Semi-Markov Processes and Reliability

by Limnios, N.; Oprisan, Gheorghe

• ISBN13: 9780817641962
• ISBN10: 0817641963
• Format: Hardcover
• Copyright: 2001-03-01
• Publisher: Birkhauser

Summary

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.

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