Estimation of rare event probabilities in stochastic networks with exponential and beta probability distributions

The paper is dealing with estimation of rare event probabilities in stochastic networks. The well known variance reduction technique, called Importance Sampling (IS) is an effective tool for doing this. The main idea is to simulate the random system under a modified set of parameters, so as to make the occurrence of the rare events more likely. The major problem of the IS technique is that the optimal modified parameters, called reference parameters to be used in IS are usually very difficult to obtain. Rubinstein (1997) developed the Cross Entropy (CE) method for the solution of this problem of IS technique and then he and his collaborators applied this for the estimation of rare event probabilities in stochastic networks with exponential distribution. The main result of the paper is the extension of CE method for estimation of rare event probabilities in stochastic networks with beta distribution. In this case the calculation of reference parameters of the importance sampling distribution requires numerical solution of a nonlinear equation system. This is done by applying a Newton-Raphson iteration scheme. In this case the CPU time spent for calculation of the reference parameters values can not be neglected. Numerical results are also presented.