Nikolaos Limnios, G. Oprisan

Semi-Markov Processes and Reliability

• Broschiertes Buch

Produktbeschreibung

At first there was the Markov property. The theory of stochastic processes, which can be considered as an exten sion of probability theory, allows the modeling of the evolution of systems through the time. It cannot be properly understood just as pure mathemat ics, separated from the body of experience and examples that have brought it to life. The theory of stochastic processes entered a period of intensive develop ment, which is not finished yet, when the idea of the Markov property was brought in. Not even a serious study of the renewal processes is possible without using the strong tool of Markov processes. The modern theory of Markov processes has its origins in the studies by A. A: Markov (1856-1922) of sequences of experiments "connected in a chain" and in the attempts to describe mathematically the physical phenomenon known as Brownian mo tion. Later, many generalizations (in fact all kinds of weakenings of the Markov property) of Markov type stochastic processes were proposed. Some of them have led to new classes of stochastic processes and useful applications. Let us mention some of them: systems with complete connections [90, 91, 45, 86]; K-dependent Markov processes [44]; semi-Markov processes, and so forth. The semi-Markov processes generalize the renewal processes as well as the Markov jump processes and have numerous applications, especially in relia bility.

Produktdetails

• Statistics for Industry and Technology
• Verlag: Birkhäuser / Birkhäuser Boston / Springer, Basel
• Artikelnr. des Verlages: 978-1-4612-6640-2
• Softcover reprint of the original 1st ed. 2001
• Seitenzahl: 240
• Erscheinungstermin: 21. Oktober 2012
• Englisch
• Abmessung: 254mm x 178mm x 14mm
• Gewicht: 460g
• ISBN-13: 9781461266402
• ISBN-10: 1461266408
• Artikelnr.: 39560218

Inhaltsangabe

1 Introduction to Stochastic Processes and the Renewal Process.- 1.1 Preliminaries.- 1.2 Stopping Times.- 1.3 Important Families of Stochastic Processes.- 1.4 Renewal Processes.- 1.5 Regenerative Processes.- 2 Markov Renewal Processes.- 2.1 The Semi-Markov Kernel.- 2.2 Processes Associated to a Semi-Markov Kernel.- 2.3 Specification of a Markov Renewal Process.- 2.4 Robustness of Markov Renewal Processes.- 2.5 Korolyuk's State Space Merging Method.- 3 Semi-Markov Processes.- 3.1 Basic Definitions and Properties.- 3.2 Markov Renewal Equation.- 3.3 Functional of the Semi-Markov Process.- 3.4 Associated Markov Processes.- 3.5 Asymptotic Behavior.- 4 Countable State Space Markov Renewal and Semi-Markov Processes.- 4.1 Definitions.- 4.2 Classification of States.- 4.3 Markov Renewal Equation.- 4.4 Asymptotic Behavior.- 4.5 Finite State Space Semi-Markov Processes.- 4.6 Distance Between Transition Functions.- 4.7 Phase Type Semi-Markov Kernels.- 4.8 Elements of Statistical Estimation.- 5 Reliability of Semi-Markov Systems.- 5.1 Introduction.- 5.2 Basic Definitions.- 5.3 Coherent Systems.- 5.4 Reliability Modeling in the Finite State Space Case.- 5.5 Methods for Obtaining Transition Probabilities.- 5.6 Reliability and Performability Modeling in the General State Space Case.- 6 Examples of Reliability Modeling.- 6.1 Introduction.- 6.2 A Three-State System.- 6.3 A System with Mixed Constant Repair Time.- 6.4 A System with Multiphase Repair.- 6.5 Availability of a Series System.- 6.6 A Maintenance Model.- 6.7 A System with Nonregenerative States.- 6.8 A Two-Component System with Cold Standby.- 6.9 Markov Renewal Shock Models.- 6.10 Stochastic Petri Nets.- 6.11 Monte Carlo Methods.- A Measures and Probability.- A.I Fundamentals.- A.2 Conditional Distributions.- A.3 FundamentalFormulas.- A.4 Examples.- B Laplace-Stieltjes Transform.- C Weak Convergence.- References.- Notation.

Rezensionen

"The book presents an introductory and at the same time rather comprehensive treatment of semi-Markov processes and their applications to reliability theory. It also provides some general background (like measure theory, Markov processes and Laplace transform), which makes it accessible to a broader audience.... The book may be a useful tool for researchers and students interested in the theory of semi-Markov processes or its applications to reliability problems." -Applications of Mathematics