Network instabilities

Inherent Instability of Traffic Networks

The following interactive simulation illustrates how urban traffic congestion naturally moves towards areas that are already congested; and how this surprising effect can reduce the overall flow circulating in a network.

The simulation can be downloaded as an executable JAVA file HERE. Additionally, videos of the simulation in action are provided below.

In this simulation, vehicles travel on each ring and can change between the rings at the point of tangency. Vehicles are added and taken away from the system at the tangent point and always in pairs. Eight sliders allow the user to change:

The probability that a vehicle will turn from one ring to another
The proportion of drivers that make adaptive routing decisions
The total number of vehicles in the system
The number of traffic signals on each ring
The cycle length, green time, and offset of these signals
The simulation speed

The buttons on the bottom allow the user to: start and pause; reset; force one vehicle from the left ring to the right; and the reverse.

The average flow of the two rings (representing how much vehicles move in the system) and the average density of the two rings (representing how many vehicles are in the system) are calculated at 1 min intervals and plotted as points on the bottom-right. For any given density, the higher the average flow, the higher the mobility provided by the network. Also plotted is a grey triangle that represents the relationship between the flow and density on each ring. This grey curve is known as the fundamental diagram. The density at the apex (12 vehicles per ring or 24 vehicles total) is called the critical density.

No Turns Allowed

To examine the behavior of the system when vehicles do not turn in the system, try the following:

Reset the system
Change the simulation settings to the following:
- Probability of Turning – 0
- Number of Signals on each Ring – 0
- Cycle Length, Green Time, Offset – N/A
- Number of Vehicles – Any number between 2 and 120
Start the simulation.

Observe that each ring is loaded evenly (has the same number of vehicles) and that this does not change with time. Note how the flow-density points fall very close to the grey triangle. This is expected. Now, with the simulation running, slowly change the number of vehicles in the system. Observe that the rings remain evenly balanced and the flow-density points still fall close to the grey triangle. Note that the network completely jams only when both rings are completely filled with vehicles.

Turns Allowed

When turning is allowed between the two rings, the number of vehicles in each ring fluctuates with time about an average number and the system exhibits three regimes. To see the curious things that happen try the following:

Reset the system
Change the simulation settings to the following:
- Probability of Turning – Any value greater than 0
- Number of Signals on each Ring – 0
- Cycle Length, Green Time, Offset – N/A
- Number of Vehicles – Any number between 2 and 24
Start the simulation.

Observe that the number of vehicles in each ring changes as the simulation runs, but over time the two rings tend to be balanced. If the number of vehicles is considerably smaller than the critical density (between 2 and 12) then the flow-density points fall fairly close to the grey triangle. Observe that neither ring exceeds the critical density at any time. If the number of vehicles is higher (between 14 and 24) then the two rings will still be balanced on average, but every once in a while one of the rings will exceed the critical density. This causes the flow-density points to be scattered and lie slightly below the gray triangle. This effect is due to the granularity of the simulation and diminishes for larger rings that can contain many vehicles. Flow is also reduced by conflicts at the point of tangency, which delay vehicles and impede their progress.

Next: increase the number of vehicles in the system to any value between 26 and 58 (less than the required to jam a single ring). Observe that the two rings become unbalanced, one ring having more vehicles than the other. The rings tend to stay that way for extended periods of time but once in a while the ring densities flip and settle into the opposite asymmetric pattern. The system spends most of the time in these asymmetric states, which have less flow than their symmetric counterparts. This is why the observed flow-density points lie so cleanly below the grey triangle. Because the frequency of flipping declines with the number of vehicles, higher densities more clearly illustrate this effect. You can use the L-to-R and R-to-L buttons to try to balance the two rings; however, the rings will become unbalanced again!

Next: increase the number of vehicles to any value greater than 60 but below 120. Observe that the system will now become completely jammed. One ring will eventually fill up with vehicles; the other will stop moving because one vehicle will be unable to turn onto the filled ring. The flow-density points will from then on lie on the x-axis showing that the average flow is zero. Again, you can use the L-to-R and R-to-L buttons to un-jam the system but observe that the system will eventually jam again!

This unstable behavior can be mitigated by the presence of drivers that make adaptive routing decisions. For the purposes of this simulation, the adaptive drivers will not turn from a less congested ring onto a more congested ring. To see how this changes the behavior of the system, try the following:

Reset the system
Change the simulation settings to the following:
- Probability of Turning – Any value greater than 0
- Proportion of Adaptive Drivers – 0.3
- Number of Signals on each Ring – 0
- Cycle Length, Green Time, Offset – N/A
- Number of Vehicles – 60
Start the simulation.

Observe that one of the rings eventually fills up with vehicles (as before) but it takes much longer than before (on average) for this to occur. Additionally, the observed flow-density pairs lie closer to the grey curve. The more adaptive drivers are, the longer it takes for the system to become jammed and the closer the flow-density points are to the grey curve. To confirm this, reset the simulation and change the Proportion of Adaptive Drivers to a very high value (0.8). Observe in this case that drivers are so adaptive that the two rings remain balanced for a very long time (perhaps so long that unbalanced rings are never observed while watching the simulation), and that the blue points are consistently located on the grey curve while the two rings are balanced.

Other Key Points

You can verify by repeating the exercises in the “Turning Allowed” link (for various values of the turning probability) that the same qualitative effects are seen for any value of the probability of turning, as long as it is greater than 0. Therefore, the flow-reduction effect can be attributed to turning. The critical number of vehicles (24) at which the two-ring system changes from being evenly loaded to unevenly loaded (providing less mobility than possible) is called a bifurcation. This bifurcation arises even if traffic signals are introduced in the system. Try it and note how for high densities the system settles into asymmetric patterns.

For more information on the simulation and other things to try see Gayah and Daganzo (2011). The asymmetry and flow-reduction observed in the simulation arises even if there is no granularity. For more information, including the theoretical background and extensions to real networks with more realistic driver behavior, see Daganzo et al. (2011). Working versions of these papers can be found here and here, respectively.

References:

Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. 0856193. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation (NSF).