  We address the question of the time needed by $N$ particles, initially
located on the first sites of a finite 1D 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 $N \ll L$, the mean
exit time of the particles is asymptotically given by $T_N(L) \sim L+\beta_N
\sqrt{L}$ for large lattices. Building upon exact results obtained for 2
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 $\beta_N$. 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 $L\rightarrow \infty$. This complements previous asymptotic
results obtained by Johansson in \cite{Johansson2000} in the limit where both
$N$ and $L$ tend to infinity while keeping the particle density $N/L$ finite.
