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We study an one-dimensional stochastic model of vehicular traffic on opensegments of a single-lane road of finite size $L$. The vehicles obey astochastic discrete-time dynamics which is a limiting case of the generalizedTotally Asymmetric Simple Exclusion Process. This dynamics has been previouslyused by Bunzarova and Pesheva [Phys. Rev. E 95, 052105 (2017)] for anone-dimensional model of irreversible aggregation. The model was shown to havethree stationary phases: a many-particle one, MP, a phase with completelyfilled configuration, CF, and a boundary perturbed MP+CF phase, depending onthe 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 thechain and the probability P(1) of a completely jammed configuration. Theunusual phase transition, characterized by jumps in both the bulk density andthe current (in the thermodynamic limit), as $\alpha$ crosses the boundary$\alpha =p$ from the MP to the CF phase, is explained by the finite-sizebehavior 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 therightmost 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 theleft chain boundary; and (iii) when $\beta =p$, the jams are divided by gapswith lifetime of the order $O(L^{1/2})$. These results are supported byextensive Monte Carlo calculations.

A model of irreversible jam formation in dense traffic