Optimality of periodic control for fluid models of polling systems with setups*

A generic time-invariant polling system is modeled as a deterministic hybrid dynamical system in the form of a fluid fabrication facility. The system consists of finitely many infinite size buffers receiving constant-rate inflows from outside the system and a single source of service (finite capacity server). The server can withdraw the buffer’s content but is able to serve at most one buffer at a time, and has to switch among them from time to time, with any switch consuming a nonzero switch-over period. With respect to the long-run maximal scaled wip (work in progress) performance metric, sufficiency of periodic control protocols is established: It is shown that the deepest optimum (that is over all processes in the system, irrespective of their genesis, initial state, and features) is furnished by such a protocol with as high precision as desired. Moreover, sufficiency of special periodic control protocols is established. These protocols prescribe to serve any buffer at the maximal rate until its size reduces to a pre-specified percent of its initial size at the beginning of the visit. This percent is set to be zero for any buffer such that its service at the maximal rate implies reduction of the scaled wip. This is related to another result of the paper that shows optimality of the exhaustive policy for such buffers.