Question:

Please note that this problem has four parts $($a$,$ b$,$ c$,$ and d$).$

You may need to scroll the screen down and navigate left and right to see the whole problem.

Clearly label your parts when answering.

You are allowed a Maximum of $3$ Files for this problem.

Show all calculations$/$results to at least FOUR PLACES AFTER THE DECIMAL POINT

$($except when all succeeding digits are zeros$).$

Must show work completely $($including all calculations$/$steps$).$ Failure to do so will result in

a score of $"0,"$ even if your final answer is correct.

Rock Property Management $($RPM$)$ team win the contract of managing the parking building close to

the new Mega Outlets shopping complex. Currently the parking building is operating with one lane

and one parking validation window, where one parking staff operated on a first$-$come first served

basis. There are variety of stores in this shopping complex and the area surrounded by the office

buildings. Due to shortage of the parking spaces in the surrounding areas, the customer population is

fairly large. The parking staff can validate the parking passes$/$tickets and issue receipts at a rate of

fifty seconds per car on average following a negative exponential distribution. Following

a Poisson distribution, sixty six cars will arrive at the entrance every hour on average during business

hour.

$($a$)$ What is the average number of cars waiting in line?

$($b$)$ What is the average time in minutes that a car has to spend waiting in line at the parking

entrance?

$($c$)$ What is the probability that there will be more than two cars at the parking entrance?

RPM noticed that the number of customers increased and consequently the waiting times to validate

parking passes$/$tickets started to increase. RPM want to manage the system in a faster way they

decided that the average waiting time in the line should not be more than three minutes, and hence

opened two additional parking validation windows, increasing the number of windows to serve

customers to three. Customers arrive following a Poisson distribution, $202$ cars will arrive at the

entrance every hour on average during business hour. They wait in a single waiting line to validate

the parking passes and get a receipt where the queue discipline is FCFS$.$ Service times at each

attendant are identical and distributed exponentially; each attendant spends an average of fifty

seconds per customer.

$($d$)$ What is the average length of the waiting line with three validation windows? Is RPM achieving

their goal of having the average waiting time in the line les s than three minutes?

$($A simple Yes$/$No answer will not get any mark$/$s$.)$ Please show allyour work in full steps.Explain clearly your work