Asymptotic Optimality of the Binomial-Exhaustive Policy for Polling Systems with Large Switchover Times

TL;DR

This paper establishes that a binomial-exhaustive policy is asymptotically optimal for polling systems with large switchover times by solving a fluid model and translating its solution to the stochastic system.

Contribution

It introduces an asymptotically optimal control policy for polling systems with large switchover times, based on a fluid model and a novel translation method.

Findings

The binomial-exhaustive policy is asymptotically optimal under large switchover times.

The UI condition holds for systems with linear or polynomial growth costs and finite moments.

The approach bridges fluid models and stochastic control for complex queueing networks.

Abstract

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.