Using minimal cuts to study the system capacity for a stochastic-flow network in two-commodity case

Traditionally, the flow network is assumed to allow a single type of commodity to transmit through it. The system capacity is the maximum value of flow from the source to the sink. It is trivial that the system capacity is fixed for a deterministic flow network. However, for a stochastic-flow network (the capacity of each arc may have several possible values), the system capacity is not fixed. Hence, many authors proposed methods to calculate two performance indices, the probability that the system capacity is greater than d and the system capacity is less than d, for a level d in terms of minimal paths and minimal cuts, respectively. In the case that two types of commodities are transmitted through the stochastic-flow network, this paper uses the properties of minimal cuts to define the system capacity as a two-tuple vector and to propose an algorithm in order to evaluate a new performance index, the probability that the upper bound of system capacity is (d1, d2), for a level (d1, d2).