Modeling Road Traffic

Greg PinkelBrad Ross

Math 341 – Differential EquationsDecember 1, 2008

Contents• Introduction

• Model

• Revised Model

• Solving our model

• Conclusions

Introduction

• Modeling the behavior of cars in traffic

• Car Following Model

• Quick-Thinking-Driver (QTD) Car Following Model

• Major Results:– If lead car stops suddenly, system looks like a

damped harmonic oscillator– Temporal Following Spacing should be ≥ 2 sec

Overview• Purpose – traffic modeling• Models

– 2-car model, x0 position of leading car, x1 position of following car (meters)

– Acceleration of following car (m/s2)– Sensitivity coefficients, λ (1/sec.) and β (1/sec.2)– Reaction time T (seconds)– Preferred separation, d (meters) or temporal

separation, τ (seconds)

Original Model

dt

tdx

dt

tdx

dt

Ttxd )()()( 102

12

Quick Thinking Driver Model dtxtx

dt

txd )()(

)(102

12

Original Model

• λ : sensitivity coefficient, units s-1

• Impact of and

• Addition of T is a Delay Differential Equation, very difficult to solve analytically

dt

dx

dt

dx 10

dt

tdx

dt

tdx

dt

Ttxd )()()( 102

12

dt

dx

dt

dx 10

Quick Thinking Driver Model

dtxtxdt

txd )()(

)(102

12

• Ignores reaction time for driver

• β : sensitivity coefficient, units s-2

• Assuming d is a constant “preferred separation”

• Impact of x0 – x1<d and x0 – x1>d

Preferred Separation• Tends to depend on speed (Think of real world situation)

• Relate preferred spatial separation to temporal separation by

with a preferred temporal separation of τ seconds.

dt

tdxd

)(1

Review (and the 2nd order D.E.)

dt

tdx

dt

tdx

dt

Ttxd )()()( 102

12

)()()()(

011

21

2

txtxdt

tdx

dt

txd

dtxtxdt

txd )()(

)(102

12

1)

2)

Physically Reasonable Stopping

With initial conditions ,

Dxdt

dx

dt

xd 1

12

12

Udt

dx

)0(10)0(1 x

D = Separation of vehicles, x0 –x1, at time of x0 sudden stop

m/sm

Solving: Laplace Transforms and Mathematica!

The Solution: X1(t)

Major Results: Simple Harmonic Motion • Have three distinct cases for different values of τ:

)4( 2t

Overdamped,

(τ=1s)

Underdamped,

(τ=3s)

Critically Damped,

(τ=2s)

42

42

42

42

U=10 m/s, D=10m, β=1s-2

Significance of Results

• Always have collision with the underdamped situation

•

• To avoid collision: τ2 ≥ 4, τ ≥ 2 seconds

42 42

Minimum Stopping Distance• Collision is possible but will not necessarily

happen: a βτ2 > 4, βτ2 = 4

• Collision can be avoided by ensuring initial separation greater than Dstop

• Derived by setting the solution of x(t) equal to zero with given conditions, D becomes Dstop

• Having initial separation greater than this ensures that the collision will be avoided

4,4)(

2 2

2

UDstop

Conclusion

• Based on conditions: βτ2 > 4, βτ2 = 4, and βτ2 < 4 – collision is only avoidable for βτ2 ≥ 4

– collision only avoided if x0 – x1 > Dstop

– Because βτ2 < 4 always results in collision, maintain temporal time cushion at least 2 sec behind lead car