  We consider a network whose links have random capacities and in which a
certain target amount of flow must be carried from some source nodes to some
destination nodes. Each destination node has a fixed demand that must be
satisfied and each source node has a given supply. We want to estimate the
unreliability of the network, defined as the probability that the network
cannot carry the required amount of flow to meet the demand at all destination
nodes. When this unreliability is very small, which is our main interest in
this paper, standard Monte Carlo estimators become useless because failure to
meet the demand is a rare event. We propose and compare two different methods
to handle this situation, one based on a conditional Monte Carlo approach and
the other based on generalized splitting. We find that the first is more
effective when the network is highly reliable and not too large, whereas for a
larger network and/or moderate reliability, the second is more effective.
