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
ISI
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
Platoon FormationPlatoon Formation
v2 vC
vCv1
v2vC
vC
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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.
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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’
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
Average Platoon Size Average Platoon Size
3.0
10 3
R
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
Platoon Size DistributionPlatoon Size Distribution
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
Steady State Car Velocity Steady State Car Velocity DistributionDistribution
)(0 vP
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
ISI
USC Information Sciences Institute K. LermanStochastic Model of Platoon Formation
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
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