Part $1$: Understanding ALOHA

A$)$ You studied in class that the efficiency of Slotted Aloha, with N users and a transmit probability p $,$ is given by:

Efficiency$($slotted Aloha$)=Np(1-p{)}^{N-1}$

i$.$ Prove that the probability that maximizes the efficiency of Slotted Aloha is $p^{***}=\frac{1}{N}$

ii$.$ Show that when $N$ tends to infinity, the maximum efficiency of Slotted$-$Aloha reaches the value of $\frac{1}{e}=0\mathrm{.}37.($Hint: you can use the fact that the limit when $N$ goes to infinity of $($:$(1-\frac{1}{N}{)}^N=\frac{1}{e}\}.$

B$)$ You studied in class that the efficiency of Pure Aloha, with N users and a transmit probability p $,$ is given by:

Efficiency$($pure Aloha$)=Np(1-p{)}^{2(N-1)}$

iii. Prove that the probability that maximizes the efficiency of Pure Aloha is $p^{***}=\frac{1}{2N-1}$

iv$.$ Show that when N tends to infinity, the maximum efficiency of Slotted$-$Aloha reaches the value of $\frac{1}{2e}=0\mathrm{.}18.($Hint: you can use the fact that the limit when $N$ goes to infinity of $($:$(1-\frac{1}{N}{)}^N=\frac{1}{e}\}.$

C$)$ Plot the value of the efficiency of Slotted Aloha as a function of N for:

$N=1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100(using{p}^{***}$ obtained $in$ part $A).$

On the same figure, Plot the value of the efficiency of Pure Aloha as a function of N for: $N=1,2,3,4,5,6,7,8,9,10,20,30,40,50,60,70,80,90,100($using $p^{***}$ obtained in part B$).$

Show that the results plotted of the figure should confirm the calculations of Part A and B above.