1. 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

2. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Traffic on Automated Highways Benefits increased safety increased highway capacity Ordinary highway Platoon formation on an automated highway

3. ISI USC Information Sciences Institute 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

4. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Traffic as a MAS Car = agent velocity v i drawn from a velocity distribution P 0 (v) risk factor R i : 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

5. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Passing Rules When a fast car (velocity v i ) approaches a platoon (velocity v c ), 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

6. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Platoon Formation v2v2 vCvC vCvC v1v1 v2v2 vCvC vCvC

7. ISI USC Information Sciences Institute 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.

8. ISI USC Information Sciences Institute 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

9. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Some Definitions Density of platoons of size m, velocity v Car joins platoon at rate Initial conditions: where P 0 (v) is the initial distribution of car velocities Individual cars enter and leave highway at rate  for v>v’

10. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Master Equation for Platoon Formation loss due to collisionsmerging of smaller platoons outflow of carsinflow of cars Inflow and outflow drive the system into a steady state

11. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Average Platoon Size

12. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Platoon Size Distribution

13. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Steady State Car Velocity Distribution

14. ISI USC Information Sciences Institute 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

15. ISI USC Information Sciences Institute K. Lerman Stochastic Model of Platoon Formation Future work Multi-lane model for each lane i, P m 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