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Date:  Thursday, April 21, 2022
Location:  Zoom: Virtual (9:00 AM to 11:00 AM)

Title:  Dissertation Defense: Macroscopic Traffic Models with Behavior Variation Driven by Noise

Abstract:   Hyperbolic PDE can be used to describe the macroscopic dynamics of traffic flow. Equilibrium models (also called first order models) are scalar conservation laws expressing conservation of mass

ρt + (ρv)x = 0

where ρ denotes the traffic density and v the flow velocity. The closure relation v = V (ρ), called the Fundamental Diagram (FD), describes a velocity that instantaneously adjusts itself to flow density. Driver behavior, however, differs among drivers and over time; this variability is not captured by deterministic models. Indeed, real data suggests that while one may identify 'mean' driver behavior, non-equilibrium effects and general variability in driving style results in some distribution around the mean. To model driver variability, we introduce a (small) driver-related parameter z that describes deviation from the mean and create a family of fundamental diagrams V (ρ, z). z is modeled by an advection diffusion equation with white noise forcing and a relaxation to mean (z = 0) behavior. The resulting models adhere to accepted principles for traffic modeling and are capable of reproducing a richer
set of traffic flow phenomena. Most notably, they illustrate that small perturbations may grow into large coherent wave structures, including the formation of jams and emergence of stop-and-go flow patterns, in equilibrium (i.e. first order) models. Dynamic generalizations (also called second order models) have been proposed by numerous authors and have the general form

ρt + (ρv)x = 0
vt + (v − g(ρ)) vx =

V (ρ) − v

where the velocity v = v(x, t) does not adjust instantaneously to traffic density, but instead is governed by an equation that describes rules for acceleration. In addition to modeling driver variation as an auxiliary variable, these models also permit variation of velocity through a direct modification to the velocity equation. In the present work equilibrium and non-equilibrium traffic models with an stochastic behavior variable are presented, as is a direct stochastic velocity perturbation.

Jack's advisors are Smadar Karni and Romesh Saigal.


Speaker:  John (Jack) Wakefield
Institution:  UM

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