  We define and study one-dimensional model of irreversible aggregation of
particles obeying a discrete-time kinetics which is a special limit of the
generalized Totally Asymmetric Simple Exclusion Process (gTASEP) on open
chains. The model allows for clusters of particles to translate as a whole
entity one site to the right with the same probability as single particles do.
A particle and a cluster, as well as two clusters, irreversibly aggregate
whenever they become nearest neighbors. Nonequilibrium stationary phases appear
under the balance of injection and ejection of particles. By extensive Monte
Carlo simulations it is established that the phase diagram in the plane of the
injection-ejection probabilities consists of three stationary phases: a
multi-particle (MP) one, a completely filled (CF) phase and a 'mixed' (MP+CF)
one. The transitions between these phases are: an unusual transition between MP
and CF with jump discontinuity in both the bulk density and the current, a
conventional first-order transition with a jump in the bulk density between MP
and MP+CF, and a continuous clustering-type transition from MP to CF, which
takes place throughout the MP+CF phase between them. By the data collapse
method a finite-size scaling function for the current and bulk density is
obtained near the unusual phase transition line. A diverging correlation length
associated with that transition is identified and interpreted as the size of
the largest cluster. The model allows for a future extension to account for
possible cluster fragmentation.
