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.

Chapter2: Introduction To Spreadsheet Modeling

Section: Chapter Questions

Problem 20P: Julie James is opening a lemonade stand. She believes the fixed cost per week of running the stand...

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V5.
On a toll road, there are 2 lanes for drivers to pay their toll. Customer arrival times are constant, with a degenerate distribution. Service times are random, with an exponential distribution. What is the proper description for this queueing system?
Consider a system of two infinite queues in series, where each service facility has a server. All service times are independent and have an exponential distribution with a mean of 3 min at facility 1 and 4 min at facility 2. Facility 1 has Poisson inputs with a mean rate of 10 per hour. Find the total expected number of customers in the system and the total expected waiting time (with service) for a customer.
Suppose that 8 customers per hour arrive at a bank during the 11a-1p. "lunch-hour", period. It is observed that there are, on average, 6 customers in the bank during this same "lunch-hour" period. Which of the following statements are true assuming the system is in steady- state during that period? (Hint: What law govern's the behavior of systems in "steady-state"?) The average customer spends 45 minutes in the bank. The bank must have at least 2 bank tellers. There is always a line of customers waiting during the 11a-1p period. The average service time (not including waiting) at the bank is 3/4 of an hour. Exactly two answers are correct. None of the answers are correct.
8. Jim McDonald, manager of the fast-food hamburger restaurant McBurger, realizes that providing fast service is a key to the success of the restaurant. Customers who have to wait very long are likely to go to one of the other fast-food restaurants in town next time. He estimates that each minute a customer has to wait in line before completing service costs him an average of 30 cents in lost future business. Therefore, he wants to be sure that enough cash registers always are open to keep waiting to a minimum. Each cash register is operated by a part- time employee who obtains the food ordered by each customer and collects the payment. The total cost for each such employee is $9 per hour. During lunch time, customers arrive according to a Poisson process at a mean rate of 66 per hour. The time needed to serve a customer is estimated to have an exponential distribution with a mean of 2 minutes. Determine how many cash registers Jim should have open during lunch time to minimize his…
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 15 customers per hour or 0.25 customers per minute. Let's assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 21 customers per hour, or 0.35 customers per minute. Determine the following operating characteristics for the system. (Round your answers to four decimal places.) (a) The probability that no customers are in the system (b) The average number of customers waiting (c) The average number of customers in the system (d) The average time (in min) a customer spends waiting min (e) The average time (in min) a customer spends in the system min (f) The probability that arriving customers will have to wait for service
41) Burger Cart April has a popular burger cart where an average of 12 customers per hour arrive to buy a burger. April can serve an average of 20 customers per hour. Layla runs a coffee stand across the street from April. As Layla looks over at April's burger cart, what is the probability she will see 3 customers at the cart (in line or being served)? Group of answer choices 0.4219 0.0156 0.7500 0.0117 0.0720 0.1055 0.2500 42) Burger Cart The average time required to serve a customer = _______ minutes. Group of answer choices 2.50 minutes 1.67 minutes 3.00 minutes 20.00 minutes 4.00 minutes 5.00 minutes 2.00 minutes
c1 need urgent in 15 minutes
In queueing theory, λ represents the mean arrival rate; for example, the number of peoplearriving at the service counter per hour. What does 1/λ represent?o Service rateo Interarrival timeo Service time None
An M/M/1 queueing system has that customers arrive to it at a rate of 5 per hour, i.e., its interarrival times between two consecutive arrivals follows an exponential distribution with parameter 5 per hour. This question will ask you to evaluate two options for designing the server in the system. Both options for the single server provide the same service for the customers in the queueing system but they cost different amounts to implement. In addition, the total costs of the queuing system are the implementation costs plus the customer costs. Currently, the customer costs are $100 per hour per customer in the queueing system. In comparing these options, it costs us $100 per hour per customer in the queueing system. (a) The first option for the server is one that has a service time that follows an exponential distribution with mean 10 minutes. Determine L, W, Lq, and Wq for this system. (b) The second option for the server is one that has a service time that follows an…
The manager of a bank wants to use an MyMys queueingmodel to weigh the costs of extra tellers against the costof having customers wait in line. The arrival rate is 60customers per hour, and the average service time is fourminutes. The cost of each teller is easy to gauge at the$11.50 per hour wage rate. However, because estimatingthe cost per minute of waiting time is difficult, the bankmanager decides to hire the minimum number of tellersso that a typical customer has probability 0.05 of waitingmore than five minutes in line.a. How many tellers will the manager use, given thiscriterion?b. By deciding on this many tellers as “optimal,” themanager is implicitly using some value (or some range of values) for the cost per minute of wait-ing time. That is, a certain cost (or cost range) would lead to the same number of tellers as sug-gested in part a. What is this implied cost (or cost range)?
Analytical queueing models assume a Poisson probability distribution for _____ .

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