# Question

The traffic managers of toll roads and toll bridges need specific information to properly staff the toll booths so that the queues are minimized (i. e., the waiting time is minimized).

(a) Assume that there is one toll booth on a busy interstate highway and that the number of cars per minute approaching the toll booth follows a Poisson PMF with a = 10. The traffic manager wants you to determine the probability that exactly 11 cars approach this toll booth in the minute from noon to 1 min past noon.

(b) Now assume that there are toll booths at a toll plaza and that the number of cars per minute approaching the plaza follows the same Poisson PMF with a = 30. The traffic manager wants you to calculate the minimum number of toll booths that need to be staffed if the probability that more than five cars approach any toll booth in one minute is no more than 0.05. For this part, assume the traffic approaching the plaza divides evenly among the N booths such that the traffic approaching each booth is independent and follows a Poisson PMF with a = 30 / N.

(a) Assume that there is one toll booth on a busy interstate highway and that the number of cars per minute approaching the toll booth follows a Poisson PMF with a = 10. The traffic manager wants you to determine the probability that exactly 11 cars approach this toll booth in the minute from noon to 1 min past noon.

(b) Now assume that there are toll booths at a toll plaza and that the number of cars per minute approaching the plaza follows the same Poisson PMF with a = 30. The traffic manager wants you to calculate the minimum number of toll booths that need to be staffed if the probability that more than five cars approach any toll booth in one minute is no more than 0.05. For this part, assume the traffic approaching the plaza divides evenly among the N booths such that the traffic approaching each booth is independent and follows a Poisson PMF with a = 30 / N.

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