Technical note: the expected length of an orderly path

Abstract

Logistics planners often need to estimate length of a path through n random points. The problem has connections to geographical surveys, Steiner tree problems, traveling salesman problems, Amazon.com and Alibaba.cn drone aircraft multi-package outbound delivery paths, and aircraft sorties. In this effort, we utilize two applied probability tenets in concert to derive a closed form model for the expected distance between orderly pairs of randomly distributed delivery locations across some region. Indeed, by themselves each of the two tenets offer little help but when utilized simultaneously they elucidate the problem. Then, this expected distance is multiplied by (n − 1) to estimate the total length of the associated orderly path (albeit not the optimal path) through the n points. This total path length estimate, in turn, provides decision makers with a better information for distribution and logistics planning.