Problem $2$:

Customers arrive at a shop at an average rate of $10$ per hour $($Poisson$).$ There is only one worker

who can serve at an average rate of $14$ customers per hour. Service times are exponentially

distributed.

Part A$.$ What is this waiting line model? What is the average length of the queue?

Part B$.$ What is the system utilization?

Part C$.$ What is the probability that when a customer arrives, she will see two or more customers

ahead of her? Note: $$two or more$$ includes the customer who is receiving service.

Part D$.$ When customers wait longer than half an hour in the line, they will complain about the

excessive wait times. On average, how many complaints can be expected in an hour? Hint: Try to

relate this question to one of the probability equations.

In the upcoming sales event of the year, the average arrival rate is expected to be $21$ per hour

$($Poisson$).$ Management wants to keep the probability that a new arrival will have to wait for service

at below $10\%$ by hiring additional workers $($working independently$)$ for the event.

Part E$.$ What is the minimum number of additional workers needed for the shop during the event

to achieve this goal?