  Two cellular automata models with directed mass flow and internal time scales
are studied by numerical simulations. Relaxation rules are a combination of
probabilistic critical height (probability of toppling $p$) and deterministic
critical slope processes with internal correlation time $t_c$ equal to the
avalanche lifetime, in Model A, and $t_c\equiv 1$, in Model B. In both cases
nonuniversal scaling properties of avalanche distributions are found for $p\ge
p^\star $, where $p^\star$ is related to directed percolation threshold in
$d=3$. Distributions of avalanche durations for $p\ge p^\star $ are studied in
detail, exhibiting multifractal scaling behavior in model A, and finite size
scaling behavior in model B, and scaling exponents are determined as a function
of $p$. At $p=p^\star$ a phase transition to noncritical steady state occurs.
Due to difference in the relaxation mechanisms, avalanche statistics at
$p^\star$ approaches the parity conserving universality class in Model A, and
the mean-field universality class in Model B. We also estimate roughness
exponent at the transition.
