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We consider random rectangles in R 2 that are distributed according to a Poisson random measure, i.e., independently and uniformly scattered in the plane. The distributions of the length and the width of the rectangles are heavy-tailed with different parameters. We investigate the scaling behaviour of the related random fields as the intensity of the random measure grows to infinity while the mean edge lengths tend to zero. We characterise the arising scaling regimes, identify the limiting random fields and give statistical properties of these limits.2010 Mathematics Subject Classification: 60G60 (primary); 60F05, 60G55 (secondary).Keywords: Gaussian random field, generalised random field, Poisson point process, Poisson random field, random balls model, random grain model, random field, stable random field.Since our purpose is to deal with centred random fields, we introduce the notation for the corresponding centred Poisson random measure N := N − n and centred integral J(µ) := J(µ) − EJ(µ).The goal of this paper is to obtain scaling limits for the random field J. By scaling, we mean that the length and the width of the boxes are shrinking to zero, i.e., the scaled edge lengths are ρu i with scaling parameter ρ → 0, and that the expected number of boxes is increasing, i.e., the intensity λ of the Poisson point process is tending to infinity as a function of ρ. The precise behaviour of λ = λ(ρ) → ∞ is specified in the different scaling regimes below. Following the notational convention from above, we denote by J ρ the centred random field corresponding to the Poisson random measure N ρ with the modified intensity λ ρ := λ(ρ) and scaled edge lengths, i.e., F ρ is the image measure of F by the change u → ρu.Next, we want to say a few words about the applications of random balls models. The motivation comes from models from telecommunication networks. A list of some references can be found at the beginning of Chapter 3

We consider polling models in the sense of Takagi [18]. In our case, the feature of the server is that it may be forced to wait idly for new messages at an empty queue instead of switching to the next station. We propose four different wait-and-see strategies that govern these waiting periods. We assume Poisson arrivals for new messages and allow general service and switchover time distributions. The results are formulas for the mean average queueing delay and characterisations of the cases where the wait-and-see strategies yield a lower delay compared to the exhaustive strategy.2010 Mathematics Subject Classification: 60K25 (primary); 90B22, 68M20 (secondary).

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