11. Automobiles arrive at a tollbooth on a highway completely at random according to aPoisson process with rate À = 4 cars per hour. Determine the following:a. The probability that the time between any two successive arrivals exceeds20 minutes.b. The probability that exactly four cars arrive in any given hour.c. The probability that more than five cars arrive in any given hour.d. The probability that more than 10 cars arrive in any given two-hour period.e. A person working for the transportation department begins counting arrivals atthe tollbooth at 8 A.M. What is the probability that he counts 20 arrivals before12 noon? (Use a normal approximation for your calculations.)

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Answered: 11. Automobiles arrive at a tollbooth…

Practical Management Science

6th Edition

ISBN:9781337406659

Author:WINSTON, Wayne L.

Publisher:WINSTON, Wayne L.

Chapter12: Queueing Models

Section12.5: Analytic Steady-state Queueing Models

Problem 11P

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Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive-up teller window occur at random, with an arrival rate of 24 customers per hour or 0.4 customers per minute. What is the mean or expected number of customers that will arrive in a five-minute period?λ = fill in the blank 1 per five minute period Assume that the Poisson probability distribution can be used to describe the arrival process. Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a five-minute period. If required, round your answers to four decimal places. x P(x) 0 fill in the blank 2 1 fill in the blank 3 2 fill in the blank 4 3 fill in the blank 5 Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur? If required, round your…
48. Ships arrive at a port facility at an average rate of twoships every three days. On average, it takes a single crewone day to unload a ship. Assume that interarrival andservice times are exponential. The shipping companyowns the port facility as well as the ships using that facility. The companyestimates that it costs $1000 per dayfor each day that a ship spends in port. The crew servicing the ships consists of 100 workers, each of whom ispaid an average of $30 per day. A consultant has recommended that the shipping company hire an additional40 workers and split the employees into two equal-sizecrews of 70 each. This would give each crew an averageunloading or loading time of 1.5 days. Which crew arrangement would you recommend to the company?
Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cars. On weekday mornings, arrivals to the drive- up teller window occur at random, with an arrival rate of 30 customers per hour or 0.5 customers per minute. Assume the Poisson probability distribution can be used to describe the arrival process. (a) What is the mean or expected number of customers that will arrive in a six-minute period? (b) Use the arrival rate in part (a) and compute the probabilities that exactly 0, 1, 2, and 3 customers will arrive during a six-minute period. (Round your answers to four decimal places.) X 0 1 2 3 P(x) (c) Delays are expected if more than three customers arrive during any six-minute period. What is the probability that delays will occur? (Round your answer to four decimal places.)
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