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Semi-Markov models are common modeling tools in the analysis of machines subject to stochastic failures. In this book, we consider systems with finite number of states and random holding times in each state. This consideration relaxes the exponential assumption and provides a rich class of models applicable in reliability, maintenance studies and survival analysis. In practice, data analysis for semi-Markov processes can be quite difficult. The problem of statistical inference for finite state semi-Markov processes is studied using maximum likelihood and empirical estimators of nonlinear functions including Markov kernel, Markov renewal and semi-Markov transition matrices and reliability and availability functions, the rate of occurrence of failure function, the failure rate of a semi-Markov system and other main indicators (MTTF, MDT, MUT...). Moreover, the estimators of peformability functions of such systems are given. The properties of those estimators are studied and some applications are presented.
In several applications of stochastic modeling, semi-Markov processes are useful in modeling a system evolution, so their statistical inference is important. A random function for semi-Markov systems that is of main interest is the time to failure. Despite of developments of theory and applications from the point of view of probability and stochastic processes, no book on statistical studies of semi-Markov process was performed up to now concerning reliability and performability in a general setting. In this book we present a statistical study of the nonparametric estimation of reliability and related quantities of a finite state space semi-Markov system. We propose estimators and give consistency and asymptotic normality results for such quantities, as well a numerical examples.
