For each of the following queuing systems, indicate whether it is a single- or multiple-server model, the queue discipline, and whether its calling population is infinite or finite.
In the Fast Shop Market example in this chapter, Alternative II was to add a new checkout counter at the market. This alternative was analyzed using the single-server model. Why was the multiple-server model not used?
Why do waiting lines form at a service facility even though there may be more than enough service capacity to meet normal demand in the long run?
Provide an example of when a first-in, first-out (FIFO) rule for queue discipline would not be appropriate.
Under what conditions will the single-server queuing model with Poisson arrivals and undefined service times provide the same operating characteristics as the basic model with exponentially distributed service times?
A single-server queuing system with an infinite calling population and a first-come, first- served queue discipline has the following arrival and service rates:
l = 16 customers per hour
µ = 24 customers per hour
Determine P , P 3 , L , L q , W , W q , and U .
The ticket booth on the Tech campus is operated by one person, who is selling tickets for the annual Tech versus State football game on Saturday. The ticket seller can serve an average of 12 customers per hour; on average, 10 customers arrive to purchase tickets each hour. Determine the average time a ticket buyer must wait and the portion of time the ticket seller is busy.
The Petroco Service Station has one pump for regular unleaded gas, which (with an attendant) can service 10 customers per hour. Cars arrive at the regular unleaded pump at a rate of 6 per hour. Determine the average queue length, the average time a car is in the system, and the average time a car must wait. If the arrival rate increases to 12 cars per hour, what will be the effect on the average queue length?
The Dynaco Manufacturing Company produces a particular product in an assembly line operation. One of the machines on the line is a drill press that has a single assembly line feeding into it. A partially completed unit arrives at the press to be worked on every 7.5 minutes, on average. The machine operator can process an average of 10 parts per hour. Determine the average number of parts waiting to be worked on, the percentage of time the operator is working, and the percentage of time the machine is idle.
The management of Dynaco Manufacturing Company (see Problem 10) likes to have its operators working 90% of the time. What must the assembly line arrival rate be for the operators to be as busy as management would like?
The Peachtree Airport in Atlanta serves light aircraft. It has a single runway and one air traffic controller to land planes. It takes an airplane 12 minutes to land and clear the runway. Planes arrive at the airport at the rate of four per hour.
The First American Bank of Rapid City has one outside drive-up teller. It takes the teller an average of 4 minutes to serve a bank customer. Customers arrive at the drive-up window at a rate of 12 per hour. The bank operations officer is currently analyzing the possibility of adding a second drive-up window, at an annual cost of $20,000. It is assumed that arriving cars would be equally divided between both windows . The operations officer estimates that each minute's reduction in customer waiting time would increase the bank's revenue by $2,000 annually. Should the second drive-up window be installed?
During registration at State University every semester, students in the college of business must have their courses approved by the college adviser. It takes the adviser an average of 2 minutes to approve each schedule, and students arrive at the adviser's office at the rate of 28 per hour.
All trucks traveling on Interstate 40 between Albuquerque and Amarillo are required to stop at a weigh station. Trucks arrive at the weigh station at a rate of 200 per 8-hour day, and the station can weigh, on the average, 220 trucks per day.
In Problem 15, suppose passing truck drivers look to see how many trucks are waiting to be weighed at the weigh station. If they see four or more trucks in line, they will pass by the station and risk being caught and ticketed. What is the probability that a truck will pass by the station?
In Problem 14, the dean of the college of business at State University is considering the addition of a second adviser in the college advising office to serve students waiting to have their schedules approved. This new adviser could serve the same number of students per hour as the present adviser. Determine L , L q , W , and W q for this altered advising system. As a student, would you recommend adding the adviser?
The Dynaco Manufacturing Company has an assembly line that feeds two drill presses. As partially completed products come off the line, they are lined up to be worked on as drill presses become available. The units arrive at the workstation (containing both presses) at the rate of 100 per hour. Each press operator can process an average of 60 units per hour. Compute L , L q , W , and W q .
The Acme Machine Shop has five machines that periodically break down and require service. The average time between breakdowns is 4 days, distributed according to an exponential distribution. The average time to repair a machine is 1 day, distributed according to an exponential distribution. One mechanic repairs the machines in the order in which they break down.
McBurger's fast-food restaurant has a drive-through window with a single server who takes orders from an intercom and also is the cashier. The window operator is assisted by other employees who prepare the orders. Customers arrive at the ordering station prior to the drive-through window every 4.5 minutes (Poisson distributed), and the service time is 2.8 minutes (exponentially distributed). Determine the average length of the waiting line and the waiting time. Discuss the quality of the service.
Game World, a video game arcade at Tanglewood Mall, has just installed a new virtual reality battle game. It requires exactly 2.7 minutes to play. Customers arrive, on average, every 2.9 minutes (Poisson distributed) to play the game. How long will the line of customers waiting to play the game be, and how long, on average, must a customer wait?
Drivers who come to get their licenses at the Department of Motor Vehicles have their photograph taken by an automated machine that develops the photograph onto the license card and laminates the complete license. The machine requires a constant time of 4.5 minutes to prepare a completed license. If drivers arrive at the machine at the mean rate of 10 per hour (Poisson distributed), determine the average length of the waiting line and the average waiting time.
A vending machine at City Airport dispenses hot coffee, hot chocolate, or hot tea, in a constant service time of 20 seconds. Customers arrive at the vending machine at a mean rate of 60 per hour (Poisson distributed). Determine the average length of the waiting line and the average time a customer must wait.
Norfolk, Virginia, a major seaport on the East Coast , has a ship coal-loading facility. Coal trucks filled with coal arrive at the port facility at the mean rate of 149 per day (Poisson distributed). The facility operates 24 hours a day. The coal trucks are unloaded one at a time, on a first-come, first-served basis, by automated mechanical equipment that empties the trucks in a constant time of 8 minutes per truck, regardless of truck size . The port authority is negotiating with a coal company for an additional 30 trucks per day. However, the coal company will not use this port facility unless the port authority can assure it that its coal trucks will not have to wait to be unloaded at the port facility for more than 12 hours per truck, on average. Can the port authority provide this assurance?
The Bay City Police Department has eight patrol cars that are on constant call 24 hours per day. A patrol car requires repairs every 20 days, on average, according to an exponential distribution. When a patrol car is in need of repair, it is driven into the motor pool, which has a repair person on duty at all times. The average time required to repair a patrol car is 18 hours (exponentially distributed). Determine the average time a patrol car is not available for use and the average number of patrol cars out of service at any one time. Indicate whether the repair service seems adequate.
The Rowntown Cab Company has four cabs that are on duty during normal business hours. The cab company dispatcher receives requests for service every 8 minutes, on average, according to an exponential distribution. The average time to complete a trip is 20 minutes (exponentially distributed). Determine the average number of customers waiting for service and the average time a customer must wait for a cab.
The Riverton Post Office has four stations for service. Customers line up in single file for service on a FIFO basis. The mean arrival rate is 40 per hour, Poisson distributed, and the mean service time per server is 4 minutes, exponentially distributed. Compute the operating characteristics for this operation. Indicate whether the operation appears to be satisfactory in terms of the following: (a) postal workers' (servers') idle time; (b) customer waiting time and/or the number waiting for service; and (c) the percentage of the time a customer can walk in and get served without waiting at all.
In Problem 18, the Dynaco Company has found that if more than three units (average) are waiting to be processed at any one workstation, then too much money is being tied up in work-in-process inventory (i.e., units waiting to be processed). The company estimates that (on average) each unit waiting to be processed costs $50 per day. Alternatively, operating a third press would cost $150 per day. Should the company operate a third press at this workstation?
Cakes baked by the Freshfood Bakery are transported from the ovens to be packaged by one of three wrappers. Each wrapper can wrap an average of 200 cakes per hour. The cakes are brought to the wrappers at the rate of 500 per hour. If a cake sits longer than 5 minutes before being wrapped, it will not be fresh enough to meet the bakery's quality control standards. Does the bakery need to hire another wrapper?
The associate dean of the college of business at Tech is determining which of two copiers he should lease for the college's administrative suite. A regular copier leases for $8 per hour, and it takes an employee an average of 6 minutes (exponentially distributed) to complete a copying job. A deluxe, high-speed copier leases for $16 per hour, and it requires an average of 3 minutes to complete a copying job. Employees arrive at the copying machine at a rate of 7 per hour (Poisson distributed), and an employee's time is valued at $10 per hour. Determine which copier the college should lease.
The Quick Wash 24-hour Laundromat has 16 washing machines. A machine breaks down every 20 days (exponentially distributed). The repair service with which the laundromat contracts takes an average of 1 day to repair a machine (exponentially distributed). A washing machine averages $5 per hour in revenue. The laundromat is considering a new repair service that guarantees repairs in 0.50 days, but it charges $10 more per hour than the current repair service. Should the laundromat switch to the new repair service?
The Regency Hotel has enough space at its entrance for six taxicabs to line up, wait for guests, and then for loading passengers. Cabs arrive at the hotel every 8 minutes; if a taxi drives by the hotel and the line is full, then it must drive on. Hotel guests require a taxi every 5 minutes, on average. It takes a cab driver an average of 3.5 minutes to load passengers and luggage and leave the hotel (exponentially distributed).
The local Burger Doodle fast-food restaurant has a drive-through window. Customers in cars arrive at the window at the rate of 10 per hour (Poisson distributed). It requires an average of 4 minutes (exponentially distributed) to take and fill an order. The restaurant chain has a service goal of an average waiting time of 3 minutes.
From 3:00 P.M. to 8:00 P.M. the local K-Star Supermarket has a steady stream of customers. Customers finish shopping and arrive at the checkout area at a rate of 70 per hour (Poisson distributed). It is assumed that when customers arrive at the cash registers, they will divide themselves relatively evenly so that all the checkout lines contain an equal number. The average checkout time at a register is 7 minutes (exponentially distributed). The store manager's service goal is for customers to be out of the store within 12 minutes (on average) after they complete their shopping and arrive at a cash register. How many cash registers must the store have open to achieve the manager's service goal?
Customers arrive to check in at the exclusive and expensive Regency Hotel's lobby at a rate of 40 per hour (Poisson distributed). The hotel normally has three clerks available at the desk to check in guests. The average time for a clerk to check in a guest is 4 minutes (exponentially distributed). Clerks at the Regency are paid $24 per hour, and the hotel assigns a goodwill cost of $2 per minute for the time a guest must wait in line. Determine whether the present check-in system is cost-effective . If it is not, recommend what hotel management should do.
The Riverview Clinic has two general practitioners who see patients daily. An average of six patients arrive at the clinic per hour. Each doctor spends an average of 15 minutes with a patient. The patients wait in a waiting area until one of the two doctors is able to see them. However, because patients typically do not feel well when they come to the clinic, the doctors do not believe it is good practice to have a patient wait longer than an average of 15 minutes. Should this clinic add a third doctor, and, if so, will this alleviate the waiting problem?
The Footrite Shoe Company is going to open a new branch at a mall, and company managers are attempting to determine how many salespeople to hire. Based on an analysis of mall traffic, the company estimates that customers will arrive at the store at a rate of 10 per hour, and from past experience at its other branches, the company knows that salespeople can serve an average of 6 customers per hour. How many salespeople should the company hire to uphold a company policy that on average the probability of a customer having to wait for service be no more than .30?
When customers arrive at Gilley's Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers, the store's staff knows that customers arrive at a rate of 40 per hour on summer days between 3:00 P.M. and 10:00 P.M. , and a server can serve 15 customers per hour on average. Gilley's wants to make sure that customers wait no longer than 10 minutes for service. Gilley's is contemplating keeping three servers behind the ice cream counter during the peak summer hours. Will this number be adequate to meet the waiting time policy?
Moore's television repair service receives an average of six TV sets per 8-hour day to be repaired. The service manager would like to be able to tell customers that they can expect 1-day service. What average repair time per set will the repair shop have to achieve to provide 1-day service, on average? (Assume that the arrival rate is Poisson distributed and repair times are exponentially distributed.)
In Problem 39, suppose that Moore's television repair service cannot accommodate more than 30 TV sets at a time. What is the probability that the number of TV sets on hand (under repair and waiting for service) will exceed the shop capacity?
Maggie Attaberry is a nurse on the evening shift from 10:00 P.M. to 6:00 A.M. at Community Hospital. She has 15 patients for whom she is responsible in her area. She averages two calls from each of her patients every evening, on average (Poisson distributed), and she must spend an average of 10 minutes (negative exponential distribution) with each patient who calls. Nurse Attaberry has indicated to her shift supervisor that, although she has not kept records, she believes her patients must wait about 10 minutes, on average, for her to respond, and she has requested that her supervisor assign a second nurse to her area. The supervisor believes 10 minutes is too long for a patient to wait, but she does not want her nurses to be idle more than 40% of the time. Determine what the supervisor should do.
The Escargot is a small French restaurant with six waiters and waitresses. The average service time at the restaurant for a table (of any size) is 85 minutes (Poisson distributed). The restaurant does not take reservations , and parties arrive for dinner (and stay and wait) every 18 minutes (negative exponential distribution). The restaurant owner is concerned that a lengthy waiting time might hurt its business in the long run. What are the current waiting time and queue length for the restaurant? Discuss the business implications of the current waiting time and any actions the restaurant owner might take.
Hudson Valley Books is a small, independent publisher of fiction and nonfiction books. Each week the publisher receives an average of seven unsolicited manuscripts to review (Poisson distributed). The publisher has 12 freelance reviewers in the area who read and evaluate manuscripts. It takes a reviewer an average of 10 days (exponentially distributed) to read a manuscript and write a brief synopsis. (Reviewers work on their own, 7 days a week.) Determine how long the publisher must wait, on average, to receive a reviewer's manuscript evaluation, how many manuscripts are waiting to be reviewed, and how busy the reviewers are.
Amanda Fall is starting up a new house painting business, Fall Colors. She has been advertising in the local newspaper for several months. Based on inquiries and informal surveys of the local housing market, she anticipates that she will get painting jobs at the rate of four per week (Poisson distributed). Amanda has also determined that it will take a four-person team of painters an average of 0.7 week (exponentially distributed) for a typical painting job.
Partially completed products arrive at a workstation in a manufacturing operation at a mean rate of 40 per hour (Poisson distributed). The processing time at the workstation averages 1.2 minutes per unit (exponentially distributed). The manufacturing company estimates that each unit of work-in-process inventory at the workstation costs $31 per day (on average). However, the company can add extra employees and reduce the processing time to 0.90 minute per unit, at a cost of $52 per day. Determine whether the company should continue the present operation or add extra employees.
The Atlantic Coast Shipping Company has a warehouse terminal in Spartanburg, South Carolina. The capacity of each terminal dock is three trucks. As trucks enter the terminal, the drivers receive numbers ; and when one of the three dock spaces becomes available, the truck with the lowest number enters the vacant dock. Truck arrivals are Poisson distributed, and the unloading and loading times (service times) are exponentially distributed. The average arrival rate at the terminal is five trucks per hour, and the average service rate per dock is two trucks per hour (30 minutes per truck).
The Waterfall Buffet in the lower level of the National Art Gallery serves food cafeteria-style daily to visitors and employees. The buffet is self-service. From 7:00 A.M. to 9:00 A.M. customers arrive at the buffet at a rate of 10 per minute; from 9:00 A.M. to noon, at 4 per minute; from noon to 2:00 P.M. , at 14 per minute; and from 2:00 P.M. to closing at 5:00 P.M. , at 8 per minute. All the customers take about the same amount of time to serve themselves from the buffet. Once a customer goes through the buffet, it takes 0.4 minute to pay the cashier. The gallery does not want a customer to have to wait longer than 4 minutes to pay. How many cashiers should be working at each of the four times during the day?
The Clip Joint is a hairstyling salon at University Mall. Four stylists are always available to serve customers on a first-come, first-served basis. Customers arrive at an average rate of five per hour, and the stylists spend an average of 35 minutes on each customer.
The Delacroix Inn in Alexandria is a small, exclusive hotel with 20 rooms. Guests can call housekeeping from 8:00 A.M. to midnight for any of their service needs. Housekeeping keeps one person on duty during this time to respond to guest calls. Each room averages 0.7 call per day to housekeeping (Poisson distributed), and a guest request requires an average of 30 minutes (exponentially distributed) for the staff person to respond to and take care of. Determine the portion of time the staff person is busy and how long a guest must wait for his or her request to be addressed. Does the housekeeping system seem adequate?
Jim Carter builds custom furniture, primarily cabinets , bookcases, small tables, and chairs. He works on only one piece of furniture for a customer at a time. It takes him an average of 5 weeks (exponentially distributed) to build a piece of furniture. An average of 14 customers approach Jim to order pieces of furniture each year (Poisson distributed); however, Jim will take only a maximum of 8 advance orders. Determine the average time a customer must wait to receive a furniture order once it is placed and how busy Jim is. What is the probability that a customer will be able to place an order with Jim?
Judith Lewis is a doctoral student at State University, and she also works full-time as an academic tutor for 10 scholarshiped student athletes. She took the job, hoping it would leave her free time between tutoring to devote to her own studies. An athlete visits her for tutoring an average of every 16 hours, and she spends an average of 1.5 hours (exponentially distributed) with him or her. She is able to tutor only one athlete at a time, and athletes study while they are waiting.
TSA security agents at the security gate at the Tri-Cities Regional Airport are able to check passengers and process them through the security gate at a rate of 50 per hour (Poisson distributed). Passengers arrive at the gate throughout the day at a rate of 40 per hour (Poisson distributed). Determine the average waiting time and waiting line.
In Problem 52 the passenger traffic arriving at the airport security gate varies significantly during the day. At times close to flight takeoffs, the traffic is very heavy, while at other times there is little or no passenger traffic through the security gate. At the times leading up to flight takeoffs, passengers arrive at a rate of 110 per hour (Poisson distributed). Suggest a waiting line system that can accommodate this level of passenger traffic.
The inland port at Front Royal, Virginia, is a transportation hub that primarily transfers shipping containers from trucks coming from eastern seaports to rail cars for shipment to inland destinations. Containers arrive at the inland ports at a rate of eight per hour. It takes 28 minutes (exponentially distributed) for a crane to unload a container from a truck, place it on a flatbed railcar, and secure it. Suggest a waiting line system that will effectively handle this level of container traffic at the inland port.
The Old Colony theme park has a new ride, the Double Cyclone. The ride holds 30 people in double roller coaster cars, and it takes 3.8 minutes to complete the ride circuit. It also takes the ride attendants another 3 minutes (with virtually no variation) to load and unload passengers. Passengers arrive at the ride during peak park hours at a rate of 4 per minute (Poisson distributed). Determine the length of the waiting line for the ride.