A Note on Asymptotic Formulae for One-Dimensional Network Flow Problems

This note develops asymptotic formulae for single-commodity network flow problems with random inputs. The transportation linear programming problem (TLP) where N points lie in a region of R1 is one example. It is found that the average distance traveled by an item in the TLP increases with N1/2; i.e., the unit cost is unbounded when N and the length of the region are increased in a fixed ratio. Further, the optimum distance does not converge in probability to the average value. These one-dimensional results are a useful stepping stone toward a network theory for two and higher dimensions.