Suppose that are to design a wireless network with the following constraints and properties:

Cells are arranged in a conventional hexagonal layout, with equal sized hexagons.

Each cell has an area of $1k{m}^2.$

The number of cells per frequency$-$reuse cluster may be $N=1,3,4,7$ or $12.$

The path$-$loss exponent is $a=3\mathrm{.}5.$

The minimum acceptable SIR is $11$ dB $.$

There are $144$ available channels $($i$.$e$.,144$ calls can be supported at any given time$).$

Each user offers traffic with an intensity of $0\mathrm{.}1$ Erlangs.

The call blocking probability must not exceed $0\mathrm{.}01.$

Answer the following:

$($a$)$ Suppose that the system does not use sectorization $($i$.$e$,$ the base stations use omnidirectional antennas$).$ What is the minimum value of $N$ that satisfies the given SIR requirement? In computing the SIR, you only need to consider the first tier of interference and you may assume that all interferers are distance $D=R\sqrt[\phantom{2}]{3N}$ away $($i$.$e$.,$ the "rough" approximation$),$ where $R$ is the radius of the cell.

Answer: The minimum acceptable value of $N$ is $4.$

$($b$)$ Using the value of $N$ found in part $($a$),$ determine the user density $($i$.$e$.,$ the number of users per square kilometer$)$ that can be supported by your network at the specified call blocking probability.

Answer: $255$ subscribers can be supported per cell, and since each cell is $1$ square kilometer, it follows that the system supports $255$ subscribers per square kilometer

$($c$)$ Repeat parts $($a$)$ and $($b$),$ but this time assume that the system uses ideal $120-$degree sectorized antemnas $($i$.$e$.,$ each cell is divided into three sectors$).$ Again, only considering first$-$tier interferers, and assuming they are all at distance $D=R\sqrt[\phantom{2}]{3N},$ determine the minimum value of $N$ that satisfies the SIR requirement. Then determine the user density that can be supported by the network.

Answer: The minimum acceptable value of $N$ is now $3,$ and the total number of subscribers supported by each cell is $264$ subscribers. Thus, the network can support $264$ subscribers per square kilometer.