Dynamic reliability via computational solution of generalized state-transition equations for entry-time processes

Entry-time processes are finite-state continuous-time jump processes with transition rates depending only on the two states involved, the calendar time, and the most recent arrival time (entry time). Entry-time processes are transformed into Markov processes via the standard technique of incorporating entry time into the state variables. It is shown that the associated state-transition (Chapman–Kolmogorov) equations can be written as a coupled pair of integrodifferential equations. A finite-difference approximation to these equations is developed. This computational approach is verified, and some of its properties delineated, via two hypothetical examples. One of these examples admits a semi-analytic solution, while simulations provide the base of comparison for the other.