Traffic Modeling - Phantom Traffic Jams and Traveling Jamitons
Research Group
Morris R. Flynn (University of Alberta)
Aslan R. Kasimov (MIT/KAUST)
Jean-Christophe Nave (MIT)
Rodolfo Ruben Rosales (MIT)
Benjamin Seibold (Temple University)
Overview
This web site presents theoretical results about special traveling wave solutions of continuum
traffic models. We consider mathematical equations that model traffic similar to the equations of fluid flow.
Specifically, we consider the Payne-Whitham model, the Aw-Rascle model, and generalizations thereof.
In the simplest case, a single-lane, straight, and uniform road is considered. The models are purely
deterministic. All drivers behave according to the same laws, and fully predictably. The considered
traffic models predict a nice, uniform traffic flow at low traffic densities. However, above a critical
threshold density (that depends on the model parameters) the flow becomes unstable, and small disturbances
amplify. This phenomenon is typically addressed as a model for phantom traffic jams, i.e. jams that
arise in the absence of any obstacles. The instabilities are observed to grow into traveling waves, which
are local peaks of high traffic density, although the average traffic density is still moderate
(the highway is not fully congested). Cars are forced to brake when they run into such waves.
In analogy to other traveling waves, so called solitons, we have decided to call such traveling
traffic waves jamitons.
Our research is based on the observation that the considered traffic models are similar to the equations
that describe detonation waves produced by explosions. Employing the theory of denotation waves, we have
found a way to analytically predict the exact shape and the speed of propagation of those jamitons.
Numerical simulations of the considered traffic models show that the predicted jamiton solutions are in
fact achieved, if the initial traffic density is sufficiently dense. The considered jamitons can
qualitatively be found both in observed real traffic as well as in experiments.
The theoretical description of the jamiton solution admits a better understanding of their behavior.
Another finding of the research are trains of multiple jamitons that can occur on long roads.
In the language of detonation theory, such traffic roll waves are very similar to roll waves
in shallow water flows.
Theoretical Results
We consider continuum two-equation ("second order") traffic models, such as the Payne-Whitham
or the Aw-Rascle equations for traffic flow. The traffic flow is not modeled as individual cars.
Instead, the evolution of a continuous car density function and a continuous velocity function is
described. We consider inviscid models, i.e. any smoothing exchange of momentum between neighboring
cars is neglected. While in reality a small amount of viscosity is obviously present, the inviscid
model can be interpreted as a limiting case that admits a simpler analysis. The considered models
are purely deterministic, and all drivers behave according to the same laws.
Phantom traffic jams
It is well known that two-equation traffic models are linearly unstable for sufficiently
large densities. In other words: A chain of equidistant cars that move all with the same velocity
will not remain in this nice configuration. Instead, a small perturbation grows, and builds up to
become a wave of high car density. This phenomenon is called phantom traffic jam, since it
arises in free flowing traffic, without any obvious reason, such as obstacles, bottlenecks, etc.
Instabilities in traffic flow and the onset of phantom traffic jams have been studied extensively
in various types of traffic models. In continuum traffic models, there are two competing effects.
On the one hand there is a stabilizing traffic pressure due to preventive driving. On the other
hand, there is a destabilizing effect, which comes from the combination of drivers slowing down
when car density is higher and a delay in the adjustment of drivers to new conditions
(the adjustment time is inverse to the "aggressiveness" of the drivers). If the density is
above a certain threshold, then the destabilizing effect outweighs the stabilizing pressure,
and small disturbances grow.
Jamitons
While the instability that leads to a local concentration of cars is understood and reported
in many papers, the exact shape of the final traffic jam wave has not been addressed in
traffic literature. Our studies show that in inviscid Payne-Whitham type traffic models,
instabilities grow into traveling detonation waves. These consist of a sharp jump in
car density (a shock) on one side, and a smooth decay in density on the other side. These
detonation waves are stable structures that travel unchanged with a constant velocity along the
road. In analogy to traveling waves in other fields, solitons, we decided to christen the
traveling traffic waves jamitons.
Properties of jamitons
Our analysis is able to predict fundamental properties of such jamitons. A central result is
that sharp shocks must always face towards incoming cars. Furthermore it can be proved that
jamitons always travel slower than the individual cars. Hence, cars run into a sharp and sudden
increase in density (the end of a phantom traffic jam), which forces each car to break very
suddenly. Then, cars accelerate again our of the jamiton. Our analysis also shows that jamitons
are stable structures. They can only vanish by strong smoothing effects (extremely cautious
drivers) or a lowering of density (a widening road, cars exiting).
Jamitinos
A growing jamiton may trigger a new instability downstream the road. This instability can
also grow and become another traveling wave. A jamiton has given birth to another traffic
wave: a jamitino. In a similar fashion, the second traveling wave may trigger a third
wave, and so on. Thus, a single instability can trigger an infinitely growing sequence of
jamitinos. This phenomenon is visible in the videos below. It resembles roll waves in
shallow water flows.
Conclusions for traffic modeling
For simple traffic laws, the shape of jamitons can be described exactly, allowing a precise
prediction of the maximum traffic density that is achieved in the presence of instabilities.
This result is fundamentally based on the exact shape of the traveling traffic waves, and
traditional analysis of continuum traffic models has not been able to make such predictions.
Furthermore, having a description of the nonlinear traffic waves allows a study of the traffic
outcome in dependence on the model parameters, such as anticipation and aggressiveness of the
drivers. While one jamiton does not delay the travel time of individual cars
significantly (cars travel through a jamiton very quickly), the sharp jump in car density is
a potential hot spot for accidents. In addition, the results indicate that jamitons rarely
stay alone. In practice, whole trains of jamitons can be expected in long stretches of heavy
traffic. Thus, a detailed understanding of the structure of and the interaction between
jamitons can be a fundamental step in understanding the mechanics of traffic flow.
Results for a Circular Road
A circular road is a particular friendly case for an analysis, since the total number of cars
is exactly conserved. If the road is not too long, traffic will in general form one single
traveling wave, i.e. a single jamiton, and thus a single shock is observed.
Below figures and videos show the results of simulations and theoretical predictions for the
case of a circular road of length 230m.
In the case of inviscid equations a sharp shock is realized. Here, the final solution is
predicted theoretically. The match between theory and numerical results is generally very
good. While the inviscid equations allow a simple analysis, using the Rankine-Hugoniot
conditions at the shock, the resulting car behavior is somewhat extreme. Cars slow down
from high to low velocity in zero time.
In real traffic flow, a small viscosity is present. The physical rationale is that a fast
car running towards a slow car results in the fast car slowing down, and (to some extent)
the slow car speeding up, before a minimum distance is reached. Unlike above described
traffic parameters (adjustment time, preventive driving), the nature of viscosity is
difficult to model and extract from real traffic flow. The inviscid equations are expected
to be a good approximation to real traffic with a small viscosity.
For comparison, below are simulation results for viscous traffic equations. The general
behavior is similar to the inviscid case. In particular, again a traveling wave solution is
obtained. However, now with continuous density and velocity functions (the shock is
smoothed out). As a result, cars break earlier and more smoothly.
Comparison with Experimental Results
In March 2008,
Sugiyama et al.
have published an article
Traffic jams without
bottlenecks - Experimental evidence for the physical mechanism of the formation of a jam
in the
New Journal of Physics, in which they report experimental results of traffic
waves. On a circular road of 230m length, 22 cars were placed equidistantly,
and the drivers were instructed to drive, trying to preserve a fixed distance and fixed velocity.
As the video below shows, small instabilities amplify, and a traveling arises that moves
backwards on the road. Sugiyama et al. conclude:
Finally, a jam cluster appears and propagates backward like a solitary wave with the same
speed as that of a jam cluster on a highway.
In our simulations, a circular road of 230m length is considered. The desired velocity resembles
the velocity in the experiment. A comparison of our computational results for 22 cars with the
experimental result reveals strong similarities. While there is no one-to-one match in every
detail, we do believe that the solitary wave found in the experiment is the same as the
jamiton we find analytically and by numerical experiment.
Results for an Infinite Road
Below videos show simulations of a long road with a small initial disturbance. The instability
grows into a jamiton. The shape of the jamiton converges to the theoretically predicted
shape. In addition, a train of jamitinos is triggered, each of which grows to an independently
traveling jamiton.
Comparison with Real Traffic
A nice
video of emergent phantom traffic jams
(copyright:
Dirk Helbing) shows a long straight
road in Cairo, Egypt. Strong similarities with the above simulations are present. The typical shape of
jamitons is visible, as well as their ability to travel backwards on the road. Unfortunately, too many
disturbances (on- and off ramps, interactions between multiple lanes) prevent an undisturbed evolution.
In particular, roll waves, as predicted by our simulations, are not clearly visible here. We would
love to hear of any observations of traffic roll waves.
Conclusions
The observation that simple, purely deterministic traffic models possess jamiton solutions indicates
that phantom traffic jams are not necessarily caused by individual drivers behaving in
a "wrong" way. In fact, they can even occur if all drivers behave by the exact same laws.
In the considered traffic models, two key effects work towards the occurrence of phantom traffic jams:
first, denser traffic travels slower; and second, it takes a certain "adjustment time" for drivers to
react to new traffic conditions. These effects are counter-acted by a certain tendency of the drivers
to drive preventively. In light traffic, the good effects dominate. In heavy traffic, the bad effects
prevail. Hence, phantom traffic jams are probably an unavoidable feature of traffic flow.
Benefits of a better understanding of jamitons
Real traffic possesses jamitons. Hence, a better understanding of their structure can be beneficial
for the simulation and prediction of real highway traffic. Furthermore, the research can be one step
towards answering the key question "how can the occurrence of phantom traffic jams be avoided".
The occurrence of jamitons depends on the model parameters, such as road capacities, speed limits,
and driving behavior. A deeper understanding of jamitons may give indications on how to lower peak
densities, and how to shift the critical threshold density at which jamitons occur upwards. The
latter may be achieved by electronic devices that assist the drivers' behavior in a subtle fashion.
Press Coverage
- Traffic jams follow explosive
pattern, says researcher (PhysOrg.com, June 5, 2009)
- Mathematicians
take aim at 'phantom' traffic jams (MIT news, June 9, 2009)
- Mathematicians
Take Aim At 'Phantom' Traffic Jams: New Model Could Help Design Better Roads (ScienceDaily, June 9, 2009)
- Explaining
Shadow Traffic Jams with Gas Flows (softpedia.com, June 9, 2009)
- MIT
Works to Solve Traffic Jam Problem (DailyTech, June 10, 2009)
- Mathematicians
take aim at 'phantom' traffic jams: beware 'the event horizon of a black hole' (National Post, June 10, 2009)
- 'Jamitons'
slow traffic flow (ft.com, June 11, 2009)
- New Hope
for Ending Pointless Traffic Jams (livescience.com, June 12, 2009)
- Math model
may decrease phantom traffic jams (msnbc.msn.com, June 12, 2009)
- Mathematicians
Take Aim At 'Phantom' Traffic Jams (usnews.com, June 15, 2009)
- MIT Hopes to
Exorcise 'Phantom' Traffic Jams (wired.com, June 17, 2009)
- Traffic
blows up (channel19.blogspot.com, June 17, 2009)
- Study looks to find
reason behind phantom jams (wtop.com, June 18, 2009)
- MIT
working to eliminate phantom traffic jams (autobloggreen.com, June 18, 2009)
- Mathematically
modelling phantom traffic jams (boingboing.net, June 23, 2009)
Scientific Publications
Scientific publications:
- M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales, and B. Seibold.
Self-sustained nonlinear waves in traffic flow.
Physical Review E, 2009; 79 (5)
Scientific paper on the considered traffic models, the prediction of jamitons using the ZND theory of self-sustained detonation waves, and comparisons with computational results.
- M. R. Flynn, A. R. Kasimov, J.-C. Nave, R. R. Rosales, and B. Seibold.
On jamitons, self-sustained nonlinear traffic waves.
A brief phenomenological description of the phenomenon of jamitons.
Links
Movies of real-life phantom traffic jams:
Contact
This web site is currently managed by Benjamin Seibold, MIT applied math.
Contact by email: seibold(at)math.mit.edu
