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2011 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Stochastic Processes

with Applications to Reliability Theory

verfasst von: Toshio Nakagawa

Verlag: Springer London

Buchreihe : Springer Series in Reliability Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Reliability theory is of fundamental importance for engineers and managers involved in the manufacture of high-quality products and the design of reliable systems. In order to make sense of the theory, however, and to apply it to real systems, an understanding of the basic stochastic processes is indispensable.

As well as providing readers with useful reliability studies and applications, Stochastic Processes also gives a basic treatment of such stochastic processes as:

the Poisson process,the renewal process,the Markov chain,the Markov process, andthe Markov renewal process.

Many examples are cited from reliability models to show the reader how to apply stochastic processes. Furthermore, Stochastic Processes gives a simple introduction to other stochastic processes such as the cumulative process, the Wiener process, the Brownian motion and reliability applications.

Stochastic Processes is suitable for use as a reliability textbook by advanced undergraduate and graduate students. It is also of interest to researchers, engineers and managers who study or practise reliability and maintenance.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
The importance of reliability will be greatly enhanced by environmental considerations, and moreover, for the protection of natural resources and the earth. Reliability techniques have to be developed and expanded as objective models become more complex and large-scale. They also will be applied not only to daily life, but also to a variety of other fields because consumers, workers, and managers must make, buy, sell, and use, and handle articles and products with a sense of safety and security. A wide knowledge of probabilities, statistics, and stochastic processes are needed for learning reliability theory mathematically.
Toshio Nakagawa
Chapter 2. Poisson Processes
Abstract
It is well-known that most units operating in a useful life period, and complex systems that consist of many kinds of components, fail normally due to random causes independently over the time interval. Then, it is said in technical terms of stochastic processes, that failures occur in a Poisson process that counts the number of failures through time. This is a natural modeling tool in reliability problems. Some reliability measures such as MTTF (Mean Time To Failure), availability, and failure rate are estimated statistically from life data and are in practical use under such modelings without much theoretical arguments. Furthermore, because a Poisson process has stationary and independent properties, it is much convenient for formulating stochastic models in mathematical reliability theory.
Toshio Nakagawa
Chapter 3. Renewal Processes
Abstract
A renewal process in this chapter is the most fundamental process, and a renewal theory is an important theory in stochastic processes. Many basic stochastic systems form renewal processes essentially. Renewal processes are the largest and most crucial chapter in stochastic processes for studying reliability theory because learning a reliability theory is learning a renewal theory:
Toshio Nakagawa
Chapter 4. Markov Chains
Abstract
In Sect.​ 3.​4, we consider the system that repeats up and down such as operating and failed states alternately. Next, as one example of extended models, we take up the system with repair maintenance: The system begins to operate at time 0 and undergoes repair according to two types of failures such as minor and major failures. After the repair completion, the system becomes like new and begins to operate again.
Toshio Nakagawa
Chapter 5. Semi-Markov and Markov Renewal Processes
Abstract
State space is usually defined by the number of units that are working satisfactorily. As far as the applications to reliability theory is concerned, we consider only a finite number of states, contrast with a queueing theory. We mention only the theory of stationary Markov processes with a finite-state space. It is shown that transition probabilities, first-passage distributions, and renewal functions are given by forming renewal equations. Furthermore, some limiting properties are summarized when all states communicate.
Toshio Nakagawa
Chapter 6. Cumulative Processes
Abstract
Failures of units or systems are generally classified into two failure modes: Catastrophic failure in which units fail by some sudden shock, and degradation failure in which units fail by physical deterioration suffered from some damage. In the latter case, units fail when the total damage due to shocks has exceeded a critical failure level. This is called a cumulative damage model or shock model with additive damage and can be described theoretically by a cumulative process [1] in stochastic processes. Damage models can be applied to actual units that are working in architecture, industry, service, information, and computers, and were summarized [2].
Toshio Nakagawa
Chapter 7. Brownian Motion and Lévy Processes
Abstract
Wiener (1923) and Lévy (1939) mathematically gave the theoretical foundation and construction of the Brownian Motion based on the phenomenon of small particles observed by Brown (1827) and Einstein (1905). This is called the Brownian Motion or Wiener process, and is an example of a Markov process with continuous time and state space. This is now one of the most useful stochastic processes in applied sciences such as physics, economics, communication theory, biology, management science, and mathematical statistics, and recently, it is also used to make several valuable models in finance.
Toshio Nakagawa
Chapter 8. Redundant Systems
Abstract
High system reliability can be achieved by providing redundancy and maintenance. It is not too much to say that learning reliability is analyzing redundant systems and deriving optimum maintenance policies. In the preceding chapters, we have already given many useful examples of reliability models to understand naturally stochastic processes, and conversely, to apply the theory of stochastic processes to actual reliability models. As final examples of reliability models, we take up standard redundant systems and show systematically how to use well the techniques of stochastic processes to analyze them and to obtain their reliability properties theoretically. This would be greatly helpful for understanding stochastic processes and learning reliability theory [1, p. 160].
Toshio Nakagawa
Backmatter
Metadaten
Titel
Stochastic Processes
verfasst von
Toshio Nakagawa
Copyright-Jahr
2011
Verlag
Springer London
Electronic ISBN
978-0-85729-274-2
Print ISBN
978-0-85729-273-5
DOI
https://doi.org/10.1007/978-0-85729-274-2

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