Multiple Commodity Flow Problem with Unreliable Nodes subject to Budget Constraints

Many real life systems such as manufacturing, telecommunication, computers, and urban traffic systems can be modeled as capacitated-flow networks. The reliability and unreliability are two important indices to measure the quality level for a capacitated-flow network. For a multiple commodity (different types of commodity) capacitated-flow network with unreliable nodes, the arcs and nodes all have several possible capacities and may fail. Different types of commodities that are transmitted through the same network simultaneously compete for the available capacities of arcs and nodes. In this paper, we first define the system capacity as a vector composed by the capacity of each component (arc or node). Then we propose a performance index, which is the probability that the upper bound of the system capacity equals a given pattern, subject to the budget constraint. The performance index can be applied to evaluate the quality level for such a network. A simple algorithm based on minimal cuts is thus presented to evaluate the performance index.