Question

Trucks are required to pass through a weighing station so that they can be checked for weight violations. Trucks arrive at the station at the rate of 40 an hour between 7:00 p.m. and 9:00 p.m. Currently two inspectors are on duty during those hours, each of whom can inspect 25 trucks an hour. Use Table 1.

a. How many trucks would you expect to see at the weighing station, including those being inspected? (Round your answer to 3 decimal places.)
Ls trucks
b. If a truck was just arriving at the station, about how many minutes could the driver expect to be at the station? (Round your answer to 2 decimal places.)
Ws min.
c. What is the probability that both inspectors would be busy at the same time? (Round your answer to 4 decimal places.)
Pw d. How many minutes, on average, would a truck that is not immediately inspected have to wait? (Round your answer to the nearest whole number.)
Wa min.
f. What is the maximum line length for a probability of .97? (Round up your answer to the next whole number.)
Lmax

Answer 1

a. Ls = 45.000 trucks

b. Ws = 0.030 min

c. Pw = 0.0000

d. Wa = 0 min

f. Lmax = 43 trucks

Step-by-step explanation:

The expected number of trucks at the weighing station, including those being inspected, is 45.000 trucks per hour. This is calculated by multiplying the arrival rate of trucks (40 trucks/hour) by the average service rate of both inspectors (2 inspectors * 25 trucks/hour/inspector).

For a truck just arriving, the expected time at the station is 0.030 minutes. This is found by taking the reciprocal of the total service rate (1/40 trucks/minute).

The probability that both inspectors are busy simultaneously is essentially zero (0.0000). This is because the arrival rate is lower than the combined service rate of both inspectors, making it highly unlikely for both to be occupied at the same time.

Trucks that are not immediately inspected have zero waiting time on average (Wa = 0 min) because the service rate exceeds the arrival rate.

To achieve a probability of .97, the maximum line length (Lmax) is calculated to be 43 trucks. This ensures that 97% of the time, the number of trucks in the system is below this threshold, preventing excessive queuing.

Answer 2

• a. We would expect to see approximately 0.8 trucks at the weighing station, including those being inspected.
• b. The driver can expect to be at the station for approximately 1.2 minutes.
• c.The probability that both inspectors would be busy at the same time is approximately 0.2143.
• d. A truck that is not immediately inspected would have to wait, on average, for approximately 1.2 minutes.
• f.The maximum line length for a probability of 0.97 is 3 trucks.

Step-by-step explanation:

a. To determine the number of trucks expected to be at the weighing station, including those being inspected, we need to consider the arrival rate of trucks and the inspection rate of the inspectors.

The arrival rate of trucks is given as 40 per hour. The inspection rate of each inspector is 25 trucks per hour. Since there are two inspectors, the total inspection rate is 2 * 25 = 50 trucks per hour.

To find the number of trucks expected to be at the weighing station, we can use Little's Law, which states that the average number of trucks in the system (including those being inspected) is equal to the arrival rate multiplied by the average time spent in the system.

The average time spent in the system can be calculated by dividing the number of trucks in the system by the inspection rate. In this case, the inspection rate is 50 trucks per hour.

Therefore, the average time spent in the system is 1/50 hours per truck.

Now, we can calculate the average number of trucks in the system:

Average number of trucks = Arrival rate * Average time spent in the system

= 40 trucks/hour * 1/50 hour/truck

= 0.8 trucks

Therefore, we would expect to see approximately 0.8 trucks at the weighing station, including those being inspected.

b. If a truck has just arrived at the station, the driver can expect to be at the station for the average time spent in the system, which is 1/50 hours per truck. To convert this to minutes, we multiply by 60:

Average time spent at the station = 1/50 hour/truck * 60 minutes/hour

= 1.2 minutes

Therefore, the driver can expect to be at the station for approximately 1.2 minutes.

c. The probability that both inspectors would be busy at the same time can be calculated using the formula:

Probability of both inspectors being busy = (Inspection rate * Inspection rate) / (Arrival rate * (Arrival rate + Inspection rate))

In this case, the inspection rate is 25 trucks per hour, and the arrival rate is 40 trucks per hour.

Probability of both inspectors being busy = (25 trucks/hour * 25 trucks/hour) / (40 trucks/hour * (40 trucks/hour + 25 trucks/hour))

Using the above formula, the probability that both inspectors would be busy at the same time is approximately 0.2143.

d. The average waiting time for a truck that is not immediately inspected can be calculated by dividing the average number of trucks in the system (0.8 trucks) by the arrival rate (40 trucks per hour), and then converting it to minutes:

Average waiting time = (0.8 trucks) / (40 trucks/hour) * 60 minutes/hour

= 1.2 minutes

Therefore, a truck that is not immediately inspected would have to wait, on average, for approximately 1.2 minutes.

f. The maximum line length for a probability of 0.97 can be found using queuing theory formulas. In this case, we can use the formula:

Maximum line length = (Square root of (2 * Arrival rate * Inspection rate) + 1) - 1

Using the given arrival rate of 40 trucks per hour and inspection rate of 25 trucks per hour, we can calculate the maximum line length as follows:

Maximum line length = (Square root of (2 * 40 trucks/hour * 25 trucks/hour) + 1) - 1

The maximum line length is approximately 3.

Therefore, the maximum line length for a probability of 0.97 is 3 trucks.