Large deviations for spatial telecommunication systems : The Boolean model

There have been some concerns that telecommunication companies are facing today which are spatial in nature. Systems operated by these telecommunication companies have evolved along the years. The main concern of these companies is the ability to provide quality service to customers or users in a dense regime. Therefore, they sought to answer questions such as (i) what is the best possible configurations of base stations and users that maximizes quality service? (ii) can one estimate and control the probability of bad service, which may be seen as a rare event? and many more which may arise during companies operations. To answer these questions one will often have to estimate the tail distribution of events, which may be considered under the realm of large deviations. In this article, we define the empirical marked measure of the Boolean model, which will serve estimate the intensity measure of the Marked Poisson Point Process of devices or users, and the empirical connectivity measure of the Boolean Model which will serve estimate the coverage probability density of the spatial telecommunication area. For these empirical measures, we establish large deviation principle (LDP) in the τ – topology.