  We use a discrete-time formulation to study the asymmetric avalanche process
[Phys. Rev. Lett. vol. 87, 084301 (2001)] on a finite ring and obtain an exact
expression for the average avalanche size of particles as a function of
toppling probabilities depending on parameters $\mu$ and $\alpha$. By mapping
the model below and above the critical line onto driven interface problems, we
show how different regimes of avalanches may lead to different types of
critical interface behavior characterized by either annealed or quenched
disorders and obtain exactly the related critical exponents which violate a
well-known scaling relation when $\alpha \ne 2$.
