This is an interesting mathematical challenge to help solve a big business problem. Where to put the cell towers to achieve (say) a 95% coverage of a specific area. By coverage, I mean that 95% of people have signal in the area in question. Also, cell towers do not need to have the same power, and we can play with this to reduce costs.

Typically, such problems are solved using Monte-Carlo simulations, and are considered to be research operations problems. Yet a mathematical solution might be available as well, based on stochastic geometry. For instance, in the picture below, a boolean model for wireless network coverage and connectivity is constructed from randomly sized disks placed at random locations. More can be found here.

The picture below shows SINR cells of a wireless network model expanding as the transmitter powers increase (source: click here.)

Finally, below is a simulation of four Poisson–Boolean (constant-radius or Gilbert disk) models as the density increases with largest clusters in red. Click here for details.

Note that earlier studies used to approximate coverage using hexagons instead of circles, see picture below. However, this does not take into account cell towers with different power, or elevation which indirectly has an impact on how far the signal can reach.

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