3. Vehicles arriving at a toll booth at a mean rate of 6 per 10 min and departing at a mean rate of 8 per 10 min. Suppose that both the arrivals and departures follow a Poisson distribution. Compute the average queue length (Q) at the toll booth. Suppose that the arrival is Poisson and the departure is uniform, compute the average time spent in the system. What is the probability of having 10 [Q] arrivals in [0,20] minutes ([Q] is the largest integer that is smaller than Q, for example, if Q-4.8, then [Q]-4)?
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![3. Vehicles arriving at a toll booth at a mean rate of 6 per 10 min and departing at a mean
rate of 8 per 10 min. Suppose that both the arrivals and departures follow a Poisson
distribution. Compute the average queue length (Q) at the toll booth. Suppose that the
arrival is Poisson and the departure is uniform, compute the average time spent in the
system. What is the probability of having 10*[Q] arrivals in [0,20] minutes ([Q] is the
largest integer that is smaller than Q, for example, if Q=4.8, then [Q]=4)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb6fa7e8-9d31-42f7-a5f5-e025c05cdfb2%2F1d3cd185-ae79-41ac-a091-bfdc84721276%2F4yg879d_processed.jpeg&w=3840&q=75)
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