  We study a totally asymmetric simple exclusion process (TASEP) with one
defect site, hopping rate $q<1$, near the system boundary. Regarding our system
as a pair of uniform TASEP's coupled through the defect, we study various
methods to match a \emph{finite} TASEP and an \emph{infinite} one across a
common boundary. Several approximation schemes are investigated. Utilizing the
finite segment mean-field (FSMF) method, we set up a framework for computing
the steady state current $J$ as a function of the entry rate $% \alpha $ and
$q$. For the case where the defect is located at the entry site, we obtain an
analytical expression for $J(\alpha, q) $ which is in good agreement with Monte
Carlo simulation results. When the defect is located deeper in the bulk, we
refined the scheme of MacDonald, et.al. [Biopolymers, \textbf{6}, 1 (1968)] and
find reasonably good fits to the density profiles before the defect site. We
discuss the strengths and limitations of each method, as well as possible
avenues for further studies.
