$13\mathrm{.}8$ More Open versus Closed Systems

Figure $13\mathrm{.}9$ shows a closed queueing network and an open one. Job sizes are Exponentially distributed, where the service rate is shown for each server, and $\backslash ($ p $\backslash )$ denotes a probability.

Figure $13\mathrm{.}9.$ Figure for Exercise $13\mathrm{.}8.$

$($i$)$ For the closed system, what value of $\backslash ($ p $\backslash )$ minimizes mean response time, $\backslash (\backslash$mathbf$\{$E$\}[$T$]\backslash )?$

$($ii$)$ Derive an expression for $\backslash (\backslash$mathbf$\{$E$\}[$T$]\backslash )$ for the open system.

$($iii$)$ Does the value of $\backslash ($ p $\backslash )$ that you found in $($i$)$ minimize $\backslash (\backslash$mathbf$\{$E$\}[$T$]\backslash )$ in the open system? If yes, give a proof. If no$,$ give a counterexample.

$($iv$)$ Under the optimal $\backslash ($ p $\backslash )$ for the closed system, what is the $($approximate$)$ throughput $\backslash ($ X $\backslash )$ for the closed system?

$($v$)$ Under the optimal $\backslash ($ p $\backslash )$ for the open system, what is $\backslash ($ X $\backslash )$ for the open system?