Exact and approximate improvement to the throughput of a stochastic network

This paper presents a model for throughput-improvement in stochastic networks. It synthesizes and extends the state-of-knowledge on determining the mean value, the lower bound (LB), and upper bound (UB) for the maximum-flow network-design problem. Through the use of an artificial-intelligence programming-language such as PROLOG, the LB was solved very efficiently the first time. The key lies in using `depth-first search´ to generate the flow paths, which are then directly fed into a linear programming (LP) or mixed-integer mathematical programming (MIP) package. In all networks, the LB solution is a better approximation to the exact solution than the UB, which is consistent with theoretical findings. This finding is important in that the 7 acyclic networks represent a diversity of geometry and of survival probabilities. Another advantage of using the LB model and the path-enumeration solution-strategy has to do with an approximation of network reliability and vulnerability in that these approximate measures of performance are readily available from the solutions. Built upon the idea of most-probable states, reliability UB and vulnerability are related to the paths actually used. The paths used are but a small fraction of the paths generated, and the number of generated paths is again tiny compared to the 2ⁿ probability states. A minute fraction of the total number of failure states accounts for most of the state space-a fact that is further reinforced by the prevalence of a zero-throughput state in the flow distributions