  In this paper, we study a two-lane totally asymmetric simple exclusion
process (TASEP) coupled with random attachment and detachment of particles
(Langmuir kinetics) in both lanes under open boundary conditions. Our model can
describe the directed motion of molecular motors, attachment and detachment of
motors, and free inter-lane transition of motors between filaments. In this
paper, we focus on some finite-size effects of the system because normally the
sizes of most real systems are finite and small (e.g., size $\leq 10,000$). A
special finite-size effect of the two-lane system has been observed, which is
that the density wall moves left first and then move towards the right with the
increase of the lane-changing rate. We called it the jumping effect. We find
that increasing attachment and detachment rates will weaken the jumping effect.
We also confirmed that when the size of the two-lane system is large enough,
the jumping effect disappears, and the two-lane system has a similar density
profile to a single-lane TASEP coupled with Langmuir kinetics. Increasing
lane-changing rates has little effect on density and current after the density
reaches maximum. Also, lane-changing rate has no effect on density profiles of
a two-lane TASEP coupled with Langmuir kinetics at a large
attachment/detachment rate and/or a large system size. Mean-field approximation
is presented and it agrees with our Monte Carlo simulations.
