Proposed translation of Helmholtz's “Handbuch der physiologischen Optik”

Abstract

The splitting rate model proposed by Smith and Mounce (2011) establishes a traffic evolution process on a link-node network representation, which overcomes the difficulties in applying traditional path-based models and provides the ease of implementing controls at nodes. While their model offers a new method for modeling traffic evolution, it contains an ad-hoc step of flow adjustment to preserve the flow conservation. This flow adjustment step leads to difficulties in analyzing the system properties. This paper proposes a generalized flow splitting model for day-to-day traffic assignment based on the concept of splitting flow at nodes. The proposed model preserves the flow conservation endogenously by introducing the inflow variable into the formulation. The generalized formulation provides the ease to construct a variety of day-to-day traffic assignment models, and serves as a framework for analyzing the models’ properties, such as the invariance property and the preservation of the Lipschitz continuity and strong monotonicity. Specifically, a proportional-adjustment model and a projection-type model are developed based on the proposed generalized formulation. A numerical example demonstrates the ease of implementing the proposed generalized model, as well as its convergence to user equilibrium.