  We investigate a non-equilibrium one-dimensional model known as the raise and
peel model describing a growing surface which grows locally and has non-local
desorption. For specific values of adsorption ($u_a$) and desorption($u_d$)
rates the model shows interesting features. At $u_a = u_d$, the model is
described by a conformal field theory (with conformal charge $c=0$) and its
stationary probability canbe mapped to the ground state of the XXZ quantum
chain. Moreover, for $u_a \geq u_d$, the model shows a phase in which the the
avalanche distribution is scale invariant. In this work we study the surface
dynamics by looking at avalanche distributions using Finite-size Scaling
formalism and explore the effect of adding a wall to the model. The model shows
the same universality for the cases with and without a wall for an odd number
of tiles removed, but we find a new exponent in the presence of a wall for an
even number of avalanches released. We provide new conjecture for the
probability distribution of avalanches with a wall obtained by using exact
diagonalization of small lattices and Monte-Carlo simulations.
