  We study a model of aggregation and fragmentation of clusters of particles on
an open segment of a single-lane road. The particles and clusters obey the
stochastic discrete-time discrete-space kinetics of the Totally Asymmetric
Simple Exclusion Process (TASEP) with backward ordered sequential update
(dynamics), endowed with two hopping probabilities, p and pm. The second
modified probability, pm, models a special kinematic interaction between the
particles belonging to the same cluster. This modification is called
generalized TASEP (gTASEP) since it contains as special cases TASEP with
parallel update and TASEP with backward ordered sequential update for specific
values of the second hopping probability pm. We focus here on exemplifying the
effect of the additional attraction interaction on the system properties in the
non-equilibrium steady state. We estimate various physical quantities (bulk
density, density distribution, and the current) in the system and how they
change with the increase of pm (p < pm<1). Within a random walk theory we
consider the evolution of the gaps under different boundary conditions and
present space-time plots generated by MC simulations, illustrating the
applicability of the random walk theory for the study of gTASEP.
