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Abstract. We obtain a complete description of anisotropic scaling limits and the existence of

We obtain a complete description of anisotropic scaling limits of random grain model on the plane with heavy tailed grain area distribution. The scaling limits have either independent or completely dependent increments along one or both coordinate axes and include stable, Gaussian and some 'intermediate' infinitely divisible random fields. Asymptotic form of the covariance function of the random grain model is obtained.Application to superposed network traffic is included.