Resolving traffic congestion is one of the important issues related to urban planning. TASEP-LKF (Totally Asymmetric Simple Exclusion Process with Langmuir Kinetics depending on the occupancy of Forward sites) has been proposed as a model for analyzing the phenomenon of traffic congestion itself. Consider a graph paper with only one row of squares. This is considered to be a road, and is filled in black when there are cars and left white when there are no cars. As time progresses, the black squares randomly jump in one direction (the original square is made white and the adjacent square is made black). However, if the square to which they jump is already black, this movement does not occur. If the movement of the black squares stagnates as a result of observing the time evolution, it can be said that a traffic jam is occurring. This type of mathematical model is called TASEP. TASEP-LK, which further takes lane changes into account, is called TASEP. In other words, black squares randomly become white, and vice versa. The method that changes the probability of lane changing depending on the vehicle ahead is called TASEP-LKF.
In the first half of this year, we carried out the following research activities in order to analyze traffic flow using TASEP-LKF.

Ⅰ. Discussion at Chiba University of Commerce

Arai, the principal investigator, met with collaborator Ueshima (National Center for Theoretical Sciences, Research Center for Sustainability) at Chiba University of Commerce to discuss mathematical and numerical calculations regarding the TASEP-LKF model. In addition, during the above activities, related researchers Jun Sato (Tokyo Polytechnic University) and Zhihuang Li (National Taiwan University) visited us and discussed the transition probability and asymptotic analysis of TASEP.

II. Participation in the research conference "2023 Young Summer Seminar on Probability Theory"

Research leader Arai and co-researcher Ueshima (National Center for Theoretical Science Research, Research Center for Sustainability) participated in the research conference "2023 Young Summer Seminar on Probability Theory" to collect information from researchers with knowledge of the TASEP model and related stochastic processes. Arai and Ueshima also discussed the asymptotic stability of the differential equation describing the particle density of TASEP.

III. Presentation at the 2023 NCTS Postdoc Symposium

Based on the results obtained to date in this project, collaborator Ueshima (National Center for Theoretical Sciences, Research Center for Sustainability) gave an international lecture at a symposium hosted by his organization. Specifically, he presented a differential equation for the particle number density to which the mean field approximation is applied, and spoke about the existence of its equilibrium point and its asymptotic stability. He also proved the monotonicity of the steady-state particle number density at the right end point. He presented the mathematically established parts and the results of numerical calculations to support them.

Reference site National Theoretical Science Research Center