A Stochastic Model of Platoon Formation in Traffic Flow

A Stochastic Model of Platoon A Stochastic Model of Platoon Formation in Traffic FlowFormation in Traffic Flow

USC/Information Sciences Institute

K. Lerman and A. GalstyanUSC

M. Mataric and D. GoldbergTASK PI Meeting, Santa Fe, NM

April 17-19 2001

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USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation

Traffic on Automated HighwaysTraffic on Automated Highways

• Benefits• increased safety • increased highway capacity

Ordinary highway

Platoon formation on an automated highway

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Our ApproachOur Approach

• Traffic as a MAS• each car is an agent with its own velocity• simple passing rules based on agent

preference• distributed mechanism for platoon formation

• MAS is a stochastic system• stochastic Master Equation describes the

dynamics of platoons• study the solutions

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Traffic as a MASTraffic as a MAS

• Car = agent• velocity vi drawn from a velocity distribution P0(v)

• risk factor Ri : agent’s aversion to passing• desire for safety (no passing)• desire to minimize travel time (passing)

• Traffic = MAS• heterogeneous system (velocity distribution)• on- and off-ramps• distributed control – platoons arise from local

interactions among cars

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Passing RulesPassing Rules

• When a fast car (velocity vi) approaches a platoon (velocity vc), it

• maintains its speed and passes the platoon with probability W

• slows down and joins platoon with probability 1-W

• Passing probability W

• (x) is a step function• R is the same for all agents

)()( iavecici RvvvvvW

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Platoon FormationPlatoon Formation

v2 vC

vCv1

v2vC

vC

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MAS as a Stochastic SystemMAS as a Stochastic System

Behavior of an individual agent in a MAS is very complex and has many influences:

• external forces – may not be anticipated• noise – fluctuations and random events • other agents – with complex trajectories• probabilistic behavior – e.g. passing probability

While the behavior of each agent is very complex, the collective behavior of a MAS is described very simply as a stochastic system.

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Physics-Based Models of Traffic Physics-Based Models of Traffic FlowFlow

• Gas kinetics models• similarities between behavior of cars in traffic and

molecules in dilute gases• state of the system given by distribution funct

P(v,x,t)

• Hydrodynamic models• can be derived from the gas kinetic approach• computationally more efficient• reproduce many of the observed traffic phenomena

free flow, synchronous flow, stop & go traffic• valid at higher traffic densities

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Some DefinitionsSome Definitions

),( tvPm

)1(),( vvWvvvvU

Density of platoons of size m, velocity v

Car joins platoon at rate

Initial conditions: )()0,( 01, vPvP mm

where P0(v) is the initial distribution of car velocities

Individual cars enter and leave highway at rate

for v>v’

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Master Equation for Platoon Master Equation for Platoon FormationFormation

1,01

0

)()()()1(

)()()()()(

mmm

mjk vjk

kkm

m

vPvmPvPm

UvPvPvdUvPvdvPtvP

loss due to collisions merging of smaller platoons

outflow of cars

,...2,1m

inflow of cars

Inflow and outflow drive the system into a steady state

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Average Platoon Size Average Platoon Size

3.0

10 3

R

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Platoon Size DistributionPlatoon Size Distribution

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Steady State Car Velocity Steady State Car Velocity DistributionDistribution

)(0 vP

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ConclusionConclusion

• Platoons form through simple local interactions

• Stochastic Master Equation describes the time evolution of the platoon distribution function

• Study platoon formation mathematicallyBut,• Does not take into account spatial

inhomogeneities• Need a more realistic passing mechanism

• effect of the passing lane

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Future workFuture work

• Multi-lane model• for each lane i, Pm

i(v,t)

• Passing probability depends on density of cars in the other lane, and on platoon size

• Microscopic simulations of the system• Particle hopping (stochastic cellular automata)• What are the parameters that optimize

• average travel time• total flow

A Stochastic Model of Platoon Formation in Traffic Flow

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