Polling Systems with Two-Phase Gated Service: Heavy Traffic Results for the Waiting Time Distribution

We study an asymmetric cyclic polling system with Poisson arrivals, general service-time and switch-over time distributions, and with so-called two-phase gated service at each queue, an interleaving scheme that aims to enforce some level of "fairness" among the different customer classes. For this model, we use the classical theory of multi-type branching processes (MTBPs) to derive closed-form expressions for the Laplace-Stieltjes Transform (LST) of the waiting-time distributions when the load tends to 1, in a general parameter setting and under proper heavy-traffic (HT) scalings. This result is strikingly simple and provides new insights in the behavior of two-phase polling systems. In particular, the result provides insight in the waiting-time performance, and the tradeoff between efficiency and fairness of two-phase gated polling compared to the classical one-phase gated service policy.