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Traffic Assignment Problem for Footpath Networks with Bidirectional Links

Tanapon Lilasathapornkit, David Rey, Wei Liu, Meead Saberi

TL;DR

The paper develops a macroscopic user equilibrium pedestrian TAP (UE-pTAP) framework that incorporates bidirectional flow, self-organization, and stochastic walking times through four pVDF variants (deterministic/stochastic, symmetric/asymmetric). It formalizes network representation, derives theoretical properties (existence/uniqueness) for symmetric pVDFs and discusses limitations for asymmetric forms, and calibrates the models against controlled experiments using quasi-density to handle oversaturation. Numerical results on a toy network and a large Sydney CBD network demonstrate how bidirectionality affects path choice, travel times, and computational demands, with scenario analyses showing planning utility for closures and demand changes. The work offers a practical, scalable tool for forecasting pedestrian traffic and informs infrastructure planning by capturing realistic bidirectional dynamics and variability. Key contributions include the four pVDF formulations, stochastic symmetric guarantees, and empirical calibration supporting application to large urban networks.

Abstract

The estimation of pedestrian traffic in urban areas is often performed with computationally intensive microscopic models that usually suffer from scalability issues in large-scale footpath networks. In this study, we present a new macroscopic user equilibrium traffic assignment problem (UE-pTAP) framework for pedestrian networks while taking into account fundamental microscopic properties such as self-organization in bidirectional streams and stochastic walking travel times. We propose four different types of pedestrian volume-delay functions (pVDFs), calibrate them with empirical data, and discuss their implications on the existence and uniqueness of the traffic assignment solution. We demonstrate the applicability of the developed UE-pTAP framework in a small network as well as a large scale network of Sydney footpaths.

Traffic Assignment Problem for Footpath Networks with Bidirectional Links

TL;DR

The paper develops a macroscopic user equilibrium pedestrian TAP (UE-pTAP) framework that incorporates bidirectional flow, self-organization, and stochastic walking times through four pVDF variants (deterministic/stochastic, symmetric/asymmetric). It formalizes network representation, derives theoretical properties (existence/uniqueness) for symmetric pVDFs and discusses limitations for asymmetric forms, and calibrates the models against controlled experiments using quasi-density to handle oversaturation. Numerical results on a toy network and a large Sydney CBD network demonstrate how bidirectionality affects path choice, travel times, and computational demands, with scenario analyses showing planning utility for closures and demand changes. The work offers a practical, scalable tool for forecasting pedestrian traffic and informs infrastructure planning by capturing realistic bidirectional dynamics and variability. Key contributions include the four pVDF formulations, stochastic symmetric guarantees, and empirical calibration supporting application to large urban networks.

Abstract

The estimation of pedestrian traffic in urban areas is often performed with computationally intensive microscopic models that usually suffer from scalability issues in large-scale footpath networks. In this study, we present a new macroscopic user equilibrium traffic assignment problem (UE-pTAP) framework for pedestrian networks while taking into account fundamental microscopic properties such as self-organization in bidirectional streams and stochastic walking travel times. We propose four different types of pedestrian volume-delay functions (pVDFs), calibrate them with empirical data, and discuss their implications on the existence and uniqueness of the traffic assignment solution. We demonstrate the applicability of the developed UE-pTAP framework in a small network as well as a large scale network of Sydney footpaths.

Paper Structure

This paper contains 24 sections, 3 theorems, 30 equations, 14 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Traffic assignment problem for vehicular traffic
  4. Link performance function
  5. Bidirectional pedestrian stream
  6. Methodology
  7. Network representation
  8. Pedestrian VDF
  9. Deterministic pVDF
  10. Stochastic pVDF
  11. pTAP with stochastic symmetric pVDF
  12. pTAP with deterministic asymmetric pVDF
  13. pTAP with stochastic asymmetric pVDF
  14. Calibration
  15. Deterministic pVDF
  16. ...and 9 more sections

Key Result

Proposition 1

Consider a pair of links $a, a' \in A$ which belongs to the same bidirectional stream. If the link travel times follow a symmetric pVDF as given by Equation eq:vdf_sym, then From equations eq:variance_link and eq:covariance_link, if standard deviation of two links are equal, variance and covariance of both links are equal as well.

Figures (14)

  • Figure 1: Comparison between physical infrastructure and network representation of footpath
  • Figure 1: The breakdown of the deterministic asymmetric pVDF shows the relationship between travel time and flow of the reference direction across various flow ratios.
  • Figure 2: An overview of the empirical data (a) Speed and flow relationship from Equation \ref{['eq:tregenza']} (b) Travel time and flow comparison between observed flow and quasi-density
  • Figure 3: Calibrated deterministic symmetric pVDF: (a) peredicted link travel time as a function of flow of the reference and counter directions and (b-j) estimated link travel time as a function of flow of the reference direction for different flow ratios $r$. Measurements are based on a 10m long and 4m wide corridor.
  • Figure 4: Calibrated deterministic asymmetric pVDF: (a) estimated link travel time as a function of flow of the reference and counter directions and (b-j) estimated link travel time as a function of flow of the reference direction for different flow ratios $r$. Measurements are based on a 10m long and 4m wide corridor.
  • ...and 9 more figures

Theorems & Definitions (12)

  • Definition 1: Bidirectional stream
  • Definition 2: Symmetric pVDF
  • Definition 3: Deterministic Symmetric pVDF
  • Proposition 1: Covariance of links on the same bidirectional stream
  • proof
  • Definition 4: Stochastic Symmetric pVDF
  • Proposition 2: Existence condition for pTAP with stochastic symmetric pVDF
  • proof
  • Proposition 3: Uniqueness condition
  • proof
  • ...and 2 more