Mathematical Models Of Traffic Flow Research Articles

Simulation of Vehicular Traffic using Macro- and Microscopic Models

To effectively regulate traffic on highways and networks of modern megacities, it is necessary to introduce Intelligent Transport Systems, which include many innovative solutions, in particular, mathematical models for describing the dynamics of traffic flows.The article is devoted to a brief description of the current state in this area in its development — from the simplest macroscopic and microscopic models that have become classic to modern developments.Special attention is paid to the original multilane models developed by the authors of the article within both approaches.The macroscopic model is based on the quasigasdynamic approach, while the microscopic one uses the ideology of cellular automata and constitutes a generalization of the Nagel-Schreckenberg model for the multilane case.The difference in the representation method and the mathematical apparatus for the mac-roscopic and microscopic description of traffic flows is briefly described, followed by the review of the main models at different stages of their development, presented by foreign and Russian authors.Special attention is paid to the three-phase theory of Boris Kerner and models built in the framework of this theory.Examples of modern software for traffic modeling are given.The original quasigasdynamic model of traffic flows, which uses the continuum approximation and is constructed by analogy with the well-known model of gas dynamics, is briefly described. Due to the introduction of the lateral speed, the model is generalized to the multilane case.An original microscopic model based on the cellular automata theory and representing a generalization of Nagel- Schreckenberg model for the multilane case is described. The model has been further developed by taking into account various driving strategies and behavioral aspects.The article presents a brief overview of the state of the art in the field of mathematical modeling of traffic flows, as well as original macroscopic and microscopic models developed by the authors for the case of multilane traffic.

The main directions for optimizing traffic flows to increase their throughput

This paper analyzes various problems of mathematical modeling of traffic flows that are relevant for optimizing and reducing the congestion of road networks in large cities and metropolitan areas. Various types of relationships between traffic lights on a controlled section of the road have been studied, formulas have been proposed for calculating the optimal joint mode of operation and minimizing the formation of traffic jams in front of the intersection. A model for calculating the optimal traffic flow on a section of the road network and its implementation at the program level are proposed.

Клеточные автоматы в приложении к моделированию транспортных потоков

The preprint summarizes the experience of mathematical modeling of traffic flows on urban road networks using the theory of cellular automata. The proposed multi-lane model allows the description of traffic on complex road networks and takes into account drivers with different lane change strategies. The model is the basis for a set of programs that can be used to obtain various average traffic characteristics, such as, for example, the throughput of an intersection. In addition, the software package allows you to predict the consequences of various decisions regarding changes in the road infrastructure.

MODELS OF TRAFFIC FLOW DYNAMICS ON HIGHWAYS

The paper is an analytical review of the currently existing methods of traffic flows modeling. The movement of vehicles on the road can be modeled in different ways. Mathematical models as tools that allow us to study complex processes in the real world, including transport infrastructure, without capital expenditures, are a popular tool for solving many problems in various spheres of the national economy. There are several approaches to mathematical modeling of traffic flows. In microscopic models, the law of motion of each car is set, depending on its current position, speed, characteristics of the movement of neighboring cars, and other factors. Microscopic models, in turn, can be divided into models that are continuous in space and time, and into models that are discrete in space and time, the so-called cellular automata. In macroscopic models, the transport flow is considered as a fluid flow with special properties. The equations of the macroscopic model establish the relationship between the flow, density, speed of movement, possibly acceleration, and so on. Macroscopic models can also be continuous or discrete. In continuous models, the change in the state of a road section without branches and intersections is usually described by partial differential equations. Modeling traffic flows is necessary because active experiments in the existing transport network are fraught with unpredictable consequences, and in many cases are not feasible at all. The work presents a description and analysis of the models, and of their advantages and disadvantages.

Rule 184 fuzzy cellular automaton as a mathematical model for traffic flow

The rule 184 fuzzy cellular automaton is regarded as a mathematical model of traffic flow because it contains the two fundamental traffic flow models, the rule 184 cellular automaton and the Burgers equation, as special cases. We show that the fundamental diagram (flux–density diagram) of this model consists of three parts: a free-flow part, a congestion part and a two-periodic part. The two-periodic part, which may correspond to the synchronized mode region, is a two-dimensional area in the diagram, the boundary of which consists of the free-flow and the congestion parts. We prove that any state in both the congestion and the two-periodic parts is stable, but is not asymptotically stable, while that in the free-flow part is unstable. Transient behaviour of the model and bottle-neck effects are also examined by numerical simulations. Furthermore, to investigate low or high density limit, we consider ultradiscrete limit of the model and show that any ultradiscrete state turns to a travelling wave state of velocity one in finite time steps for generic initial conditions.

TO QUESTION OF APPLICATION OF HYDRODYNAMIC ANALOGY FOR PROCEDURE OF CALCULATION OF PARAMETERS OF TRANSPORT STREAMS

A mathematical modeling of traffic flow for today is rather intricate and actual problem. The most perfect mathematical models of traffic flow are described by equations of mathematical physics. In the theory of traffic flow there are various approaches to classification of their mathematical models. One of widespread is classification on macroscopic models and microscopic models. The whole group of transport vehicles, that is described by the corresponding parameters of motion, is examined in macroscopic models. Microscopic models are based on conception of safe distance to the leader. The most known models are: model of optimal speed, following by a leader model, Treibe’s model of clever driver. Historically one of the first macroscopic models is a model of Lighthill – Whitham – Richards (LWR). In it the stream of motor-car transport vehicles is considered as an unidimensional stream of compressible liquid. In a model LWR is accepted, that an unambiguous interconnection exists between speed and fluid flow density, and the laws of maintenance of mass are satisfied. Other macroscopic models based on analogues of a traffic flow to the hydrodynamic features of compressible liquid flow exist. It is a model of Tanaka, Whitham. Payne et al. The model of Helbing Eeler Navier Stokes is proposed in 1995. In this model to the system of Payne’s equations the third equation that represents the law of conservation of energy for variation of speed is added. In the second equation (law of momentum conservation) an additional component is considered that allows to take into account variation of speed. It should be noted that for the system of Navier – Stokes equations it is not known how to formulate the initial Cauchy boundary value problem in order the global solution is unique for all times. The main difference between the hydrodynamic models of traffic flows from the corresponding hydrodynamic analogs is the formation of the right-hand side of the equations. This refers to the correct notation, as a rule, of hyperbolic systems of equations and their diffusion analogs. The difficulties arising in the description of the traffic flow are similar to those that arise in the description of the turbulent motion of a liquid. The aim of work is a construction of mathematical model, numerical method, algorithm for obtaining numerical solutions and creation of software for studying the dynamics of traffic flows. The task of modeling of traffic flows of motor-car transport vehicles is examined in the article. For description of physical process the system of equations of Navier – Stokes is used. Methodology, numerical algorithm, and software, is worked out. For numerical integration of the systems of differential equations, finite volume method is used. The developed methodology was tested. Based on the results of numerical calculations, a fundamental traffic flow diagram has been constructed.

RETRACTED: Fatigue reliability assessment for hangers of a special-shaped CFST arch bridge

Abstract The main bridge of the Yingzhou Bridge is a special -shaped concrete-filled steel tubular (CFST) arch bridge. As a key load-carrying member, the safety of a hanger affects the service life of arch bridges. This paper presents a comprehensive frame work for fatigue reliability analysis of hangers in CFST arch bridges. A simplified finite element (FE) model is established to satisfy the fatigue analysis accuracy. The mathematical model for traffic flow of the Yingzhou Bridge is simulated using the Monte Carlo method. Then the stress time history of the hanger is calculated. The rain flow counting method is used to obtain the stress range values and the number of cycles. Based on the Miner cumulative damage rule, considering the effect of mean stress, the fatigue reliability for hangers with and without corrosion is calculated using the Monte Carlo Importance Sampling method. It is found that the reliability index of inclined hangers is lower than that of vertical hangers. Inclined hangers have a higher probability of failure earlier. The reliability index of the inclined hanger X3 in the position where the stiffness changes is the lowest, which should be paid greater attention to in daily maintenance. The reliability index of hangers with corrosion is significantly lower than that of hangers without corrosion, and the decline rate of the reliability index is increased with time.

Numerical Research of the Mathematical Model for Traffic Flow

Numerical Research of the Mathematical Model for Traffic Flow

Плотность городской ткани и население жилых кварталов

Современная градостроительная практика планирова­ния развития транспортных систем городов основывается на математических расчётах транспортных потоков. Когда планируется реконструкция или строительство достаточно крупной транспортной системы, специалисты применяют математическое моделирование транспортных потоков. Важ­ными исходными данными для моделирования транспортных потоков служат количество людей, живущих или работающих на определённой территории. Последнее десятилетие градо­строители сталкиваются с проблемой определения числен­ности населения квартала или микрорайона. В настоящей статье описана методика определения численности людей, проживающих в жилых районах, основанная на анализе типа застройки. Данная методика может быть использована в градостроительстве при прогнозировании и моделировании транспортных потоков[1].
  
 [1] Данное исследование частично проведено в рамках НИР Минстроя РФ «Формирование базовых стратегических направлений развития транспортной инфраструктуры городов России», выполняемом в ФГБУ «Центральный на­учно-исследовательский и проектный институт Министерства строительства и жилищно-коммунального хозяйства Российской Федерации» ( ФГБУ «ЦНИИП Минстроя России»).

Mathematical Model for Traffic Flow

Every year countless hours are lost in traffic jams. When the density of traffic is sufficiently high small disturbances in vehicle’s accelerations can cause phantom traffic jams. We can relate the traffic flow to mathematics and physics like that of liquids and gases. This paper presents mathematical model for phantom jams and Gauss Jordan elimination for traffic flow.  

Finite volume method for simulations of traffic dynamics with exits and entrances

We discuss a macroscopic study of vehicle traffic dynamics in which traffic flow is considered as a continuum governed by the kinematic Lighthill--Whitham--Richard model, with a source term accounting for traffic entries and exits through a junction. The kinematic model is solved using finite volume method, with the flux function is approximated using the upwind method. In order to prevent flows which exceed road capacity and to preserve the positivity of traffic density in simulations with entries and exits, a stop-go procedure is adopted. The resulting scheme is used to simulate the responses of traffic density to light and heavy entrances from a junction, and the dynamics predicted by the simulations are shown to conform well to analytical solutions. With a validated numerical algorithm at hand, we simulate traffic dynamics for several scenarios on a roadway with entrances and exits. Depending on the rate of exit or entrance and the initial condition, density profiles in the form of shock wave and rarefaction wave may appear. References P. Bagnerini, R. M. Colombo, A. Corli, S. Pedretti, Conservation Versus Balance Laws in Traffic Flow, Traffic and Granular Flow '03 , 235–240, (2005). doi:10.1007/3-540-28091-X_22 C. F. Daganzo, The cell transmission model: a dynamic representation of highway traffic consistent with hydrodynamic theory, Transportation Research B , 28 (4), 269–287 (1994). doi:10.1016/0191-2615(94)90002-7 C. F. Daganzo, The cell transmission model II: Network traffic, Transportation Research B , 29 (2), 79–93, (1995). doi:10.1016/0191-2615(94)00022-R N. D. Fowkes, J. J. Mahony, An Introduction to Mathematical Modelling . John Wiley and Sons, England, (1994). R. Haberman, Mathematical Model , Prentice–Hall, Inc., New York, (1977). doi:10.1137/1.9781611971156 H. Holden and N. H. Risebro. A mathematical model of traffic flow on a network of unidirectional roads, SIAM Journal on Mathematical Analysis , 26 (4):999–1017, (1995). doi:10.1137/S0036141093243289 J. P. Lebacque, The godunov scheme and what it means for first order traffic flow models, In The International Symposium on Transportation and Traffic Theory , Lyon, France, (1996). http://worldcat.org/isbn/0080425860 R. M. M. Mattheij, ten Thije Boonkkamp, S. W. Rienstra, Partial Differential Equation: Modelling, Analysis, Computation, The Netherlands Society for Industrial and Applied Mathematics , Philadelphia, (2005). doi:10.1137/1.9780898718270 M. Mercier, Traffic flow modelling with junctions, Jour. of Math. Analysis and Applications , 350 (1), 369–383, (2009). doi:10.1016/j.jmaa.2008.09.040 G. Orosz, R. E. Wilson, G. Stepana, Traffic jams: dynamics and control, Phil. Trans. R. Soc. A 368, 2010, pp. 4455–4479. doi:10.1098/rsta.2010.0205 M. Papageorgiou, Dynamic modelling, assignment and route guidance in traffic networks, Transportation Research B , 24(6), (1990), pp. 471–495. doi:10.1016/0191-2615(90)90041-V S.R. Pudjaprasetya, J. Bunawan, C. Novtiar, Traffic light or roundabout? Analysis using the modified kinematic LWR model, East Asian Journal of Applied Mathematics , 6 (1), 80–88, (2016). doi:10.4208/eajam.210815.281215a

Traffic Flow Modeling on Road Networks Using Cellular Automata Theory

The paper deals with mathematical modeling of traffic flows on urban road networks. The original model is based on the cellular automata theory and presents a generalization of Nagel-Schreckenberg model to a multilane case.Numerical realization of the model is represented in a form of the program package that consists of two modules: User Interface and Visualization module (for setting initial conditions and modelling parameters and visual representation of calculations) and Computation module (for calculations).Computations are carried out for each element of the road (i.e. T or X type intersection, straight road fragment) separately and in parallel, that allows performing calculations on various complex road networks. Different kinds of average characteristics (e.g. the capacity of the crossroad) can be also obtained using the program package. 

Quality of Traffic Observability on Highways With Lagrangian Sensors

Observability is essential to estimate the states of a system from sensor measurements. Observability indicates the sufficiency of the sensors for state estimation. In a highway traffic setting, using global positioning system-enabled smartphones as sensors sets this observability problem in the Lagrangian dynamics framework, in which the moving vehicle that follows the microscopic dynamics of the system senses its own current state. This paper addresses the question of how different numbers of sensor vehicles in the traffic stream affect the observability of the system. This was accomplished by performing spatiotemporal discretization of the traffic hydrodynamics to obtain a system on which the rank observability condition was applied. Then, the observability problem for a generic system was studied in terms of the average penetration of these mobile devices in the traffic stream. Simulation results are provided to support the developments in this paper. Note to Practitioners —This paper is motivated by the problem of observing the state of traffic in a highway setting from Lagrangian sensors. These are sensors that travel with individual vehicles rather than the traditional sensors that remain at fixed positions. There is much literature written on traditional sensors, but currently, there is development on what additional value Lagrangian sensors can provide. Lagrangian sensors recently became plausible, and readily available with the growth of smartphones. The problem that is studied is how much of the traffic behavior can be inferred from different numbers of the Lagrangian sensors. A traditional mathematical model for traffic flow is studied in the Lagrangian coordinates. The concept of observability and condition number is used to analyze how much and how well the sensors can capture traffic information. This problem is studied with idealized vehicle behavior, but this paper will be extended to cope with more realistic driving behaviors in the future.

Multilane Traffic Flow Modeling Using Cellular Automata Theory

The paper deals with the mathematical modeling of traffic flows on urban road networks using microscopic approach. The model is based on the cellular automata theory and presents a generalization of the Nagel-Schreckenberg model to a multilane case. The created program package allows to simulate traffic on various types of road fragments (T or X type intersection, strait road elements, etc.) and on road networks that consist of these elements. Besides that, it allows to predict the consequences of various decisions regarding road infrastructure changes, such as: number of lanes increasing/decreasing, putting new traffic lights into operation, building new roads, entrances/exits, road junctions.

Model matematyczny przepływu ruchu ulicznego w skali makroskopowej

Model matematyczny przepływu ruchu ulicznego w skali makroskopowej

Construction and simulation of a novel continuous traffic flow model

In this paper, we aim to propose a novel mathematical model for traffic flow and apply a newly developed characteristic particle method to solve the associate governing equations. As compared with the existing non-equilibrium higher-order traffic flow models, the present one is put forward to satisfy the following three conditions:1.Preserve the equilibrium state in the smooth region.2.Yield an anisotropic propagation of traffic flow information.3.Expressed with a conservation law form for traffic momentum.These conditions will ensure a more practical simulation in traffic flow physics: The current traffic will not be influenced by the condition in the behind and result in unambiguous condition across a traffic shock. Through analyses of characteristics, stability condition and steady-state solution adherent to the equation system, it is shown that the proposed model actually conform to these conditions. Furthermore, this model can be cast into its characteristic form which, incorporated with the Rankine-Hugoniot relation, is appropriate to be simulated by the characteristic particle method to obtain accurate computational results.

Bifurcation structure of a car-following model with nonlinear dependence on the relative velocity

ABSTRACTUnderstanding the stability of solutions of mathematical models of traffic flow is important for alleviating jams as these are considered stable inhomogeneous solutions of traffic models. Traffic jams can be alleviated by destabilizing these solutions. Solution stability can be studied with the aid of bifurcation analysis, which has been used to describe the global bifurcation structure of a car-following model that exhibits bistable behavior and loss of stability due to Hopf bifurcations. However, previous studies on bifurcation analysis for traffic models have not considered the relative velocity effect, which is important in real-world traffic scenarios. This study analytically derives linear stability conditions and numerically investigates the global bifurcation structure of a car-following model with nonlinear dependence on the relative velocity (the STNN model), which exhibits multistable states. Moreover, the relative velocity drastically changes the bifurcation structure. This supports implementation of (semi-)automatic driving systems as a means to alleviate traffic jams.

МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ТРАНСПОРТНОГО ПОТОКА НА УЛИЧНО-ДОРОЖНОЙ СЕТИ ГОРОДА

This article describes a practical method of constructing mathematical models of traffic flow, the most suitable for a particular city highway. The initial data are: instant speed, time, distance, flux density, intensity of movement of vehicles. Using the obtained data, built regression model, and conducted correlation analysis. The choice of the mathematical model that most faithfully describes the transport process was made on the basis of the correlation coefficient

Development of Mathematical Model for Traffic Flow within Speed Limitation

The paper discusses a symbiotic model of the traffic flow, based on the classical model of the longitudinal dynamic gauge of the automobile and an alternative model developed by the authors. The presented mathematical model makes it possible to quantify the dynamic characteristics of the traffic flow for various levels of speed limitation and for a wide range of road conditions. The problem of structural and parametric identification of the mathematical model is challenging. A solution to this problem is based on the traffic flow theory, regression analysis and dynamic programming.

Numerical Methods for Modeling of Traffic Flows at Research and Optimization of Traffic on the Signal-controlled Road Intersections

Abstract This article describes the numerical methods of mathematical modeling of traffic flows to meet the challenges of optimizing the traffic light phases at signal-controlled road intersections. The flow of vehicles on multilane roads is described by Poisson process. The paper introduced, justified and calculated concept «effective number of lanes», characterized by a maximum flow of vehicles with different phases of traffic lights.