Abstract

We will analyze the asymptotic properties of an Euclidean optimization problem on the plane. Specifically, we consider a network with 3 bins and n objects spatially uniformly distributed, each object being allocated to a bin at a cost depending on its position. Two allocations are considered: the allocation minimizing the bin loads and the allocation allocating each object to its less costly bin. This model is motivated by issues in wireless cellular networks. We will aim at the asymptotic properties of these allocations as the number of objects grows to infinity. Using the symmetries of the problem, we will derive a law of large numbers, a central limit theorem and a large deviation principle for both loads with explicit expressions. In particular, we prove that the two allocations satisfy the same law of large numbers, but they do not have the same asymptotic fluctuations and rate functions.