Final answer:
The average length of the waiting line at a border inspection station, where vehicles arrive at a Poisson rate of 12 per hour and inspections occur at an exponential rate of 13 per hour, is approximately 10.16 vehicles.
Step-by-step explanation:
The subject of the student's question deals with the Poisson distribution to model vehicle arrivals and an exponentially distributed inspection time to model the service process. With vehicles arriving at a rate of 12 per hour and being inspected at a rate of 13 per hour, we use queueing theory to find the average length of the waiting line. The questions provided in the Try It examples and the additional context given about other rates and probabilities establish a background for understanding these types of stochastic processes.
The arrival rate (λ) is 12 vehicles per hour, and the service rate (μ) is 13 vehicles per hour. Since this is an M/M/1 queue system (Poisson arrivals with an exponential service time and one server), we can use the formula L = λ^2 / μ(μ - λ) to find the average number of vehicles in the system (L). Substituting the given rates, L = 12^2 / 13(13 - 12) = 144 / 13 ≈ 11.08. However, this number includes the vehicle being serviced; to find the average length of the waiting line (Lq), we use the formula Lq = λ^2 / μ(μ - λ), which in this case simplifies to Lq = L - (λ/μ). So Lq = 11.08 - (12/13) ≈ 10.16. Therefore, the average length of the waiting line is approximately 10.16 vehicles.