Routing Problems on Networks

We consider the problem of optimizing a polling system, i.e., of optimally sequencing a server in a multi-class queueing system with switch-over times in order to minimize a linear objective function of the waiting times. The problem has important applications in computer, communication, production and transportation networks. We propose nonlinear programming relaxations that provide strong lower bounds to the optimal cost for all static policies. We also obtain lower bounds for dynamic policies as well, which are primarily useful under light traffic conditions and/or small switch-over times. We conjecture that the lower bounds developed in this paper for the class of static policies are also valid for dynamic policies under heavy traffic conditions. We use the information from the lower bound and integer programming techniques to construct static policies that are very close (0-3%) to the lower bounds. We compare numerically our proposed policies with static policies proposed in the literature as well as with dynamic policies and find that the policies we propose outperform all static policies proposed in the literature and at least in heavier traffic outperform dynamic policies as well.