Consider a satellite communication network where users transmit data packets to a satellite

using a Slotted ALOHA protocol with a slot duration of $25ms.$ The satellite acts as a relay, forwarding the

received packets to a base station on Earth. The base station has a single$-$server queuing system that

handles the incoming packets. The queuing system follows an $\frac{M}{M}\frac{?}{?}\frac{1}{K}$ model, with a finite buffer size

$($K$)$ that can hold up to $20$ packets. The service rate $()$ at the base station is $50$ packets per second. Users

generate packets according to a Poisson process with a mean rate $()$ of $40$ packets per second.

a$.$ What is the probability that a packet is successfully transmitted to the satellite using the Slotted

ALOHA protocol?

b$.$ Calculate the traffic intensity $()$ for the queuing system at the base station.

c$.$ What is the probability $(Pk)$ that all $20$ buffers in the base station are full?

d$.$ Given the above system parameters, what is the average time a packet spends in the system and

the queue?

e$.$ If the satellite network is to be designed to ensure the average delay for a packet is not more than

$10$ milli seconds, what changes would you recommend to the system parameters, if any?