# Quantitative probabilities that exactly 0, 1, 2, and

### Quantitative probabilities that exactly 0, 1, 2, and

Quantitative Analysis BA 452 Supplemental Questions 9 This document contains practice questions that supplement review questions for Lessons II-7 and II-8. This document first identifies the learning objectives of solving supplemental questions. The document then lists 35 questions and answers. All questions can be helpful. Questions marked with an asterisk * are similar to review questions.

Tip: Supplemental questions are grouped into sets of similar type. Once you have mastered the questions in a set, you can skip the rest of the questions in that set. 1Quantitative Analysis BA 452 Supplemental Questions 9 Objectives By working through the homework questions and the supplemental questions, you will: 1. Be able to identify where waiting line problems occur and realize why it is important to study these problems. Know the difference between single-channel and multiple-channel waiting lines. Understand how the Poisson distribution is used to describe arrivals and how the exponential distribution is used to describe services times.

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Learn how to use formulas to identify operating characteristics of the following waiting line models: a.Single-channel model with Poisson arrivals and exponential service times b. Multiple-channel model with Poisson arrivals and exponential service times 5. Know how to incorporate economic considerations to arrive at decisions concerning the operation of a waiting line. Understand the following terms: queuing theory queue single-channel multiple-channel service rate queue discipline steady state utilization factor operating characteristics arrival rate 2. 3. 4.

6. 2 Quantitative Analysis BA 452 Supplemental Questions 9Supplemental Questions 9 1. 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. c. Delays are expected if more than three customers arrive during any five-minute period. What is the probability that delays will occur? 2. In the Willow Brook National Bank waiting line system (see Problem 1), assume that the service times for the drive-up teller follow an exponential probability distribution with a service rate of 36 customers per hour, or 0. 6 customer per minute.

Use the exponential probability distribution to answer the following questions: a. What is the probability the service time is one minute or less? b. What is the probability the service time is two minutes or less? c. What is the probability the service time is more than two minutes? 3. Use the single-channel drive-up bank teller operation referred to in Problems 1 and 2 to determine the following operating characteristics for the system: 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 a customer spends waiting e. The average time a customer spends in the system f. The probability that arriving customers will have to wait for service 4.

Use the single-channel drive-up bank teller operation referred to in Problems 1-3 to determine the probabilities of 0, 1, 2, and 3 customers in the system. What is the probability that more than three customers will be in the drive-up teller system at the same time? 3 Quantitative Analysis BA 452 Supplemental Questions 9 5. 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? b. What is the average number of requests that will be waiting for service? c. What is the average waiting time in minutes before service begins? d. What is the average time at the reference desk in minutes (waiting time plus service time)? e.What is the probability that a new arrival has to wait for service? 6.

Movies Tonight is a typical video and DVD movie rental outlet for home viewing customers. During the weeknight evenings, customers arrive at Movies Tonight with an arrival rate of 1. 25 customers per minute. The checkout clerk has a service rate of 2 customers per minute. Assume Poisson arrivals and exponential service times. a. What is the probability that no customers are in the system? b.

What is the average number of customers waiting for service? c. What is the average time a customer waits for service to begin? . What is the probability that an arriving customer will have to wait for service? e. Do the operating characteristics indicate that the one-clerk check out system provides an acceptable level of service? 7. Speedy Oil provides a single-channel automobile oil change and lubrication service.

Customers provide an arrival rate of 2. 5 cars per hour. The service rate is 5 cars per hour. Assume that arrivals follow a Poisson probability distribution and that service times follow an exponential probability distribution.

a. What is the average number of cars in the system? b.What is the average time that a car waits for the oil and lubrication service to begin? c. What is the average time a car spends in the system? d. What is the probability that an arrival has to wait for service? 8.

For the Burger Dome single-channel waiting line in Section 11. 2, assume that the arrival rate is increased to 1 customer per minute and that the service rate is increased to 1. 25 customers per minute. Compute the following operating characteristics for the new system: P0, Lq, L, Wq, W, and Pw. Does this system provide better or poorer service compared to the original system?Discuss any differences and the reason for these differences. 4 Quantitative Analysis BA 452 Supplemental Questions 9 9. Marty’s Barber Shop has one barber.

Customers have an arrival rate of 2. 2 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? b. What is the probability that one customer is receiving a haircut and no one is waiting? c. What is the probability that one customer is receiving a haircut and one customer is waiting? d.

What is the probability that one customer is receiving a haircut and two customers are waiting? e. What is the probability that more than two customers are waiting? f. What is the average time a customer waits for service? 10.

Trosper Tire Company decided to hire a new mechanic to handle all tire changes for customers ordering a new set of tires. Two mechanics applied for the job. One mechanic has limited experience, can be hired for \$14 per hour, and can service an average of three customers per hour. The other mechanic has several years of experience, can service an average of four customers per hour, but must be paid \$20 per hour.Assume that customers arrive at the Trosper garage at the rate of two customers per hour. a.

What are the waiting line operating characteristics using each mechanic, assuming Poisson arrivals and exponential service times? b. If the company assigns a customer waiting cost of \$30 per hour, which mechanic provides the lower operating cost? 11. Agan Interior Design provides home and office decorating assistance to its customers.

In normal operation, averages of 2. 5 customers arrive each hour. One design consultant is available to answer customer questions and make product recommendations.The consultant averages 10 minutes with each customer. a. Compute the operating characteristics of the customer waiting line, assuming Poisson arrival and exponential service times.

b. Service goals dictate that an arriving customer should not wait for service more than an average of 5 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 8 minutes, what is the mean service rate? Will the service goal be met? 12. Pete’s market is a small local grocery store with only one checkout counter.Assume that shoppers arrive at the checkout lane according to a Poisson probability distribution, with an arrival rate of 15 customers per hour.

The checkout service times follow an exponential probability distribution, with a service rate of 20 customers per hour. a. Compute the operating characteristics for this waiting line. b. If the manager’s service goal is to limit the waiting time prior to beginning the check out process to no more than five minutes, what recommendations would you provide regarding the current checkout system? 5 Quantitative Analysis BA 452 Supplemental Questions 9 3. After reviewing the waiting line analysis of Problem 12, the manager of Pete’s Market wants to consider one of the following alternatives for improving service.

What alternative would you recommend? Justify your recommendation. a. Hire a second person to bag the groceries while the cash register operator is entering the cost data and collecting money from the customer. With this improved single channel operation, the service rate could be increased to 30 customers per hour. b.

Hire a second person to operate a second checkout counter.The two-channel operation would have a service rate of 20 customers per hour for each channel. 14. Ocala Software Systems operates a technical support center for its software customers.

If customers have installation or use problems with Ocala software products, they may telephone the technical support center and obtain free consultation. Currently, Ocala operates its support center with one consultant. If the consultant is busy when a new customer call arrives, the customer hears a recorded message stating that all consultants are currently busy with other customers.The customer is then asked to hold and a consultant will provide assistance as soon as possible. The customer calls follow a Poisson probability distribution with an arrival rate of five calls per hour.

On average, it takes 7. 5 minutes for a consultant to answer a customer’s questions. The service time follows an exponential probability distribution.

a. What is the service rate in terms of customers per hour? b. What is the probability that no customers are in the system and the consultant is idle? c. What is the average number of customers waiting for a consultant? d.What is the average time a customer waits for a consultant? e.

What is the probability that a customer waits for a consultant? f. Ocala’s customer service department recently received several letters from customers complaining about the difficulty in obtaining technical support. If Ocala’s customer service guidelines state that no more than 35% of all customers should have to wait for technical support and that the average waiting time should be two minute s or less, does your waiting line analysis indicate that Ocala is or is not meeting its customer service guidelines? What action, if any, would you recommend? 5.

To improve customer service, Ocala Software Systems (see Problem 14) wants to investigate the effect of using a second consultant at its technical support center. What effect would the additional consultant have on customer service? Would two technical consultants enable Ocala to meet its service guidelines with no more than 35% of all customers having to wait for technical support and an average customer waiting time of two minutes or less? Discuss. 6 Quantitative Analysis BA 452 Supplemental Questions 9 16. The new Fore and Aft Marina is to be located on the Ohio River near Madison, Indiana.Assume that Fore and Aft decides to build a docking facility where one boat at a time can stop for gas and servicing.

Assume that arrivals follow a Poisson probability distribution, with an arrival rate of 5 boats per hour, and that service times follow an exponential probability distribution, with a service rate of 10 boats per hour. Answer the following questions: a. What is the probability that no boats are in the system? b. What is the average number of boats that will be waiting for service? c. What is the average time a boat will spend waiting for service? d.What is the average time a boat will spend at the dock? e.

If you were the manager of fore and Aft Marina, would you be satisfied with the service level your system will be providing? Why or why not? 17. The manager of the fore and Aft Marina in Problem 16 wants to investigate the possibility of enlarging the docking facility so that two boats can stop for gas and servicing simultaneously. Assume that the arrival rate is 5 boats per hour and that the service rate for each channel is 10 boats per hour. a. What is the probability that the boat dock will be idle? b.What is the average number of boats that will be waiting for service? c.

What is the average time a boat will spend waiting for service? d. What is the average time a boat will spend at the dock? e. If you were the manager of Fore and Aft Marina, would you be satisfied with the service level you system will be providing? Why or why not? 18. All airplane passengers at the Lake City Regional Airport must pass through a security screening are before proceeding to the boarding area. The airport has three screening stations available, and the facility manager must decide how many to have open at any particular time.

The service rate for processing passengers at each screening station is 3 passengers per minute. On Monday morning the arrival rate is 5. 4 passengers per minute.

Assume that processing times at each screening station follow an exponential distribution and that arrivals follow a Poisson distribution. a. Suppose two of the three screening stations are open on Monday morning. Compute the operation characteristics for the screening facility. b. Because of space considerations, the facility manager’s goal is to limit the average number of passengers waiting in line to 10 or fewer.Will the two-screening-station system be able to meet the manager’s goal? c.

What is the average time required for a passenger to pass through security screening? 7 Quantitative Analysis BA 452 Supplemental Questions 9 19. Refer again to the Lake City Regional Airport described in Problem 18. When the security level is raised to high, the service rate for processing passengers is reduced to 2 passengers per minute at each screening station. Suppose the security level is raised to high on Monday morning.

The arrival rate is 5. 4 passengers per minute. a.

The facility manager’s goal is to limit the average number of passengers waiting in line to 10 or fewer. How many screening stations must be open in order to satisfy the manager’s goal? b. What is the average time required for a passenger to pass through security screening? 20. A Florida coastal community experiences a population increase during the winter months with seasonal residents arriving from northern states and Canada. Staffing at a local post office is often in a state of change due to the relatively low volume of customers in the summer months and the relatively high volume of customers in the winter months.The service rate of a postal clerk is 0.

75 customers per minute. The post office counter has a maximum of three work stations. The target maximum time a customer waits in the system is five minutes. a.

For a particular Monday morning in November, the anticipated arrival rate is 1. 2 customers per minute. What is the recommended staffing for this Monday morning? Show the operating characteristics of the waiting line. b. A new population growth study suggests that over the next two years the arrival rate at the post office during the busy winter months can be expected to be 2.

1 customers per minute.Use a waiting line analysis to make a recommendation to the post office manager. 21. Refer to the Agan Interior Design situation in Problem 11.

Agan’s management would like to evaluate two alternatives: • Use one consultant with an average service time of 8 minutes per customer. • Expand to two consultants, each of whom has an average time of 10 minutes per customer. If the consultants are paid \$16 per hour and the customer waiting time is valued at \$25 per hour for waiting time prior to service, should Agan expand to the two-consultant system? Explain. 8 Quantitative Analysis BA 452 Supplemental Questions 9 2.

A fast-food franchise is considering operating a drive-up window food-service operation. Assume that customer arrivals follow a Poisson probability distribution, with an arrival rate of 24 cars per hour, and that service times follow an exponential probability distribution. Arriving customers place orders at an intercom station at the back of the parking lot and them drive to the service window t5o pay for and receive their orders.

The following three service alternatives are being considered: a. A single-channel operation in which one employee fills the order and takes the money from the customer.The average service time for this alternative is 2 minutes. b. A single-channel operation in which one employee fills the order while a second employee takes the money from the customer. The average service time for this alternative is 1.

25 minutes. c. A two-channel operation with two service windows and two employees. The employee stationed at each window fills the order and takes the money from customers arriving at the window. The average service time for this alternative is 2 minutes for each channel. Answer the following questions and recommend an alternative design for the fast-food franchise: a.What is the probability that no cars are in the system? b.

What is the average number of cars waiting for service? c. What is the average number of cars in the system? d. What is the average time a car waits for service? e. What is the average time in the system? f. What is the probability that an arriving car will have to wait for service? 23. The following cost information is available for the fast-food franchise in problem 22: • Customer waiting time is valued at \$25 per hour to reflect the fact that waiting time is costly to the fast-food business.

• The cost of each employee is \$6. 50 per hour. To account for equipment and space, an additional cost of \$20 per hour is attributable to each channel. What is the lowest-cost design for the fast-food business? 24.

Patients arrive at a dentist’s office with an arrival rate of 2. 8 patients per hour. The dentist can treat patients at a service rate of 3 patients per hour.

A study of patient waiting times shows that a patient waits o average of 30 minutes before seeing the dentist. a. What are arrival and service rates in terms of patients per minute? b. What is the average number of patients in the waiting room? c. If as patient arrives at 10:10A. M.

at what time is the patient expected to leave the office? 9 Quantitative Analysis BA 452 Supplemental Questions 9 25. A study of the multiple-channel food-service operation at the Red Birds baseball park shows that the average time between the arrival of a customer at the food-service counter and his or her departure with a filled order is 10 minutes. During the game, customers arrive at the rate of four per minute. The food-service operation requires an average of 2 minutes per customer order. a. What is the service rate per channel in terms of customers per minute? b. What is the average waiting time in the line prior o placing an order? c.

On average, how many customers are in the food-service system? 26. Manning Auto operates an automotive service counter. While completing the repair work, Manning mechanics arrive at the company’s parts department counter with an arrival rate of four per hour. The parts coordinator spends an average of 6 minutes with each mechanic, discussing the parts the mechanic needs and retrieving the parts from inventory. a. Currently, Manning has one parts coordinator.