  In a recent study [C Arita and D Yanagisawa: J. Stat. Phys. 141, 829 (2010)]
the stationary state of a parallel-update TASEP with varying system length,
which can be regarded as a queueing process with excluded-volume effect
(exclusive queueing process, EQP), was obtained. We analyze the dynamical
properties of the number of particles $< N_t>$ and the position of the last
particle (the system length) $< L_t>$, using an analytical method (generating
function technique) as well as a phenomenological description based on domain
wall dynamics and Monte Carlo simulations. The system exhibits two phases
corresponding to linear convergence or divergence of $< N_t>$ and $< L_t>$.
These phases can both further be subdivided into high-density and
maximal-current subphases. The predictions of the domain wall theory are found
to be in very good agreement quantitively with results from Monte Carlo
simulations in the convergent phase. On the other hand, in the divergent phase,
only the prediction for $< N_t>$ agrees with simulations.
