The effect of incorporating a priority scheme into a data flow computer is studied in this paper. Specifically, we deal with the scheduling of instructions in a data flow program, and the mechanisms by which such scheduling may be implemented within a data flow computer. We show that the assignment of priorities to data flow operations is a special case of a problem in scheduling theory, and also belongs to the NP-complete class of problems. Therefore, we develop a heuristic approach, based on the well-known Critical Path algorithm, as a basis for determining instruction priorities. Our conclusions, based on the simulation of programs executed in a modified data flow computer, show that adding a priority mechanism is not justifiable in the general case. This is due mostly to the inability to reach the potential improvement offered by scheduling operations, because of implementation restrictions. Nevertheless, certain algorithms (e. g., DFT) can still benefit from the proposed scheme, mainly because of their highly regular, static structure.