The typical behavior of a queueing system leading to a rare event can be described by minimizing paths. Such paths are defined as the solutions of variational problems. Large deviations theory justifies that thay represent the most probable behavior in an asymptotic sense. New analytical and algorithmical developments allow the automatic calculation of minimizing paths of queueing systems in a number of new and interesting situations. As examples we present several minimizing paths of multi-class feedforward queueing networks, generalized processor sharing disciplines and networks with priority service disciplines. The calculated minimizing paths allow new and deep insights to the complicated behavior of such systems. The common root of these examples is the possibility to model them by a Skorohod map. Based on sample path large deviation principles and the continuity of the Skorohod map we obtain the variational problems which characterize the minimizing paths. A branch-and-bound algorithm is used to approximate a minimizing path by restricting the search space to piecewise linear paths. Numerically we observe that very few pieces are sufficient to obtain a minimizing path.