$2.$ In the small network depicted below, two external Poisson sources generate packets with rates $4$ and $3$ packets per second, respectively. All sources generate packets with exponentially distributed lengths with the same mean. All nodes have infinite queue capacity and can process $8$ packets per second each. A packet leaving node $1$ joins node $2$ with a probability of $0\mathrm{.}6$ and leaves the system with probability $0\mathrm{.}4.$ Similarly, a packet leaving node $2$ joins node $1$ with probability $0\mathrm{.}5$ and leaves the system with probability $0\mathrm{.}5.$

$($a$)$ Calculate steady state probability $\backslash ($ P$\backslash$left$($n$\_\{1\},$ n$\_\{2\}\backslash$right$)\backslash )$ that queues $1$ and $2$ have $\backslash ($ n$\_\{1\}\backslash )$ and $\backslash ($ n$\_\{2\}\backslash )$ packets, respectively. $(10$pt$)$

$($b$)$ What is the average packet delay in the system? $(8$pt$)$

$($c$)$ Assume that a packet enters the system at queue $1.$ How many transmissions $($on the average$)$ does it undergo before it leaves the system? Express your answer in terms of $\backslash ($ p$\_\{12\},$ p$\_\{1$ d$\},$ p$\_\{21\}\backslash ),$ and $\backslash ($ p$\_\{2$ d$\}.(7\backslash$mathrm$\{$pt$\})\backslash )$