This book provides a modern view of discrete state space and continuous time semi-Markov processes and their applications in reliability and maintenance. It explains how to construct semi-Markov models and discusses the different reliability parameters and characteristics that can be obtained from those models. Topics include: properties and theorems from discrete state Semi-Markov Process (SMP) theory; method behind constructing Semi-Markov (SM) models and SM decision models in the field of reliability and maintenance; numerous individual versions of SM models, including the most recent and their impact on system reliability and maintenance. -- Edited summary from book.
Contents
Front Cover; Semi-Markov Processes: Applications in System Reliability and Maintenance; Copyright; Dedication; Contents; Preface; Chapter 1: Discrete state space Markov processes; 1.1 Basic definitions and properties; 1.2 Homogeneous Markov chains; 1.2.1 Basic definitions and properties; 1.2.2 Classification of states; 1.2.3 Limiting distribution; 1.3 Continuous-time homogeneous Markov processes; 1.4 Important examples; 1.4.1 Poisson process; 1.4.2 Furry-Yule process; 1.4.3 Finite state space birth and death process; 1.5 Numerical illustrative examples; Chapter 2:Semi-Markov process
2.1 Markov renewal processes2.2 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; 2.7.1 Observation of SMP sample path; 2.7.2 Empirical estimators; 2.7.3 Nonparametric estimators of kernel elements densities; 2.8 Nonhomogeneous Semi-Markov process; Chapter 3:Characteristics and parameters of SMP; 3.1 First passage time to subset of states; 3.2 Interval transition probabilities
3.2.1 Interval transition probabilities for alternating process3.2.2 Interval transition probabilities for Poisson process; 3.2.3 Interval transition probabilities for Furry-Yule process; 3.3 The limiting probabilities; 3.4 Reliability and maintainability characteristics; 3.4.1 Reliability function and parameters of the system; 3.4.2 Pointwise availability; 3.4.3 Maintainability function and parameters of the system; 3.5 Numerical illustrative example; 3.5.1 Description and assumptions; 3.5.2 Model construction; 3.5.3 Reliability characteristics and parameters
3.5.4 Numerical illustrative exampleChapter 4:Perturbed Semi-Markov processes; 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; 4.5.1 Numerical illustrative example; 4.6 State space aggregation method; 4.7 Remarks on advanced perturbed Semi-Markov processes; Chapter 5:Stochastic processes associated with the SM process; 5.1 The renewal process generated by return times; 5.1.1 Characteristics and parameters; 5.2 Limiting distribution of the process
5.3 Additive functionals of the alternating process5.4 Additive functionals of the Semi-Markov process; Chapter 6:SM models of renewable cold standby system; 6.1 Two different units of cold standby system with switch; 6.1.1 Introduction; 6.1.2 Description and assumptions; 6.1.3 Construction of Semi-Markov reliability model; 6.1.4 Reliability characteristics; 6.1.5 An approximate reliability function; 6.1.6 Illustrative numerical examples; 6.1.7 Conclusions; 6.2 Technical example; 6.2.1 Assumptions; 6.2.2 Model construction; 6.2.3 Reliability characteristic
