Slow-to-start traffic models, coalescing Brownian motions and M/M/1 queues
Seminar Room 1, Newton Institute
"Cars" are points in the real line that move to the left and have two possible velocities: 0 or 1. Cars maintain order, so that when a moving car is blocked by a stopped car, it has to stop too. Unblocked cars with zero velocity wait a random time with exponential distribution to adquire velocity one (this is the slow-to-start rule). The initial distribution of cars is a Poisson process of parameter $\l$. There are three regimes: supercritical (\l>1) where cars condensate in a random set of initial positions; critical (\l = 1) when there is condensation but the asymptotic speed of the cars is 1, and subcritical (\l<1) where each car get eventually unblocked. The rescaling of this process in the critical and supercritical regimes is related with coalescing Brownian motions. In the subcritical regime the final relative positions of the cars is given by a Poisson process. In this case the car trajectories can be mapped to the loading process of a M/M/1 queue. Joint work with Leo Rolla, Fredy Caceres and Eugene Pechersky.