$(77$ points$)$ A communication channel of capacity $CMbps($megabits per second$)$ is

shared by a group of $N$ users. Here, the channel capacity is the rate at which infor$-$

mation can be transmitted over the channel. Each user, User $i,$ either transmits at a rate

of $T_{i}Mbps$ or becomes idle with no transmission, where $i=1,2,3,$cdots,$N.$ Unless stated

otherwise, the transmission activity of a user is independent with that of another. In a

certain time period, the probability that User i transmits is $p_i,$ where $ASC{T}_i=p_i=pi=1,2,3,$cdots,$N{f}_A()A=1,2,3,$cdots,$NA{F}_A()={\displaystyle \underset{i=0}{\overset{||}{}}}(\frac{N}{i})p^i(1-p{)}^{N-i}AAinR{f}_S()SinRA,M_A(t)P(Aa){}_A^{2}AS,M_S\frac{t}{b}ar(S){}_S^{2}SAxYx{p}_{x}xY{p}_{Y}xY{q}_Y{S}_x{S}_{Y}xY{}_{S_x{S}_Y}{S}_x{S}_{Y}0.$

The activation score $of$ a user $is$ defined $as$ the indicator function $of$ the transmission

activity. That $is,$ the score $is$ one when the user transmits, $or$ zero otherwise. Let

$S_x$ and $S_{Y}be$ the activation scores $of$ Users $x$ and $Y$ during the time period.

Evaluate ${}_{S_x{S}_Y},$ the correlation coefficient $of{S}_x$ and $S_Y\mathrm{.}0.On$ the other hand, $if$

User $x$ does not transmit, User $Yis$ also idle with probability $q_Y,$ where $0.$

The activation score $of$ a user $is$ defined $as$ the indicator function $of$ the transmission

activity. That $is,$ the score $is$ one when the user transmits, $or$ zero otherwise. Let

$S_x$ and $S_{Y}be$ the activation scores $of$ Users $x$ and $Y$ during the time period.

Evaluate ${}_{S_x{S}_Y},$ the correlation coefficient $of{S}_x$ and $S_Y\mathrm{.}0.$ When User $x$ transmits,

User $Y$ also transmits with probability $p_Y,$ where $0.On$ the other hand, $if$

User $x$ does not transmit, User $Yis$ also idle with probability $q_Y,$ where $0.$

The activation score $of$ a user $is$ defined $as$ the indicator function $of$ the transmission

activity. That $is,$ the score $is$ one when the user transmits, $or$ zero otherwise. Let

$S_x$ and $S_{Y}be$ the activation scores $of$ Users $x$ and $Y$ during the time period.

Evaluate ${}_{S_x{S}_Y},$ the correlation coefficient $of{S}_x$ and $S_Y\mathrm{.}0.$ The

channel $is$ saturated when the total transmission rate $of$ all users equals $to$ the channel

capacity $or$ higher.

Let A and $Sbe$ the number $of$ users who are transmitting concurrently and the total

transmission rate $of$ all users, respectively. There $isno$ overhead $in$ sharing the channel.

$(a)(53$ points$)$ Suppose that $Cis$ a constant, and $T_i=$ and $p_i=p$ for all $i=1,2,3,$cdots,$N.$

$i.(4$ points$)$ Find $f_A(),$ the probability mass function $of$ A for $=1,2,3,$cdots,$N.$ Hence,

show that the cumulative distribution function $ofA,$

$F_A()={\displaystyle \underset{i=0}{\overset{||}{}}}(\frac{N}{i})p^i(1-p{)}^{N-i}AAinR.$

$ii.(5$ points$)$ Determine $f_S(),$ the probability mass function $ofS,$ where $inR.$

iii. $(3$ points$)$ Compute the moment$-$generating function $ofA,M_A(t).$

$iv.(3$ points$)$ Using Chernoff bound, find $an$ upper bound $onP(Aa).$

$v.(8$ points$)By$ using the result $of(iii),$ derive ${}_A^2,$ the variance $ofA.$

$vi.(3$ points$)$ Find the moment$-$generating function $ofS,M_S(t).$

vii. $(8$ points$)By$ using the results $of(v)$ and $(vi),or$ otherwise, compute $\frac{?}{\text{b}\text{a}\text{r}}(S)$ and ${}_S^2,$ the

mean and variance $ofS,$ respectively.

viii. $(5$ points$)$ Determine whether the probability distribution $ofAis$ memoryless.

$ix.(4$ points$)$ Calculate the probability that the channel $is$ saturated. State the condi$-$

tion $in$ which the probability can $be$ zero.

$x.(10$ points$)$ There are two users, $x$ and $Y,$ using the same channel. User $x$ transmits

with probability $p_x$ during a time period, where $0.$ When User $x$ transmits,

User $Y$ also transmits with probability $p_Y,$ where $0.On$ the other hand, $if$

User $x$ does not transmit, User $Yis$ also idle with probability $q_Y,$ where $0.$

The activation score $of$ a user $is$ defined $as$ the indicator function $of$ the transmission

activity. That $is,$ the score $is$ one when the user transmits, $or$ zero otherwise. Let

$S_x$ and $S_{Y}be$ the activation scores $of$ Users $x$ and $Y$ during the time period.

Evaluate ${}_{S_x{S}_Y},$ the correlation coefficient $of{S}_x$ and $S_Y.$