Reliabilities for feedback systems and their saddlepoint approximation

It is shown how saddlepoint methods may be used to approximate reliabilitie s and failure rates of finite stochastic systems with feedback loops. Some countably infinite state systems including birth-death processes are also c onsidered. The use of saddlepoint methods requires as input the moment gene rating functions (MGFs) for the system failure time distributions. Some new explicit formulas for these MGFs are given that are amenable to symbolic c omputation and which also make the numerical computation of saddlepoint app roximations quite simple and convenient.