 9.1 Introduction Modelling Framework Dependability Measures Methods of Analysis Continuous-parameter models Discrete--parameter models Equations for the Dependability Measures Numerical Solution Techniques Solving the integral equations Discrete-parameter approximations: Recent Developments, Conclusions and Further Work Software Reliability Models S. Yamada 10.1 Introduction Definitions and Software Reliability Model Software Reliability Growth Modeling

10 Contents XI 10.4 Imperfect Debugging Modeling Imperfect debugging model with perfect correction rate Imperfect debugging model for introduced faults Software Availability Modeling Model description Software availability measures Application of Software Reliability Assessment Optimal software release problem Maintenance cost model Maintenance cost model with reliability requirement Statistical software testing-progress control Optimal testing-effort allocation problem Reliability Models in Data Communication Systems K. Yasui, T. Nakagawa, H. Sandoh 11.1 Introduction SW ARQ Model with Intermittent Faults Intermittent faults ARQ policy Optimal retransmission number Numerical examples and remarks SR ARQ Model with Retransmission Number Model and analysis Optimal policy Numerical examples and remarks Hybrid ARQ Models with Response Time Type-I hybrid ARQ Type-II hybrid ARQ Comparison of type-i and type-ii hybrid ARQs Numerical examples and remarks Quick Monte Carlo Methods in Stochastic Systems and Reliability C. Papadopoulos, N. Limnios 12.1 Introduction The Problem with Direct Simulation Importance Sampling The Optimal Change of Measure Remarks Preliminary definitions The recursive approach Exact calculat ion of "((x) Cases of Application of the Recursive Approach System Model

11 XII Contents 12.7 Regenerative Simulation Failure Biasing Methods Simple failure biasing (SFB) Balanced failure biasing (BFB) Bias2 failure biasing Failure distance biasing (FDB) Balanced 1 failure biasing (B1FB) Balanced 2 failure biasing (B2FB) Bounded relative error and failure biasing Unreliability Estimation One-component system General case Example Analytical-Statistical Methods... " Concluding Remarks Index