A Nonparametric Approach to Performability Analysis in Semi-Markov Systems

This work introduces a nonparametric estimator for evaluating the performability of semi-Markov systems, formulated as the sum or integral of a real-valued functional stochastic process. The concept of performability, originally proposed by Meyer, extends classical reliability measures by incorporating performance-related aspects of system behavior. It represents a unified and comprehensive measure that captures both the reliability and performance of a stochastic system as it evolves over time under uncertainty.

For a homogeneous continuous-time semi-Markov process with a given state space and reward rate function, we develop empirical nonparametric estimators for key quantities such as the semi-Markov kernel, renewal matrix, semi-Markov transition matrix, and the mean performance of the system. These estimators are constructed without imposing restrictive parametric assumptions on the sojourn-time distributions, thereby offering improved flexibility, robustness, and adaptability in practical and theoretical applications.

Furthermore, the asymptotic properties of the proposed estimators are rigorously analyzed. In particular, we establish their strong consistency and asymptotic normality, providing a solid theoretical foundation for nonparametric inference in homogeneous continuous-time semi-Markov process. Finally, the usefulness and effectiveness of the theoretical results are demonstrated through a numerical example, confirming the practical relevance and applicability of the proposed approach to performability analysis in complex stochastic systems.