Problem $1$: A packet$-$switched Jackson network routes packets among two routers ac$-$

cording to the routing probabilities shown in Fig. $1.$ Notice that there are two points at

which packets enter the network and two points at which they can depart.

$($a$)$ What is the maximum allowable rate $r_1$ that the network can tolerate? Call this $r_1^{\text{m}\text{a}\text{x}}.$

$($b$)$ Set $r_1=0\mathrm{.}9r_1^{\text{m}\text{a}\text{x}}.$ What is the mean response time for a packet entering at the router

$1$ queue?

$($c$)$ How does the mean response time change if the rates of both servers increase by two

times?

Figure $1$: Jackson network from Problem $1.$