Graph searching games involve a team of searchers that aims at capturing a fugitive in a graph. These games have been widely studied for their relationships with the tree-and the path-decomposition of graphs. In order to define de-compositions for directed graphs, similar games have been proposed in directed graphs. In this paper, we consider a game that has been defined and studied in the context of routing reconfiguration problems in WDM networks. Namely, in the processing game, the fugitive is invisible, arbitrarily fast, it moves in the opposite direction of the arcs of a digraph, but only as long as it can access to a strongly connected component free of searchers. We prove that the processing game is monotone which leads to its equivalence with a new digraph decomposition.