  We study an optimal-control problem of polling systems with large switchover
times, when a holding cost is incurred on the queues. In particular, we
consider a stochastic network with a single server that switches between
several buffers (queues) according to a pre-specified order, assuming that the
switchover times between the queues are large relative to the processing times
of individual jobs. Due to its complexity, computing an optimal control for
such a system is prohibitive, and so we instead search for an asymptotically
optimal control. To this end, we first solve an optimal control problem for a
deterministic relaxation (namely, for a fluid model), that is represented as a
hybrid dynamical system. We then "translate" the solution to that fluid problem
to a binomial-exhaustive policy for the underlying stochastic system, and prove
that this policy is asymptotically optimal in a large-switchover-time scaling
regime, provided a certain uniform integrability (UI) condition holds. Finally,
we demonstrate that the aforementioned UI condition holds in the following
cases: (i) the holding cost has (at most) linear growth, and all service times
have finite second moments; (ii) the holding cost grows at most at a polynomial
rate (of any degree), and the service-time distributions possess finite moment
generating functions.
