System optimization subject:

$2)$ A small petrol station with only one petrol pump is operating eight hours per day. The

customers arrive according to a Poisson distribution with an average time of $5$ minutes

between arrivals. The service time is exponentially distributed with mean time of $2$

minutes per customer. Based on this information:

$($a$)$ Calculate the expected average queue length.

$(6$ marks$)$

$($b$)$ Calculate the expected average number of customers in the system.

$(2$ marks$)$

$($c$)$ Calculate the expected average time spent by a car in the petrol station.

$(2$ marks$)$

$($d$)$ Calculate the expected average waiting time of a car before able to use the petrol

pump.

$(2$ marks$)$

$($e$)$ The owner of the pump station is contemplating to add another petrol pump, similar

to the current one, to reduce customer waiting time. The estimated cost of waiting

in queue for the customer is RM$10/$hour$.$ The installation of another petrol pump

will cost the owner RM $50$ per day.

Evaluate with analysis if the owner should add the new petrol pump.

$(13$ marks$)$