  Abelian sandpile models, both deterministic, such as the Bak, Tang,
Wiesenfeld (BTW) model [P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett.
{\bf 59}, 381 (1987)], and stochastic, such as the Manna model [S.S. Manna, J.
Phys. A {\bf 24}, L363 (1991)], are studied on finite square lattices with open
boundaries. The avalanche size distribution $P_L(n)$ is calculated for a range
of system sizes, $L$. The first few moments of this distribution are evaluated
numerically and their dependence on the system size is examined. The sandpile
models are conservative in the sense that grains are conserved in the bulk and
can leave the system only through the boundaries. It is shown that the
conservation law provides an interesting connection between sandpile models and
random walk models. Using this connection, it is shown that the average
avalanche sizes, $<n >_L$, for the BTW and the Manna models are equal to each
other, and both are equal to the average path-length of a random walker
starting from a random initial site on the same lattice of size $L$. This is in
spite of the fact that sandpile models with deterministic (BTW) and stochastic
(Manna) toppling rules exhibit different critical exponents, indicating that
they belong to different universality classes.
