Mathematical model of the distribution of cars on the road network according to the first Wardrop principle based on the Greenshields model
https://doi.org/10.52170/1815-9265_2023_67_13
Abstract
The issue of distribution of car flows along the transport network has not lost its relevance in recent decades. Increasing levels of motorization lead to congestion, the different sections of the road network are being filled to varying degrees. The article presents a four-stage model for the formation of traffic flows. Particular attention is paid to the stage of travel route choice. The article provides a review of mathematical models of traffic flow, including 3 types: predictive, simulation and optimization. The model discussed in the article is of the predictive type. The main classical macroscopic models showing the influence of traffic flow intensity on the flow time are shown: Greenshields, Greenberg, BPR. The most famous classic Greenshields model is taken as a basis; it shows that as the density of traffic flow increases, its speed decreases. The principles of Wardrop, which describe the distribution of traffic flow along the road network, are considered. The first Wardrop principle was chosen – the socalled equilibrium distribution of flows, in which each passenger chooses the best one from the available alternatives. Transformations have been made in the Greenshields model: the dependence of speed on the intensity of traffic flow has been replaced by the dependence of travel time on the intensity of traffic flow. A formula has been obtained showing at what intensity of traffic flow vehicles will use a longer road. For two roads, a quadratic equation is obtained that allows one to find the equilibrium distribution of traffic flows. A numerical example is considered showing changes in the behavior of passengers with an increase in the intensity of traffic flow. The prospects for using and generalizing these models are determined.
About the Authors
M. E. Koryagin
Siberian Transport University
Russian Federation
Russian Federation
Mark E. Koryagin – Professor of the Higher Mathematics Department, Doctor of Engineering
Novosibirsk
V. I. Medvedev
Siberian Transport University
Russian Federation
Russian Federation
Vladimir I. Medvedev – Professor of the Life Safety Department, Doctor of Engineering
Novosibirsk
Yu. V. Shvets
Siberian Transport University;
Siberian State University of Water Transport
Russian Federation
Russian Federation
Yulia V. Shvets – Associate Professor of the Higher Mathematics Department,
Associate Professor of the Higher Mathematics and Informatics Department, Candidate of Pedagogical Sciences
Novosibirsk
N. B. Popova
Siberian Transport University
Russian Federation
Russian Federation
Natalya B. Popova – Professor of the Transport Economics Department, Doctor of Geographical Sciences
Novosibirsk
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Review
For citations:
Koryagin M.E., Medvedev V.I., Shvets Yu.V., Popova N.B. Mathematical model of the distribution of cars on the road network according to the first Wardrop principle based on the Greenshields model. Bulletin of Siberian State University of Transport. 2023;(4):13-20. (In Russ.) https://doi.org/10.52170/1815-9265_2023_67_13
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