Exercise $2.$ The server of a queue completes service time according to a random variable denoted as $x($in

$ms,$ whose probability density function is

$f_x(x)=\{\begin{array}{cc}-\frac{2}{81}x+\frac{2}{9},& 0x9\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

The server utilization is $30\%.$ The average time spent by a customer in the queue $($inclusive of the service time$)$

is $20ms.$

A$)$ Compute $E[x]=\stackrel{}{x},$ defined as the average service time $[$pt$.10].$

B$)$ Compute $,$ defined as the customer arrival rate into the queue. $[$pt$.10].$

C$)$ Compute $N,$ defined as the average number of customers in the queue. $[$pt$.10].$

Exercise $3.$ The following string of $5$ data bits is transmitted $($from left to right$)"11111".$ A CRC is

attached at the end of the string during transmission. The CRC is computed using the generator polynomial

$g(D)=D^9+D^2+D+1.$

A$)$ Compute $c(D),$ defined as the remainder when $D^{3}s(D)$ is divided by $g(D),$ using modulo $2$ arithmetic,

where $s(D)$ is the polynomial representing the string of data bits. Write down the sequence of bits as they

are transmitted inclusive of CRC$,$ starting left with the first bit to be transmitted $[$pt$.10].$

B$)$ Assume that at the receiver the sequence of bits is affected by an error, described by $e(D)=D^6+D^6.$

Compute $r(D),$ defined as the remainder when $D^{3}s(D)+c(D)+e(D)$ is divided by $g(D),$ using modulo

$2$ arithmetic. Is the error detected by the recelver, and if so$,$ why $[$pt$.10]?$

C$)$ What is the minimum distance of the used code and why $[$pt$.10]?$