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We present an algorithm to compute the rank of a configuration of thesandpile model for the complete bipartite graph K_{m,n} of complexity O(m+n).Furthermore, we provide a formula for the generating function of parking sortedconfigurations on complete bipartite graphs K_{m,n} according to rank, degree,and the sizes m and n.  The results in the present paper are similar to those found by Robert Coriand the second named author for the complete graph K_{n+1}, and they rely onthe analysis of certain operators on the stable sorted configurations ofK_{m,n} developed in a previous work by the authors together withJean-Christophe Aval and Mark Dukes.

The sandpile model on K_{m,n} and the rank of its configurations