Why do MFDs exhibit different hysteresis patterns depending on how network productivity is measured?
Macroscopic Fundamental Diagrams (MFDs) are relationships between network productivity and density or accumulation averaged across an entire network. The following interactive simulation illustrates how MFDs may exhibit hysteresis patterns due to different traffic patterns as congestion forms and dissipates in the network and how these hysteresis patterns change based on what metric is used to quantify traffic network productivity.
The simulation can be downloaded as an executable JAVA file HERE
In this simulation, vehicles travel counterclockwise on a ring divided into four even blocks by traffic signals. The ring is composed of many cells, each the size of a vehicle. Each vehicle enters the ring at a signal, travels some number of blocks in the ring, and then exits the ring at a signal. Five sliders allow the user to control:
- How quickly new vehicles and are created and enter the network, with higher values being more frequent trip generation.
- The average trip length of vehicles in the network (note 1 block = 30 cells)
- How evenly the new trips are distributed between the different signals. 25% represents perfectly even trip distribution where vehicles are created uniformly across the four blocks while 100% means that every trip starts at the right-most traffic signal.
- Traffic signal cycle length.
- How quickly (compared to real time) the simulation runs
The buttons at the bottom of allow the user to start/pause and reset the simulation.
The two plots located in the bottom right (below the ring) represent two versions of the MFD that allow us to observed how congestion might influence relationships between network productivity and accumulation. On the left is a traditional MFD that provides the relationship between how well vehicles are able to move through a system (average flow) and the spatial rate of vehicles in the network (average density). Also plotted is a grey trapezoid that represents the ideal flow-density relationship based on the signal and geometric settings (Daganzo and Geroliminis, 2008). On the right is the relationship between how well vehicles are able to leave the system (trip completion rate) and the spatial rate of vehicles in the network (average density). This is known more commonly as the production MFD or Network Exit Function (NEF). A grey trapezoid is also included that represents the ideal relationship. Note that this is based on the network’s MFD and average trip length, and thus shifts when the trip length slider is adjusted.
Both graphs plot points at regular one-minute (in simulation time) intervals, using changing colors so that time-dependent trends can be observed. The colors used are the same for both plots and points plotted at the same time will be the same color on each graph. The colors do change gradually and loop in the following order:
- Red
- Orange
- Yellow
- Green
- Blue
- Violet
The color cycle (starting with red) repeats after 100 simulation minutes.
In order to see the desired behavior (which will be observed using the MFD and NEF plots in the bottom right), do the following things in the simulation.
Setup
- Set the simulation speed to 360 (default)
- Set average trip length to 60 cells (default)
- Set trip generation to 56 (default)
- Leave other sliders as is
Run
(Note: all times refer to simulation time)
- Press start and allow the simulation to run for about 10 minutes, and press pause.
- Move the trip generation slider up one tick to 57. This can be done while the simulation is running or while paused.
- Run the simulation for another 10 minutes, then increase generation slider to 58.
- Run the simulation for another 10 minutes, then increase the generation slider to 59. You should be seeing the points creeping up along the theoretical lines on both charts.
- Run the simulation for a further 10 minutes, and press pause.
- Approximately 40 minutes should have passed. Increase the trip generation slider to 60. Serious congestion will begin when the simulation is restarted. Both graphs will begin plotting points along the flat part of their curves and to the right.
- Run the simulation with the generation slider set to 60 for 20 minutes (for 60 minutes in total). The ring should be pretty congested at this point but there should still be a little bit of open space (system should not be jammed). There should be some light blue plot points on the right side.
- Decrease the trip generation slider to 59, and let the simulation run for 10 minutes (or until the ring looks fairly empty) before pressing pause. Depending on how congested the ring got, this dissipation may take some time.
Findings
It may be a little difficult to see since the simulation is random, but if you look closely at both plots, you’ll notice two things about them
- Both plots are exhibiting a loop. The older, green points are not following the same path as the newer, blue ones.
- These loops go in different directions. One the MFD (left) plot, the loop is very small and relatively clockwise. The newer plot points are below the older ones. On the NEF (right) plot, the loop is larger and relatively counterclockwise. While it may be difficult to ser through the scatter, the newer blue points are above the green older ones.
These loops show an important characteristic of traffic congestion; it behaves differently when it forms (green) than it does when it dissipates (blue). During recovery, the flow at given densities is lower than it was at those same densities when the congestion was forming. The trip completion rate shows the opposite effect. During recovery, the trip completion rate is generally higher than it was at the same densities during congestion formation. An example is shown below.
The clockwise hysteresis loop in the MFD occurs because congestion is naturally more even distributed as density increases than density decreases (see Gayah and Daganzo, 2011). The counterclockwise in the NEF occurs because while flow measures how well vehicles travel within the network, the trip completion rate only measures how quickly vehicles can get to the end of their trip. When congestion builds, vehicles approach their destination more slowly than when congestion does not exist, and the congestion keeps vehicles from reaching their destination. As congestion dissipates, these vehicles that were previously traveling slowly in the network but are near their destination are able to more rapidly reach their destination and exit the network, which causes the trip completion rate metric to temporarily spike. For this reason, this counterclockwise loop will not be observed if the network does not get congested and vehicles always travel at about the same speed in the network (e.g., if the density never exceeds 24 vehicles per ring).
Acknowledgment
This research was supported by NSF Grant CMMI-1749200, the Penn State College of Engineering REU program and the Dr. and Mrs. David and Shirley Wormley REU Scholarship.