Semi-Markov Processes

1. Discrete state space Markov processes

• Abstract
• 1.1 Basic definitions and properties
• 1.2 Homogeneous Markov chains
• 1.3 Continuous-time homogeneous Markov processes
• 1.4 Important examples

2. Semi-Markov process

• Abstract
• 2.1 Markov renewal processes
• 2.2 Definition of discrete state space SMP
• 2.3 Regularity of SMP
• 2.4 Other methods of determining the SMP
• 2.5 Connection between Semi-Markov and Markov process
• 2. 6 Illustrative examples
• 2.7 Elements of statistical estimation
• 2.8 Nonhomogeneous Semi-Markov process

3. Characteristics and parameters of SMP

• Abstract
• 3.1 First passage time to subset of states
• 3.2 Interval transition probabilities
• 3.3 The limiting probabilities
• 3.4 Reliability and maintainability characteristics
• 3.5 Numerical illustrative example

4. Perturbed Semi-Markov processes

• Abstract
• 4.1 Introduction
• 4.2 Shpak concept
• 4.3 Pavlov and Ushakov concept
• 4.4 Korolyuk and Turbin concept
• 4.5 Exemplary approximation of the system reliability function
• 4.6 State space aggregation method
• 4.7 Remarks on advanced perturbed Semi-Markov processes

5. Stochastic processes associated with the SM process

• Abstract
• 5.1 The renewal process generated by return times
• 5.2 Limiting distribution of the process
• 5.3 Additive functionals of the alternating process
• 5.4 Additive functionals of the Semi-Markov process

6. SM models of renewable cold standby system

• Abstract
• 6.1 Two different units of cold standby system with switch
• 6.2 Technical example
• 6.3 Cold standby system with series exponential subsystems

7. SM models of multistage operation

• Abstract
• 7.1 Introduction
• 7.2 Description and assumptions
• 7.3 Construction of Semi-Markov model
• 7.4 Illustrative numerical examples
• 7.5 Model of multimodal transport operation

8. SM model of working intensity process

• Abstract
• 8.1 Introduction
• 8.2 Semi-Markov model of the ship engine load process
• 8.3 SM model for continuous working intensity process

9. Multitask operation process

• Abstract
• 9.1 Introduction
• 9.2 Description and assumptions
• 9.3 Model construction
• 9.4 Reliability characteristics
• 9.5 Approximate reliability function
• 9.6 Numerical example

10. Semi-Markov Failure Rate Process

• Abstract
• 10.1 Introduction
• 10.2 Reliability function with random failure rate
• 10.3 Semi-Markov Failure Rate Process
• 10.4 Random Walk Failure Rate Process
• 10.5 Alternating failure rate process
• 10.6 Poisson failure rate process
• 10.7 Furry-Yule failure rate process
• 10.8 Failure rate process depending on random load
• 10.9 Conclusions

11. Simple model of maintenance

• Abstract
• 11.1 Introduction
• 11.2 Description and assumptions
• 11.3 Model
• 11.4 Characteristics of operation process
• 11.5 Problem of time to preventive service optimization
• 11.6 Example

12. Semi-Markov model of system component damage

• Abstract
• 12.1 Semi-Markov model of multistate object
• 12.2 General Semi-Markov model of damage process
• 12.3 Multistate model of two kinds of failures
• 12.4 Inverse problem for simple exponential model of damage
• 12.5 Conclusions

13. Multistate systems with SM components

• Abstract
• 13.1 Introduction
• 13.2 Structure of the system
• 13.3 Reliability of unrepairable system components
• 13.4 Binary representation of MMSs
• 13.5 Reliability of unrepairable system
• 13.6 Numerical illustrative example
• 13.7 Renewable multistate system
• 13.8 Conclusions

14. Semi-Markov maintenance nets

• Abstract
• 14.1 Introduction
• 14.2 Model of maintenance net
• 14.3 Model of maintenance net without diagnostics
• 14.4 Conclusions

15. Semi-Markov decision processes

• Abstract
• 15.1 Introduction
• 15.2 Semi-Markov decision processes
• 15.3 Optimization for a finite states change
• 15.4 SM decision model of maintenance operation
• 15.5 Optimal Strategy for the Maintenance Operation
• 15.6 Optimization Problem for Infinite Duration Process
• 15.7 Decision problem for renewable series system
• 15.8 Conclusions