Towards a unifying theory on branching-type polling systems in heavy traffic

Abstract For a broad class of polling models the evolution of the system at specific embedded polling instants is known to constitute a multi-type branching process (MTBP) with immigration. In this paper it is shown that for this class of polling models the vector that describes the state of the system at these polling instants, say X=(X1,…,XM), satisfies the following heavy-traffic behavior (under mild assumptions): 1$$\label{eq01}(1-\rho)\underline{X}\rightarrow_{d}\underline{\gamma}~\Gamma(\alpha,\mu)\quad (\rho\uparrow1),$$where γ is a known M-dimensional vector, Γ(α,μ) has a gamma-distribution with known parameters α and μ, and where ρ is the load of the system. This general and powerful result is shown to lead to exact—and in many cases even closed-form—expressions for the Laplace-Stieltjes Transform (LST) of the complete asymptotic queue-length and waiting-time distributions for a broad class of branching-type polling models that includes many well-studied polling models policies as special cases. The results generalize and unify many known results on the waiting times in polling systems in heavy traffic, and moreover, lead to new exact results for classical polling models that have not been observed before. To demonstrate the usefulness of the results, we derive closed-form expressions for the LST of the waiting-time distributions for models with cyclic globally-gated polling regimes, and for cyclic polling models with general branching-type service policies. As a by-product, our results lead to a number of asymptotic insensitivity properties, providing new fundamental insights in the behavior of polling models.