$4.$ Design a queue management system which can reduce delay during the packet arriving at the buffer. Assumed now a group of router in WAN, there is a buffer at one router for only three packet waiting to be forwarded at one time. On average it takes a packet two minutes to settle the forwarding process. Packet arrive at an average of $1$ per $5$ minutes. If the buffer is full, the packet will be discarded immediately.

Using the M$/$M$/1/$K model, find the following:

$($a$)$ The probability that the buffer of a router will be idle.

P$0=(1\backslash$rho $)$ K $=(14/5)3=(1/5)3=1/125$

$($b$)$ What is the mean number of packet at the buffer?

Lq

$($c$)$ The probability of packet will be discarded per hour.

$($d$)$ What is the expected total waiting time for a packet?