Question

All trucks traveling on Interstate 40 between Alburquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 200 per 8-hour day, and the station can weigh, on the average, 220 trucks per day.

1. Determine the average number of trucks waiting, the average time spent waiting and being weighed at the weigh station by each truck, and the average waiting time before being weighed for each truck.
2. If the truck drivers find out they must remain at the weigh station longer than 15 minutes on the average, they will start taking a different route or traveling at night, thus depriving the state of taxes. The state of New Mexico estimates it loses $10,000 in taxes per year for each extra minute that trucks must remain at the weigh station. A new set of scales would have the same service capacity as the present set of scales, and it is assumed that arriving trucks would line up equally behind the two sets of scales. It would cost $50,000 per year to operate the new scales. Should the state install the new set of scales?
3. Suppose passing truck drivers look to see how many trucks are waiting to be weighed at the weigh station. If they see four or more trucks in line, they will pass by the station and risk being caught and ticketed. What is the probability that a truck will pass by the station?

Answer 1

This is the problem based on queueing theory. In this problem there is one server.

Arrival rate λ = 200 trucks per day

Service rate µ = 220 trucks per day

(a)

Average number of trucks waiting is Lq

Average time spent waiting is Ws

Average waiting time before being weighed

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(b)

Current loss = Waiting time * loss per minute per year = (24 – 15) * 10000 = 9*10000 = 90000

New system will reduce the arrival rate to 100 per station.

The waiting time will reduce to 4 minutes, so the government can save 90,000. The cost of new system is 50,000, which is lower than the savings. So the state should install the system.

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(c)

The probability of having four or more trucks:

P4ormore = (λ/µ) ^4+1 = (200/220) ^5 = 0.6209

The probability that a truck will pass by station = 0.6209