Exercise $1.$ Consider the open network $($referred to as the system$)$ of three queues, i$.$e$.,Q_1,Q_2,$ and $Q_3.$

New customers enter $Q_1$ at rate $.$ Customers leaving $Q_1$ will choose to enter either $Q_2$ with probability $p_1,$

or $Q_3$ with probability $1-p_1.$ Customers leaving $Q_2$ will depart from the system. Customers leaving $Q_3$ will

choose to either depart from the system with probability $p_2$ or $($re$)$enter $Q_1$ with probability $1-p_2.$ The average

time spent by a customer in $Q_1$ while visiting this queue once is $T_1.$ The average time spent by a customer in

$Q_2$ while visiting this queue once is $T_2.$ The average time spent by a customer in $Q_3$ while visiting this queue

once is $T_3.$

A$)$ Compute $N_1,N_2,$ and $N_3,$ defined as the average number of customers in $Q_1,Q_2,$ and $Q_3,$ respectively

$[$pt$.10].$

B$)$ Compute $T,$ defined as the total time spent in the system by a customer $[$pt$.10].$

Exercise $2.$ An Ethernet switch transmits frames at a transmission rate of $1$ gigabit per second $($Gbps$).$

The frame average length is $887$ bytes. The transmitter utilization is $30\%.$ The average time a frame stays

stored in the switch $($including its transmission time$)$ is estimated to be $473s.$

A$)$ Compute $x,$ defined as the average frame transmission time $[$pt$.10j.$

B$)$ Compute $N,$ defined as the average number of frames stored in the blade at steady$-$state $[$pt$.10].$

Exercise $3.$ The following string of $6$ data bits is transmitted $($from left to right$)"101011".$ A CRC is

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

$g(D)=D^3+D+1.$

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 $e(D)=D^5.$ 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 receiver, and if so$,$ why $[$pt$.10]?$