Flow assignment of a stochastic flow network with multiple node pairs

This article mainly generalizes the flow problem for a stochastic flow network from one node pair case to a multiple node pairs case, where the capacity of each arc has several possible values. Given the demands for multiple node pairs simultaneously, we propose a performance index, the probability that all demands are fulfilled simultaneously. Such an index is named system reliability. A simple algorithm is proposed first to find out the family of all lower boundary points for such demands in terms of minimal paths. Such lower boundary points are vectors representing the current capacity of each arc. The system reliability can be calculated in terms of such lower boundary points. The stochastic flow model can be applied to computer and telecommunication networks. Significance: This paper discusses the flow assignment problem for a stochastic flow network in multiple node pairs case. We propose an index to evaluate the performance for such networks.