Two types of packets, namely voice packets and data packets, arrive at a single chan$-$

nel transmission link according to Poisson processes with rates $10$ and $15$ per minute,

respectively. The voice packets are always accepted into the buffer for transmission but

the data packets are only accepted when the total number of packets in the system is

$($strictly$)$ less than $3($a packet arriving at the link will leave if it is not accepted into

the buffer$).$ A packet $($of either type$)$ needs an exponentially distributed transmission

$($service$)$ time with mean of $\frac{1}{20}$ minute.

$($i$)$ Draw the birth$-$death rate diagram. $($Note: A packet being accepted into the buffer

means a birth. A packet completing its transmission means a death.$)$

$($ii$)$ Find the steady$-$state probability $p_0$ of having $0$ packets in the system.

$($iii$)$ Find the expected number of packets in the system.

$($iv$)$ Find the expected waiting time in the system.