A Stochastic Model of Platoon
Formation in Traffic Flow
USC/Information Sciences Institute
K. Lerman and A. Galstyan
USC
M. Mataric and D. Goldberg
TASK PI Meeting, Santa Fe, NM
April 17-19 2001
Traffic on Automated Highways
Ordinary highway
Platoon formation on an automated highway
• Benefits
• increased safety
• increased highway capacity
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Our 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
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Traffic 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
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Passing 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
W (vi  vc )  Q(vi  vc  vave Ri )
• Q(x) is a step function
• R is the same for all agents
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Platoon Formation
v1
vC
vC
v2
vC
vC
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v2
K. Lerman
Stochastic Model of Platoon Formation
MAS 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.
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Physics-Based Models of Traffic Flow
• 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
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Some Definitions
Density of platoons of size m, velocity v
Pm (v, t )
Initial conditions: Pm (v,0)   m,1P0 (v)
where P0(v) is the initial distribution of car velocities
Car joins platoon at rate
U (v, v)  v  v(1  W v  v)
for v>v’
Individual cars enter and leave highway at rate g
USC Information Sciences Institute
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K. Lerman
Stochastic Model of Platoon Formation
Master Equation for Platoon Formation
loss due to collisions
merging of smaller platoons


Pm (v)
  Pm (v)  dvPk (v)U    dvPk (v) Pj (v)U
t
k 0
k  j m v
 g (m  1) Pm1 (v)  mPm (v)  gP0 (v) m,1 m  1,2,...
outflow of cars
inflow of cars
Inflow and outflow drive the system into a steady state
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K. Lerman
Stochastic Model of Platoon Formation
Average Platoon Size
g  103
R  0 .3
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ISI
K. Lerman
Stochastic Model of Platoon Formation
Platoon Size Distribution
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K. Lerman
Stochastic Model of Platoon Formation
Steady State Car Velocity Distribution
P0 (v)
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation
Conclusion
• Platoons form through simple local interactions
• Stochastic Master Equation describes the time
evolution of the platoon distribution function
• Study platoon formation mathematically
But,
• Does not take into account spatial
inhomogeneities
• Need a more realistic passing mechanism
• effect of the passing lane
USC Information Sciences Institute
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K. Lerman
Stochastic Model of Platoon Formation
Future work
• Multi-lane model
• for each lane i, Pmi(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
USC Information Sciences Institute
ISI
K. Lerman
Stochastic Model of Platoon Formation