  Uncertainty often plays an important role in dynamic flow problems. In this
paper, we consider both, a stationary and a dynamic flow model with uncertain
boundary data on networks. We introduce two different ways how to compute the
probability for random boundary data to be feasible, discussing their
advantages and disadvantages. In this context, feasible means, that the flow
corresponding to the random boundary data meets some box constraints at the
network junctions. The first method is the spheric radial decomposition and the
second method is a kernel density estimation. In both settings, we consider
certain optimization problems and we compute derivatives of the probabilistic
constraint using the kernel density estimator. Moreover, we derive necessary
optimality conditions for the stationary and the dynamic case. Throughout the
paper, we use numerical examples to illustrate our results by comparing them
with a classical Monte Carlo approach to compute the desired probability.
