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Priority Queue Formulation of Agent-Based Bathtub Model for Network Trip Flows in the Relative Space

Irene Martinez, Wen-long Jin

TL;DR

This paper addresses privacy and computational challenges in simulating network trip flows by developing an agent-based bathtub model (AB2M) in the relative space, where each traveler is represented by their remaining distance to destination. A key innovation is the SCDFO-based priority-queue formulation, which uses a sorted set of characteristic distances Θ(t,n) to determine trip completions without updating all active trips each step. The authors analyze scalability through flow-based downscaling (and discuss distance-based scaling), derive complexity bounds, and examine numerical errors associated with scaling, including discrete versus stochastic demands. The ABD2M demonstrates the ability to simulate millions of trips rapidly while preserving privacy, enabling large-scale studies of travel time reliability, scheduling, and mode choices, with potential extensions to multi-region and multi-modal systems.

Abstract

Agent-based models have been extensively used to simulate the behavior of travelers in transportation systems because they allow for realistic and versatile modeling of interactions. However, traditional agent-based models suffer from high computational costs and rely on tracking physical locations, raising privacy concerns. This paper proposes an efficient formulation for the agent-based bathtub model (AB2M) in the relative space, where each agent's trajectory is represented by a time series of the remaining distance to its destination. The AB2M can be understood as a microscopic model that tracks individual trips' initiation, progression, and completion and is an exact numerical solution of the bathtub model for generic (time-dependent) trip distance distributions. The model can be solved for a deterministic set of trips with a given demand pattern (defined by the start time of each trip and its distance), or it can be used to run Monte Carlo simulations to capture the average behavior and variation stochastic demand patterns, described by probabilistic distributions of trip distances and departure times. To enhance the computational efficiency, we introduce a priority queue formulation, eliminating the need to update trip positions at each time step and allowing us to run large-scale scenarios with millions of individual trips in seconds. We systematically explore the scaling properties and discuss the introduction of biases and numerical errors. The systematic exploration of scaling properties of the modeling of individual agents in the relative space with the AB2M further enhances its applicability to large-scale transportation systems and opens up opportunities for studying travel time reliability, scheduling, and mode choices.

Priority Queue Formulation of Agent-Based Bathtub Model for Network Trip Flows in the Relative Space

TL;DR

This paper addresses privacy and computational challenges in simulating network trip flows by developing an agent-based bathtub model (AB2M) in the relative space, where each traveler is represented by their remaining distance to destination. A key innovation is the SCDFO-based priority-queue formulation, which uses a sorted set of characteristic distances Θ(t,n) to determine trip completions without updating all active trips each step. The authors analyze scalability through flow-based downscaling (and discuss distance-based scaling), derive complexity bounds, and examine numerical errors associated with scaling, including discrete versus stochastic demands. The ABD2M demonstrates the ability to simulate millions of trips rapidly while preserving privacy, enabling large-scale studies of travel time reliability, scheduling, and mode choices, with potential extensions to multi-region and multi-modal systems.

Abstract

Agent-based models have been extensively used to simulate the behavior of travelers in transportation systems because they allow for realistic and versatile modeling of interactions. However, traditional agent-based models suffer from high computational costs and rely on tracking physical locations, raising privacy concerns. This paper proposes an efficient formulation for the agent-based bathtub model (AB2M) in the relative space, where each agent's trajectory is represented by a time series of the remaining distance to its destination. The AB2M can be understood as a microscopic model that tracks individual trips' initiation, progression, and completion and is an exact numerical solution of the bathtub model for generic (time-dependent) trip distance distributions. The model can be solved for a deterministic set of trips with a given demand pattern (defined by the start time of each trip and its distance), or it can be used to run Monte Carlo simulations to capture the average behavior and variation stochastic demand patterns, described by probabilistic distributions of trip distances and departure times. To enhance the computational efficiency, we introduce a priority queue formulation, eliminating the need to update trip positions at each time step and allowing us to run large-scale scenarios with millions of individual trips in seconds. We systematically explore the scaling properties and discuss the introduction of biases and numerical errors. The systematic exploration of scaling properties of the modeling of individual agents in the relative space with the AB2M further enhances its applicability to large-scale transportation systems and opens up opportunities for studying travel time reliability, scheduling, and mode choices.
Paper Structure (14 sections, 4 theorems, 29 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 14 sections, 4 theorems, 29 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model formulation
  3. Review of existing models
  4. Naive Formulation of AB$^2$M
  5. Priority Queue Formulation of AB$^2$M
  6. Shorter-Characteristic-Distance-First-Out principle
  7. Priority queue definition and efficient algorithm
  8. Scalability Property of the Bathtub Models
  9. Scaling of bathtub models
  10. Numerical errors induced by flow-based scaling in the AB$^2$M
  11. Discrete and deterministic demands
  12. Speed variation
  13. Complexity analysis
  14. Conclusion

Key Result

Lemma 1

The upper bound complexity of Part 1 and Part 3 of the algorithms to model AB$^2$M are $\mathcal{O} \left( I log(I) \right)$ for Part 1, and $\mathcal{O} \left( I log (\frac{I}{\bar{e} \Delta t L_N} ) \right)$ for Part 3, where $\bar{e}$ is the average inflow.

Figures (8)

  • Figure 1: Overview of the main steps of the AB$^2$M.
  • Figure 2: Example with 10 agents in 200 seconds, NFD as $V(\rho) = 50(1- \rho/10)^2$, $L_N=1$ km, $\Delta t =25$ s, simulation stopped at $t=150$ s.
  • Figure 3: Representation of characteristic trip distance $\theta(i)$ from the trajectories in relative space.
  • Figure 4: Evolution of $\Theta(t,x)$. When the characteristic trip distance of the new trips is the largest of all the current circulating agents, it is added at the end of the queue. If a new trip $\theta(i)$ is not the longest, it is added between active trips.
  • Figure 5: downscaling of demand: (a) For continuous $E(t)$. (b) For piece-wise constant $E(t)$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 3.1