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 $p,$ or they leave the system otherwise $($with probability $1-p).$ Customers leaving $Q_2$ choose to

reneter $Q_2$ with probability $q,$ or they enter $Q_1$ otherwise with probability $1-q.$ The known paramenters are:

$,p,q,N_1$ and $N_2($defined as the average number of customers in $Q_1$ and $Q_2,$ respectively$).$ 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 $T_2$ defined as the customer's average time spent in $Q_2$ during one single visit $[$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$)[$pt$.10].$

Exercise $2.$ A Wi$-$Fi gateway supports $802\mathrm{.}11g,$ achieving $54Mbps$ radio data rate. The gateway radio

transmitter has a utilization of $=0\mathrm{.}2.$ The average data frame contains $800B.$ The average number of frames

stored at the gateway is $N=\frac{1}{4}.$

A$)$ Compute $\stackrel{}{x},$ defined as the average time a frame remains in transmission $[$pt$.10].$

B$)$ Compute $,$ defined as the frame arrival rate at the gateway $[$pt$.10].$

C$)$ Compute $T$ defined as the average total time a frame spends in the gateway $[$pt$.10].$