Multiclass queueing systems. Polymatroidal structure and optimal scheduling control

In many multiclass queueing systems, certain performance measures of interest satisfy strong conservation laws. That is, the total performance over all job types is invariant under any nonidling service control rule, and the total performance over any subset (say A) of job types is minimized or maximized by offering absolute priority to the types in A over all other types. We develop a formal definition of strong conservation laws, and show that as a necessary consequence of these strong conservation laws, the state space of the performance vector is a (base of a) polymatroid. From known results in polymatroidal theory, the vertices of this polyhedron are easily identified, and these vertices correspond to absolute priority rules. A wide variety of multiclass queueing systems are shown to have this polymatroidal structure, which greatly facilitates the study of the optimal scheduling control of such systems. When the defining set function of the performance space belongs to the class of generalized symmetric functions, additional applications are discussed, which include establishing convexity and concavity properties in various queueing systems.