  We present a numerical study of a two-lane version of the stochastic
non-equilibrium model known as the totally asymmetric simple exclusion process.
For such a system with open boundaries, and suitably chosen values of
externally-imposed particle injection ($\alpha$) and ejection ($\beta$) rates,
spontaneous symmetry breaking can occur. We investigate the statistics and
internal structure of the stochastically-induced transitions, or "flips", which
occur between opposite broken-symmetry states as the system evolves in time.
From the distribution of time intervals separating successive flips, we show
that the evolution of the associated characteristic times against
externally-imposed rates yields information regarding the proximity to a
critical point in parameter space. On short time scales, we probe for the
possible existence of precursor events to a flip between opposite
broken-symmetry states. We study an adaptation of domain-wall theory to mimic
the density reversal process associated with a flip.
