Semi-Markov Processes and Reliability

1. Front Matter

   Pages i-xii

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2. Introduction to Stochastic Processes and the Renewal Process

   • N. Limnios, G. Oprişan

   Pages 1-29

3. Markov Renewal Processes

   • N. Limnios, G. Oprişan

   Pages 31-49

4. Semi-Markov Processes

   • N. Limnios, G. Oprişan

   Pages 51-83

5. Countable State Space Markov Renewal and Semi-Markov Processes

   • N. Limnios, G. Oprişan

   Pages 85-120

6. Reliability of Semi-Markov Systems

   • N. Limnios, G. Oprişan

   Pages 121-151

7. Examples of Reliability Modeling

   • N. Limnios, G. Oprişan

   Pages 153-176

8. Back Matter

   Pages 177-222

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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.

• Brownian motion
• Engineering reliability
• Markov
• Markov kernel
• Markov process
• Simulation
• Stochastic processes
• algorithm
• algorithms
• data analysis
• modeling
• probability theory
• risk engineering
• statistics
• stochastic process
• quality control, reliability, safety and risk

"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

• Division Mathématiques Appliquées, Université de Technologie de Compiègne, Compiègne Cedex, France

  N. Limnios

• University of “Politehnica” of Bucharest 313, Bucharest, Romania

  G. Oprişan

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