$($a$)$ An Ethernet router has a single outgoing transmission link of $12$ Mbps$.$ The average length of each packet is $720$ bytes. If the average packet delay through the switch, is to be less than $15$ ms$,$ assuming an M$/$M$/1$ queue determine: $($i$)$ the departure rate in packets$/$s; $($ii$)$ the maximum gross input packet rate to the switch in packets$/$s; $($iii$)$ the average length of the queue.