The Transportation Security Administration $($TSA$)$ operates ten "checkpoints" to screen departing passengers at the St$.$ Paul International Airport. Each checkpoint consists of a baggage conveyor belt, a CTX scanning$/$imaging machine, and a walk$-$through metal detector through which passengers pass. However, all ten checkpoints are not necessarily in operation at all times. This is because each checkpoint requires a team of three TSA officers to staff it$,$ and there is not always a need to have ten checkpoints operating simultaneously. Passengers arriving at the airport are first greeted by a "greeter" who directs each passenger to a particular checkpoint that is in operation. Once "greeted," each passenger then joins the single line at their assigned checkpoint to be screened, and it takes an average of four minutes to actually screen each passenger through the checkpoint. The greeter's job is to distribute the arriving passengers evenly among the operating checkpoints. However, the rate at which passengers arrive at the airport and approach the greeter varies with the time of day:

Time Period Average Number of Passengers Arriving per Hour

$5$:$00$ AM to $6$:$00$ AM $60$

$6$:$00$ AM to $8$:$00$ AM $90$

TSA must maintain enough checkpoints so that the average delay at checkpoint screening is less than equal to $9$ minutes $($W $=9$ minutes$).$ This delay includes the time the passenger may wait to be screened. Recommend the number of checkpoints TSA should operate during the various periods of the day first described in the table above, keeping in mind that TSA would prefer to minimize the required personnel.

Answer:

Time Period Number of CheckPoints Average Waiting Time $($Wq$)$

$5$:$00$ AM to $6$:$00$ AM

$6$:$00$ AM to $8$:$00$ AM