Ring Car Following Models

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Ring Car Following Models

• by
• Sharon Gibson and Mark McCartney
• School of Computing Mathematics, University of
  Ulster at Jordanstown

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Car Following Models

• Mathematical models which describe how individual
  drivers follow one another in a stream of
  traffic.
• Many different approaches, including
• Fuzzy logic
• Cellular Automata (CA)
• Differential equations
• Difference equations

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Car Following Models

• Classical stimulus response model (GHR model)
• where
• xi(t) is the position of the ith vehicle at time
  t
• T is the reaction or thinking time of the
  following driver
• and the sensitivity coefficient ? is a measure
  of how strongly the following driver responds to
  the approach/recession of the vehicle in front.

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Car Following Models

• A simpler linear form of the GHR model (SGHR) can
  be expressed in terms of vehicle velocities as
• where
• ui(t) is the velocity of the ith vehicle at time
  t.

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Ring Models

• A model in which the last vehicle in the stream
  is itself being followed by the lead (first)
  vehicle
• Motivation
• Real simulations re-use data
• Idealised as a representation of outer rings
• Mathematically interesting

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A Simple Ring Model

• If the driver of each vehicle has zero reaction
  time model simplifies to
• Implication
• The steady state velocity of all vehicles can be
  found immediately once we have been given initial
  velocities.

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A Simple Ring Model

• Need to give the lead car a preferred velocity
  profile, w0(t)
• where
• the sensitivity coefficient ? is a measure of
  how strongly
• the lead driver responds to his/her preferred
  velocity.

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A Simple Ring Model

• For n 2, the transient velocity of the ith
  vehicle is of the form
• where
• and
• The post transient velocity of the ith vehicle is
  dependent on the form of the preferred velocity.

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A Simple Ring Model

• Three forms of preferred velocity considered
• Constant velocity,
• Linearly increasing velocity,
• Sinusoidal velocity,
• NB. The post transient results hold for a general
  n vehicles in the system.

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Ring Model with Time Delay

• This new ring model when the drivers reaction
  times are included can be expressed as
• We solve this system of Time Delay Differential
  Equations (TDDE) numerically using a RK4 routine

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Approximating Time Delay

• An approximate solution to the Time Delay
  Differential Equation (TDDE) form of the Ring
  Model can be found using a Taylors series
  expansion in time delay, T

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Approximating Time Delay

• For n 2, the transient velocity of the ith
  vehicle is of the form
• where
• and
• If system is to reach steady state then

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Comparison of Zero Time Delay, Taylors Series
Approximation RK4 Numerical Methods
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Stability of the Ring Model

• System is locally stable if each car in the
  system eventually reaches a steady state
  velocity.
• Non-oscillatory motion
• Damped oscillatory motion
• Stability criteria is dependent upon the number
  of vehicles in the system.
• General criteria for n 2
• Criteria for n gt 2 currently under investigation
  One of the boundaries obtained for n 3
• Hypothesis The stable region for each value of n
  gt 2 is bounded by exactly 2 boundaries.

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Stability of the Ring Model (n2)
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Stability of the Ring Model (n3)
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Stability of the Ring Model (n5)
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Future Work

• Investigate discrete time models, as
• Easier to implement (Computationally faster)
• Arguably more realistic
• More likely to give rise to chaotic behaviour

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Stability of the Ring Model (n2)(Euler Method)
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Questions?