Exit times of totally asymmetric simple exclusion processes

We address the question of the time needed by N particles, initially located on the first sites of a finite one-dimensional lattice of size L, to exit that lattice when they move according to a TASEP transport model. Using analytical calculations and numerical simulations, we show that when N≪L, the mean exit time of the particles is asymptotically given by T_{N}(L)∼L+β_{N}sqrt[L] for large lattices. Building upon exact results obtained for two particles, we devise an approximate continuous space and time description of the random motion of the particles that provides an analytical recursive relation for the coefficients β_{N}. The results are shown to be in very good agreement with numerical results. This approach sheds some light on the exit dynamics of N particles in the regime where N is finite while the lattice size L→∞. This complements previous asymptotic results obtained by Johansson [Commun. Math. Phys. 209, 437 (2000)0010-361610.1007/s002200050027] in the limit where both N and L tend to infinity while keeping the particle density N/L finite.