SIAM News Blog
SIAM News
Print

Mathematical Models of Traffic Flow

By Helge Holden and Nils Henrik Risebro

Preliminary mathematical models of traffic flow date back to at least the early 1950s [10]. Two distinct classes of models prevail: follow-the-leader (FtL) models and traffic hydrodynamics. The former tracks individual vehicles, while the sufficiently dense traffic in the latter justifies a continuum approach wherein vehicle density is the fundamental quantity. Given the prevalence of traffic in modern society, the development of unprecedented computer power and progressive tracking devices, and continued advances in mathematical research, mathematical traffic modeling has become increasingly important in recent years.

Here we describe a novel mathematical model that allows for the analysis of multilane traffic [9]. But first we start with the basics. Consider dense unidirectional traffic on a single lane. At the most fundamental level, your velocity is determined by your distance from the vehicle just ahead — the closer you are, the slower you drive. If \(z_i\) is the position of the \(i\)th vehicle on a single-lane road, we can model this simple point by

\[\frac{d}{dt} z_i = v\Bigl(\frac{\ell}{z_{i+1}-z_i}\Bigr).\tag1\]

Here, \(z_{i+1}\) is the position of the vehicle directly in front of you, \(\ell\) is the vehicle length, and \(v\) is a decreasing velocity function. This calculation amounts to the FtL model and generates a system of ordinary differential equations with a size that is equal to the number of vehicles. However, society is getting too accustomed to (prohibitively) dense traffic, for which a “particle” description becomes inadequate. It is thus natural to wonder whether we can take advantage of available mathematical technology to study the continuum limit of particle models in the context of traffic dynamics. It turns out that we can.

Figure 1. A simulation of traffic on a periodic road. The number of vehicles is 200, their length is \(\ell=1/500\), and the velocity function is \(v(\rho)=1-\rho\). 1a. Initially, the vehicles are equally spaced on half of the road so that the density \(\ell/(x_{i+1}-x_i)=0.8\). 1b. As time begins to pass, the first vehicle instantly approaches the maximum velocity. Vehicles at the tail move more slowly and are eventually approached by those that have completed one loop (at \(t=1\)). 1c. Vehicle density approaches the familiar N-wave from conservation laws, wherein a shock is sandwiched between two rarefaction waves. This graph is virtually indistinguishable from the solution of the corresponding Lighthill-Whitham-Richards (LWR) model. See Animation 1 for more detail. Figure courtesy of the authors.

Define \(\rho_i=\ell/(z_{i+1}-z_i)\); a straightforward calculation then yields

\[\frac{d}{dt} \Bigl(\frac{1}{\rho_i}\Bigr) - \frac{1}{\ell} \Bigl( v(\rho_{i+1})-v(\rho_i)\Bigr)=0.\tag2\]

Now we can let \(\ell\to 0\) and \(\#\text{(vehicles)}\to\infty\) with \(z_i=i\ell\) fixed to obtain \(\rho_i(t)\to \rho(t,z)\), where \(\rho\) satisfies

\[\frac{\partial}{\partial t}\Bigl(\frac{1}{\rho}\Bigr) - \frac{\partial}{\partial z} v(\rho)=0,\tag3\]

Animation 1. A simulation of traffic on a periodic road. The number of vehicles is 200, their length is \(\ell=1/500\), and the velocity function is \(v(\rho)=1-\rho\).
since \((2)\) is a first-order (in \(\ell\)) semi-discrete scheme for \((3)\). In this case, the continuum limit follows from a result in finite difference approximations of nonlinear partial differential equations. Equation \((3)\) is an example of a first-order hyperbolic conservation law. Solutions of the Cauchy problem for this equation develop singularities in finite time that are independent of the initial data’s smoothness. We therefore must develop the machinery of weak solutions and entropy conditions to single out unique solutions [6]. Because \((3)\) is not in the standard form, we introduce the classical transformation—well-known from fluid dynamics—from Lagrangian to Eulerian variables. This transformation yields

\[\rho_t + (\rho v(\rho))_x = 0,\]

which is the celebrated Lighthill-Whitham-Richards (LWR) model for traffic flow. In the simplest case, we assume that the velocity depends only on density. If we scale the maximum density to unity and assume a linear dependence in the velocity—that is, \(v=1-\rho\)—we obtain the equivalent of the inviscid Burgers’ equation

\[\rho_t +(\rho(1-\rho))_x=0.\]

We have thus connected the two most fundamental traffic models by establishing the convergence of a numerical scheme. By examining more general velocity functions and allowing these functions to depend on time and position, we see that the “hydrodynamic” approach to traffic on a single-lane road is a rich source of interesting mathematical problems — even in this very simple case.

The aforementioned reasoning is formal and assumes the differentiability of all quantities, but one can rigorously establish that the limit \(\rho\) exists and is an entropy solution [3, 7, 8] (see Figure 1).

Figure 2. The result of a simulation of two-lane traffic on a periodic road. The number of vehicles is 200 and their length is \(\ell = 1/500\). Their initial locations are such that on half of the road, \(x\in [0, 1/2)\) and density \(\rho=0.8\). 2a. Initially, the vehicles are equally spaced on half of lane 1; the other lane (lane 2) is empty. 2b. As time passes, some vehicles switch to the outer lane. The tendency for vehicles to change lanes is determined randomly, and probability is proportional to the gain in velocity that a vehicle obtains by changing lanes. The velocity function \(v(\rho)=1-\rho\) is the same in both lanes. Vehicles begin moving to lane 2 and traffic becomes denser there. The distance between vehicles in lane 1 starts to increase, which makes the vehicles’ velocities increase. The image depicts vehicle positions at \(t=1\). 2c. The density for the two lanes at \(t=1\) again displays an N-wave, as in Figure 1. See Animation 2 for more detail. Figure courtesy of the authors.

Two Lanes

Animation 2. The result of a simulation of two-lane traffic on a periodic road. The number of vehicles is 200 and their length is \(\ell = 1/500\). Their initial locations are such that on half of the road, \(x\in [0, 1/2)\) and density \(\rho=0.8\).
We model two lanes of traffic as two individual roads, where vehicles move according to the FtL model \((1)\) and are allowed to change lanes. Our basic assumption is that the likelihood of a driver changing lanes is zero if doing so would lead to a decrease in speed, and is otherwise proportional to the potential gain in speed. This simple idea is a bit complicated to describe mathematically.

Let \(\{z_i\}\) and \(\{y_i\}\) denote the vehicle positions in the two lanes \(z\) and \(y\) respectively. We assume that the drivers continuously monitor the prospective speeds (and thereby positions) of their vehicles if they were to move to the other lane. As such, \(\tilde{z}_i\) signifies the dynamics of vehicle \(z_i\) if it were in the other lane. The probability that the vehicle changes lanes within a small time interval is then given by

\[\phi\left(\tilde{z}_i(t+\Delta t) - z_i(t+\Delta t)\right),\]

where \(\phi\) is an increasing smooth function with \(\phi(s)=0\) for \(s\le 0\) and \(\phi(\infty)=1\). In this model, drivers behave rather selfishly and do not consider the consequences of their lane changing for other drivers (in particular, for vehicle \(y_{j-1}\)). 

If we again take the formal limit of reducing the time interval and \(\ell \to 0\), 

\[\frac{\ell}{z_{i+1}-z_i} \to \rho_1(x,t),\ \ \ \frac{\ell}{y_{j+1}-y_j} \to \rho_2(x,t),\]

Figure 3. Simulation that uses model (4) in the situation that corresponds to that in Figure 2. Here we utilize \(\rho_1(x,0)=0.8\) for \(x\le 1/2\)—and zero otherwise—and \(\rho_1(x,0)=0\). Furthermore, \(v_1=v_2=1-\rho\) and \(K=1\). Figure courtesy of the authors.
just as in the single lane case. Here, \(\rho_1\) is the density of vehicles in lane \(z\) and \(\rho_2\) is the density of vehicles in lane \(y\). The lane-changing model leads to a flux from lane \(z\) to lane \(y\). This flux allows for different velocity functions in the two lanes:

\[S\left(\rho_1,\rho_2\right)= K[(v_2(\rho_2)-\]

\[v_1(\rho_1))^+\rho_1 -\left(v_2(\rho_2)-v_1(\rho_1)\right)^-\rho_2],\]

where \(a^{\pm}=(|a|\pm a)/2\) and \(K\) is a constant. Therefore, conservation of vehicles reads as

\[\frac{\partial \rho_1}{\partial t} + \frac{\partial}{\partial x} \left(\rho_1 v_1(\rho_1)\right) =-S(\rho_1,\rho_2)\]

\[\frac{\partial \rho_2}{\partial t} + \frac{\partial}{\partial x} \left(\rho_2 v_2(\rho_2)\right) =S(\rho_1,\rho_2).\tag4\]

This is a weakly coupled system of conservation laws. Its special structure allows for the sharp estimate of the difference between two solutions [9], stating that the sum over all lanes of the \(L^1\) norm of the difference between two solutions does not exceed the initial difference, again measured in the \(L^1\) norm. In contrast to the single lane case, the scaling limits that lead to \((4)\) are not rigorously established (see Figure 2).

Figure 3 compares the numerical solution of \((4)\) to the same initial data and reveals some similarities to the data in Figure 2.

Figure 4. Experimental data for occupancy (i.e., density) versus speed. Figure courtesy of [2].
We can expand this analysis to arbitrarily many lanes. The density \(\rho_i\) of vehicles in lane \(i\) with velocity function \(v_i\) satisfies

\[\partial_t \rho_i + \partial_x \left(\rho_iv_i(\rho_i)\right) = S_{i-1}(\rho_{i-1},\rho_i) - S_i(\rho_i,\rho_{i+1}),\] \[\qquad i=1,\ldots,N,\tag5\]

with \(S_0=S_N=0\). 

It is tempting to mathematically scale the lane “width” to allow for infinitely many lanes. Even if the connection to traffic flow is absent, doing so gives rise to an interesting non-heterogeneous diffusion model with Neumann boundary conditions [1, 9]. One can also extend the LWR model to a network of roads [4, 5].

Modeling traffic flow provides a treasure chest for interesting mathematical problems. Much of the work assumes that the velocity is a decreasing function of the density, but the experimental data in Figure 4 indicates that this is not always the case.

It is also wise to recall the advice of Sherlock Holmes, courtesy of Arthur Conan Doyle: “It is a capital mistake to theorize before one has data,” he says. “Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.”

Acknowledgments: This work is supported in part by the Waves and Nonlinear Phenomena (WaNP) grant from the Research Council of Norway.

References
[1] Coclite, G.M., Holden, H., & Risebro, N.H. (2021). Singular diffusion with Neumann boundary conditions. Nonlin., 34(3), 1633.
[2] Delle Monache, M.L., Chi, K., Chen, Y., Goatin, P., Han, K., Qiu, J.-M., & Piccoli, B. (2021). A three-phase fundamental diagram from three-dimensional traffic data. Axioms, 10(1), 17. 
[3] Di Francesco, M., & Rosini, M.D. (2015). Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Arch. Ration. Mech. Anal., 217(3), 831-871.
[4] Garavello, M., Han, K., & Piccoli, B. (2016). Models for vehicular traffic on networks. In AIMS series on applied mathematics (Vol. 9). Springfield, MO: American Institute of Mathematical Sciences.
[5] Holden, H., & Risebro, N.H. (1995). A mathematical model of traffic flow on a network of unidirectional roads. SIAM J. Math. Anal., 26(4), 999-1017.
[6] Holden, H., & Risebro, N.H. (2015). Front tracking for hyperbolic conservation laws (2nd ed.). In Applied mathematical sciences (Vol. 152). New York, NY: Springer.
[7] Holden, H., & Risebro, N.H. (2018). The continuum limit of follow-the-leader models — a short proof. Discrete Contin. Dyn. Syst., 38(2), 715-722.
[8] Holden, H., & Risebro, N.H. (2018). Follow-the-leader models can be viewed as a numerical approximation to the Lighthill-Whitham-Richards model for traffic flow. Netw. Heterog. Media, 13(3), 409-421.
[9] Holden, H., & Risebro, N.H. (2019). Models for dense multilane vehicular traffic. SIAM J. Math. Anal., 51(5), 3694-3713.
[10] Knoop, V.L (2018). Introduction to traffic flow theory (2nd ed.). The Netherlands: Delft University of Technology.
[11] Lighthill, M.J., & Whitham, G.B. (1955). On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. A, 229(1178), 317-345.
[12] Richards, P.I. (1956). Shockwaves on the highway. Oper. Res., 4(1), 42-51.

Helge Holden is a professor of mathematics at the Norwegian University of Science and Technology. Nils Henrik Risebro is a professor of mathematics at the University of Oslo.

blog comments powered by Disqus