ISBN: 3540672753 TITLE: Reliability Theory AUTHOR: Gertsbakh, Ilya TOC: Preface vii 1 System Reliability as a Function of Component Reliability 1 1.1 The System and Its Components 1 1.2 Independent Components: System Reliability and Stationary Availability 5 1.3 Lifetime Distribution of a System without Component Renewal 10 1.4 Exercises 14 2 Parametric Lifetime Distributions 17 2.1 Poisson Process - Exponential and Gamma Distributions 17 2.2 Aging and Failure Rate 24 2.3 Normal, Lognormal and Weibull Families 29 2.3.1 Normal and Lognormal Distributions 29 2.3.2 The Weibull distribution 32 2.4 Exercises 38 3 Statistical Inference from Incomplete Data 41 3.1 Kaplan-Meier Estimator of the Reliability Function 41 3.2 Probability Paper 44 3.3 Parameter Estimation for Censored and Grouped Data 50 3.3.1 The Maximum Likelihood Method 50 3.3.2 Maximum Likelihood Function for Censored and Grouped Data 51 3.3.3 Finding Maximum Likelihood Estimates for a Censored Sample: Weibull and lognormal Distribution 54 3.3.4 Point and Confidence Estimation of Location and Scale Parameters Based on Linear Combination of Order Statistics 57 3.3.5 Large-Sample Maximum Likelihood Confidence Intervals 62 3.4 Exercises 63 4 Preventive Maintenace Models Based on the Lifetime Distribution 67 4.1 Basic Facts from Renewal Theory and Reward Processes 67 4.1.1 Renewal Process 67 4.1.2 Renewal Reward Process 72 4.1.3 Alternating Renewal Process. Component Stationary Importance Measure for a Renewable System 74 4.2 Principal Models of Preventive Maintenance 77 4.2.1 Periodic (Block) Replacement - Cost-type Criterion 77 4.2.2 Block Replacement: Availability Criterion 78 4.2.3 Periodic Group Repair - Operation Tie Based Criterion 79 4.2.4 Periodic Preventive Maintenance with Minimal Repair 81 4.2.5 Age Replacement - Cost-type Criterion 84 4.2.6 Age Replacement - Availability-type Criterion 85 4.3 Qualitative Investigation of Age and Block Preventive Maintenance 86 4.3.1 The Principal Behavior of r)(T) 86 4.3.2 Which is Better: Age or Block Replacement? 88 4.3.3 Finding the Optimal Maintenance Period (Age) T^* 88 4.3.4 Contamination of F(t) by Early Failures 89 4.3.5 Treating Uncertainty in Data 89 4.3.6 Age Replacement for the IFR Family 91 4.3.7 Age Replacement with Cost Discounting 92 4.3.8 Random Choice of Preventive Maintenance Periods 94 4.4 Optimal Opportunistic Preventive Maintenance of a Multielement System 96 4.4.1 Introduction 96 4.4.2 System Description - Coordination of Maintenance Actions - Cost Structure 97 4.4.3 Optimization Algorithm. Basic Sequence 99 4.4.4 Discussion - Possible Generalizations 101 4.5 Exercises 102 5 Preventive Maintenance Based on Parameter Control 107 5.1 Introduction - System Parameter Control 107 5.2 Optimal Maintenance of a Multiline System 109 5.3 Preventive Maintenance in a Two-Stage Failure Model 111 5.3.1 The Failure Model 111 5.3.2 Preventive Maintenance. Costs 112 5.3.3 Some Facts about the Nonhomogeneous Poisson Process 112 5.3.4 Finding the Optimal Location of the PM Points 114 5.4 Markov-Type Processes with Rewards (Costs) 116 5.4.1 Markov Chain 116 5.4.2 Semi-Markov Process 117 5.4.3 SMP with Rewards (Costs) 119 5.4.4 Continuous-Tie Markov Process 120 5.5 Opportunistic Replacement of a Two-Component System 122 5.5.1 General Description 122 5.5.2 Formal Setting of the Problem 123 5.5.3 Numerical Example 124 5.6 Discovering a Malfunction in an Automatic Machine 127 5.6.1 Problem Description 127 5.6.2 Problem Formalization 127 5.6.3 Numerical Example 129 5.7 Preventive Maintenance of Objects with Multidimensional State Description 136 5.7.1 General Description 130 5.7.2 The Best Scalarization 132 5.7.3 Preventive maintenance: multidimensional state description 135 6 Best Time Scale for Age Replacement 139 6.1 Introduction 139 6.2 Finding the Optimal Tie Scale 141 6.3 Optimal Time Scale for Age Heplacement: Fatigue Data 142 6.4 Algorithm for Finding the Optimal Tau_alpha 145 6.4.1 Complete Samples 145 6.4.2 Bight-Censored Samples 146 6.4.3 Quantal-Type Data: Parametric Approach 146 6.5 Optimal Age Replacement for Fatigue Test Data 147 7 Preventive Maintenance with Learning 151 7.1 Information Update: Learning from New Data 151 7.2 Maintenance-Inspection Model with No Information Update 154 7.3 Rolling Horizon and Information Update 155 7.4 Actual and "Ideal" Rewards 156 7.5 Numerical Example 156 7.6 Exercises to Chapters 5-7 158 Solutions to Exercises 161 Appendix A: Nonhomogeneous Poisson Process 187 A.1 Definition and Basic Properties 187 A.2 Parametric estimation of the intensity function 190 Appendix B: Covariances and Means of Order Statistics 195 Appendix C: The Laplace Transform 199 Appendix D: Probability Paper 201 Appendix E: Renewal Function 205 References 207 Glossary of Notation 213 Index 215 END