Introduction

Traffic jams are an unavoidable inconvenience when it comes to travel. Even more frustrating than a jam caused by an accident or a narrowing roadway is congestion that occurs in the absence of any type of bottleneck. These buildups without seemingly any cause are known as phantom traffic jams, and they have been widely studied in both mathematical and physical realms. Specifically, mathematicians have over time developed many models simulating traffic patterns that vary in tractability–how easy the model is to solve–and feasibility–how accurate the model is in portraying reality. For traffic flow, there are two main types of models, mirroring the way physicists look at gases: particle and fluid dynamical models [2]. A particle model may look at individual cars, just as one would look at individual gas molecules, in order to learn more about how the parts interact in a system. In our case of traffic flow, this can create a huge system of equations that is largely complicated and not very tractable.

The fluid dynamical approach has been used since about the 1950’s [2]. Rather than looking at individual vehicles, this model looks at the entire system, evaluating the flow continuously rather than discretely. This is similar to a physicist describing a gaseous system as a fluid rather than as discrete molecules. There are many advantages of this model, including that it is much more tractable than other approaches. More recently, nonlinear dynamics have been applied to these models as well to widen the scope of the patterns they can replicate [2]. By better understanding traffic patterns and being able to effectively simulate traffic dynamics, we can change the way we deal with jams, making travel more efficient. Modeling traffic patterns can also allow us to add provisional laws and regulations to keep drivers safe and in fewer accidents.

Approaches to the Traffic Flow Modeling

Particle Models

As mentioned above, particle models imitate the way kinetic gas theory looks at individual molecules to describe the entire system. In the view of traffic flow, these models create systems of equations that describe how individual drivers respond to surrounding traffic. One major example of a particle model is car-following theory [7]. This model treats each vehicle as a discrete point, indexed by a counting variable. Using physical intuition and mathematically realized relationships, an equation for each vehicle is crafted that explicates the motion of the vehicle due to its response to the distance between it and the car in front of it. These indexed ordinary differential equations produces a large system of equations, one for each car, which can be computationally complicated.

Fluid Dynamical Models

The simplest model that uses fluid dynamical techniques is called the continuum model of traffic flow, a first-order partial differential equation model that was popularized by Lighthill and Whitham [4]. It uses the idea that a one-lane roadway of traffic with discrete vehicles traveling in one direction can be extrapolated to an unbroken flow with continuous properties, such as density and velocity as continuous functions of space and time. By using the continuum model as opposed to car-following theory, mathematicians can generalize the model by extending it to the continuum limit (thus allowing for the use of differential calculus) for different parameters of the model. An important insight of this model is that the individual vehicles are not as important as the overall flow of the cars, so the traffic flow can be viewed macroscopically rather than microscopically.

A more complicated version of this model is the Payne-Whitham model [9]. It again uses partial differential equations to simulate traffic patterns, but it is a nonlinear second order model. Payne and Whitham added a second equation to Whitham’s original model by accounting for the acceleration of the vehicles, using physical intuition to relate variables, so the model is a system of two differential equations.

Continuum Model of Traffic Flow

Formulation

The continuum model of traffic flow extrapolates the discrete variables associated with individual variables into fields defined by continuous functions. The simplest way to formulate this model is by imagining a one-lane highway with all the cars moving in the same direction, and no on-off ramps. Thus, the number of vehicles is constant, and there are no external factors that might confound the effects of a phantom traffic jam. We can also imagine that each car $i$ has some velocity, $v_i(t)$, and a location $x_i(t)$ at time $t$. This coordinate of a particle vehicle is often called the Lagrangian coordinate [7]. To an observer of the entire roadway, the individual cars are not as important as the overall flow of the cars. Thus a general principle can be used in developing this model: the closer together the cars are, the slower they will move [6]. The continuum model tells us that we can replace the discrete objects with a moving continuum, so that the discrete velocities associated with each car can be replaced by a ‘‘Eulerian velocity field ‘’[7]. This field is represented by the function $u(x,t)$, a continuous function with a value at every point in space and time (rather than only a value at the locations $x_i$ of each car). However, by using this Eulerian field, we can "go backwards" in the calculation and still find the velocity of a particular car by specifying a time and location at which we would like to know the velocity. Thus $u(x,t)=\frac{dx}{dt}$.

A second parameter we can define for this model is the traffic density, $\rho(x,t)$, which described the average number of vehicles per unit length of road at a specified position $(x)$ and time $(t)$ [7]. If all the vehicles have the same length $L$, for example, and the space between the cars is the distance $d$, then $\frac{1}{L+d}$ vehicles will be present per unit length of road. Thus, in this simplistic calculation, the density of the road is constant: $\rho(x,t)=\frac{1}{L+d}$. Another parameter used in the continuum model is the traffic flux, or flow of traffic, which is represented by $q(x,t)$. This is the number of cars per unit time that cross a specified location, and is meant to parallel the flux of a fluid through a volume. Physics intuition allows these three variables to be related in the model by the equation $q(x,t)=\rho(x,t) \cdot u(x,t)$ [6]. Thus the rate of cars that pass a point is equal to the velocity of the cars at that point times the density of cars that pass that location.

As is the case with mathematical models of physically realized phenomena, physical laws can help to elucidate the relationships between variables of the model. An important idea for deriving our partial differential equations is the conservation of the number of vehicles [3]. If the focus is placed on a certain sublength of the road between two points (call them $x= A$ and $x=B$), it is clear that the number of cars in this length depends on time. The flux, or traffic flow, at the endpoints A and B affect the number of cars in the length; for example, if the flux is greater at A than B (and more cars flow into the length), the number will increase. More explicitly, the rate of change in the number of vehicles in the segment is equal to the different in fluxes at the endpoints. We also know that the number of vehicles in this length comes from "adding up" the density at each point in the interval: \(\label{1} N_{AB}(t)= \int_A^B \rho(x,t)dx.\) The derivative with respect to time of this integral thus equals the flow out (flux at B) subtracted from the flow in (flux at A): \(\label{2} \frac{d}{dt} \int_A^B \rho(x,t)dx=\int_A^B \frac{\partial \rho}{\partial t}dx=-q(B,t)+q(A,t).\) This is the global conservation law for the traffic flow [7]. There is another relation that we can derive from the fundamental theorem of calculus: \(\label{3} \int_A^B \frac{\partial q(x,t)}{\partial x}dx=q(B,t)-q(A,t).\) The right hand side of this equation is then the additive inverse of the right hand side of equation $\ref{2}$, so we can add these two equations to get \(\label{4} \int_A^B \left ( \frac{\partial \rho(x,t)}{\partial t} + \frac{\partial q(x,t)}{\partial x} \right ) dx=0.\)

This is crucial for the continuum model. Because the interval from $x=A$ to $x=B$ can be any interval along the road, and thus the integral is 0 for whatever interval chosen, it must be true that the integrand itself is 0 for all $x$ and $t$ on the road, according to the Vanishing Theorem [10]. Thus, we arrive at our partial differential equation: \(\label{5} \frac{\partial \rho}{\partial t} + \frac{\partial (\rho u)}{\partial x}=0.\)

::: center {#vd} :::

This differential equation is the basis for the continuum model. More information is needed for this model to accurately replicate what is seen in reality, because right now the partial different equation has two seemingly unrelated variables: $\rho$ and $u$. However, most mathematical models for traffic flow make the important assumption that velocity is only a function of density; in other words, the velocity field is entirely dependent on the density at each location [7]. This is a reasonable assumption because it is well known that the closer the vehicles are (high density), the slower the cars move (low velocity), as shown in figure $\ref{vd}$. Therefore equation ($\ref{5}$) is sometimes rewritten as \(\label{6} \frac{\partial \rho}{\partial t} + \frac{dQ(\rho)}{d\rho} \frac{\partial (\rho)}{\partial x}=0,\) where $Q(\rho)=\rho u(\rho).$ (Often $\frac{dQ}{d\rho}$ will be written as $c(\rho)=u_{max}(1-2\rho/\rho_{max}$, for reasons explained below.) But how is this velocity dependent on density? We could assume a velocity that is constant with respect to density, but this model is not very realistic (except possibly for low densities). A common approximation used for $u(\rho)$ is called Greenshields model: \(\label{7} u(\rho)=u_{max}(1-\rho/\rho_{max}),\) with $u_{max}$ and $\rho_{max}$ equaling the upper limits on the velocity and density, respectively [8]. They both must be nonnegative of course, but they must have upper bounds as well. The maximum speed, $u_{max}$ can be restricted by a number of things, including a speed limit or road conditions. The maximum density, $\rho_{max}$, occurs in bumber to bumber traffic, when the velocity of the cars is zero. If the length of the vehicles is on average $L$, this maximum density can reasonably be $\rho_{max}=1/L$.

Solutions to the Model

This model can help explain a variety of patterns in traffic flow. Specifically, we are interested in this model for its application to traffic jams. If, on a one-lane highway, the density in vehicles is some constant of freely-flowing traffic, then there are no jams where the density is extremely high and the velocity is close to zero. Thus we can imagine the case where traffic density is slightly perturbed from an equilibrium state, $\rho_0$. Thus $\rho(x,t)=\rho_0+\tilde{\rho}(x,t)$, where $\tilde{\rho}(x,t)$ is the small addition to the equilibrium density [3]. By this assumption, where $\rho_0$ is a constant, we obtain: \(\label{12} \frac{\partial \tilde{\rho}(x,t)}{\partial t} + (Q'(\rho_0)+\tilde{\rho}(x,t)Q''(\rho_0))\frac{\partial \tilde{\rho}(x,t)}{\partial x} =0.\)

and the equation, after keeping the linear terms in $\tilde{\rho}$, becomes [3]: \(\label{8} \frac{\partial \tilde{\rho}(x,t)}{\partial t} + Q'(\rho_0) \frac{\partial \tilde{\rho}(x,t)}{\partial x} =0.\)

By using a wave ansatz, we find the exponential solution:

\(\label{9} \tilde{\rho}(x,t)=Ae^{ik(x-ct)}\) with $c=Q’(\rho_0)$ [3]. A few things become clear from this solution. If there is a slight perturbation in the density of the traffic flow (such as a jam), it will travel as a linear wave propogation of higher density through the line of traffic (of some constant density) [5]. This wave, under linearization of the equation, is stable, as the amplitude of the distubance is constant (the wave does not grow or shrink as it travels in time). Thus, it is sometimes possible for a traffic jam to remain stable and thus virtually unchanged for hours. According to some models, jams are even more stable occurences in simulations upon introduction of key phenomena observed experimentally. For example, the concept of inertia can be applied to traffic flow in that vehicles tend to keep the same velocity in the presence of external disturbances (like a car slowing down in front of them).

Problems with the Model

As with any model detailing real-life phenomena, there are problems and limitations to the continuum approach to traffic flow. The first problem is evident in the name: this model is only an approximation, because discrete traffic flow has been simulated as a flow with a continuous, averaged density. The model also runs into complications at low densities, where the relationship between the velocity and density breaks down [8]. At low densities, when traffic flow is in a free-flow state, small perturbations in density can easily be smoothed out, so there is no lasting effect of a traveling wave as was a solution above. This shows the stability of traffic flow at low densities; however, when it reaches a critical density, the traffic flow changes to a congested regime nonlinearly. This is demonstrated in a "fundamental diagram," which shows the relationship between flux and density for a one-lane roadway, as in figure ($\ref{fd}$) [3].

::: center {#fd} :::

Lastly, the model may be too restrictive in definition of velocity, and its ability to account for external factors. Velocity may not be linearly dependent on density, and it may be dependent on other factors (such as the density derivatice with respect to position). The continuum model cannot account for different numbers of lanes, merging, and on/off ramps. It also does not take the length of the roadway into account, as it assumes a one-directional, one-lane infinitely-long roadway. This model is therefore very limited in scope. However, it is an attractive starting place for more complex models, such as the Payne-Whitham Model, because of its high tractability.

Revised Continuum Model

The Lighthill and Whitham continuum model, detailed above, is a first order model that restricts the traffic modeled to equilibrium states, with small perturbations. In reality, traffic is generally observed in disequilibrium. Thus the continuum model, though not necessarily inaccurate, is restricted in scope and cannot fully explicate the event of a traffic jam. The Payne-Whitham model adds another equation alongside the first order partial differential equation in the continuum model (equation [5]{reference-type=”ref” reference=”5”}) to define traffic acceleration. This model approximates $u(x,t)$ by the equilibrium velocity $u_0$, which is the "fundamental diagram." The second equation, which defines acceleration, is a momentum equation for $u(x,t)$, giving the completed model: \(\frac{\partial \rho}{\partial t}+ \frac{\partial (\rho u)}{\partial x}=0\) \(\label{10} \frac{\partial u}{\partial t} +u \frac{\partial u}{\partial x} +\frac{a(\rho)}{\rho}\frac{\partial \rho}{\partial x}= \frac{u_0(\rho)-u}{T(\rho)},\) where $a(\rho)$ is the "anticipation coefficient" and $T(\rho)$ accounts for human relaxation time [1]. With $q=\rho u$, the model can be rewritten as

\[\left( \begin{array}{cc} \rho \\ q \end{array} \right )_{t}+ \left( \begin{array}{cc} q \\ \frac{q^2}{\rho}+a(\rho)\rho \end{array} \right )_{x} = \left( \begin{array}{cc} 0 \\ \frac{f_0(\rho)-q}{T(\rho)} \end{array} \right ).\]

This model can be written in a conservative form \(\label{11} w_t+f(w)_x= s(w)\) where $w$ is the vector $(\rho,q)$ [9]. The model can be solved numerically, but we may also care to investigate the homogeneous solution to this system of equations, where the right hand side of the system $s(u)=0$. The continuum model of traffic flow is actually a degenerate system of this model, and thus this model "absorbs" the characteristics of the simpler continuum model [9]. The homogeneous system of equations can be solved with the "frozen characteristic speeds" of the system, $\lambda_{1,2}$. These are the eigenvalues of the homogeneous system, where $\lambda_{1,2} = u \mp \sqrt{a}$ [9]. More complicated calculations can be derived for the solution to the inhomogenous Payne-Whitham model. The solutions fall into two regimes: free-flow solutions and congested-flow solutions. When the traffic densities are low and the velocities are approximately constant, the solution to the model falls within the free-flow regime. Around a critical threshold density of traffic, this solution becomes unstable, and small perturbations will amplify and grow into travelling waves (thus, a traffic jam) [4].

For larger densities, the system is linearly unstable. There are two effects on the traffic flow that influence the stability of the system (and work against each other). The first is a stabilizing effect that arises from preventative driving; this effect clearly would decrease the likelihood of a compounding traffic jam. A second effect is more destabilizing and causes the variation in traffic density to increase, thus enlarging the jam: drivers tend to slow down when density is higher, and there is a delay in adjustment of drivers to new conditions [5]. Above the critical density threshold, the destabilizing effect will outweigh the stabilizing effect, and the perturbation will build up into a travelling wave. These instabilities looks like a sharp jump in vehicle density on the side of the wave facing incoming traffic and a smooth decay on the other side, which travel unchanged along the road. Because each vehicle reacts the cars around it, information cannot travel faster than the speed of the cars themselves, and the travelling waves move at a constant velocity slower than the vehicle velocities [3]. Once these jams are formed, they are relatively stable structures, to which anyone who has sat in hours of traffic can attest. They can only vanish by diminishing effects on the waves, such as those of cautious drivers and and those that lower density (adding more lanes, for example).

Further Commentary on the Continuum Model

::: center {#pic} :::

The predictions that come from the Payne-Whitham model of traffic flow regarding traffic jams are easily tested on a circular road. Using a circular road instead of an "infinitely long" straight roadway allows for conservation in number of vehicles and for the experiment to remain localized. Sugiyama et al. conducted such an experiment in 2008 to test the behavior of drivers that causes phantom traffic jams, and to compare model simulations with real-life occurences [11]. In this experiment, 22 vehicles were told to maintain a constant speed of 30 kilometers per hour after being spaced evenly around a 230-meter circular track. The velocity had to be appropriately high in order to create the unstable conditions that cause traffic jams in a parallel model simulation. The vehicles began in the free-flow regime, and are told to maintain safety by following the car ahead of them while still trying to remain at a constant speed. The cars remain in this regime for some time, until it breaks down into the congested-flow regime, as reported by Sugiyama et al.:

After some time, small fluctuations appeared in the headway distances and developed as time progressed. It takes a period of time for the fluctuations to grow enough to break down the free flow. The free flow has been disturbed and the vehicles cannot move homogeneously any more. Finally, several vehicles are forced to stop completely for a moment (figure 3{reference-type=”ref” reference=”pic”}(b)). A clear ‘stop-and-go wave’ is observed, and furthermore, it propagates in the opposite direction to the movement of vehicles, which is nothing but a jam. The jam cluster moves as a solitary wave and maintains its size and velocity [11].

Thus a traffic jam can be created organically by the behavior of drivers. The effect of collective motion caused by the interaction among vehicles creates the dynamic solution of the model and real-life experience of traffic jams. Because traffic flow is a physical system in disequilibrium that consists of moving particles with asymmetric interaction, small perturbations can move the system from a state of free-flow to one of a jam, where there are travelling clusters on low velocity in the flow. Thus, by both the Payne-Whitham model and the experimental evidence provided by Sugiuama et al., phantom traffic jams are a culminating result of the small fluctuations present in traffic density when that density is near a critical value. The free-flow state becomes unstable, fluctuations grow steadily and the break the free flow, and the jam is created to travel along the road without decay.

Conclusion

When a traffic jam occurs without any dependence on a bottleneck, such as narrowing roadways or obstacles, a phantom traffic jam is born. These jams can be explained by a variety of models that investigate traffic flow analogously to gas theories. Traffic can be modeled microscopically as a system of discrete particles, with rules of interaction that define the motion of each vehicle. A different and more recent approach is a macroscopic view, which mirrors fluid dynamics of a many-particle system. The individual vehicles in this system are ignored, and generalized formulations of variables such as traffic density and flux are modeled.

Lighthill and Whitham proposed an early model of this type, known as a continuum model of traffic flow, which describes the dynamics of the system with a first-order partial differential equation. This equation can be solved given initial conditions, and can help to elucidate the mechanism of linear traveling waves that arise from variations in traffic density (i.e. a traffic jam formation). However, despite the model’s tractability, it is often regarded as too simplistic to describe more complex phenomena of traffic. Thus, Whitham improved his model with a momentum differential equation that defines traffic acceleration in what is now the Payne-Whitham model. This second order system of equations accounts for the complexity of driver’s reactions to the traffic around them more acutely than the continuum model. In the case of traffic jams, this model can help explain the shape of the higher-density congestions, and how they evolve over time. While much of travel occurs as free flow, where density is low and velocity is high, the solutions that fall in this regime are unstable. If the density of traffic nears a critical value, the instability cause small fluctuations in density to grow steadily and break the free flow, resulting in a jam. The jam then travels along the roadway, without decaying, in a nonlinear traveling wave. Sugiyama et al. [11] tested the predictions of this model, and the predictions of the model have been verified.

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