Consider a four server system with exponential interarrival and service times; no room for a queue; and a controller that adjusts the arrival rate depending on how busy the system is$.$ More specifically, when $)=(0,1,2,3$ servers are busy, the controller sets the arrival rate to $15-i.$ Moreover, the servers distract each other when more than one of them are busy which causes their service rate to slow down. More specifically, when $)=(1,2,3,4$ servers are busy, each busy server's rate is $\frac{12}{i}.$

$($a$)$ Model this process as a birth$-$death process by drawing the transition diagram.

$($b$)$ Calculate the long run fraction of time that the system is empty. $(0\mathrm{.}1456)$

$($c$)$ Calculate the long run fraction of time that the system is full. $(0\mathrm{.}23)$

$($d$)$ Calculate the long run average rate at which customers enter the system. $(\frac{10\mathrm{.}25}{h}r)$

$($e$)$ If dividers are installed to eliminate distractions so that each server always works at rate $12,$ how many more customers can be served per unit time? $(\frac{3\mathrm{.}36}{h}r)$