On the existence of a feasible flow in a stochastic transportation network

Many transportation networks, e.g., networks of cooperating power systems, and hydrological networks involve a real-valued demand function, defined on the set of nodes, and it is said to be feasible if there exists a flow such that at each node the sum of the incoming flow values is greater than or equal to the demand assigned to this node. By the theorem of D. Gale and A. Hoffman, a system of linear inequalities involving the demand and the arc capacity functions, gives necessary and sufficient condition for the feasibility of the demand. If the demands and/or the arc capacities are random, then an important problem is to find the probability that all these inequalities are satisfied. This paper proposes a new method to eliminate all redundant inequalities for given lower and upper bounds of the demand function, and finds sharp lower and upper bounds for the probability that a feasible flow exists. The results can be used to support transportation network analysis and design.