Point Processes for Reliability Analysis: Shocks and Repairable Systems

✍ Scribed by Cha, Ji Hwan.; Finkelstein, Maxim

• Publisher: Springer International Publishing : Imprint : Springer
• Year: 2018
• Tongue: English
• Leaves: 427
• Series: Springer series in reliability engineering
• Category: Library

✦ Synopsis

Introduction -- Preliminaries: Reliability and Point Processes -- Renewal Processes and Applications -- Poisson Process -- Advanced Poisson Shock Models -- Advanced Poisson Shock Models -- Generalizations of Renewal Process -- Generalized Polya Process -- Applications of the Generalized Polya Process -- Multivariate Generalized Polya Process -- Applications of the Mixed Poisson Process -- Shocks as the Discrete Scale.;Focusing on the theory and applications of point processes, Point Processes for Reliability Analysis naturally combines classical results on the basic and advanced properties of point processes with recent theoretical findings of the authors. It also presents numerous examples that illustrate how general results and approaches are applied to stochastic description of repairable systems and systems operating in a random environment modelled by shock processes. The real life objects are operating in a changing, random environment. One of the ways to model an impact of this environment is via the external shocks occurring in accordance with some stochastic point processes. The Poisson (homogeneous and nonhomogeneous) process, the renewal process and their generalizations are considered as models for external shocks affecting an operating system. At the same time these processes model the consecutive failure/repair times of repairable engineering systems. Perfect, minimal and intermediate (imperfect) repairs are discussed in this respect. Covering material previously available only in the journal literature, Point Processes for Reliability Analysis provides a survey of recent developments in this area which will be invaluable to researchers and advanced students in reliability engineering and applied mathematics.

✦ Table of Contents

Preface......Page 7
Contents......Page 9
1.1 Aim and Scope of the Book......Page 13
1.2 Brief Overview......Page 16
References......Page 20
2.1 Distribution Function and Failure Rate......Page 21
2.2 Mean Remaining Lifetime......Page 26
2.3 Monotonicity of the Failure Rate and the MRL Function......Page 29
2.4 Stochastic Ordering......Page 33
2.5 Point Processes and Their Characterization......Page 36
2.6 Perfect and Minimal Repair......Page 42
2.7 Shocks and Extreme Shock Model......Page 43
References......Page 46
3.1 Definition and Main Properties......Page 48
3.2 Limiting Properties......Page 54
3.3 Alternating Renewal and Renewal Reward Processes......Page 58
3.4.1 Renewal Reward Process with Continuous Output......Page 62
3.4.2 Alternating Renewal Process with Gradual Repair......Page 64
3.4.3 Classical Optimal Replacement Problem and Its Generalization......Page 70
3.4.4 Using Renewal Equations in Shocks Modeling......Page 75
3.4.5 Renewal in Populations of Technical Items and Organisms......Page 78
References......Page 82
4.1 Homogeneous Poisson Process......Page 83
4.2 Nonhomogeneous Poisson Process......Page 86
4.3 Minimal Repair and Applications to Optimal Maintenance......Page 96
4.4.1 ‘Classical’ Extreme Shock Model......Page 103
4.4.2 Direct Impact on the Failure Rate......Page 105
4.5 General Compound Poisson Process and a Shot Noise Process......Page 108
4.6 Mixed Poisson Process......Page 115
References......Page 119
5.1 The Terminating Shock Process with Independent Wear Increments......Page 121
5.1.1 General Setting......Page 122
5.1.2 Exponentially Distributed Boundary......Page 124
5.1.3 Deterministic Boundary......Page 127
5.2 History-Dependent Termination Probability......Page 130
5.3.1 Shot Noise Process Without Critical Shocks......Page 137
5.3.2 Shot Noise Process with Critical Shocks and Deterioration......Page 138
5.4 Extreme Shock Model with Delayed Termination......Page 149
5.5 Cumulative Shock Model with Initiated Wear Processes......Page 152
5.6 ‘Curable’ Shock Processes......Page 156
5.7 Stress-Strength Model with Delay and Cure......Page 161
5.8 Survival of Systems with Protection Subject to Two Types of External Attacks......Page 163
5.9.1 General Setting......Page 170
5.9.2 Formal Description of the Information-Dependent Thinning......Page 173
5.9.3 Stress-Strength Type Classification Model......Page 174
References......Page 177
6.1.1 Conditional Characteristics......Page 179
6.1.2 A Specific Case......Page 185
6.1.3 The Dynamics of Conditional Distributions......Page 187
6.2.1 The Failure Model and Supplementary Results......Page 193
6.2.2 The PM Models for Systems Under Shocks......Page 196
6.3 Bivariate Preventive Maintenance Policy......Page 200
6.4 Bivariate Preventive Maintenance Policy with Minimal Repair......Page 208
References......Page 218
7.1 Virtual Age......Page 220
7.2 G-Renewal Process......Page 226
7.3 General Repair Process......Page 228
7.4 Equilibrium Virtual Age......Page 231
7.5 Aging and Limiting Properties......Page 235
7.6 Application: Optimal Degree of Repair......Page 240
7.7 Renewal Equations......Page 244
7.8 Failure Rate Reduction Models......Page 246
7.9 Geometric and Geometric-Type Processes......Page 249
References......Page 254
8.1 Introduction and Fundamental Property......Page 256
8.2 Conditional Distribution of Arrival Times......Page 269
8.3 Compound Generalized Polya Process......Page 276
8.4 Reliability Application......Page 279
8.5 Characterization as a Subclass of Mixed Poisson Process......Page 284
References......Page 286
9 Applications of the Generalized Polya Process......Page 288
9.1.1 Stochastic Model and Main Result......Page 289
9.1.2 Stochastic Analysis and Interpretation: A Point Process Point of View......Page 294
9.2.1 Properties of the Shot Noise Process......Page 302
9.2.2 Stochastic Failure Model......Page 309
9.2.3 History-Dependent Residual Lifetime......Page 316
9.3 Bivariate Replacement Policy for a System Subject to the GPP Repair Process......Page 321
9.4.1 Interpretation of Modelling Parameters......Page 324
9.4.2 Two Periodic Preventive Maintenance Policies......Page 325
9.4.3 Optimization of PM Models......Page 332
References......Page 334
10.1 Definition and Basic Properties: Bivariate Case......Page 337
10.2 Characterization and Further Properties......Page 346
10.3 Generalization to Multivariate Processes......Page 355
References......Page 358
11.1.1 Motivation and Heterogeneous Replacement Policy......Page 359
11.1.2 Optimal Replacement Policy......Page 369
11.2.1 Information-Based Burn-in Procedure......Page 372
11.2.2 Optimal Burn-in Parameters......Page 380
11.3.1 Information-Based Warranty Policy......Page 385
11.3.2 Generalized Warranty Policy Model......Page 395
References......Page 397
12 Shocks as the Discrete Scale......Page 399
12.1 Time-Free Modeling......Page 400
12.1.1 Shocks with Delay......Page 401
12.1.2 Discrete Shot Noise......Page 402
12.2 Shocks and Signatures......Page 406
12.3 Optimal Mission Duration for Systems Affected by Shocks......Page 409
12.3.1 Optimal Mission Duration with Major and Minor Failures......Page 410
12.3.2 Optimal Mission Duration for Non-repairable Systems......Page 417
References......Page 426

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