Reference no: EM133743072
Assignment: Customer Waiting Times in Different Scenarios Problems
Problem I: Algorithmic
Marty's Barber Shop has one barber. Customers have an arrival rate of 2.1 customers per hour, and haircuts are given with a service rate of 5 per hour. Use the Poisson arrivals and exponential service times model to answer the following questions:
A. What is the probability that no units are in the system? If required, round your answer to four decimal places.
B. What is the probability that one customer is receiving a haircut and no one is waiting? If required, round your answer to four decimal places.
C. What is the probability that one customer is receiving a haircut and one customer is waiting? If required, round your answer to four decimal places.
D. What is the probability that one customer is receiving a haircut and two customers are waiting? If required, round your answer to four decimal places.
E. What is the probability that more than two customers are waiting? If required, round your answer to four decimal places.
F. What is the average time a customer waits for service? If required, round your answer to four decimal places.
Problem II: Algorithmic
Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average of 2.9 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations. The consultant averages 12 minutes with each customer.
A. Compute the operating characteristics of the customer waiting line, assuming Poisson arrivals and exponential service times. If required, round your answers to four decimal places.
B. Service goals dictate that an arriving customer should not wait for service more than an average of 9 minutes. Is this goal being met? If not, what action do you recommend?
C. If the consultant can reduce the average time spent per customer to 9 minutes, what is the mean service rate? If required, round your answer to one decimal place.
D. Will the service goal be met?
Problem III
The reference desk of a university library receives requests for assistance. Assume that a Poisson probability distribution with an arrival rate of 10 requests per hour can be used to describe the arrival pattern and that service times follow an exponential probability distribution with a service rate of 12 requests per hour.
A. What is the probability that no requests for assistance are in the system? If required, round your answer to four decimal places.
B. What is the average number of requests that will be waiting for service? If required, round your answer to four decimal places.
C. What is the average waiting time in minutes before service begins? If required, round your answer to four decimal places.
D. What is the average time at the reference desk in minutes (waiting time plus service time)? If required, round your answer to one decimal place.
E. What is the probability that a new arrival has to wait for service? If required, round your answer to four decimal places.
Problem IV: Algorithmic
Gubser Welding, Inc., operates a welding service for construction and automotive repair jobs. Assume that the arrival of jobs at the company's office can be described by a Poisson probability distribution with an arrival rate of three jobs per 8-hour day. The time required to complete the jobs follows a normal probability distribution, with a mean time of 1.9 hours and a standard deviation of 1 hour. Answer the following questions, assuming that Gubser uses one welder to complete all jobs:
A. What is the mean arrival rate in jobs per hour? If required, round your answer to two decimal places.
B. What is the mean service rate in jobs per hour? If required, round your answer to four decimal places.
C. What is the average number of jobs waiting for service? If required, round your answer to three decimal places.
D. What is the average time a job waits before the welder can begin working on it? If required, round your answer to one decimal place.
E. What is the average number of hours between when a job is received and when it is completed? If required, round your answer to one decimal place.
F. What percentage of the time is Gubser's welder busy?
Problem V
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.
A. What is the mean or expected number of customers that will arrive in a five-minute period?
B. 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.
C. 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 answer to four decimal places.
Problem VI: Algorithmic
Burger Dome sells hamburgers, cheeseburgers, French fries, soft drinks, and milk shakes, as well as a limited number of specialty items and dessert selections. Although Burger Dome would like to serve each customer immediately, at times more customers arrive than can be handled by the Burger Dome food service staff. Thus, customers wait in line to place and receive their orders. Suppose that Burger Dome analyzed data on customer arrivals and concluded that the arrival rate is 24 customers per hour and 1 customer processed per minute.
Compare a multiple-server waiting line system with a shared queue to a multiple-server waiting line system with a dedicated queue for each server. Suppose Burger Dome establishes two servers but arranges the restaurant layout so that an arriving customer must decide which server's queue to join. Assume that this system equally splits the customer arrivals so that each server sees half of the customers. How does this system compare with the two-server waiting line system with a shared queue? Compare the average number of customers waiting, average number of customers in the system, average waiting time, and average time in the system. If required, round your answers to four decimal places.Shared single queueDedicated queuesNumber of customers waitingfill in the blank 1fill in the blank 2Average number of customers in the systemfill in the blank 3fill in the blank 4Average waiting timefill in the blank 5minutesfill in the blank 6minutesAverage time in the systemfill in the blank 7minutesfill in the blank 8minutes
Comparing these numbers, it is clear that the
A. Shared single queuetwo dedicated queuesresults in better process performance than the
B. Shared single queuetwo dedicated queues.
Problem VII: Algorithmic
Mid-West Publishing Company publishes college textbooks. The company operates an 800 telephone number whereby potential adopters can ask questions about forthcoming texts, request examination copies of texts, and place orders. Currently, two extension lines are used, with two representatives handling the telephone inquiries. Calls occurring when both extension lines are being used receive a busy signal; no waiting is allowed. Each representative can accommodate an average of 15 calls per hour. The arrival rate is 30 calls per hour.
A. How many extension lines should be used if the company wants to handle 90% of the calls immediately?
B. What is the probability that a call will receive a busy signal if your recommendation in part (a) is used? If required, round your answer to four decimal places.
C. What number of calls, in decimal form, receive a busy signal for the current telephone system with two extension lines? If required, round your answer to four decimal places.
Problem VIII: Algorithmic
Kolkmeyer Manufacturing Company is considering adding two machines to its manufacturing operation. This addition will bring the number of machines to nine. The president of Kolkmeyer asked for a study of the need to add a second employee to the repair operation. The arrival rate is 0.06 machines per hour for each machine, and the service rate for each individual assigned to the repair operation is 0.5 machines per hour.
A. Compute the operating characteristics if the company retains the single-employee repair operation. If required, round your answers to four decimal places.
B. Compute the operating characteristics if a second employee is added to the machine repair operation. If required, round your answers to four decimal places.
C. Each employee is paid $15 per hour. Machine downtime is valued at $85 per hour. From an economic point of view, should one or two employees handle the machine repair operation? Explain. If required, round your answers to two decimal places.