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.

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.

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?

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 issue dependent on queueing hypothesis. In this issue there is one server.

Appearance rate λ = 200 trucks for each day

Administration rate µ = 220 trucks for every day

(a)

Normal number of trucks holding up is Lq

2002 40000 220(20) = 9.0909 trucks (u - 2) 220(220-200)

Normal time spent holding up is Ws

Ws= 1 u - a 220 - 200 = 20 = 0.05 day = 0.05 * 60 = 3 minutes out of every hour * 8 hours = 24 minutes

Normal holding up time before being gauged

La Wq=7 9.0909 - = 0.04545 = 0.04545*60*8 = 21.816 minutes 200

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(b)

Current misfortune = Waiting time * misfortune every moment every year = (24 – 15) * 10000 = 9*10000 = 90000

New framework will diminish the appearance rate to 100 for every station.

Ws = _1_-_1 u-1 220 - 100 = 4 minutes 1 120 = 0.008333 day = 0.00833* 60 * 8 hours

The holding up time will diminish to 4 minutes, so the legislature can spare 90,000. The expense of new framework is 50,000, which is lower than the reserve funds. So the state ought to introduce the framework.

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(c)

The likelihood of having at least four trucks:

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

The likelihood that a truck will pass by station = 0.6209

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