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Semi-Markov Processes: Applications in System Reliability and Maintenance

Book Description

Semi-Markov Processes: Applications in System Reliability and Maintenance is a modern view of discrete state space and continuous time semi-Markov processes and their applications in reliability and maintenance. The book explains how to construct semi-Markov models and discusses the different reliability parameters and characteristics that can be obtained from those models. The book is a useful resource for mathematicians, engineering practitioners, and PhD and MSc students who want to understand the basic concepts and results of semi-Markov process theory.



  • Clearly defines the properties and theorems from discrete state Semi-Markov Process (SMP) theory.
  • Describes the method behind constructing Semi-Markov (SM) models and SM decision models in the field of reliability and maintenance.
  • Provides numerous individual versions of SM models, including the most recent and their impact on system reliability and maintenance.

Table of Contents

  1. Cover image
  2. Title page
  3. Table of Contents
  4. Copyright
  5. Dedication
  6. Preface
    1. Acknowledgments
  7. 1. Discrete state space Markov processes
    1. Abstract
    2. 1.1 Basic definitions and properties
    3. 1.2 Homogeneous Markov chains
    4. 1.3 Continuous-time homogeneous Markov processes
    5. 1.4 Important examples
  8. 2. Semi-Markov process
    1. Abstract
    2. 2.1 Markov renewal processes
    3. 2.2 Definition of discrete state space SMP
    4. 2.3 Regularity of SMP
    5. 2.4 Other methods of determining the SMP
    6. 2.5 Connection between Semi-Markov and Markov process
    7. 2. 6 Illustrative examples
    8. 2.7 Elements of statistical estimation
    9. 2.8 Nonhomogeneous Semi-Markov process
  9. 3. Characteristics and parameters of SMP
    1. Abstract
    2. 3.1 First passage time to subset of states
    3. 3.2 Interval transition probabilities
    4. 3.3 The limiting probabilities
    5. 3.4 Reliability and maintainability characteristics
    6. 3.5 Numerical illustrative example
  10. 4. Perturbed Semi-Markov processes
    1. Abstract
    2. 4.1 Introduction
    3. 4.2 Shpak concept
    4. 4.3 Pavlov and Ushakov concept
    5. 4.4 Korolyuk and Turbin concept
    6. 4.5 Exemplary approximation of the system reliability function
    7. 4.6 State space aggregation method
    8. 4.7 Remarks on advanced perturbed Semi-Markov processes
  11. 5. Stochastic processes associated with the SM process
    1. Abstract
    2. 5.1 The renewal process generated by return times
    3. 5.2 Limiting distribution of the process
    4. 5.3 Additive functionals of the alternating process
    5. 5.4 Additive functionals of the Semi-Markov process
  12. 6. SM models of renewable cold standby system
    1. Abstract
    2. 6.1 Two different units of cold standby system with switch
    3. 6.2 Technical example
    4. 6.3 Cold standby system with series exponential subsystems
  13. 7. SM models of multistage operation
    1. Abstract
    2. 7.1 Introduction
    3. 7.2 Description and assumptions
    4. 7.3 Construction of Semi-Markov model
    5. 7.4 Illustrative numerical examples
    6. 7.5 Model of multimodal transport operation
  14. 8. SM model of working intensity process
    1. Abstract
    2. 8.1 Introduction
    3. 8.2 Semi-Markov model of the ship engine load process
    4. 8.3 SM model for continuous working intensity process
  15. 9. Multitask operation process
    1. Abstract
    2. 9.1 Introduction
    3. 9.2 Description and assumptions
    4. 9.3 Model construction
    5. 9.4 Reliability characteristics
    6. 9.5 Approximate reliability function
    7. 9.6 Numerical example
  16. 10. Semi-Markov Failure Rate Process
    1. Abstract
    2. 10.1 Introduction
    3. 10.2 Reliability function with random failure rate
    4. 10.3 Semi-Markov Failure Rate Process
    5. 10.4 Random Walk Failure Rate Process
    6. 10.5 Alternating failure rate process
    7. 10.6 Poisson failure rate process
    8. 10.7 Furry-Yule failure rate process
    9. 10.8 Failure rate process depending on random load
    10. 10.9 Conclusions
  17. 11. Simple model of maintenance
    1. Abstract
    2. 11.1 Introduction
    3. 11.2 Description and assumptions
    4. 11.3 Model
    5. 11.4 Characteristics of operation process
    6. 11.5 Problem of time to preventive service optimization
    7. 11.6 Example
  18. 12. Semi-Markov model of system component damage
    1. Abstract
    2. 12.1 Semi-Markov model of multistate object
    3. 12.2 General Semi-Markov model of damage process
    4. 12.3 Multistate model of two kinds of failures
    5. 12.4 Inverse problem for simple exponential model of damage
    6. 12.5 Conclusions
  19. 13. Multistate systems with SM components
    1. Abstract
    2. 13.1 Introduction
    3. 13.2 Structure of the system
    4. 13.3 Reliability of unrepairable system components
    5. 13.4 Binary representation of MMSs
    6. 13.5 Reliability of unrepairable system
    7. 13.6 Numerical illustrative example
    8. 13.7 Renewable multistate system
    9. 13.8 Conclusions
  20. 14. Semi-Markov maintenance nets
    1. Abstract
    2. 14.1 Introduction
    3. 14.2 Model of maintenance net
    4. 14.3 Model of maintenance net without diagnostics
    5. 14.4 Conclusions
  21. 15. Semi-Markov decision processes
    1. Abstract
    2. 15.1 Introduction
    3. 15.2 Semi-Markov decision processes
    4. 15.3 Optimization for a finite states change
    5. 15.4 SM decision model of maintenance operation
    6. 15.5 Optimal Strategy for the Maintenance Operation
    7. 15.6 Optimization Problem for Infinite Duration Process
    8. 15.7 Decision problem for renewable series system
    9. 15.8 Conclusions
  22. Summary
  23. Bibliography
  24. Notation