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Semi-Markov Models

Book Description

Featuring previously unpublished results, Semi-Markov Models: Control of Restorable Systems with Latent Failures describes valuable methodology which can be used by readers to build mathematical models of a wide class of systems for various applications. In particular, this information can be applied to build models of reliability, queuing systems, and technical control.

Beginning with a brief introduction to the area, the book covers semi-Markov models for different control strategies in one-component systems, defining their stationary characteristics of reliability and efficiency, and utilizing the method of asymptotic phase enlargement developed by V.S. Korolyuk and A.F. Turbin. The work then explores semi-Markov models of latent failures control in two-component systems. Building on these results, solutions are provided for the problems of optimal periodicity of control execution. Finally, the book presents a comparative analysis of analytical and imitational modeling of some one- and two-component systems, before discussing practical applications of the results



  • Reflects the possibility and effectiveness of this method of modeling systems, such as phase merging algorithms developed by V.S. Korolyuk, A.F. Turbin, A.V. Swishchuk, little covered elsewhere
  • Focuses on possible applications to engineering control systems

Table of Contents

  1. Cover
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Preface
  6. List of Notations and Abbreviations
  7. Introduction
  8. Chapter 1: Preliminaries
    1. Abstract
    2. 1.1. Strategies and characteristics of technical control
    3. 1.2. Preliminaries on renewal theory
    4. 1.3. Preliminaries on semi-Markov processes with arbitrary phase space of states
  9. Chapter 2: Semi-Markov Models of One-Component Systems with Regard to Control of Latent Failures
    1. Abstract
    2. 2.1. The system model with component deactivation while control execution
    3. 2.2. The System Model Without Component Deactivation While Control Execution
    4. 2.3. Approximation of Stationary Characteristics of One-Component System Without Component Deactivation
    5. 2.4. The System Model With Component Deactivation and Possibility Of Control Errors
    6. 2.5. The System Model With Component Deactivation and Preventive Restoration
  10. Chapter 3: Semi-Markov Models of Two-Component Systems with Regard to Control of Latent Failures
    1. Abstract
    2. 3.1. The model of Two-Component serial system with immediate control and restoration
    3. 3.2. The model of Two-Component parallel system with immediate control and restoration
    4. 3.3. The model of Two-Component serial system with components deactivation while control execution, the distribution of components operating TF is exponential
    5. 3.4. The model of Two-Component parallel system with components deactivation while control execution, the distribution of components operating TF is exponential
    6. 3.5. Approximation of stationary characteristics of Two-Component serial systems with components deactivation while control execution
  11. Chapter 4: Optimization of Execution Periodicity of Latent Failures Control
    1. Abstract
    2. 4.1. Definition of optimal control periodicity for One-Component systems
    3. 4.2. Definition of optimal control periodicity for Two-Component systems
  12. Chapter 5: Application and Verification of the Results
    1. Abstract
    2. 5.1. Simulation models of systems with regard to latent failures control
    3. 5.2. The structure of the automatic decision system for the management of periodicity of latent failures control
  13. Chapter 6: Semi-Markov Models of Systems of Different Function
    1. Abstract
    2. 6.1. Semi-Markov model of a queuing system with losses
    3. 6.2. The system with cumulative reserve of time
    4. 6.3. Two-phase system with a intermediate buffer
    5. 6.4. The model of technological cell with nondepreciatory failures
  14. Appendix A: The Solution of the System of Integral Equations (2.24)
  15. Appendix B: The Solution of the System of Integral Equations (2.74)
  16. Appendix C: The Solution of the System of Integral Equation (3.6)
  17. Appendix D: The Solution of the System of Equation (3.34)
  18. References
  19. Index