The splitting method in rare event simulation

The past fifty years the field of the estimation of rare event probabilities has grown considerably, partly because of the enormous growth in computing power during this period. Because it still is and will not ever be possible to estimate these probabilities efficiently using standard techniques a multitude of methods has been developed and described in the literature and incorporated into simulation packages. The single two most important techniques are based on simple principles. Importance Sampling Change the system in a way that makes the probabilities to estimate become large, so that standard methods can be applied again. One can get the original probabilities back by accounting for the system transformation used. Importance Splitting Change the paths traversed in the simulation in a way that the promising paths are split into a multitude of lightweight paths. In this manner one obtains more activity in the interesting area and one will see the rare event happening more frequently, making the estimates better. Both techniques are about fifty years old and have evolved in several directions. Here we have limited the research to the latter method. The splitting method has been reinvented a number of times in the literature and it has only reached maturity during the last decade. Before, the method was limited to simple models and it was not asymptotically efficient. In this thesis a mathematical foundation is developed for the used methods; the efficiency and complexity of the basic algorithm are also derived. New techniques are developed and compared based on efficiency measures on a broad range of reference models. We see that all known problems and limitations can be dealt with using new techniques that enrich the splitting method. The combination of the analytical approach of the proposed methods and the validation in practice produces a strategy whose efficiency is optimal for a broad class of models and problems. The collection of methods and techniques is gathered in a tool designed explicitly to solve a broad range of rare event problems. The practical application of the work reported here in this thesis can be found in modern communication networks where one is interested in the quality of service delivered to the customer. The (hopefully) rare event in such a setting is the probability of loss of data which the customer wishes to transfer over the network. The presented method will apply to many rare event problems; the currently most common practical use for the method is probably the telecommunications area