Novel and faster ways for solving semi-markov processes: mathematical and numerical issues

AUTOR(ES)

MÃrcio Josà das Chagas Moura

DATA DE PUBLICAÇÃO

2009

RESUMO

Continuous-time semi-Markov processes (SMP) are important stochastic tools for modeling reliability metrics over time for systems where the future behavior depends on the current and next states as well as on sojourn times. The classical approach for solving the interval transition probabilities of SMP consists of directly applying any general quadrature method to the integral equations. However, this approach has a considerable computational effort. Namely N2 coupled integral equations must be solved, where N is the number of states. Therefore, this thesis proposes more efficient mathematical and numerical treatments for SMP. The first approach, which is called 2N-method, is based on transition frequency densities and general quadrature methods. Basically, it consists of only solving N coupled integral equations and N straightforward integrations. Another proposed method, named Lapmethod, is based on the application of Laplace transforms that are inverted by the Gauss quadrature method known as Gauss Legendre to obtain the state probabilities on the time domain. Mathematical formulation of these approaches as well as descriptions of their numerical treatment, including accurateness and time convergence issues, are developed and provided with details. The effectiveness of the novel 2N- and Lap-developments will be compared against the results provided by the classical method by using examples in the context of reliability engineering. From these examples, it is showed that the 2N- and the Laplace-based approach are significantly less time-consuming and have accuracy comparable to the classical method

ASSUNTO(S)

transition frequency densities availability assessment laplace transforms densidades de frequÃncia de transiÃÃo confiabilidade quadrature methods mÃtodos de quadratura processos semi-markovianos avaliaÃÃo da disponibilidade gauss quadrature reliability semi-markov process quadratura gaussiana transformadas de laplace engenharia de producao

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