Table Of ContentFluctuation
Mechanism and
Control on System
Instantaneous
Availability
© 2016 by Taylor & Francis Group, LLC
© 2016 by Taylor & Francis Group, LLC
Fluctuation
Mechanism and
Control on System
Instantaneous
Availability
YI YANG
Yong-Li Yu • Li-Chao Wang
© 2016 by Taylor & Francis Group, LLC
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Contents
Preface ................................................................................................................xi
Authors ..............................................................................................................xv
1 Introduction ...........................................................................................1
1.1 Research Background ........................................................................1
1.2 Research Status at Home and Abroad ................................................3
1.3 Main Content and Structure .............................................................8
2 Concepts Related to System Availability ..............................................11
2.1 Basic Knowledge about Reliability ...................................................11
2.1.1 Basic Concept of Reliability ................................................11
2.1.2 Main Parameters Describing Reliability ..............................12
2.1.2.1 Mean Time to Failure ..........................................13
2.1.2.2 Mean Time to the First Failure ............................13
2.1.2.3 Mean Failure Frequency and Steady-State
Failure Frequency .................................................14
2.1.2.4 Mean Time between Failures ...............................14
2.2 Basic Knowledge about Maintenance and Supportability ................14
2.2.1 Concept and Main Parameters of Maintenance ..................14
2.2.1.1 Product Repair Rate .............................................15
2.2.1.2 Mean Time to Repair ...........................................15
2.2.2 Concept and Main Parameters of Supportability ................15
2.2.2.1 Probability Distribution Function of
Logistic Delay ......................................................16
2.2.2.2 Probability Density Function of Logistic
Delay Time ..........................................................16
2.2.2.3 Switch-to-Repair Rate ..........................................16
2.2.2.4 Mean Logistic Delay Time ...................................16
2.3 Frequently Used Probability Distributions .......................................17
2.3.1 Geometric Distribution .......................................................17
2.3.2 Discrete Weibull Distribution .............................................17
2.4 Concept of Availability ....................................................................17
2.4.1 Basic Concept of Availability ..............................................17
v
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vi ◾ Contents
2.4.2 Major Parameters of Availability .........................................18
2.4.2.1 Instantaneous Availability ....................................18
2.4.2.2 Inherent Availability (A) ......................................18
i
2.4.2.3 Achieved Availability (A) ....................................18
a
2.4.2.4 Operational Availability (A ) ................................19
0
2.5 Common Problems with System Instantaneous Availability ............19
2.5.1 Single-Unit Repairable System ............................................19
2.5.2 Multi-Unit Repairable System .............................................20
2.6 Summary .........................................................................................21
3 Approximate Solution and Its Problem for General Continuous
Availability Model ................................................................................23
3.1 Availability Equation Analysis of Single-Unit System with
General Probability Distribution .....................................................24
3.2 Approximate Solution of System Instantaneous Availability ............25
3.2.1 Modeling ............................................................................26
3.2.2 Physical Significance of Instantaneous Availability
Approximate Solution .........................................................28
3.2.3 Error Analysis of System Approximate Availability .............31
3.2.4 Numerical Example ............................................................32
3.3 Instantaneous Availability Model of Single-Unit Repairable
System with Partial Repair Time Negligible ....................................35
3.3.1 Problem Proposed ...............................................................35
3.3.2 Mathematical Model ...........................................................36
3.3.3 Establishment of Instantaneous Availability Model ............37
3.3.4 Numerical Example ............................................................39
3.4 Summary ........................................................................................42
4 Instantaneous Availability Model of Discrete-Time System under
General Probability Distribution .........................................................45
4.1 Single-Unit Repairable System under Discrete Time ......................46
4.1.1 Modeling ...........................................................................46
4.1.2 Main Results .......................................................................49
4.1.2.1 System Reliability .................................................49
4.1.2.2 System Availability ...............................................50
4.1.2.3 Mean Failure Frequency .......................................50
4.1.3 Numerical Example ............................................................51
4.2 Single-Unit Repairable System with Repair Delay ...........................54
4.2.1 Modeling ............................................................................54
4.2.2 Main Results .......................................................................56
4.2.2.1 System Reliability .................................................56
4.2.2.2 System Availability ...............................................57
4.2.3 Numerical Example ............................................................59
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Contents ◾ vii
4.3 Series System ...................................................................................60
4.3.1 Modeling ............................................................................60
4.3.2 Common Reliability Index .................................................65
4.3.2.1 System Reliability .................................................65
4.3.2.2 Mean Time to the First Failure of System ............65
4.3.2.3 System Instantaneous Availability .......................66
4.3.3 Numerical Example ...........................................................66
4.4 Repairable System with Preventive Maintenance Considered ..........68
4.4.1 Modeling ............................................................................68
4.4.2 System Instantaneous Availability .......................................71
4.4.3 Numerical Example ............................................................72
4.5 Parallel System .................................................................................74
4.5.1 Modeling ............................................................................74
4.5.2 System Instantaneous Availability .......................................78
4.6 Summary .........................................................................................78
5 Instantaneous Availability Model of System with Finite Time
Restrained and Proof of Its Stability ...................................................81
5.1 Introduction ....................................................................................81
5.2 General Repairable System ..............................................................81
5.2.1 Instantaneous Availability Model ........................................81
5.2.2 Stability Proving of Instantaneous Availability ....................83
5.3 Repairable System with Repair Delay ..............................................93
5.3.1 System Instantaneous Availability Model ............................93
5.3.2 Stability Proving of Instantaneous Availability ....................96
5.4 Repairable System with Preventive Maintenance Considered ........113
5.4.1 System Instantaneous Availability Model ..........................113
5.4.2 Stability Proving of Instantaneous Availability ..................115
5.5 Repairable System with Complex Structure ...................................132
5.5.1 System Instantaneous Availability Model ..........................132
5.5.2 Stability Proving of Instantaneous Availability ..................133
5.6 Summary .......................................................................................133
6 Analysis and Control Method for Volatility of System
Instantaneous Availability .................................................................135
6.1 Introduction ..................................................................................135
6.2 Volatility Parameters of System Instantaneous Availability ............137
6.2.1 Indicative Function ...........................................................137
6.2.2 System Matching and System Adaptation Time ...............139
6.2.3 Availability Amplitude and Occurrence Time of
Minimum Availability ......................................................140
6.2.4 Interval Availability ...........................................................140
6.2.5 Parameter of Stable Volatility Speed .................................141
6.2.5.1 Parameter Stability Concept ...............................141
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viii ◾ Contents
6.2.5.2 Conditions for Determining Parameter Stability ...144
6.2.5.3 Parameter Steady Speed Measurement ..............146
6.2.5.4 Definition of Instantaneous Availability
Volatilities Stable Rate .......................................147
6.2.5.5 Instantaneous Availability Volatility Stability
Simulation Analysis ............................................150
6.3 Mechanism Analysis on Volatility of System Instantaneous
Availability ....................................................................................153
6.3.1 Theoretical Analysis on Volatility of System
Instantaneous Availability .................................................153
6.3.1.1 Introduction of the Instantaneous
Availability Model ..............................................153
6.3.1.2 Both X and Y Obey Uniform Distribution
(M = M = M) ...................................................154
1 2
6.3.1.3 Both X and Y Obey Uniform Distribution
(M < M ) ...........................................................156
1 2
6.3.1.4 Both X and Y Obey Exponential Distribution .....160
6.3.1.5 Both X and Y Obey Exponential and
Uniform Distribution .........................................163
6.3.1.6 Summary of Theoretical Analysis of Volatility ....167
6.3.2 Numerical Methods on the Volatility of Instantaneous
Availability ........................................................................167
6.3.2.1 Numerical Algorithm .........................................167
6.3.2.2 Numerical Calculation Solutions and Results .....171
6.4 Volatility Control of System Instantaneous Availability during
Equipment Life Cycle ....................................................................180
6.4.1 State Equation and Constraint Equation .........................181
6.4.1.1 Demonstration Phase .........................................181
6.4.1.2 Development Phase ...........................................182
6.4.1.3 Operation Phase ................................................182
6.4.2 Control Problem ...............................................................183
6.5 Summary .......................................................................................184
7 Comparative Analysis on Discrete-Time Instantaneous
Availability Models under Exponential Distribution .........................185
7.1 Analysis of Errors between Discrete-Time and
Continuous-Time Distributions ....................................................186
7.2 Comparative Analysis on Analytical Solutions and
Numerical Solutions of Continuous-Time Models under
Exponential Distribution ...............................................................189
7.3 Comparison between Analytical Solution of
Continuous-Time Equation and Discrete Equation
Solution Obtained via Sampling ................................................192
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Contents ◾ ix
7.4 Comparison between Numerical Solution of Continuous-Time
Equation and Discrete-Time Equation Solution under
Exponential Distribution ...............................................................196
7.5 Comparison of Instantaneous Availability Models for Systems
with Repair Delay ..........................................................................198
7.5.1 Numerical Method for Continuous-Time System .............198
7.5.2 Description of Discrete-Time Models ..............................200
7.5.3 Model Comparison ...........................................................201
7.6 Comparison of Availability Models for Series
Repairable Systems ....................................................................204
7.6.1 Numerical Method for Continuous-Time System ............204
7.6.2 Discrete Model..................................................................205
7.6.3 Model Comparison ..........................................................206
7.7 Comparison of Availability Models for Systems Considering
Preventive Maintenance ................................................................208
7.7.1 Numerical Solution of Continuous-Time Model ..............208
7.7.2 Discrete-Time Model ........................................................210
7.7.3 Model Comparison ...........................................................210
7.8 Parallel System with Two Different Models of Units .....................212
7.8.1 Continuous-Time Model ..................................................212
7.8.2 Discrete-Time Model ........................................................214
7.8.3 Model Comparison ...........................................................214
7.9 Summary .......................................................................................215
8 Analysis of Instantaneous Availability Fluctuation under
Truncated Discrete Weibull Distribution ..........................................217
8.1 Overview .......................................................................................217
8.2 Weibull Truncated and Discrete Distribution ................................218
8.3 Analysis of Instantaneous Availability Fluctuation for General
Repairable System ..........................................................................219
8.3.1 Effects of Scale and Shape Parameters in Service Life
Distribution on Instantaneous Availability Fluctuation ....219
8.3.2 Effects of Scale and Shape Parameters in Repair Time
Distribution on Instantaneous Availability Fluctuation ......226
8.4 Analysis of Instantaneous Availability Fluctuation for
Repairable System with Repair Delay ............................................232
8.5 Analysis of Instantaneous Availability Fluctuation for
Repairable System with Preventive Maintenance ...........................242
8.5.1 Effects of Repair Time Distribution for Preventive
Maintenance on System Availability Fluctuation ..............242
8.5.2 Effects of Preventive Maintenance Cycle on System
Availability Fluctuation ....................................................246
8.6 Summary .......................................................................................248
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x ◾ Contents
9 Optimal Design of Instantaneous Availability Fluctuation under
Truncated and Discrete Weibull Distribution ...................................251
9.1 Overview .......................................................................................251
9.2 Algorithm Design ..........................................................................252
9.3 Demonstration Phase ....................................................................254
9.3.1 Design of System Minimum Availability Amplitude .........256
9.3.2 Optimal System Matching Design ....................................256
9.3.3 Simulation Experiment .....................................................257
9.4 Development Phase .......................................................................261
9.4.1 Optimal System Matching Design ....................................262
9.4.2 Simulation Experiment .....................................................263
9.5 Operation Phase ...........................................................................266
9.5.1 Optimal Preventive Maintenance Cycle ...........................268
9.5.2 Simulation Experiment ....................................................268
9.6 Summary .......................................................................................269
10 Conclusions and Prospects .................................................................271
References ...................................................................................................273
Index ...........................................................................................................279
© 2016 by Taylor & Francis Group, LLC
