Traffic flow on networks

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Traffic flow on networks

• Benedetto Piccoli
• Istituto per le Applicazioni del Calcolo Mauro
  Picone CNR Roma
• Joint work with G. Bretti, Y. Chitour, M.
  Garavello, R.Natalini , A. Sgalambro

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Macroscopic models

• Vehicular traffic can be treated in different
  ways with microscopic,mesoscopic or macroscopic
  models.
• Macroscopic models mimic some phenomena such as
  the creation ofshocks and their propagation
  backwards along the road, since they candevelop
  discontinuities in a finite time even starting
  from smooth data.

Representation of intersections - Backward
propagation of queues - Distribution of flow
capacity resource on the downstream links of a
node to its upstream links
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Fluidodynamical models for traffic flow

• LWR model

M.J. Lighthill, G.B. Whitham, Richards 1955

• Example

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Traffic features difficult to reproduce traffi
c jams

• Empirical Evidences
• Once created, jams are stable and can move for
  hours against the flow of traffic
• The flow out of a jam is a stable, reproducible
  quantity

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Bilinear model

• Simple model with reasonable properties

• Two characteristic velocities

• Respect phenomenon of backward moving clusters

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Fluidodynamical models for traffic flow

• Aw Rascle model

Aw Rascle model solves typical problems of second
order models Cars going
backwards!
Other models Greenberg, Helbing, Klar, Rascle,
Benzoni - Colombo, etc.
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Road networks

• Road networds consists of a finite set of roads
  with junctions connecting roads

Problem how to define a solution at junctions.
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Solutions at junctions

• Solve the Riemann problem at junctions

• There are prescribed preference of drivers, i.e.
  traffic from incoming roads
• distribute on outgoing roads according to
  fixed (probabilistic) coefficients

(B) Respecting rule (A) drivers behave so as to
maximize flow.

• REMARK
• The only conservation of cars does not give
  uniqueness
• Rule (A) implies conservation of cars
• The only rule (A) does not give uniqueness

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Rules (A) and (B)
Rule (A) corresponds to fix a traffic
distribution matrix
Using rule (A) and (B) (under generic assumption
on the matrix A), it is possible to define a
unique solution to Riemann problems at junctions.
Remark other definition given by Holden-Risebro
95.
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LP problem at junctions

• It is enough to solve a LP problem at junctions
  for incoming fluxes!
• Then other fluxes and densities are determined.

Other way of looking Demand Supply of J.P.
Lebacque
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Solutions via WFT for 2x2 junctions
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Continuous dependence

• Lipschitz continuous dependence does not hold
  explicit counterexample.

• Open problem continuous dependence.

• Lipschitz continuous dependence holds only for
• Single crossing with assumptions on initial data.
• Small BV perturbations of generic equilibria.

• Open problems control the BV norm of density,
  extension to networks with any junction.

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Numerical approximation

• Approximation schemes (explicit schemes)
• Godunovs scheme (first order)
• Kinetic scheme (kinetic scheme with 2 or 3
• velocities) of first order (Aregba-Driollet
  Natalini)
• Kinetic scheme with 3 velocities of second order
  
• (Aregba-Driollet Natalini)

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Godunov scheme
Construct piecewise constant approximation of the
initial data
The scheme defines recursively
starting from .
CFL-like condition
The projection of the exact solution on a
piecewise constant function is
These values are computed with the Gauss-Green
formula.
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Godunovs scheme
The scheme reads
with the numerical flux (associated to flux
function )
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Kinetic scheme

• Drawback
• Diffusivity at first order for 2
• velocities (Lax-Friedrichs scheme).

• Advantages of kinetic scheme
• Extension to High order
• No instability at the boundary

Godunov scheme
Kinetic scheme
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Simulation on Rome road network
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Simulation on Rome road network
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LP solvers
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Running times
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Large simulations

• Necessity of simulating networks with thousands
  of arcs and nodes
• Fast simulation to apply for optimization
  problems
• Elaboration of big data bases for network
  characteristic
• Visualization time

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Modified Godunov
IDEA Use bilinear model to have simplified
choices of Numerical fluxed
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FVST scheme
1. Use simplified flux function with two
characteristic speed
2. Make use of theoretical results to bound the
number of regimes changes
3. Track exactly regimes changes or separating
shocks and use simple dynamics for one-sided
zones
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FVST scheme

• Finite volumes shock tracking scheme
• Strongly bounded computational times
• Error due only to initial data rounding and
  junction data rounding
• Bounded to solutions for empty initial network

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Comparison of schemes
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Salerno network simulation
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• Networks and
• Heterogeneous Media

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Packets flow on telecommunication networks

• Benedetto Piccoli
• Istituto per le Applicazioni del Calcolo Mauro
  Picone CNR Roma
• Joint work with C. DApice, R. Manzo, A. Marigo

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Packets flow on telecommunication networks

• Telecommunication networks as Internet no
  conservation of packets at small time scales.

Assume there exists a loss probability function
and packets are re-sent if lost.
Then at 1st step (1-p) packets sent, p
lost at 2nd step p(1-p) packets
sent, p2 lost . at kth step p(k-1)
(1-p) sent, pk lost Finally the
average transmission time and velocity are
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Riemann problems at junctions

• Maximize the fluxes over incoming and outgoing
  lines
• remove rule (A)

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Riemann problems at junctions
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BV Estimates
For interactions with a junction we get
For special flux fuction we get BV estimates on
the densities.
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Lipschitz continuous dependence
Lemma