Maximum flow in networks with a small number of random arc capacities

In a directed, capacitated network with single source and sink in which the capacities of some of the arcs are random variables, the max‐flow value is also a random variable. I propose a new approach to the problem of computing its distribution and expected value via a corollary of the max‐flow min‐cut theorem, and implement it explicitly for small numbers of random capacities. I consider the error obtained in estimating the expected max‐flow value by setting some of the random capacities equal to their means. When the flows are restructed by lower bounds, I give the probability of existence of a feasible flow.