  We study an one-dimensional stochastic model of vehicular traffic on open
segments of a single-lane road of finite size $L$. The vehicles obey a
stochastic discrete-time dynamics which is a limiting case of the generalized
Totally Asymmetric Simple Exclusion Process. This dynamics has been previously
used by Bunzarova and Pesheva [Phys. Rev. E 95, 052105 (2017)] for an
one-dimensional model of irreversible aggregation. The model was shown to have
three stationary phases: a many-particle one, MP, a phase with completely
filled configuration, CF, and a boundary perturbed MP+CF phase, depending on
the values of the particle injection ($\alpha$), ejection ($\beta$) and hopping
($p$) probabilities.
  Here we extend the results for the stationary properties of the MP+CF phase,
by deriving exact expressions for the local density at the first site of the
chain and the probability P(1) of a completely jammed configuration. The
unusual phase transition, characterized by jumps in both the bulk density and
the current (in the thermodynamic limit), as $\alpha$ crosses the boundary
$\alpha =p$ from the MP to the CF phase, is explained by the finite-size
behavior of P(1). By using a random walk theory, we find that, when $\alpha$
approaches from below the boundary $\alpha =p$, three different regimes appear,
as the size $L\rightarrow \infty$: (i) the lifetime of the gap between the
rightmost clusters is of the order $O(L)$ in the MP phase; (ii) small jams,
separated by gaps with lifetime $O(1)$, exist in the MP+CF phase close to the
left chain boundary; and (iii) when $\beta =p$, the jams are divided by gaps
with lifetime of the order $O(L^{1/2})$. These results are supported by
extensive Monte Carlo calculations.
