Abstract
The link transmission model (LTM) has great potential for simulating traffic flow in large-scale networks since it is much more efficient and accurate than the Cell Transmission Model (CTM). However, there lack general continuous formulations of LTM, and there has been no systematic study on its analytical properties such as stationary states and stability of network traffic flow. In this study we attempt to fill the gaps. First we apply the Hopf–Lax formula to derive Newell’s simplified kinematic wave model with given boundary cumulative flows and the triangular fundamental diagram. We then apply the Hopf–Lax formula to define link demand and supply functions, as well as link queue and vacancy functions, and present two continuous formulations of LTM, by incorporating boundary demands and supplies as well as invariant macroscopic junction models. With continuous LTM, we define and solve the stationary states in a road network. We also apply LTM to directly derive a Poincaré map to analyze the stability of stationary states in a diverge-merge network. Finally we present an example to show that LTM is not well-defined with non-invariant junction models. We can see that Newell’s model and continuous LTM complement each other and provide an alternative formulation of the network kinematic wave theory. This study paves the way for further extensions, analyses, and applications of LTM in the future.
Highlights
Cell Transmission Model (CTM) is a discrete Godunov version of the hyperbolic conservation law formulation of the network kinematic wave model, in which a link is divided into cells, a time duration discretized into time steps, boundary fluxes calculated from upstream demands and downstream supplies according to macroscopic junction models, and densities updated from the conservation law
In link transmission model (LTM), macroscopic junction models are used to determine boundary fluxes, but the demand and supply functions are defined from cumulative flows based on Newell’s formulation of the LWR model (Newell, 1993); Newell’s model was variational principle solutions to the Hamilton-Jacobian formulation of the LWR model in (Daganzo, 2005a,b, 2006)
We show that non-invariant junction models lead to ill-defined LTM
Summary
An understanding of congestion patterns in a road network is critical for developing efficient control, management, planning, and design strategies to improve safety, mobility, and. In this study we attempt to fill the gaps: we will first derive two continuous formulations of LTM under arbitrary initial conditions and apply them to solve stationary states and analyze their stability property in a road network. To derive the continuous formulations, we first apply the Hopf-Lax formula to derive Newell’s model under general initial conditions (Evans, 1998; Claudel and Bayen, 2010), and define link demand and supply functions as well link queue and vacancy sizes. In this study we systematically apply the Hopf-Lax formula to derive Newell’s model and two continuous formulations of LTM under general initial conditions.
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