Polling systems with periodic server routing in heavy traffic

We consider polling systems with mixtures of exhaustive and gated service, and in which the server visits the queues according to a general polling table, in heavy traffic. We derive exact expressions for the expected delay at each of the queues (under heavy-traffic scalings), requiring the solution of a set of M – N linear equations, where M is the length of the polling table and N is the number of queues. The results lead to closed-form expressions for the scaled expected delay under the commonly used star and elevator routing schemes, in a general parameter setting. In addition, the results reveal several insensitivity properties of the scaled expected delay with respect to the system parameters, providing new insights into the behavior of periodic polling systems in heavy traffic. The results also suggest simple and fast-to-evaluate approximations for the expected delay at each of the queues in stable polling systems. Numerical examples show that the approximations are very accurate in practical heavy-traffic scenarios.