Polling Models with Two-Stage Gated Service: Fairness Versus Efficiency

We consider an asymmetric cyclic polling system with general service-time and switchover time distributions with so-called two-stage gated service at each queue, an interleaving scheme that aims to enforce fairness among the different customer classes. For this model, we (1) obtain a pseudo-conservation law, (2) describe how the mean delay at each of the queues can be obtained recursively via the so-called Descendant Set Approach, and (3) present a closed-form expression for the expected delay at each of the queues when the load tends to unity (under proper heavy-traffic scalings), which is the main result of this paper. The results are strikingly simple and provide new insights into the behavior of two-stage polling systems, including several insensitivity properties of the asymptotic expected delay with respect to the system parameters. Moreover, the results provide insight in the delay-performance of two-stage gated polling compared to the classical one-stage gated service policies. The results show that the two-stage gated service policy indeed leads to better fairness compared to onestage gated service, at the expense of a decrease in efficiency. Finally, the results also suggest simple and fast approximations for the expected delay in stable polling systems. Numerical experiments demonstrate that the approximations are highly accurate for moderately and heavily loaded systems.