Deck 11: Probability and Statistics 

Question 1

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 service rate is increased to 1.25 customers per minute. Computer the following operating characteristics for the new system: P 0 , L q , L, W q , W, and P w. Dose this system provide better or poorer service compared to the original system? Discuss any differences and the reason for these differences.

Answer 1

Single-channel waiting line model with Poisson arrivals and exponential service times
Determine the operating characteristics for the system
(a)The probability that no customers are in the system :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute.
Now, calculate the probability that no customers are in the system as follows: …… (1)(b)The average number of customers waiting :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute.
Now, calculate the average number of customers in the waiting line as follows: …… (2)(c)The average number of customers in the system :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute, and the calculated is 3.2 [refer to Equation (2)].
Now, calculate the average number of customers in the system as follows: …… (3)(d)The average time a customer spends waiting :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute, and the calculated is 3.2 [refer to Equation (2)].
Now, calculate the average time a customer spends waiting as follows: …… (4)(e)The average time a customer spends waiting :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute, and the calculated is 3.2 [refer to Equation (4)].
Now, calculate the average time a customer spends waiting as follows: …… (5)(f)The probability that arriving customers will have to wait for service :
It is given that the average arrival rate of customer is 1 customer per minute and the average service rate of customer (µ) is 1.25 customers per minute.
Now, calculate the probability that arriving customers will have to wait for service as follows: …… (6)Comparison between the original system and the new system:
Justification:
• The service rate (µ) is increased from 1 to 1.25 customers per minute.
• The arrival rate of customer is increased from 0.75 to 1 customer per minute even if the service rate is high. It shows the poor quality of service to the customers because of higher arrival rate.
• The probability of average waiting time increases in the new system when compared to the original system.

Question 2

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 of waiting times shows that a patient waits an average of 30 minutes be ore seeing the dentist.
a. What are the arrival and service rates in terms of patients per minute?
b. What is the average number of patients in the waiting room?
c. If a patient arrives at 10:10 a.m. , at what time is the patent expected to leave the office?

Answer 2

(a)Arrival and service rates in terms of patients per minute:
Arrival rate in terms of patients per minute:
It is given that the arrival rate of patients per hour is 2.8.
Now, calculate the arrival rate of patients per minute as follows: …… (1)Hence, the arrival rate in terms of patients per minute is 0.0467.
The service rate in terms of customers per hour:
It is given that the service rate of patients per hour is 3.
Now, calculate the service rate of patients per minute as follows: …… (2)Hence, the service rate in terms of patients per minute is 0.0500.
(b)Average number of patients in the waiting room:
It is given that a patient waits for 30 minutes on an average, and the calculated arrival rate is 0.0467 per minute [refer to Equation (1)].
Now, calculate average number of patients in the waiting room as follows: Hence, the average number of patients in the waiting room is 1.4 patients.
(c)The average time a customer spends in the system:
It is given that a patient waits for 30 minutes on an average, and the calculated service rate is 0.0500 per minute [refer to Equation (2)].
Now, calculate the average time a customer spends in the system as follows: If the patient arrives at 10:10 A.M, the patient is expected to leave the office at 11:00 A.M .

Question 3

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?

Answer 3

Single-channel waiting line model with Poisson arrivals and exponential service times
Determine the operating characteristics for the system:
(a)The probability that no units are in the system ( ):
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour.
Now, calculate the probability that no customers are in the system as follows: …… (1)(b)The probability that one customer is receiving a haircut and no one is waiting ( ):
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour, and the calculated is 0.56 [refer to Equation (1)].
Now, calculate the probability that one customer is receiving and no one is waiting as follows: …… (2)(c)The probability that one customer is receiving a haircut and one customer is waiting ( ):
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour, and the calculated is 0.56 [refer to Equation (1)].
Now, calculate the probability that one customer is receiving and one customer is waiting as follows: …… (3)(d)The probability that one customer is receiving a haircut and two customers are waiting ( ):
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour, and the calculated is 0.56 [refer to Equation (1)].
Now, calculate the probability that one customer is receiving and two customers are waiting as follows: …… (4)(e)The probability that more than two customers are waiting
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour, and the calculated is 0.56 [refer to Equation (1)], is 0.2464 [refer to Equation (2)], is 0.1084 [refer to Equation (3)], and is 0.0477 (refer to Equation (4)].
Now, calculate the probability that more than two customers are waiting as follows: (f)The average number of customers in the system ( )
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour.
Now, calculate the average number of customers in the waiting line as follows: …… (5)The average time a customer spends in the waiting line ( ):
It is given that the average arrival rate of customer ( ) is 2.2 customer per hour and the average service rate of customer (µ) is 5 customers per hour, and the calculated is 0.3457 [refer to equation (5)].
Now, calculate the average time a customer spends waiting as follows:

Question 4

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 to placing an order?
c. On average, how many customers are in the food-service system?

Question 5

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 experiences, 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?

Question 6

Manning Autos 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. On average, each mechanic waits 4 minutes before the parts coordinator is available to answer questions or retrieve pans from inventory. Find L q , W , and L for this single-channel parts operation.
b. A trial period with a second parts coordinator showed that, on average, each mechanic waited only 1 minute before a parts coordinator was available. Find L q , W , and L for this two-channel parts operation.
c. If the cost of each mechanic is $20 per hour and the cost of each parts coordinator is $12 per hour, is the one-channel or the two-channel system more economical?

Question 7

Agan Interior Design provides home and office decorating assistance to its customers. In normal operation, an average 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 arrivals 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?

Question 8

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 two jobs per 8-hour day. The time required to complete the jobs follows a normal probability distribution with a mean time of 3.2 hours and a standard deviation of 2 hours. 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?
b. What is the mean service rate in jobs per hour?
c. What is the average number of jobs waiting for service?
d. What is the average time a job waits before the welder can begin working on it?
e. What is the average number of hours between when a job is received and when it is completed?
f. W hat percentage of the time is Gubser's welder busy?

Question 9

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 checkout process to no more than five minutes, what recommendations would you provide regarding the current checkout system?

Question 10

Jobs arrive randomly at a particular assembly plant; assume that the arrival rate is five jobs per hour. Service times (in minutes per job) do not follow the exponential probability distribution. Two proposed designs for the plant's assembly operation are shown:
a. What is the service rate in jobs per hour for each design?
b. For the service rates in part (a), what design appears to provide the best or fastest service rate?
c. What are the standard deviations of the service times in hours?
d. Use the M / G /1 model to compute the operating characteristics for each design.
e. Which design provides the best operating characteristics? Why.

Question 11

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.

Question 12

The Robotics Manufacturing Company operates an equipment repair business where emergency jobs arrive randomly at the rate of three jobs per 8-hour day. The company's repair faculty is a single-channel system operated by a repair technician. The service time varies, with a mean repair time of 2 hours and a standard deviation of 1.5 hours. The company's cost of the repair operation is $28 per hour. In the economic analysis of the waiting line system, Robotics uses $35 per hour cost for customers waiting during the repair process.
a. What, are the arrival rate and service rate in jobs per hour?
b. Show the operating characteristics including the total cost per hour.
c. The company is considering purchasing a computer-based equipment repair system that would enable a constant repair time of 2 hours. For practical purposes, the standard deviation is 0. Because of the computer-based system, the company's cost of the new operation would be $32 per hour. The firm's director of operations said no to the request for new system because the hourly cost is $4 higher and the mean repair time is the same. Do you agree? What effect will the new system have on the waiting line characteristics of the repair service?
d. Does paying for the computer-based system to reduce the variation in service time make economic sense? How much will the new system save the company during a 40-hour workweek?

Question 13

Ocala Software Systems operates a technical support center for its software customers. If customers have 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 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 million 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 probability that a customer will have to wait for a consultant?
f. Ocalas 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 minutes 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?

Question 14

A large insurance company maintains a central computing system that contains a variety of information about customer accounts. Insurance agents in a six-state area use telephone lines to access the customer information database. Currently, the company's central computer system allows three users to access the central computer simultaneously. Agents who attempt to use the system when it is full are denied access; no waiting is allowed. Management realizes that with its expanding business, more requests will be made to the central information system. Being denied access to the system is inefficient as well as annoying for agents. Access requests follow a Poisson probability distribution, with a mean of 42 calls per hour. The service rate per line is 20 calls per hour.
a. What is the probability that 0, 1, 2, and 3 access lines will be in use?
b. What is the probability that an agent will be denied access to the system?
c. What is the average number of access lines in use?
d. In planning for the future management wants to be able to handle ? = 50 calls per hour; in addition, the probability that an agent will be denied access to the system should be no greater than computed in part (b). How many access lines should this system have?

Question 15

Regional Airlines is establishing a new telephone system for handling flight reservations. During the 10:00 A.M. to 11:00 A.M. time period, calls to the reservation agent occur randomly at an average of one call every 3.75 minutes. Historical service time data show that a reservation agent spends an average of 3 minutes with each customer. The waiting line model assumptions of Poisson arrivals and exponential service times appear reasonable for the telephone reservation system.
Regional Airlines' management believes that offering that offering an efficient telephone reservation system is an important part of establishing an image as a service-oriented airline. If the system is properly implemented, Regional Airlines will establish good customer relations, which in the long run will increase business. However, if the telephone reservation system is frequently overloaded and customers have difficulty contacting an agent, a negative customer reaction may lead to an eventual loss of business. The cost of a ticket reservation agent is $20 per hour. Thus, management wants to provide good service, but it does not want to incur the cost of overstaffing the telephone reservation operation by using more agents than necessary.
At a planning meeting, Regional's management team agreed that an acceptable customer service goal is to answer at least 85% of the incoming calls immediately. During the planning meeting, Regional's vice president of administration pointed out that the data show that the average service rate for an agent is faster than the average arrival rate of the telephone calls. The vice president's conclusion was that personnel costs could be minimized by using one agent and that the single agent should be able to handle the telephone reservations and still have some idle time. The vice president of marketing restated the importance of customer service and expressed support for at least two reservation agents.
The current telephone reservation system does not allow callers to wait. Callers who attempt to reach a reservation agent when all agents are occupied receive a busy signal and are blocked from the system. A representative from the telephone company suggested that Regional Airlines consider an expanded system that accommodates waiting. In the expanded system, when a customer calls and all agents are busy, a recorded message tells the customer that the call is being held in the order received and that an agent will be available shortly. The customer can stay on the line and listen to background music while waiting for an agent. Regional's management will need more information before switching to the expanded system.
Managerial Report
Prepare a managerial report for Regional Airlines analyzing the telephone reservation system. Evaluate both the system that does not allow waiting and the expanded system that allows waiting. Include the following information in your report:
1. An analysis of the current reservation system that does not allow callers to wait. How many reservation agents are needed to meet the service goal?
2. An analysis of the expanded system proposed by the telephone company. How many agents are needed to meet the service goal?
3. Make a recommendation concerning which system to use and how many agents to hire. Provide supporting rationale for your recommendation.
4. The telephone arrival data presented are for the 10:00 A.M. to 11:00 A.M. time period; however, the arrival rate if incoming calls is expected to change from hour to hour. Describe how your waiting line analysis could be used to develop a ticket agent staffing plan that would enable the company to provide different levels of staffing for the ticket reservation system at different times during the day. Indicate the information that you would need to develop this staffing plan.

Question 16

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 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 to wait for technical support and an average customer waiting time of two minutes or less? Discuss.

Question 17

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 an average of 12 calls per hour. The arrival rate is 20 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 average number of extension lines that will be busy if your recommendation in part (a) is used?
c. What percentage of calls receive a busy signal for the current telephone system with two extension lines?

Question 18

Willow Brook National Bank operates a drive-up teller window that allows customers to complete bank transactions without getting out of their cats. 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 customer 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?

Question 19

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?

Question 20

City Cab, Inc., uses two dispatchers to handle requests for service and to dispatch the cabs. The telephone calls that are made to City Cab use a common telephone number. When both dispatchers are busy, the caller hears a busy signal; no waiting is allowed. Callers who receive a busy signal can call back later or call another cab service. Assume that the arrival of calls follows a Poisson probability distribution, with a mean of 40 calls per hour, and that each dispatcher can handle a mean of 30 calls per hour.
a. What percentage of time are both dispatchers idle?
b. What percentage of time are both dispatchers busy?
c. What is the probability callers will receive a busy signal if two, three, or four dispatchers are used?
d. If management wants no more than 12% of the callers to receive a busy signal, how many dispatchers should be used?

Question 21

Office Equipment, Inc. (OEI), leases automatic mailing machines to business customers in Fort Wayne, Indiana. The company built its success on a reputation for providing timely maintenance and repair service. Each OEI service contract states that a service technician will arrive at a customer's business site within an average of three from the time that the customer notifies OEI of an equipment problem.
Currently, OEI has 10 customers with service contracts. One service technician is responsible for handling all services calls. A statistical analysis of historical service records indicates that a customer requests a service call at an average rate of one call per 50 hours of operation. If the service technician is available when a customer calls for service, it takes the technician an average of 1 hour of travel time to reach the customer's office and an average of 1.5 hours to complete the repair service. However, if the service technician is busy with another customer when a new customer calls for service, the technician completes the current service call and any other waiting service calls before responding to the new service call. In such cases, once the technician is free from all existing service commitments, the technician takes an average of 1 hour of travel time to reach the new customer's office and an average of 1.5 hours to complete the repair service. The cost of the service technician is $80 per hour. The down-time cost (wait time and service time) for customers is $100 per hour.
OEI is planning to expand its business. Within one year, OEI projects that it will have 20 customers, and within two years, OEI projects that it will have 30 customers. Although OEI is satisfied that one service technician can handle the 10 existing customers, management is concerned about the ability of one technician to meet the average three-hour service call guarantee when the OEI customer base expands. In a recent planning meeting, the marketing manager made a proposal to add a second service technician when OEI reaches 20 customers and to add a third service technician when OEI reaches 30 customers. Before making a final decision, management would like an analysis of OEI service capabilities. OEI is particularly interested in meeting the average three-hour waiting time guarantee at the lowest possible total cost.
Managerial Report
Develop a managerial report summarizing your analysis of the OEI service capabilities. Make recommendations regarding then number of technicians to be used when OEI reaches 20 customers and when OEI reaches 30 customers. Include a discussion of the following in your report:
1. What is the arrival rate for each customer per hour?
2. What is the service rate in terms of the number of customers per hour? Note that the average travel time of 1 hour becomes part of the service time because the time that the service technician is busy handling a service call includes the travel time plus the time required to complete the repair.
3. Waiting line models generally assume that the arriving customers are in the same location as the service facility. Discuss the OEI situation in light of the fact that a service technician travels an average of 1 hour to reach each customer. How should the travel time and the waiting time predicted by the waiting line model be combined to determine the total customer waiting time?
4. OEI is satisfied that one service technician can handle the 10 existing customers. Use a waiting line model to determine the following information:
• Probability that no customers are in the system
• Average number of customers in the waiting line
• Average number of customers in the system
• Average time a customer waits until the service technician arrives
• Average time a customer waits until the machine is back in operation
• Probability that a customer will have to wait more than one hour for the service technician to arrive
• The number of hours a week the technician is not making service calls
• The total cost per hour for the service operation
Do you agree with OEI management that one technician can meet the average three-hour service call guarantee? Explain.
5. What is your recommendation for the number of service technicians to hire when OEI expands to 20 customers? Use the information that you developed in part (4) to justify your answer.
6. What is your recommendation for the number of service technicians to hire when OEI expands to 30 customers? Use the information that you developed in part (4) to justify your answer.
7. What are the annual savings of your recommendation in part (6) compared to the planning committee's proposal that 30 customers will require three service technicians? Assume 250 days of operation per year.

Question 22

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 your system will be providing? Why or Why not?

Question 23

Kolkmeyer Manufacturing Company (see Section 11.9) is considering adding two machines to its manufacturing operation. This addition will bring the number of machines to eight. The president of Kolkmeyer asked for a study of the need to add a second employee service rate for each individual assigned to the repair operation is 0.50 machine per hour.
a. Compute the operating characteristics if the company retains the single-employee repair operation.
b. Compute the operating characteristics if a second employee is added to the machine repair operation.
c. Each employee is paid $20 per hour. Machine downtime is valued at $80 per hour. From an economic point of view, should one or more two employees handle the machine repair operation? Explain.

Question 24

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 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 minute or less?
c. What is the probability the service time is more than two minutes?

Question 25

All airplane passengers at the Lake City Regional Airport must pass through a security screening area 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 operating 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?

Question 26

Five administrative assistants use an office copier. The average time between arrivals for each assistant is 40 minutes, which is equivalent to an arrival rate of 1/40 = 0.025 arrival alent to a service rate of per minute. Use the M / M /1 model with a finite calling population to determine the following:
a. The probability that the copier is idle
b. The average number of administrative assistance in the waiting line
c. The average number of administrative assistance at the copier
d. The average time an assistance spends waiting for the copier
e. The average time an assistance spends at the copier
f. During an 8-hour day, how many minutes does an assistant spend at the copier? How much of this time is waiting time?
g. Should management consider purchasing a second copier? Explain.

Question 27

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 number time a customer spends waiting
e. The average number time a customer spends win the system
f. The probability that arriving customers will have to wait for service

Question 28

Refer again to the Lake City Regional Airport described in Problem 18. When the security level is raised to high, the service rate fir 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 station 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?

Question 29

Schips Department Store operates a fleet of 10 trucks. The trucks arrive at random times throughout the day at the store's truck dock to be loaded with new deliveries or to have incoming shipments from the regional warehouse unloaded. Each truck returns to the truck hour. The service rate is 4 trucks per hour. Using the Poisson arrivals and exponential service times model with a finite calling population of trucks, determine the following operating characteristics:
a. The probability no trucks are at the truck dock
b. The average number of trucks waiting for loading/unloading
c. The average number of trucks in the truck dock area
d. The average waiting time before loading/unloading begins
e. The average waiting time in the system
f. What is the hourly cost of operation if the cost is $50 per hour for each truck and $30 per hour for the truck dock?
g. Consider a two-channel truck dock operation where the second channel could be operated for an additional $30 per hour. How much would the average number of trucks waiting for loading/unloading have to be reduced to make the two-channel truck dock economically feasible?
h. Should the company consider expanding to the two-channel truck dock? Explain.

Question 30

Use the single-channel drive-up bank teller operation referred to in Problem 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?

Question 31

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 months. The service rate of a postal clerk is 0.75 customer 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 next two years the arrival rate at the post office during the busy winter month can be expected to be 2.1 customers per minute. Use a waiting line analysis to the post office manager.

Question 32

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 wailing 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?

Question 33

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 service time of 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.

Question 34

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?
d. What is the probability that an arriving customer will have to wait for service?
e. Do the operating characteristics indicate that the one-clerk checkout system provides an acceptable level of service?

Question 35

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 customer place orders at an intercom station at the back of the parking lot and then drive to service window to pay for and receive their orders. The following three service alternatives are being considered:
• 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.
• 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.
• A two-channel operation with two service window 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 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 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?

Question 36

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 being?
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?

Question 37

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 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?