A queueing system is expected to handle $=3$ customers per minute. There are two servers available, each of

which can work at rate $=2$ customers per minute. Two arrangments are proposed:

Option A$.$ An M$/$M$/2/4$ queue, i$.$e$.$ space for two customers to wait and two to be served.

Option B$.$ Same as Option A$,$ but with unlimited waiting space, i$.$e$.$ an M$/$M$/2$ queue.

$($a$)$ Draw a transition rate diagram for Option A$.$

$($b$)$ Draw a transition rate diagram for Option B$.$

$($c$)$ Find the equilibrium distribution for Option A$.$

$($d$)$ Find the equilibrium distribution for Option B$.$

$($e$)$ Find the proportion of the time that the system is idle $($no customers$)$ under Options A and B$.$ Compare

the two options: why does providing unlimited waiting space affect this proportion in the way that it

does?

$($f$)$ Now suppose the arrival rate turns out to be $=4$ per minute. Explain why one of the options will handle

this scenario, while the other will not.