It is shown how saddlepoint methods may be used to approximate reliabilities and failure rates of finite stochastic systems with feedback loops. Some countably in .nite state systems including birth-death processes are also considered. The use of saddlepoint methods requires as input the moment generating functions (MGFs) for the system failure time distributions. Some new explicit formulas for these MGFs are given that are amenable to symbolic computation and which also make the numerical computation of saddlepoint approximations quite simple and convenient.
"Reliabilities for feedback systems and their saddlepoint approximation." Statist. Sci. 15 (3) 279 - 298, August 2000. https://doi.org/10.1214/ss/1009212818