1. Urban Network Gridlock: Theory, Characteristics, and Dynamics Hani Mahmassani, Meead Saberi, Ali Zockaie The 20th International Symposium on Transportation and Traffic Theory Noordwijk, the Netherlands July 17, 2013

2. Research Question 2 Exploration of the physics of traffic flow in urban networks under highly congested conditions Focus on: 1.Inhomogeneous spatial distribution of congestion 2.Modeling NFD with hysteresis and gridlock 3.Characterizing gridlock phenomena

3. Outline 3 Background Theory Non-hysteretic NFD Hysteretic NFD Two-Dimensional NFD Findings from Simulation Results NFD for the entire network and CBD sub-network Gridlock properties Effects of demand management Effects of adaptive driving Conclusion

4. Background 4

5. 5 Network Fundamental Diagram Link-based definitions of network traffic flow variables Source: Geroliminis and Daganzo (2008) Background Source: Mahmassani, Williams and Herman (1984)

6. Theory 6 Equilibrium (Non-Hysteretic) NFD Source: Daganzo (2007) g = G(n) Such function is intended as an idealized description of the equilibrium (steady-state) behavior that would be expected to hold only when the inputs change slowly in time and traffic is distributed homogenously in space. Exit rate Number of vehs in network

7. 7 Proposed Non-equilibrium (Hysteretic) NFD g = G(n) + H g = G(n) + H(n,σ) Theory H represents the deviation from steady-state conditions due to the hysteretic behavior of the network traffic flow.

8. Theory Network Flow, Density and Stdv of Density Relations For the same value of network density, there is a negative correlation between the network average flow and the standard deviation of the network density. Source: Mazloumian et al. (2010) & Knoop et al. (2011) Downtown Chicago sub-network

9. Network Simulation 9 Chicago Metropolitan Network  Large Scale Network with ~40,000 links and ~13,000 nodes  ~2,000 traffic zones  ~4 millions simulated vehicles Loading profile

10. 10 Theory Proposed Two-Dimensional NFD and Calibration Q = f(K).σ K + h(K) Calibrated Relationship for Downtown Chicago sub- network slopeY-intercept For a given network density, a linear relationship between Q and σ K is assumed.

11. 11 Calibration for downtown Chicago -α=f(K)Β=h(K) Q = f(K).σ K + h(K) Two-Dimensional NFD slopeY-intercept

12. 12 Calibration for downtown Chicago Producing hysteresis loop using two-dimensional NFD relation Two-Dimensional NFD  The simulated NFD is network average flow versus network average density which are directly obtained from simulation  In the modeled NFD density and its standard deviation are obtained from simulation. The calibrated relationship is used to estimate network average flow.

13. Network Simulation Results 13 Network-wide Relation (entire network) and Gridlock  Loading and unloading phases are shown.  After a certain time the network outflow is close to zero and there are a number of “trapped” vehicles in the network (gridlock).

14. Network Simulation Results 14 Gridlock in the CBD sub-network The long-lasting invariant large densities with very small flows suggest formation of a gridlock.

15. Gridlock 15 Gridlock evolution in the CBD sub-network (number of lane-mile jammed links) Gridlock propagation speed is much larger than gridlock dissipation speed. At the end of the simulation, more than 40% of the links are empty while the rest are jammed (significant inhomogeneity of congestion distribution )

16. Gridlock 16 Characteristics Size (# vehicles or lane-miles) Configuration (spatial form) Formation Time Formation Location Dissipation Time Propagation Duration Recovery Duration Propagation Speed Recovery Speed

17. 17 Demand Management Effects of Demand Management of NFD of CBD Sub-network Gridlock configuration at CBD at the end of simulation 100% demand 85% demand 75% demand

18. 18 Demand Management Temporal Effects of Demand Level on Gridlock Evolution

19. 19 Average Network DensityAverage Network Flow Adaptive Driving Effects of Adaptive Driving on NFD of CBD Sub-network

20. 20 Adaptive Driving Effects of Adaptive Driving on Gridlock Size & Configuration

21. Conclusion 21 1.Study of a large-scale urban network consisting of both freeways and arterials with exit flow, under highly congested conditions. 2.The existing theory of equilibrium NFD is extended to non- equilibrium conditions in order to reproduce hysteresis and gridlock phenomena. 3.Networks tend to jam at a range of densities that are considerably smaller than the theoretical average network jam density due to inhomogeneous distribution of congestion. 4.Parameters for characterizing gridlock phenomenon in urban networks are introduced; opens new direction for investigation and application to better traffic management 5.Effects of demand management and adaptive driving on gridlock and hysteresis phenomena and NDF are also studied.

22. Thank you! Meead Saberi meead@u.northwestern.edu Questions? 22 Hani Mahmassani masmah@northwestern.edu Ali Zockaie ali-zockaie@u.northwestern.edu