  A two-dimensional directed stochastic sandpile model is studied analytically
with the use of directed Abelian algebras recently introduced by Alcaraz and V.
Rittenberg [Phys. Rev. E {\bf 78}, 041126 (2008)]. Exact expressions for the
probabilities of all possible toppling events which follow the transfer of
arbitrary number of particles to a site in the stationary configuration are
derived. A description of the virtual-time evolution of directed avalanches on
two dimensional lattices is suggested. Due to intractability of the general
problem, the algebraic approach is applied only to the solution of the special
cases of directed deterministic avalanches and trivial stochastic avalanches
describing simple random walks of two particles. The study of these cases has
clarified the role of each particular kind of toppling in the process of
avalanche growth. In the general case of the quadratic directed algebra we have
determined exactly the maximum possible values of: (1) the current of particles
at any given moment of virtual time and (2) the occupation number (`height') of
each site at any moment of time.
