Distribution of the delay in polling systems in heavy traffic

We consider asymmetric cyclic polling systems with an arbitrary number of queues, with general mixtures of exhaustive and gated service and with generally distributed service-times and switch-over times, in heavy traffic. We derive closed-form expressions for the Laplace–Stieltjes transform (LST) of the steady-state delay incurred at each of the queues, under standard heavy-traffic scalings. The expressions give an explicit characterization of the complete (scaled) waiting-time distributions at each of the queues. The results are strikingly simple and provide a variety of new insights into the behavior of heavily loaded polling systems. In addition, the results lead to simple and fast-to-evaluate approximations for the waiting-time distributions in stable polling systems that are close to saturation. Numerical results demonstrate that the approximations are highly accurate in many practical heavy-traffic scenarios.