Mobility Modelling

Lecture 5 - Traffic Assignment

6 March 2023

Mozhgan Pourmoradnasseri, Ph.D.

References:

  1. Dynamic Traffic Assignment, Steve Boyles (course)
  2. Transportation Network Analysis (course)
  3. Transportation Network Analysis (e-book)
  4. Bridging the user equilibrium and the system optimum in static traffic assignment: how the cooperation among drivers can solve the congestion problem in city networks (article)

Traffic modeling summary

WHAT IS TRAFFIC ASSIGNMENT?

  • The fundamental aim of the traffic assignment process is to reproduce the pattern of vehicular movements when the travel demand represented by the trip matrix is assigned.
  • Determine the network traffic flows and conditions that result from the mutual interactions among the route choices that travelers make in traversing from their origins to their destinations and the congestion that results from their travel over the network.
  • The process of allocating a given set of trips to the specified transportation system.

Why do travelers choose the paths they do?

OD-Matrix (set of trips)

Set of routes

Common behavioral assumption

Travelers choose the available route with the least travel time between their origin and destination, reflecting the idea that travel is rarely a goal in and of itself but instead involves some time, cost, or disutility that travelers would prefer to avoid.

Dijkstra's algorithm

for Shortest Path

All-or-noting assignment

  • All the traffic demand is assigned to the shortest path between the origin and destination, without considering any other routes. The approach assumes that travelers always choose the shortest route, and all the traffic is either assigned to that route or not assigned at all.
  • You can think of this as a “target” path flow indicating how people would choose paths if the travel times were fixed at their current value.
  • While this method has some limitations, including the assumption that all travelers choose the shortest path, it is often used because it is relatively easy to implement and can provide a reasonable estimate of traffic flows on a network.

Transportation systems involve interactions among multiple agents. The basic facts are:

  • Although travel choices are made individually, they impact others as well (congestion).
  • The impacts of others’ travel choices are important as you make your choices.

“It’s the evening peak period, so the freeway will probably be congested. I’ll take another route.”

but ...

Even with a relatively simple model of behavior (choosing the fastest route), we end up with mutual dependencies and circular relationships.

Route Travel Times

Route Choices

Travel time and flow

In game theory, the Nash equilibrium, named after the mathematician John Nash, is the most common way to define the solution of a non-cooperative game involving two or more players. In a Nash equilibrium, each player is assumed to know the equilibrium strategies of the other players, and no one has anything to gain by changing only one's own strategy (Wikipedia).

  • In traffic assignment, each traveler is an agent choosing a route; the joint actions of all travelers result in congestion patterns throughout the network, which determines the travel time each traveler faces.

At the equilibrium solution, no traveler can reduce their travel time by switching to another route.

NASH EQUILIBRIUM

WARDROP'S FIRST PRINCIPLE OF ROUTE CHOICE

For each origin and destination, all used routes between those nodes have equal and minimal travel time.

USER EQUILIBRIUM

  1. All drivers are on the top link, and \(t_1(N) < t_2(0)\).
  2. All drivers are on the bottom link, and \(t_2(N) < t_1(0)\).
  3. The two links have equal travel time \(t_1(x_1) = t_2(x_2)\)
  • Note that this principle follows directly from the assumptions that
    • drivers choose minimum time paths and are well-informed about network conditions.
  • If you accept these assumptions, then you must also accept the principle of user equilibrium.
  • Equivalently, if you disagree with the principle of user equilibrium, then:
    • you must either believe that drivers do not choose minimum time paths or
    • do not know the travel times on available paths.

Remark

If 7000 vehicles are choosing these routes, how many choose the top route, and how many choose the bottom?

Example

This method can be generalized in any network with a single OD pair \((r,s)\):

  1. Select a set of paths \(\pi^{rs}\) that you think will be used.
  2. Write equations for the travel times of each path in \(\pi^{rs}\) as a function of the path demands.
  3. Solve the system of equations enforcing equal travel times on all of these paths, together with the requirement that the total path demands must equal the total demand \(d^{rs}\).
  4. Verify that this set of paths is correct; if not, refine \(\pi^{rs}\) and return to step 2.

\(d^{rs}\)

\(d^{rs}\)

\(r\)

\(s\)

General Approach

100 cars travel from A to D. Equilibrium will occur when 50 drivers travel via ABD and 50 via ACD. Every driver has a total travel time of 3.5.

Equilibrium will occur when 25 drivers travel via ABD, 50 via ABCD, and 25 via ACD. Every driver now has a total travel time of 3.75.

Braess paradox

At equilibrium, 25 vehicles choose the top route, 5 chose the bottom, and all travel times are 50 minutes.

10 vehicles choose the top route, 20 choose the bottom route, and everybody has a travel time of 50 minutes.

Now, we improve the bottom link so that its cost function is 40 + x/2.

What happens to route choices now?

Nobody has saved any time at all! What happened?

What is improved?

Downs–Thomson paradox

  • User equilibrium does not minimize congestion.
  • The “invisible hand” does not always function well in traffic networks.
  • There may be room for engineers and policymakers to “improve” route choices.

A few implications

This suggests two possible traffic assignment rules:

  1. User equilibrium (UE): Find a feasible assignment in which all used paths have equal and minimal travel times.
  2. System optimum (SO): Find a feasible assignment that minimizes the total system travel time.

System optimum

User equilibrium

minimizes the total system travel time

minimizes travel time of used routes

The user equilibrium solution is unique in link flows.

iterative FRAMEWORK OF Traffic Assignment

Equilibrium = Fixed point

Fixed Point

Find a value that is unchanged when you go around the loop.

x = 4

10 → 7 → 5.5 → 4.75 → 4.375 → . . .    which converges to the correct answer 4.

Picking a different starting value:                 1 → 2.5 → 3.25 → 3.625 → . . .       also converges to the same answer.

Fixed Point

Find a value that is unchanged when you go around the loop.

x = 8

Choosing the same starting value, we have 10 → 12 → 16 → 24 → 40 → . . . which diverges to +∞.

Consider some set X and a function f whose domain is X and whose range
is contained in X. A fixed point of f is a value x ∈ X such that x = f (x).

iterative FRAMEWORK OF Traffic Assignment

Equilibrium = Fixed point

  1. Start with some feasible link flow solution \(X\). 
  2. Calculate the link travel times using the flows \(X\).
  3. Find the shortest paths between all origins and destinations.
  4. Find the all-or-nothing link flows \(X^*\) corresponding to these shortest paths.
  5. Choose λ ∈ [0, 1] and update

                \(X\) ← λ\(X^*\) + (1 − λ)\(X\).

 6. If “close enough to equilibrium,” stop,            otherwise return to step 2.

SIMULATION-BASED TRAFFIC ASSIGNMENT

  1. Initialize the process by computing the fastest route through the empty network for each vehicle.
    • Add this route to the list of routes known by this vehicle.
    • The probability of using this route by this vehicle is set to 1.
  2. Perform the simulation using the currently chosen routes to obtain the edges’ travel times over simulation time.
  3. Compare the mean travel times against those obtained in the previous run.
    • Quit if the algorithm converges, i.e., the mean travel time reduction falls below a given threshold.
  4. Compute new routes for all vehicles using the network’s current travel times.
    • If a new route for a vehicle was found, add it to the list of routes known by the vehicle.
    • Update all known routes’ estimated travel times and their probabilities to be chosen.
    • Choose one route for this vehicle, taking into account the route choice probabilities.
    • Continue with step 2.

Traffic assignment; summary

  1. All-or-nothing assignment: all traffic is assigned to the shortest path between its origin and destination, ignoring the congestion on other routes.
  2. User equilibrium assignment: each traveler chooses the shortest route given the congestion on all routes, and no traveler can reduce their travel time by switching routes.
  3. System optimum assignment: traffic is assigned to the routes that minimize the total travel time of all travelers, taking into account the congestion on all routes.
  4. Stochastic user equilibrium assignment: each traveler chooses the shortest route given the congestion on all routes, but there is some randomness involved in their choices, which can lead to a more realistic representation of traffic flows.
  5. Dynamic traffic assignment: traffic flows are simulated over time, taking into account the effects of congestion and travelers' route choices, and the routes are adjusted dynamically based on the changing traffic conditions.

IS THE SIMULATION MODEL AN ACCURATE REPRESENTATION OF THE REAL WORLD?

Simulation can be seen as a sampling experiment on a dynamic real system through a computer model formally representing it.

remember: all models are wrong ...

Validation is an iterative process consisting of:

  • collecting the simulated data
  • comparing it to the system’s measured data based on statistical analysis methods
  • determine whether the observed and simulated data samples are significantly close enough.

How do we know the results are good enough?

Traffic data can be classified into two categories:

  1. Directly observable data, i.e., measurements of traffic variables affected by errors (flows, speeds, occupancies, travel times, etc.), which are based on available technologies.
  2. Data not directly observable, such as transport demand modeled by time-dependent origin-destination matrices. 

Ground truth

  • Validation of the model is an iterative process that calibrates the model parameters, compares the model to the actual system behavior, and uses the discrepancies between the two and the insight gained, to improve the model until the accuracy is judged to be acceptable.
  • Calibration process aims to find the values of these parameters that will produce a valid model. Calibration is the process of obtaining such values from field data in a particular setting.

CALIBRATION AND VALIDATION

Source:  Pourmoradnasseri, M., Khoshkhah, K. and Hadachi, A. "Real-Time Calibration of Disaggregated Traffic Demand." arXiv preprint arXiv:2210.17315 (2022).

Calibration diagram

Validation Measures

FIELD VALIDATION

Supilinn, Tartu

Manual counting is sometimes unavoidable :)

 The traffic heat map of simulated

  trips from 16:00 to 17:00.