Cars traveling from Canada to the United States through the Thousand Islands Bor

Question

Cars traveling from Canada to the United States through the Thousand Islands Border Crossing must stop for US Customs and Immigation. During the stop, each passenger in the car will present a passport for inspection by a US Customs Officer, answer questions, and declare certain valuables that may be in the car. On average, it takes a Customs Officer about five minutes to inspect passports and release a car for entry into the United States. Since the Thousand Islands Border Crossing is not heavily traveled at night, US Customs and Immigration only keeps one lane open at the checkpoint plaza and one officer on duty to process arriving cars. Between midnight and 6:00am (6 hours), a car arrives at the Border Crossing plaza at an average rate of 9 an hour.

On average, how many cars are present at the Thousand Islands Border Crossing plaza during that time period?

A. 4 cars, B. 3 cars, C. 5 cars, D. 2 cars

Explanation / Answer

We assume that arrival rate of cars as well as service rate of cars ( i.e to inspect passport and release the car ) follow exponential distribution.

Given are following details :

Arrival rate of cars = a = 9 per hour

Service rate of cars ( 5 minutes per inspection ) = s = 60/5 = 12 per car

Therefore,

Average number of cars are present at the border crossing plaza

= a^2/ s x ( s – a ) + a/s

= 9 x 9 / 12 x ( 12 – 9 ) + 9/12

= 81 / 36 + 9/12

= 2.25 + 0.75

= 3

AVERAGE NUMBER OF CARS PRESENT AT THE BORDER CROSSING PLAZA = 3

AVERAGE NUMBER OF CARS PRESENT AT THE BORDER CROSSING PLAZA = 3

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