  As the simplest model of interacting-particle transport in a randomly
disordered medium, we consider the asymmetric simple exclusion process (ASEP)
in which particles with hard-core interactions perform biased random walks, on
the supercritical percolation cluster. In this process, the long time
trajectory of a marked particle consists of steps on the backbone, punctuated
by time spent in side-branches. We study the probability distribution in the
steady state of the waiting time $T_w$ of a randomly chosen particle, in a
side-branch since its last step along the backbone. Exact numerical evaluation
of this on a single side-branch of lengths $L= 1$ to $9$ show that for large
fields, the probability distribution of $\log T_w $ has multiple well separated
peaks. We extend this result to a random comb, and to the ASEP on the
percolation cluster. For the latter problem, we show that in the steady state,
the fractional number of particles that have been in the same side-branch for a
time interval greater than $T_w$ varies as $exp( - c \sqrt{ln T_w})$ for large
$T_w$, where $c$ depends only on the bias field. The system shows dynamical
heterogeneity, with particles segregating into pockets of high and low
mobilities. These long timescales are not reflected in the eigenvalue spectrum
of the Markov evolution matrix.
