  We report here our preliminary results in the study of a new version of the
generalized Totally Asymmetric Simple Exclusion process (gTASEP) on open
tracks. In the gTASEP an additional interaction between the particles is
considered, besides the hard-core exclusion, which exists in the standard
TASEP. It is modeled by the introduction of a second hopping probability $p_m$
for the particles, belonging to the same cluster, in addition to the standard
hopping probability p, which now applies only for single particles and the head
(rightmost) particle of a cluster. We briefly describe how one can obtain
analytically solvable version of gTASEP by modifying the left (injection)
boundary condition. A short comparison is made with the previously studied
version of gTASEP on open tracks.
