book

Semi-Markov Processes

by Franciszek Grabski

September 2014

Intermediate to advanced

270 pages

8h 1m

English

Elsevier

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Acknowledgments
Abstract1.1 Basic definitions and properties1.2 Homogeneous Markov chains1.3 Continuous-time homogeneous Markov processes1.4 Important examples
Abstract2.1 Markov renewal processes2.2 Definition of discrete state space SMP2.3 Regularity of SMP2.4 Other methods of determining the SMP2.5 Connection between Semi-Markov and Markov process2. 6 Illustrative examples2.7 Elements of statistical estimation2.8 Nonhomogeneous Semi-Markov process
Abstract3.1 First passage time to subset of states3.2 Interval transition probabilities3.3 The limiting probabilities
3.5 Numerical illustrative example

Abstract4.1 Introduction4.2 Shpak concept4.3 Pavlov and Ushakov concept4.4 Korolyuk and Turbin concept4.5 Exemplary approximation of the system reliability function4.6 State space aggregation method4.7 Remarks on advanced perturbed Semi-Markov processes
Abstract5.1 The renewal process generated by return times5.2 Limiting distribution of the process5.3 Additive functionals of the alternating process5.4 Additive functionals of the Semi-Markov process
Abstract6.1 Two different units of cold standby system with switch6.2 Technical example6.3 Cold standby system with series exponential subsystems
Abstract7.1 Introduction7.2 Description and assumptions7.3 Construction of Semi-Markov model7.4 Illustrative numerical examples7.5 Model of multimodal transport operation
Abstract8.1 Introduction8.2 Semi-Markov model of the ship engine load process8.3 SM model for continuous working intensity process
Abstract9.1 Introduction9.2 Description and assumptions9.3 Model construction9.4 Reliability characteristics9.5 Approximate reliability function9.6 Numerical example
Abstract10.1 Introduction10.2 Reliability function with random failure rate10.3 Semi-Markov Failure Rate Process10.4 Random Walk Failure Rate Process10.5 Alternating failure rate process10.6 Poisson failure rate process10.7 Furry-Yule failure rate process10.8 Failure rate process depending on random load10.9 Conclusions
Abstract11.1 Introduction11.2 Description and assumptions11.3 Model11.4 Characteristics of operation process11.5 Problem of time to preventive service optimization11.6 Example
Abstract12.1 Semi-Markov model of multistate object12.2 General Semi-Markov model of damage process12.3 Multistate model of two kinds of failures12.4 Inverse problem for simple exponential model of damage12.5 Conclusions
Abstract13.1 Introduction13.2 Structure of the system13.3 Reliability of unrepairable system components13.4 Binary representation of MMSs13.5 Reliability of unrepairable system13.6 Numerical illustrative example13.7 Renewable multistate system13.8 Conclusions
Abstract14.1 Introduction14.2 Model of maintenance net14.3 Model of maintenance net without diagnostics14.4 Conclusions
Abstract15.1 Introduction15.2 Semi-Markov decision processes15.3 Optimization for a finite states change15.4 SM decision model of maintenance operation15.5 Optimal Strategy for the Maintenance Operation15.6 Optimization Problem for Infinite Duration Process15.7 Decision problem for renewable series system15.8 Conclusions

Content preview from Semi-Markov Processes

3.4 Reliability and maintainability characteristics

Reliability and maintainability characteristics and parameters in semi-Markov models are presented in many books: [6, 7, 11, 24, 32, 60, 72, 90, 93]. Assume that the evolution of a system’s reliability is described by a finite state space S SMP ${X (t) : t \geq 0}$ . Elements of a set S represent the reliability states of the system. Let S₊ consists of the functioning states (up states) and S_- contains all the failed states (down states). The subset S₊ and S_- form a partition of S, i.e., $S = S_{+} \cup S_{-}$ and $S$