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A Stochastic Model of Platoon Formation in Traffic Flow USC/Information Sciences Institute K. Lerman 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 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 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 Traffic as a MAS • Car = agent • • velocity vi drawn from a velocity distribution P 0(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), 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 • 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 v 1 v. C v 2 v. C USC Information Sciences Institute Platoon Formation v 1 v. C v 2 v. C USC Information Sciences Institute ISI v 2 K. Lerman Stochastic Model of Platoon Formation

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

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

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

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

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

Future work • Multi-lane model • • • for each lane i, Pmi(v, t) 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