An Improved Stirling Approximation for Trip Distribution Models

Many trip distribution models used in transport systems planning are designed to solve maximum entropy optimization problems. Discrete by nature, they must be transformed into continuous and differentiable problems, typically by applying the first-order Stirling approximation. Although it does a reasonable job for large trip flows, this approximation produces significant errors when flows are small. This paper presents two alternatives using the second-order Stirling approximation and Burnside's formula to specify new distribution models that improve prediction for small trip values. In an application to real data for the Santiago, Chile metro system, both proposed formulations obtained results with superior goodness-of-fit and predictive capacity to a traditional model using a first-order Stirling approximation. The version incorporating the second-order Stirling approximation delivered the best performance.