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

$Q_2,$ and $Q_3,$ at steady state. A first group of new customers arrive at rate $$ and enter $Q_1.$ A second group of

new customers arrive at rate $$ and enter $Q_2.$ A third group of new customers arrive at rate $$ and enter $Q_3.$

Customers departing from $Q_1$ always enter $Q_2.$ Customers departing from $Q_2$ choose to leave the system with

probability $1-p,$ or enter $Q_3$ with probability $p,$ where $Q_3{Q}_1,p{T}_i{Q}_{i}i=1,2,3{}_1,{}_2{}_3{Q}_1,Q_2{Q}_3{N}_1,N_2{N}_3{Q}_1,Q_2{Q}_3{T}_{\text{t}\text{o}\text{t}}0.$ Customers departing from $Q_3$ always enter

$Q_1.$ The known paramenters are: $,p,$ and $T_i,$ defined $as$ the average time spent $by$ a customer while visiting

$Q_i$ once, where $i=1,2,3.$ Solutions must $be$ reported $as$ functions $of$ these known paramenters only.

$A$