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
Research Question
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
2
Outline
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
3
Background
4
Background
Network Fundamental Diagram
Link-based definitions of network traffic flow variables
M

  li qi 

Q   i M1


  li 
 i 1 
Source: Geroliminis and Daganzo (2008)
M

  li k i 

K   i M1


  li 
 i 1 
Source: Mahmassani, Williams and
Herman (1984)
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Theory
Equilibrium (Non-Hysteretic) NFD
Exit rate
g = G(n)
Number of vehs in network
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.
Source: Daganzo (2007)
6
Theory
Proposed Non-equilibrium (Hysteretic) NFD
g = G(n) + H
g = G(n) + H(n,σ)
H represents the deviation from steady-state conditions due
to the hysteretic behavior of the network traffic flow.
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Theory
Network Flow, Density and Stdv of Density Relations
Network Average Flow (vph)
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.
1000
Density = 5 veh/mile
Density = 10 veh/mile
800
Density = 15 veh/mile
600
Density = 20 veh/mile
Density = 25 veh/mile
400
Density = 30 veh/mile
200
Density = 35 veh/mile
Density = 40 veh/mile
0
0
20
40
60
80
Standard Deviation of Network Density
Downtown Chicago
sub-network
Source: Mazloumian et al. (2010) & Knoop et al. (2011)
Network Simulation
Chicago Metropolitan Network
 Large Scale Network with ~40,000 links
and ~13,000 nodes
 ~2,000 traffic zones
 ~4 millions simulated vehicles
Loading profile
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Theory
Proposed Two-Dimensional NFD and Calibration
For a given network density, a linear relationship between Q and σK is assumed.
Q = f(K).σK + h(K)
slope
Y-intercept
Calibrated
Relationship
for Downtown
Chicago subnetwork
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Two-Dimensional NFD
Calibration for downtown Chicago
slope
Y-intercept
-α=f(K)
Β=h(K)
Q = f(K).σK + h(K)
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Two-Dimensional NFD
Calibration for downtown Chicago
Producing hysteresis loop using two-dimensional NFD relation
 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.
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Network Simulation Results
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).
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Network Simulation Results
Gridlock in the CBD sub-network
The long-lasting invariant large densities with very small
flows suggest formation of a gridlock.
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Gridlock
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)
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Gridlock
Characteristics
Size (# vehicles or lane-miles)
Configuration (spatial form)
Formation Time
Formation Location
Dissipation Time
Propagation Duration
Recovery Duration
Propagation Speed
Recovery Speed
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Demand Management
Effects of Demand Management of NFD of CBD Sub-network
100% demand
85% demand
75% demand
Gridlock configuration at CBD at the end of simulation
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Demand Management
Temporal Effects of Demand Level on Gridlock Evolution
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Adaptive Driving
Average Network Flow
Effects of Adaptive Driving on NFD of CBD Sub-network
Average Network Density
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Adaptive Driving
Effects of Adaptive Driving on Gridlock Size & Configuration
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Conclusion
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 nonequilibrium 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.
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Thank you!
Questions?
Hani Mahmassani
[email protected]
Meead Saberi
[email protected]
Ali Zockaie
[email protected]
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