简介
Summary:
Publisher Summary 1
The book is a valuable resource for understanding the latest developments in Semi-Markov Processes and reliability. Practitioners, researchers and professionals in applied mathematics, control and engineering who work in areas of reliability, lifetime data analysis, statistics, probability, and engineering will find this book an up-to-date overview of the field.
Publisher Summary 2
The theory of stochastic processes, for science and engineering, can be considered as an extension of probability theory allowing modeling of the evolution of systems over time. The modern theory of Markov processes has its origins in the studies of A.A. Markov (1856-1922) on sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon Brownian motion. The theory of stochastic processes entered in a period of intensive development when the idea of Markov property was brought in. This book is a modern overall view of semi-Markov processes and its applications in reliability. It is accessible to readers with a first course in Probability theory (including the basic notions of Markov chain). The text contains many examples which aid in the understanding of the theoretical notions and shows how to apply them to concrete physical situations including algorithmic simulations. Many examples of the concrete applications in reliability are given. Features:* Processes associated to semi-Markov kernel for general and discrete state spaces* Asymptotic theory of processes and of additive functionals* Statistical estimation of semi-Markov kernel and of reliability function* Monte Carlo simulation* Applications in reliability and maintenance The book is a valuable resource for understanding the latest developments in Semi-Markov Processes and reliability. Practitioners, researchers and professionals in applied mathematics, control and engineering who work in areas of reliability, lifetime data analysis, statistics, probability, and engineering will find this book an up-to-date overview of the field.
目录
Table Of Contents:
Preface xi
Introduction to Stochastic Processes and the Renewal Process 1(30)
Preliminaries 1(1)
Stopping Times 2(2)
Important Families of Stochastic Processes 4(10)
Second-Order Stochastic Processes 4(3)
Martingales 7(1)
Markov Chains 8(1)
Markov Processes 9(5)
Renewal Processes 14(14)
Introduction 14(2)
Renewal Theory 16(6)
Limit Theorems for the Counting Process 22(2)
Modified Renewal Processes 24(1)
Recurrence Times 25(1)
Stopping or Transient Renewal Processes 25(2)
Alternating Renewal Process 27(1)
Regenerative Processes 28(3)
Markov Renewal Processes 31(20)
The Semi-Markov Kernel 31(2)
Processes Associated to a Semi-Markov Kernel 33(8)
Specification of a Markov Renewal Process 41(1)
Robustness of Markov Renewal Processes 42(4)
Korolyuk's State Space Merging Method 46(5)
Semi-Markov Processes 51(34)
Basic Definitions and Properties 51(9)
Markov Renewal Equation 60(6)
Functionals of the Semi-Markov Process 66(4)
Associated Markov Processes 70(5)
Asymptotic Behavior 75(10)
Countable State Space Markov Renewal and Semi-Markov Processes 85(36)
Definitions 85(8)
Classification of States 93(3)
Markov Renewal Equation 96(3)
Asymptotic Behavior 99(10)
Limit Theorems and Distributions 99(3)
Limit Theorems for the Backward and Forward Processes 102(4)
Limit Theorems for Additive Functionals 106(3)
Finite State Space Semi-Markov Processes 109(1)
Distance Between Transition Functions 110(3)
Phase Type Semi-Markov Kernels 113(3)
Elements of Statistical Estimation 116(5)
Introduction 116(1)
Empirical Estimator 117(2)
Piecewise Exponential Estimator 119(2)
Reliability of Semi-Markov Systems 121(32)
Introduction 121(2)
Basic Definitions 123(3)
Coherent Systems 126(2)
Reliability Modeling in the Finite State Space Case 128(5)
Pointwise Availability 129(1)
Reliability 129(1)
Maintainability 130(1)
Failure Rate 131(1)
Steady-State Availability 131(1)
Hitting Times 131(2)
Methods for Obtaining Transition Probabilities 133(15)
Algebraic Method 133(4)
Complementary Variables Method 137(1)
Formal Solution Method 138(3)
Markov Renewal Equation 141(1)
PH-Distributions Method 141(5)
Equivalent Rates Method 146(1)
Statistical Estimation of Reliability 146(2)
Reliability and Performability Modeling in the General State Space Case 148(5)
Introduction 148(1)
Reliability and Performability Evaluation 148(5)
Examples of Reliability Modeling 153(24)
Introduction 153(1)
A Three-State System 153(3)
A System with Mixed Constant Repair Time 156(1)
A System with Multiphase Repair 156(1)
Availability of a Series System 157(1)
A Maintenance Model 158(2)
A System with Nonregenerative States 160(1)
A Two-Component System with Cold Standby 161(1)
Markov Renewal Shock Models 162(3)
Stochastic Petri Nets 165(2)
Monte Carlo Methods 167(10)
Algorithms 167(2)
Simulation of a Three-State System 169(1)
Simulation of a Maintained System 169(8)
A Measures and Probability 177(12)
Fundamentals 177(4)
Conditional Distributions 181(3)
Fundamental Formulas 184(1)
Examples 185(4)
B Laplace--Stieltjes Transform 189(6)
C Weak Convergence 195(6)
References 201(14)
Notation 215(3)
Index 218
Preface xi
Introduction to Stochastic Processes and the Renewal Process 1(30)
Preliminaries 1(1)
Stopping Times 2(2)
Important Families of Stochastic Processes 4(10)
Second-Order Stochastic Processes 4(3)
Martingales 7(1)
Markov Chains 8(1)
Markov Processes 9(5)
Renewal Processes 14(14)
Introduction 14(2)
Renewal Theory 16(6)
Limit Theorems for the Counting Process 22(2)
Modified Renewal Processes 24(1)
Recurrence Times 25(1)
Stopping or Transient Renewal Processes 25(2)
Alternating Renewal Process 27(1)
Regenerative Processes 28(3)
Markov Renewal Processes 31(20)
The Semi-Markov Kernel 31(2)
Processes Associated to a Semi-Markov Kernel 33(8)
Specification of a Markov Renewal Process 41(1)
Robustness of Markov Renewal Processes 42(4)
Korolyuk's State Space Merging Method 46(5)
Semi-Markov Processes 51(34)
Basic Definitions and Properties 51(9)
Markov Renewal Equation 60(6)
Functionals of the Semi-Markov Process 66(4)
Associated Markov Processes 70(5)
Asymptotic Behavior 75(10)
Countable State Space Markov Renewal and Semi-Markov Processes 85(36)
Definitions 85(8)
Classification of States 93(3)
Markov Renewal Equation 96(3)
Asymptotic Behavior 99(10)
Limit Theorems and Distributions 99(3)
Limit Theorems for the Backward and Forward Processes 102(4)
Limit Theorems for Additive Functionals 106(3)
Finite State Space Semi-Markov Processes 109(1)
Distance Between Transition Functions 110(3)
Phase Type Semi-Markov Kernels 113(3)
Elements of Statistical Estimation 116(5)
Introduction 116(1)
Empirical Estimator 117(2)
Piecewise Exponential Estimator 119(2)
Reliability of Semi-Markov Systems 121(32)
Introduction 121(2)
Basic Definitions 123(3)
Coherent Systems 126(2)
Reliability Modeling in the Finite State Space Case 128(5)
Pointwise Availability 129(1)
Reliability 129(1)
Maintainability 130(1)
Failure Rate 131(1)
Steady-State Availability 131(1)
Hitting Times 131(2)
Methods for Obtaining Transition Probabilities 133(15)
Algebraic Method 133(4)
Complementary Variables Method 137(1)
Formal Solution Method 138(3)
Markov Renewal Equation 141(1)
PH-Distributions Method 141(5)
Equivalent Rates Method 146(1)
Statistical Estimation of Reliability 146(2)
Reliability and Performability Modeling in the General State Space Case 148(5)
Introduction 148(1)
Reliability and Performability Evaluation 148(5)
Examples of Reliability Modeling 153(24)
Introduction 153(1)
A Three-State System 153(3)
A System with Mixed Constant Repair Time 156(1)
A System with Multiphase Repair 156(1)
Availability of a Series System 157(1)
A Maintenance Model 158(2)
A System with Nonregenerative States 160(1)
A Two-Component System with Cold Standby 161(1)
Markov Renewal Shock Models 162(3)
Stochastic Petri Nets 165(2)
Monte Carlo Methods 167(10)
Algorithms 167(2)
Simulation of a Three-State System 169(1)
Simulation of a Maintained System 169(8)
A Measures and Probability 177(12)
Fundamentals 177(4)
Conditional Distributions 181(3)
Fundamental Formulas 184(1)
Examples 185(4)
B Laplace--Stieltjes Transform 189(6)
C Weak Convergence 195(6)
References 201(14)
Notation 215(3)
Index 218
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