$5.$ 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 $1$ km$2$

$.$

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

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

$$ 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

$$

$3$N 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 antennas

$($i$.$e$.,$ each cell is divided into three sectors$).$ Again, only considering first$-$tier interferers, and assuming

they are all at distance D $=$ R

$$

$3$N$,$ determine the minimum value of N that satisfies the SIR requirement.

Then determine the user density that can be supported by the network.