Path probability evaluation with repeated rates

In this paper we present a method for handling big Markovian graphs for reliability studies. If the model is Markovian, it is theoretically possible to generate and quantify the state graph. However, it is often not feasible, because of the number of states. One possibility, in the case of a reliability study, is to generate paths leading from the initial state of the system to the failure state. Then we get the system unreliability summing up the probability of all these paths. One difficulty of this approach, we focus on in this paper, is the computation of the probability of each path. When the sojourn rates in two states of the path have the same value, the classical formula used for the convolution of exponential distributions does not fit any longer. We found help for a theoretical solution in a paper by P.G. Harrison (see J. Appl. Probab., vol.27, p.74-87, 1990). Two problems arose during the implementation: numerical errors, and huge computing times. Our tests show that the performance problem is very important, whereas the numerical one is not. Furthermore, we know that, in most of the cases, we do not need exact probability evaluation. Then, we are currently searching for a simplified method, which could be pessimistic, and would have to be numerically robust, and fast.