Novel and Faster Ways for Solving Semi-Markov Processes

Semi-Markov processes (SMP) are powerful stochastic tools for modeling reliability measures over time. This work precisely aims at proposing more efficient mathematical and numerical treatments for SMP in continuous time. The first approach (called 2N-) is based on transition frequency densities and general quadrature methods. The other proposed method (in short Lap-) applies Laplace transforms that are inverted by a Gaussian quadrature method known as Gauss Legendre to obtain the state probabilities in time domain. Mathematical formulation of these approaches as well as descriptions of their numerical treatment are developed and provided with details. The effectiveness of the novel 2N- and Lap- developments is validated by using examples in the context of oil industries. It is shown that the 2N- and Lap- approaches are significantly less time- consuming and have comparable accuracy to Monte Carlo simulation based solution.