The Arizona Turnpike Authority is thinking of putting in an automatic toll colle
ID: 443910 • Letter: T
Question
The Arizona Turnpike Authority is thinking of putting in an automatic toll collection machine at the 180 interchange to replace a toll-taker. They estimate that an average of 50 cars arrive to the toll station each hour. A toll-taker takes an average of one (1) minute to collect a toll from a driver, while a machine can service faster, an average of 100 cars per hour. (You can assume operations follow Model A, that is the M/M/1 waiting line model.) What is the average time a driver waits in the line and gets serviced with a toll-taker versus a machine? Average Time a driver waits in line and gets serviced with a toll-taker: Average Time a driver waits in line and gets serviced with a machine: Drivers hate wasting time on the road, so the Authority is assuming that driver time is valued at $ 10 per hour. What is the total expected cost (in $/hour) of driver time (waiting and getting serviced) if a toll-taker is used versus a machine? Total expected costs of driver time with a toll-taker. Total expected costs of driver time with a machineExplanation / Answer
The M/M/1 waiting line model assumes a Poisson arrival process.
Exected number in the waiting line = Average number of customers / (Average customers capabie of being serviced - Average number of customers)
Wating line in case or toll taker = 50/(60-50) = 5 cars
Waiting line in case of machine = 50/(100-50) = 2.5 cars
Average waiting time = Expected number in the waiting line / Average customers in an hour
Average waiting time for toll taker = 5 / 50 = 1/10 hours = 6 minutes
Average time for the machine = 2.5/50 = 1/20 hours = 3 minutes
Total service time = Waiting time + Time to get serviced
Total Service time for the toll taker = 6 + 1 = 7 minutes
Total Service time for the machine = 3 + 0.6 = 3.6 minutes
With the toll taker the car has a waiting period of 6 minutes, service period of 1 minute and the total service time is 7 minutes
With the machien the car has a waiting period of 3 minutes, service periof 0.6 minutes and the total service time 3.6 minutes
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b. If the driver's time is valued at $10 per hour.
We have seen the computation of the total service time of the driver with the toll taker and with the machine.
The expected cost shall be the cost attributable to this procedure of waiting and getting serviced.
With the toll taker the total time is 7 minutes. Therefore, the cost attributable is 7 x $10/60 = $1.17
With the machine the total time is 3.6 minutes. Thefore, the cost attributable is 3.6 x $10/60 = $0.60.
Parameter Toll-taker Machine Average number of customers in 1 hour 50 50 No. of cars that can be serviced 60 100 Time taken to service one car (minutes) 1 0.6Related Questions
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