Assume that Machine 1 needs an average of 1 minute to process one car. Furthermore, assume that the average input rate to the process is 30 cars per hour. Finally, assume that the average inventory of cars in the entire process is 12.8 cars. On average, how long does a car have to wait in line?

Note: Answer in minutes. So, if your answer is 10 minutes, then input 10.

Question

Assume that Machine 1 needs an average of 1 minute to process one car. Furthermore, assume that the average input rate to the process is 30 cars per hour. Finally, assume that the average inventory of cars in the entire process is 12.8 cars. On average, how long does a car have to wait in line?

Note: Answer in minutes. So, if your answer is 10 minutes, then input 10.

BlackTom AI · Accepted Answer

Little’s Law relates the average number in a system (L), the average arrival rate (λ), and the average time in system (W): L = λ · W. Service time and waiting time are related by W = Wq + S, where Wq is average waiting-in-queue time and S is average service time.

1) Relevant concepts
- Little’s Law (steady-state): L = λW.
- Total time in system W = waiting time in queue Wq + service time S.
- Careful unit consistency: convert hours ↔ minutes.

2) Given values and unit conversions
- Arrival rate λ = 30 cars/hour.
- Average number in system L = 12.8 cars.
- Service time S = 1 minute = 1/60 hour ≈ 0.0166667 hour.

3) Compute average time in system W using Little’s Law
W = L / λ = 12.8 cars ÷ 30 cars/hour = 0.426666... hours.

Convert to minutes:
W = 0.426666... hours × 60 minutes/hour = 25.6 minutes.

4) Compute average waiting time in queue Wq
Wq = W − S.
S = 1 minute, so Wq = 25.6 minutes − 1 minute = 24.6 minutes.

(Alternate check using average number in queue: average number in service = utilization ρ = λ·S = 30*(1/60) = 0.5 car. Then average number in queue Lq = L − ρ = 12.8 − 0.5 = 12.3 cars. Wq = Lq / λ = 12.3 / 30 hours = 0.41 hours = 24.6 minutes, same result.)

5) About other choices
- Not applicable here (no multiple-choice options). The numerical reasoning above shows why alternatives like using L/λ without subtracting service time (giving 25.6 min) would be incorrect for waiting-in-line time specifically.

6) Conclusion
The average time a car waits in line is 24.6 minutes, which matches the provided answer 24.6.

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