  We propose a modification to the study of site-wise dynamically disordered
totally asymmetric simple exclusion process (TASEP). Motivated by the process
of gene transcription, a study in ref. \cite{waclaw2019totally} introduced an
extension of TASEP, where the defects (or obstacles) bind/un-bind dynamically
to the sites of the lattice and the hopping of the particles on lattice faces a
hindrance if the arrival site is occupied by an obstacle. In addition, the
particle is only allowed to enter the lattice provided the first site is
defect-free. In our study, we propose that the particle movement at the entry
of the lattice must face an equal hindrance that is provided by the obstacles
to the rest of the particles on the lattice. For open boundaries, the continuum
mean-field equations are derived and solved numerically to obtain steady-state
phase diagrams and density profiles. The presence of obstacles produces a shift
in the phase boundaries obtained but the same three phases as obtained for the
standard TASEP. Contrary to the model introduced in ref.
\cite{waclaw2019totally}, the idea to introduce the modification at the
entrance shows that the limiting case $p_d \rightarrow 1$ converges to the
standard TASEP, where $p_d$ refers to the affected hopping rate due to presence
of obstacle. The mean-field solutions are validated using extensive Monte Carlo
simulations.
