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 24 trucks an hour.

a. How many trucks would you expect to see at the weighing station, including those being inspected?
b. If a truck was just arriving at the station, about how many minutes could the driver expect to be at the station?
c. What is the probability that both inspectors would be busy at the same time?
d. How many minutes, on average, would a truck that is not immediately inspected have to wait?
e. What condition would exist if there was only one inspector?
f. What is the maximum line length for a probability of .97?

Indicate your subject in the "School Subject" box, so those with expertise in the area will respond to the question.

a. To calculate the number of trucks expected to be at the weighing station, we need to consider the arrival rate of trucks and the inspection capacity of the inspectors.

The arrival rate of trucks is 40 per hour. This means that on average, 40 trucks arrive at the station every hour.

Each inspector can inspect 24 trucks per hour. Since there are two inspectors, the total inspection capacity is 2 * 24 = 48 trucks per hour.

The total number of trucks expected to be at the weighing station, including those being inspected, is the minimum between the arrival rate and the inspection capacity. In this case, it is limited by the inspection capacity of 48 trucks per hour.

Therefore, we would expect to see 48 trucks at the weighing station, including those being inspected.

b. To calculate the average time a truck spends at the station, we need to divide the total time spent by the number of trucks.

Since each inspector can inspect 24 trucks per hour, it takes approximately 1/24 hour (or 60/24 = 2.5 minutes) for a truck to be inspected.

Therefore, a truck driver can expect to be at the station for around 2.5 minutes.

c. The probability that both inspectors would be busy at the same time can be calculated as the product of the probabilities that each inspector is busy.

The probability that an inspector is busy can be calculated as the inspection rate divided by the arrival rate.

For each inspector, the inspection rate is 24 trucks per hour, and the arrival rate is 40 trucks per hour.

So the probability that an inspector is busy is 24/40 = 0.6.

The probability that both inspectors are busy at the same time is the product of these probabilities: 0.6 * 0.6 = 0.36.

Therefore, the probability that both inspectors would be busy at the same time is 0.36, or 36%.

d. The average waiting time for a truck that is not immediately inspected can be calculated as the average time between arrivals divided by the probability of not being inspected immediately.

The average time between arrivals is the reciprocal of the arrival rate: 1/40 hour.

The probability of not being inspected immediately is 1 minus the probability that both inspectors are free.

Since the probability that both inspectors are free is 1 - 0.36 = 0.64, the probability of not being inspected immediately is 0.64.

Dividing the average time between arrivals by the probability of not being inspected immediately gives us the average waiting time: (1/40) / 0.64 = 0.025 hour, or 1.5 minutes.

Therefore, the average waiting time for a truck that is not immediately inspected is approximately 1.5 minutes.

e. If there was only one inspector, the inspection capacity would be halved.

The inspection capacity would now be 1 * 24 = 24 trucks per hour.

Since the arrival rate is still 40 trucks per hour, the number of trucks expected to be at the weighing station, including those being inspected, would be limited by the inspection capacity of 24 trucks per hour.

Therefore, we would expect to see 24 trucks at the weighing station, including those being inspected, if there was only one inspector.

f. To determine the maximum line length for a probability of 0.97, we need to find the arrival rate that corresponds to this probability.

The probability of having a truck in the line can be calculated as 1 minus the probability of both inspectors being free.

The probability of both inspectors being free is 1 - 0.36 = 0.64.

So the probability of having a truck in the line is 1 - 0.64 = 0.36.

To find the arrival rate corresponding to a probability of 0.36, we need to solve the equation 0.36 = arrival rate / inspection capacity.

Substituting the known values, we get 0.36 = arrival rate / (2 * 24).

Simplifying, we have 0.36 = arrival rate / 48.

Solving for the arrival rate gives us arrival rate = 0.36 * 48 = 17.28 trucks per hour.

Therefore, the maximum line length for a probability of 0.97 would be 17.28 trucks.

a. To determine how many trucks would you expect to see at the weighing station, including those being inspected, we need to calculate the expected number of arrivals and the expected number of inspections.

Given:
- Arrival rate = 40 trucks per hour
- Hour of operation = 2 hours
- Number of inspectors = 2
- Inspection rate per inspector = 24 trucks per hour

The expected number of truck arrivals can be calculated using the Poisson distribution formula:

λ = arrival rate * hour of operation
λ = 40 trucks/hour * 2 hours = 80 trucks

The expected number of truck inspections can be calculated by multiplying the number of inspectors by the inspection rate per inspector:

Expected number of inspections = number of inspectors * inspection rate per inspector
Expected number of inspections = 2 inspectors * 24 trucks/hour = 48 trucks

To calculate the total number of trucks expected to be at the weighing station, including those being inspected, we add the expected number of arrivals and the expected number of inspections:

Total expected trucks = expected number of arrivals + expected number of inspections
Total expected trucks = 80 trucks + 48 trucks = 128 trucks

Therefore, you would expect to see 128 trucks at the weighing station, including those being inspected.

b. To estimate how many minutes a truck driver could expect to be at the station, we need to calculate the average time spent per truck at the weighing station.

The average time spent per truck can be calculated as the reciprocal of the inspection rate per inspector:

Average time spent per truck = 1 / inspection rate per inspector
Average time spent per truck = 1 / 24 trucks per hour

Since there are 60 minutes in an hour, we convert the result to minutes:

Average time spent per truck = (1 / 24) * 60 minutes = 2.5 minutes

Therefore, a truck driver can expect to spend approximately 2.5 minutes at the weighing station.

c. To find the probability that both inspectors would be busy at the same time, we need to use the concept of interarrival and service times.

Since the arrival rate follows a Poisson distribution with a rate of 40 trucks per hour, and the inspection rate follows a exponential distribution with a rate of 24 trucks per hour per inspector, we can model the system as a M/M/2 queue.

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

P(B) = (ρ^2) * (1 - ρ) / (1 - ρ^2)
where:
ρ = arrival rate / (number of inspectors * inspection rate per inspector)

ρ = 40 trucks/hour / (2 inspectors * 24 trucks/hour) = 0.833

P(B) = (0.833^2) * (1 - 0.833) / (1 - 0.833^2) ≈ 0.299

Therefore, the probability that both inspectors would be busy at the same time is approximately 0.299 or 29.9%.

d. To calculate the average waiting time for a truck that is not immediately inspected, we need to consider the queueing time for trucks.

Since the system can be modeled as a M/M/2 queue, we can use the Little's Law formula to calculate the average waiting time:

Average waiting time = Average number of trucks in the system / Arrival rate

The average number of trucks in the system (L) can be calculated using the formula:

L = ρ / (1 - ρ)
where:
ρ = arrival rate / (number of inspectors * inspection rate per inspector)

ρ = 40 trucks/hour / (2 inspectors * 24 trucks/hour) = 0.833

L = 0.833 / (1 - 0.833) = 4.998

Therefore, the average waiting time for a truck that is not immediately inspected is approximately 4.998 hours or 299.88 minutes.

e. If there was only one inspector, the condition would change to a M/M/1 queue. In this case, the calculation for various metrics like expected number of trucks, average waiting time, and probability of both inspectors busy would be different.

f. To determine the maximum line length for a probability of 0.97, we need to calculate the utilization factor (ρ) for the system and then find the corresponding line length (L) using queuing theory formulas.

Using the formula:
P(B) = (ρ^2) * (1 - ρ) / (1 - ρ^2)

We rearrange the formula to solve for ρ:
ρ^2 - P(B) * ρ + P(B) = 0

Substituting the given probability:
0.97ρ^2 - 0.97ρ + 0.97 = 0

We can solve this quadratic equation to find the value of ρ.

Once we have the value of ρ, we can use the Little's Law formula to calculate the line length (L):

L = ρ / (1 - ρ)

Therefore, with the value of ρ obtained from solving the quadratic equation, we can calculate the maximum line length for a probability of 0.97 using Little's Law.