Estimation of the Unidirectional Traffic Flow Velocity Limit with High Computational Efficiency

Introduction. In the modern development of intelligent transportation systems (ITS), an urgent task is the accurate estimation of the velocity limit of traffic flow on a highway. Despite existing solutions to this problem based on statistical mechanics methods and stochastic models, gaps remain in adapting these theories to real road segments of limited length. The traditional thermodynamic limit formula, used to calculate the average velocity of traffic flow, becomes inaccurate for small road segment lengths, limiting its applicability in practical traffic monitoring tasks. The aim of this study is a comparative analysis of various approaches to estimating the average velocity limit of traffic flow.

Materials and Methods. The study was conducted using the method of statistical mechanics and a stochastic model on a one-dimensional finite lattice. Numerical experiments with various parameter values (number of cells, traffic density, and movement probability) were used for analysis.

Results. The study revealed significant discrepancies between the results obtained using the statistical mechanics method and other approaches when the road segment length was small. The efficiency of the second and third approaches was confirmed for limited road segments, where they demonstrated greater accuracy and applicability.

Discussion and Conclusion. The research results have practical significance for the development of intelligent traffic management systems, especially for short road segments. The proposed approaches can be successfully integrated into modern monitoring systems to improve their accuracy. The theoretical significance of this work lies in advancing the methodology for traffic flow estimation while accounting for the specific conditions of real-world environments.

1. Femke van Wageningen-Kessels et al. Genealogy of traffic flow models. EURO Journal on Transportation and Logistics. 2015;4(4):445–473.

2. Payne H. Models of freeway traffic and control. In: Bekey, G.A. (ed.) Mathematical Models of Public Systems. Simulation Council, La Jolla, CA. 1971;1:51–61.

3. Kerner B., Konhäuser P. Structure and parameters of clusters in traffic flow. Physical Review E. 1994;50:54–83.

4. Aw A., Rascle M. Resurrection of “second order models” of traffic flow. SIAM Journal on Applied Mathematics. 2000;60:916–938.

5. Zhang H.M. A non-equilibrium traffi model devoid of gas-like behavior. Transportation Research. B. 2002;36(3):275–290.

6. Lighthill M.J., Whitham G.B. On kinematic waves II. A theory of traffic flow on long crowded roads. Proceedings of the royal society of London. series a. mathematical and physical sciences. 1955; 229(1178):317–345.

7. Richards P.I. Shock waves on the highway. Operations research. 1956;4(1):42–51.

8. Chetverushkin, B.N. Kinetic schemes and quasi-gasdynamic system of equations. Moscow: MAKS Press; 2004. 328 p.

9. Trapeznikova M.A., Chechina A.A., Churbanova N.G. Simulation of Vehicular Traffic using Macroand Microscopic Models. Computational Mathematics and Information Technologies.2023;7(2):60–72. (In Russ.) https://doi.org/10.23947/2587-8999-2023-7-2-60-72

10. Yashina M, Tatashev A. Traffic model based on synchronous and asynchronous exclusion processes. Mathematical Methods in the Applied Sciences. 2020;43(14):8136–8146.

11. Schadschneider A., Schreckenberg M. Cellular automation models and traffic flow. Journal of Physics A: Mathematical and General. 1993;26(15):L679.

12. Schreckenberg M. et al. Discrete stochastic models for traffic flow. Physical Review E. 1995;51(4):2939.

13. Kanai M., Nishinari K., Tokihiro T. Exact solution and asymptotic behavior of the asymmetric simple exclusion process on a ring. Journal of Physics A: Mathematical and General. 2006;39(29):9071.

14. Buslaev A.P., Tatashev A.G. Particles flow on the regular polygon. Journal of Concrete and Applicable Mathematics. 2011;9(4):290–303.

15. Buslaev A.P., Tatashev A.G. Monotonic random walk on a one-dimensional lattice. Journal of Concrete & Applicable Mathematics. 2012;10:71–79.

16. Daduna H. Queueing Networks with Discrete Time Scale: Explicit Expressions for the Steady State Behavior of Discrete Time Stochastic Networks. Springer. 2003; 2046.

17. Yashina M.V., Tatashev A.G. Buslaev networks: dynamical systems of particle flow on regular networks with conflict points. IKI Publishing House. 2023. (In Russ.)

18. Bugaev A.S., Buslaev A.P., Kozlov V.V., Tatashev A.G., Yashina M.V. Modelling of traffic: monotonic random walk through the network. Mathematical Modelling. 2013;25(8):3–21. (In Russ.)