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

steady state. New customers arrive at rate $.$ A new customer chooses to enter $Q_1$ with probability $p_1,Q_2$ with

probability $p_2,$ and $Q_3$ with probability $p_3=1-p_1-p_2.$ Customers leaving $Q_1$ choose to enter $Q_2.$ Customers

leaving $Q_2$ choose to enter $Q_3.$ Customers leaving $Q_3$ choose to either depart from the system with probability

$q$ or $($re$)$enter $Q_1$ with probability $1-q.$ 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 average total time spent in the system by a customer $[$pt$.10].$

C$)$ Compute the value of $p_3$ that would minimize $T[$pt$.10].$

Exercise $2.$ An Ethernet switch transmits frames at a transmission rate of $10$ gigabit per second $($Gbps$).$

The frame average length is $900$ bytes. The transmitter utilization is $20\%.$ The average number of frames stored

in the switch $($including the one in transmission$)$ is estimated to be $150.$

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

B$)$ Compute $T,$ defined as the average time a frame spends in the switch at steady$-$state $(T$ includes both

waiting and transmission time$)[$pt$.10].$