Exercise $1.$ Consider the open network $($referred to as the system$)$ of two single$-$server queues, i$.$e$.,Q_1$ and

$Q_2,$ at steady state. New customers arrive at rate $$ and enter $Q_1.$ Customers leaving $Q_1$ choose to enter $Q_2$

with probability $1-p,$ or they re$-$enter $Q_1$ otherwise $($with probability $p).$ Customers leaving $Q_2$ choose to leave

the system with probability $1-q,$ or they return to $Q_1$ otherwise $($with probability $q).$ The known paramenters

are: $,p,q,N_1($defined as the average number of customers in $Q_1)$ and $T_2($defined as the average time spent

by a customer while visiting $Q_2$ once$).$ Solutions must be reported as functions of these known paramenters

only.

A$)$ Compute ${}_1$ and ${}_2$ defined as the arrival rates into $Q_1$ and $Q_2,$ respectively $[$pt$.10].$

B$)$ Compute $T_1$ defined as the customer's average time spent in $Q_1$ during one single visit and $N_2$ defined as

the average number of customers in $Q_2[$pt$.10].$

C$)$ Compute $T_{\text{t}\text{o}\text{t}}$ defined as the average total time a customer spends in the entire system $($i$.$e$.,$ visiting the

queues before leaving the system$)[$pt$.10].$

Exercise $2.$ Consider all the possibile strings of two data bits $(K=2).$ A CRC is attached at the end

of each string during transmission to create the corresponding codework. The CRC is computed using the

generator polynomial $g(D)=D^4+D^3+D^2+1.$

A$)$ Generate all $4$ possible codewords $[$pt$.10].$

B$)$ Assume that at the receiver the received sequence of bits is affected by an error pattern described by

$e(D)=D^5+D^2+D+1.$ Regardless of what codeword is affected by this error, will the error be detected

at the receiver and why? $[$pt$.10].$

C$)$ Compute the distance between every pair of codewords, and the minimum distance of the code $d_{\text{m}\text{i}\text{n}}[$pt$.10].$

D$)$ Corroborate the $d_{\text{m}\text{i}\text{n}}$ value obtained in the previous answer using properties that you may identify in the

used generator polynomial $g(D)[$pt$.10].$

E$)$ List all the error patterns $($using polynomial notation$)$ that cannot be detected by the chosen $g(D)[$pt$.10].$

F$)$ Compute $N_4$ defined as the number of patterns of exactly $4$ errors that can be experienced by these code$-$

words, and $N_{4n}$ defined as the number of these error patters that will not be detected by the receiver?

$[$pt$.10].$