# Optimal Use of Cellphone Frequencies with Robust Graph Coloring

By Lara-Velázquez, P.; Gallardo-López, L. et al. | Fuzzy Economic Review, November 2012 | Go to article overview

# Optimal Use of Cellphone Frequencies with Robust Graph Coloring

Lara-Velázquez, P., Gallardo-López, L., de-los-Cobos-Silva, S. G., Gutiérrez-Andrade, M. Á., Rincón-García, E. A., Fuzzy Economic Review

The Robust Coloring Problem (RCP) is an NP-Hard problem for which had been developed many applications. In this work a Model based in Robust Coloring Problem is developed for the frequency assignment in cellphones that increases 25% the capacity installed nowadays. The model is tested with lifelike instances and is shown that solves efficiently the problem in less than a second.

Keywords: Robust Coloring Problem, heuristics, cellphone technology

JEL Classification: C02, C61, C65

(ProQuest: ... denotes formulae omitted.)

1. INTRODUCTION

Mexico City is the most densely populated region in the whole country, averaging 6,000 people per square kilometer according with information given by the local government, all of it is covered by cellphone services, including the subway system. If we consider that a cell covers 26 km2, each cell serves around 156,000 people approximately, although it should be mentioned that the installed capacity by companies is usually about 10%. Considering that every inhabitant has one celphone, we would require 15,600 frequencies per cell approximately; this value represents the marginal demand for using a cellular base, in a not too distant future. This value does not consider cases where a user has more than one cell in use at the same time, considering it unlikely.

In 1995, a Dynamic Assignment model that could increase the capacity of the base stations was proposed [3], this model borrows frequencies from neighboring bases when their capacity is outstripped by demand. This approach causes that the administration at the base turns really complex because of the number of frequencies involved, because the communication between bases is increased and therefore the number of frequencies involved. The model here proposed maintains the administration within the cell itself, which simplifies this problem considerably, and retains the same improvement in capacity. With this in mind we propose a model based on Robust Coloring Problem considering some properties of the base stations: 1) its geometry, 2) the distance of a user with respect to the base, 3) the threshold or range of the signal of a frequency, and 4) the distance between pairs of users. From this model, we designed a greedy algorithm that gets results equivalent to those provided by a generic algorithm, in considerably less time.

This work is organized as follows: Section 2 presents a brief introduction to the current system of frequency assignment in cellphones. Section 3 explains the Robust Coloring Problem. In Section 4 we review previous works that increase capacity in terms of number of users in saturated bases, including the aforementioned Dynamic Assignment. In Section 5 we review the Robust Coloring model proposed for the cellphone frequency assignment. In Section 6 the instances that were used to test the algorithm are described. In Section 7 we describe the greedy algorithm proposed to solve this problem. In Section 8 the experimental results are shown and finally the conclusions and perspectives are presented.

2. MOBILE TECHNOLOGY

A cellphone call involves two signals: one generated in it; and another generated in the closest base station which is usually a few kilometers away. The mobile base is responsible for managing the frequency assignment of the calls in the coverage area, this base also maintains a database of all potential users within the region, it is also responsible for transference of a call to another base station when the user makes a change of region.

Each base covers a geographic region of approximately 26km2 and usually is called a "cell". On the one hand, the distance between two cells is typically less than 5 kilometers; this short distance has two advantages: 1) allows low powered signals, which implies smaller power consumption and therefore a smaller battery size; 2) allows more people to use the service because the same frequencies can be reused by other base station. …

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