Overall-terminal reliability of a stochastic capacitated-flow network

For a stochastic and directed capacitated-flow network in which the capacity of each arc has several possible values, this article generalizes the system reliability problem from single source node and single sink node cases to an overall-terminal case. Given the demand for each node pair simultaneously, a simple algorithm is proposed first to generate all lower boundary points for such demands in terms of minimal paths. The lower boundary point is a vector denoting the current capacity of each arc. The system reliability, the probability that the system satisfies the demands simultaneously, can be calculated in terms of such lower boundary points by applying the inclusion-exclusion method.