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Modeling Transportation Network Dynamics as a Complex System

- David Levinson
- Bhanu Yerra

Objectives

- Why do networks expand and contract?
- Do networks self-organize into hierarchies?
- Roads - an emergent property?
- Can investment rules predict location of network improvement?
- How can this improved knowledge help in planning transportation networks?

How networks change with time?

- State of a network node changes
- Travel time of a link changes
- Capacity of a link changes
- Flow on a link changes
- New links and nodes are added
- Existing links are removed
- System properties, like congestion, change over time

How to model a network as a complex system?

- Links and nodes are agents
- Agent properties
- Rules of interaction that determine the state of agents in the next time step
- Spatial pattern of interaction between agents
- External forces and variables
- Initial states

Transportation network as a complex system

- Network is split into two layers
- Network layer
- Land use layer

- Network is modeled as a directed graph
- Land Use Layer has small land blocks as agents that determine the populations and land use

Models Required

- Land use and population model
- Travel demand model
- Revenue model
- Cost model
- Network investment model

Network

- Grid network
- Random networks will also be tested
- Networks under modeling stage
- Cylindrical network
- Torus network

- Initial speed distribution
- Every link with same initial speed
- Uniformly distributed speeds

Land Use and Demography

- Small Land Blocks are agents
- Population, business activity, and geographical features are attributes
- Uniformly distributed and bell shaped land use are modeled
- Land use is assumed to be static
- Dynamics of land use will be added later

Travel Demand Model

- Trip Generation
- Using Land use model trips produced and attracted are calculated for each cell
- Cells are assigned to network nodes using voronoi diagram
- Trips produced and attracted are calculated for a network node using voronoi diagram

Travel Demand Model(Contd.)

- Trip Distribution
- Calculates trips between network nodes
- Gravity model
Where,

trs is trips from origin node r to destination node s,

pr is trips produced from node r,

qsis trips attracted to node s,

drsis cost of travel between nodes r and s along shortest path

Travel Demand Model(Contd.)

- Traffic Assignment
- Path with least cost of traveling
- Cost of traveling a link is
Where, la is length of link a

va is speed of link a

is value of time

o is tax rate

1, 2 are coefficients

Travel Demand Model(Contd.)

- Traffic Assignment ( Contd.)
- Dijkstras Algorithm
- Flow on a link is
Where,

Krsis a set of links along the shortest path from node r to node s,

a,rs = 1 if a Krs, 0 otherwise

Revenue Model

- Toll is the only source of revenue
- Annual revenue generated by a link is total toll paid by the travelers
Where, coefficients are same as coefficients used in traveling cost function

- A central revenue handling agent can be modeled

Cost Model

- Assuming only one type of cost
- Cost of a link is
Where, c is cost rate,

1, 2, 3 are coefficients.

- Introducing more cost functions makes the model more complicated and probably more realistic

Network Investment Model No revenue sharing between links: Revenue from a link is used in its own investment

- A link based model
- Speed of a link improves if revenue is more than cost of maintenance, drops otherwise
Where, vat is speed of link a at time step t,

- is speed reduction coefficient.

Examples

- Base case
- Network - speed ~ U(1, 1)
- Land use ~ U(10, 10)
- Travel cost da { = 1.0, o = 1.0, 1 = 1.0, 2 = 0.0}
- Cost model {c =365, 1 = 1.0, 2 = 0.75, 3 = 0.75}
- Improvement model { = 1.0}
- Speeds on links running in opposite direction between same nodes are averaged
- Symmetric route assignment

Equilibrium network state after 8 iterations

Slow

Fast

Figure 4 Equilibrium speed distribution for the base case on a 10x10 grid network

Base Case

Initial network

Equilibrium network state after 9 iterations

Slow

Fast

Figure 5 Equilibrium speed distribution for the base case on a 15x15 grid network

Equilibrium network state after 8 iterations

Slow

Fast

Figure 6 Equilibrium speed distribution for case 2 on a 10x10 grid network

Case 2: Same as base case but initial speeds ~ U(1, 5)Equilibrium network state after 8 iterations

Slow

Fast

Figure 7 Equilibrium speed distribution for case 2 on a 15x15 grid network

Case 2: Same as base case but initial speeds ~ U(1, 5)Equilibrium network state after 7 iterations

Slow

Fast

Figure 8 Equilibrium speed distribution for case 3 on a 10x10 grid network

Case 3: Base case with a downtownEquilibrium network state after 7 iterations

Slow

Fast

Figure 9 Equilibrium speed distribution for case 3 on a 15x15 grid network

Case 3: Base case with a downtownConclusions

- Succeeded in growing transportation networks
- Sufficiency of simple link based revenue and investment rules in mimicking the network structure
- Hierarchical structure of transportation networks is a property not entirely a design
- Future work will help in better understanding of network dynamics

Future Work

- Trip distribution - An agent based combined trip distribution and traffic assignment
- Traffic assignment - Deterministic and Stochastic user equilibrium
- Revenue sharing between links
- Modeling construction of new links
- Maintenance and construction costs
- Spatial dependent revenue and investment models

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