## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 20, Number 6 (2010), 2040-2085.

### Load optimization in a planar network

Charles Bordenave and Giovanni Luca Torrisi

#### Abstract

We analyze the asymptotic properties of a Euclidean optimization problem on the plane. Specifically, we consider a network with three 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. We analyze the asymptotic properties of these allocations as the number of objects grows to infinity. Using the symmetries of the problem, we 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.

#### Article information

**Source**

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2040-2085.

**Dates**

First available in Project Euclid: 19 October 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1287494554

**Digital Object Identifier**

doi:10.1214/09-AAP676

**Mathematical Reviews number (MathSciNet)**

MR2759728

**Zentralblatt MATH identifier**

1205.60042

**Subjects**

Primary: 60F05: Central limit and other weak theorems 60F10: Large deviations

Secondary: 90B18: Communication networks [See also 68M10, 94A05] 90C27: Combinatorial optimization

**Keywords**

Euclidean optimization law of large numbers central limit theorem large deviations calculus of variations wireless networks

#### Citation

Bordenave, Charles; Torrisi, Giovanni Luca. Load optimization in a planar network. Ann. Appl. Probab. 20 (2010), no. 6, 2040--2085. doi:10.1214/09-AAP676. https://projecteuclid.org/euclid.aoap/1287494554