The US Transportation Security Administration $($TSA$)$ operates six $$checkpoints$$ for screening departing passengers at the Buffalo$/$Niagara International Airport. Each checkpoint consists of a baggage conveyor belt, a CTX scanning$/$imaging machine, and a walk$-$through metal detector for passengers to pass through, although all six checkpoints are not necessarily operating all the time. This is because each checkpoint requires a team of three TSA employees to staff it$,$ and there is not always a need to have six checkpoints running simultaneously. Passengers arriving at the airport are first met by a $$greeter$$ who directs each passenger to one particular checkpoint that is in operation at that moment. Once $$greeted$,$ each passenger then joins their assigned checkpoint$$s single line to be screened and it takes an average of one minute to screen each passenger through the checkpoint. The greeter$$s job is to divide the arriving passengers evenly between the checkpoints in operation, although the rate at which passengers arrive at the airport and approach the greeter does vary with the time of day. Recommend the number of checkpoints that TSA should operate during the various time intervals of the day first described in the table below.

$($Carry three decimal points$)$

Period Average Number of Passengers Arriving per Hour Number of Checkpoints

$5$:$00$ AM to $6$:$00$ AM $2204$

$6$:$00$ AM to $8$:$00$ AM $3506$

$8$:$00$ AM to $4$:$00$ PM $2505$

$4$:$00$ PM to $7$:$00$ PM $3306$

$7$:$00$ PM to $10$:$00$ PM $1904$

Based on the above$-$given information, what is the average length of the line in front of each checkpoint throughout the day? What is the average time a passenger spends in line $($minutes$)$ waiting for service in front of each checkpoint?

Answer:

Period Average Length of line Average waiting time $($minutes$)$

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

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

$8$:$00$ AM to $4$:$00$ PM

$4$:$00$ PM to $7$:$00$ PM

$7$:$00$ PM to $10$:$00$ PM