Question: A laundromat has five washing machines. A

> Sometimes curvature in a scatterplot can be fit adequately (especially to the naked eye) by several trend lines. We discussed the exponential trend line, and the power trend line is discussed in the previous problem. Still another fairly simple trend lin

> The file contains the monthly number of airline tickets sold by a travel agency. a. Does a linear trend appear to fit these data well? If so, estimate and interpret the linear trend model for this time series. Also, interpret the R2 and se values. b. Pr

> A fast-food restaurant has one drive-through window. On average, 40 customers arrive per hour at the window. It takes an average of one minute to serve a customer. Assume that interarrival and service times are exponentially distributed. a. On average, h

> Consider a fast-food restaurant where customers enter at a rate of 75 per hour, and three servers are working. Customers wait in a single line and go, in FCFS fashion, to the first of the three servers who is available. Each server can serve one customer

> Consider a bank where potential customers arrive at rate of 60 customers per hour. However, because of limited space, one out of every four arriving customers finds the bank full and leaves immediately (without entering the bank). Suppose that the averag

> Based on Kolesar et al. (1974), Metropolis PD Precinct 88 must determine the minimum number of police cars required to meet its needs for the next 24 hours. An average call for service requires 30 minutes. The number of calls the police department expect

> Zerox has 16 service centers throughout the United States. Zerox is trying to determine how many technicians it should assign to each service center. How would you approach this problem?

> Little’s formula applies to an entire queueing system or to a subsystem of a larger system. For example, consider a single-server system composed of two subsystems. The first subsystem is the waiting line, and the second is the service area, where servic

> Based on Quinn et al. (1991). Winter Riggers handles approximately $400 million in telephone orders per year. Winter Riggers’ system works as follows. Callers are connected to an agent if one is available. Otherwise, they are put on hold (if a trunk line

> A company manufactures mechanical heart valves from the heart valves of pigs. Different heart operations require valves of different sizes. The company purchases pig valves from three different suppliers. The cost and size mix of the valves purchased fro

> Suppose that annually an average of  library patrons want to borrow a book. A patron borrows the book for an average of 1/ years. Suppose we observe that the book is actually borrowed an average of R times per year. Explain how we can estimate , which

> Excessive delays have recently been noted on New York City’s 911 system. Discuss how you would use queueing models to improve the performance of the 911 system.

> Bloomington Hospital knows that insurance companies are going to reduce the average length of stay of many types of patients. How can queueing models be used to determine how changes in insurance policies will influence the hospital?

> The mail order firm of L. L. Pea receives an average of 200 calls per hour, where times between calls are exponentially distributed. It takes an L. L. Pea operator an average of three minutes to handle a call. If a caller gets a busy signal, L. L. Pea as

> On average, 90 patrons arrive per hour at a hotel lobby (interarrival times are exponential) waiting to check in. At present there are five clerks, and patrons wait in a single line for the first available clerk. The average time for a clerk to service a

> The Newcoat Painting Company has for some time been experiencing high demand for its automobile repainting service. Because it has had to turn away business, management is concerned that the limited space available to store cars awaiting painting has cos

> The manager of a large group of employees must decide whether she needs another photocopying machine. The cost of a machine is $40 per eight-hour day regardless of whether the machine is in use. On average, four people need to use the copying machine per

> At the Franklin Post Office, patrons wait in a single line for the first open window. On average, 100 patrons enter the post office per hour, and each window can serve an average of 45 patrons per hour. The post office estimates a cost of $0.10 for each

> On average, 40 jobs arrive per day at a factory. The time between arrivals of jobs is exponentially distributed. The factory can process an average of 42 jobs per day, and the time to process a job is exponentially distributed. a. On average, how long do

> Assume that parts arrive at a machining center at a rate of 60 parts per hour. The machining center is capable of processing 75 parts per hour—that is, the mean time to machine a part is 0.8 minute. If you are watching these parts exiting the machine cen

> A company manufactures two types of trucks. Each truck must go through the painting shop and the assembly shop. If the painting shop were completely devoted to painting type 1 trucks, 650 per day could be painted, whereas if the painting shop were comple

> A printing shop receives an average of one order per day. The average length of time required to complete an order is half a day. At any given time, the print shop can work on at most one job. Interarrival times and service times are exponentially distri

> Ships arrive at a port facility at an average rate of two ships every three days. On average, it takes a single crew one day to unload a ship. Assume that interarrival and service times are exponential. The shipping company owns the port facility as well

> On average, 300 customers arrive per hour at a huge branch of Bank 2. It takes an average of two minutes to serve each customer. It costs $10 per hour to keep a teller window open, and the bank estimates that it will lose $50 in future profits for each h

> On average, 50 customers arrive per hour at a small post office. Interarrival times are exponentially distributed. Each window can serve an average of 25 customers per hour. Service times are exponentially distributed. It costs $25 per hour to open a win

> A company’s warehouse can store up to four units of a good. Each month, an average of 10 orders for the good are received. The times between the receipts of successive orders are exponentially distributed. When an item is used to fill an order, a replace

> US Airlines receives an average of 500 calls per hour from customers who want to make reservations, where the times between calls follow an exponential distribution. It takes an average of three minutes to handle each call. Each customer who buys a ticke

> A telephone-order sales company must determine how many telephone operators are needed to staff the phones during the 9-to-5 shift. It is estimated that an average of 480 calls are received during this time period and that the average call lasts for six

> We examined whether an M/M/1 system with a single fast server is better or worse than an M/M/s system with several slow servers. Keeping the same inputs as in the example, use simulation to see whether you obtain the same type of results as with the anal

> Simulate the system in Problem 10. Make any assumptions about the warm-up and run-time periods you believe are appropriate. Try solving the problem with exponentially distributed copying times. Then try it with gamma-distributed copying times, where the

> Given the model in the Multi-server Simulation.xlsm file, what unit cost parameters should be used if we are interested in “optimizing” the system? Choose representative inputs and unit costs, and then illustrate how to use the simulation outputs to esti

> There are three factories on the Momiss River. Each emits two types of pollutants, labeled P1 and P2, into the river. If the waste from each factory is processed, the pollution in the river can be reduced. It costs $1500 to process a ton of factory 1 was

> The current spreadsheet model essentially finds the expected profit in several steps. It first finds the profit in cell B19 for a fixed value of demand. Then it uses a data table to find the profit for each of several demands, and finally it uses SUMPROD

> Do exponentially distributed random numbers have the memoryless property? Here is one way to find out. Generate many exponentially distributed random numbers with mean 5, using the formula in the previous problem. Find the fraction of them that are great

> How long does it take to reach steady state? Use simulation, with the Multiserver Simulation.xlsm file, to experiment with the effect of warm-up time and run time on the key outputs. For each of the following, assume a five-server system with a Poisson a

> The Smalltown Credit Union experiences its greatest congestion on paydays from 11:30 a.m. until 1:00 p.m. During these rush periods, customers arrive according to a Poisson process at rate 2.1 per minute. The credit union employs 10 tellers for these rus

> The small mail-order firm Sea’s Beginning has one phone line. An average of 60 people per hour call in orders, and it takes an average of one minute to handle a call. Time between calls and time to handle calls are exponentially distributed. If the phone

> Two one-barber shops sit side by side in Dunkirk Square. Each shop can hold a maximum of four people, and any potential customer who finds a shop full will not wait for a haircut. Barber 1 charges $15 per haircut and takes an average of 15 minutes to com

> Consider the following two queueing systems. ■ System 1: An M/M/1 system with arrival rate  and service rate 3µ. ■ System 2: An M/M/3 system with arrival rate  and each server working at rate µ. Which system will have the smaller W and L?

> On average, 100 customers arrive per hour at Gotham City Bank. It takes a teller an average of two minutes to serve a customer. Interarrival and service times are exponentially distributed. The bank currently has four tellers working. The bank manager wa

> The manager of a bank wants to use an M/M/s queueing model to weigh the costs of extra tellers against the cost of having customers wait in line. The arrival rate is 60 customers per hour, and the average service time is four minutes. The cost of each te

> Referring to Problem 18, suppose the airline wants to determine how many checkpoints to operate to minimize operating costs and delay costs over a 10-year period. Assume that the cost of delaying a passenger for one hour is $10 and that the airport is op

> For the M/M/1 queueing model, why do the following results hold? a. W = (L + 1) /  b. WQ = L /

> A manufacturing company makes two products. Each product can be made on either of two machines. The time (in hours) required to make each product on each machine is listed in the file. Each month, 500 hours of time are available on each machine. Each mon

> A worker at the State Unemployment Office is responsible for processing a company’s forms when it opens for business. The worker can process an average of four forms per week. Last year, an average of 1.8 companies per week submitted forms for processing

> You can easily generate random numbers in a spreadsheet that have an exponential distribution with a given mean. For example, to generate 200 such numbers from an exponential distribution with  = 1/3, enter the formula =-3*LN(RAND()) in cell A4 and co

> A bank is trying to determine which of two machines to rent for check processing. Machine 1 rents for $10,000 per year and processes 1000 checks per hour. Machine 2 rents for $15,000 per year and processes 1600 checks per hour. Assume that machines work

> Consider an airport where taxis and customers arrive (exponential interarrival times) with respective rates of one and two per minute. No matter how many other taxis are present, a taxi will wait. If an arriving customer does not find a taxi, the custome

> The limited source model can often be used to approximate the behavior of a computer’s CPU (central processing unit). Suppose that 20 terminals (assumed to always be busy) feed the CPU. After the CPU responds to a user, the user takes an average of 80 se

> On average, 40 cars per hour are tempted to use the drive-through window at the Hot Dog King Restaurant. (We assume that interarrival times are exponentially distributed.) If a total of more than four cars are in line (including the car at the window), a

> A service facility consists of one server who can serve an average of two customers per hour (service times are exponential). An average of three customers per hour arrive at the facility (interarrival times are assumed to be exponential). The system cap

> On average, 100 customers arrive per hour at the Gotham City Bank. The average service time for each customer is one minute. Service times and interarrival times are exponentially distributed. The manager wants to ensure that no more than 1% of all custo

> MacBurger’s is attempting to determine how many servers to have available during the breakfast shift. On average, 100 customers arrive per hour at the restaurant. Each server can handle an average of 50 customers per hour. A server costs $10 per hour, an

> A furniture company manufactures tables and chairs. Each table and chair must be made entirely out of oak or entirely out of pine. A total of 25,000 board feet of oak and 20,000 board feet of pine are available. A table requires either 17 board feet of o

> In this problem, assume that all interarrival and service times are exponentially distributed. a. At present, the finance department and the marketing department each has its own typists. Each typist can type 25 letters per day. Finance requires that an

> A small bank is trying to determine the number of tellers to employ. The total cost of employing a teller is $100 per day, and a teller can serve an average of 160 customers per day. On average, 210 customers arrive per day at the bank, and both service

> Explain the basic relationship between the exponential distribution and a Poisson process. Also, explain how the exponential distribution and the Poisson distribution are fundamentally different.

> A supermarket is trying to decide how many cash registers to keep open. Suppose an average of 18 customers arrive each hour, and the average checkout time for a customer is four minutes. Interarrival times and service times are exponentially distributed,

> Each airline passenger and his luggage must be checked for security. Suppose that at Gotham City Airport, 3.6 passengers per minute arrive, on average. Also, assume that interarrival times are exponentially distributed. To check passengers for security,

> Expand the MMs Template.xlsm file so that the steady state probability distribution of the number in the system is shown in tabular form and graphically. That is, enter values 0, 1, and so on (up to some upper limit you can choose) in the range from cell

> In the M/M/s model, where µ is the service rate per server, explain why  , µ is not the appropriate condition for steady state, but  , sµ is.

> Suppose that you observe a sequence of interarrival times, such as 1.2, 3.7, 4.2, 0.5, 8.2, 3.1, 1.7, 4.2, 0.7, 0.3, and 2.0. For example, 4.2 is the time between the arrivals of customers 2 and 3. If you average these, what parameter of the M/M/s model

> For an M/M/1 queueing system, L =  /(µ - ). Suppose that l and µ are both doubled. How does L change? How does W change? How does WQ change? How does LQ change?

> Expand the MM1 Template.xlsx file so that the steady-state probability distribution of the number in the system is shown in tabular form and graphically. That is, enter values 0, 1, and so on (up to some upper limit you can choose) in the range from cell

> For a telephone survey, a marketing research group needs to contact at least 600 wives, 480 husbands, 400 single adult males, and 440 single adult females. It costs $3 to make a daytime call and (because of higher labor costs) $5 to make an evening call.

> The MM1 Template.xlsx file is now set up so that when you enter any time value in cell H11, the formula in cell I11 gives the probability that the wait in queue will be greater than this amount of time. Suppose that you would like the information to go t

> The MM1 Template.xlsx file is now set up so that you can enter any integer in cell E11 and the corresponding probability of that many in the system appears in cell F11. Change this setup so that columns E and F specify the distribution of the number in t

> The Decision Sciences Department is trying to determine whether to rent a slow or a fast copier. The department believes that an employee’s time is worth $15 per hour. The slow copier rents for $4 per hour, and it takes an employee an average of 10 minut

> An extremely important concept in queueing models is the difference between rates and times. If  represents a rate (customers per hour, say), then argue why 1/ is a time and vice versa.

> Rerun the new car simulation, but now introduce uncertainty into the fixed development cost. Let it be triangularly distributed with parameters $600 million, $650 million, and $850 million. (You can check that the mean of this distribution is $700 millio

> Health care is continually in the news. Can (or should) simulation be used to help solve, or at least study, some of the difficult problems associated with health care? Provide at least two examples where simulation might be useful.

> Software development is an inherently risky and uncertain process. For example, there are many examples of software that couldn’t be “finished” by the scheduled release date—bugs still remained and features weren’t ready. (Many people believe this was th

> Suppose you are a financial analyst and your company runs many simulation models to estimate the profitability of its projects. If you had to choose just two measures of the distribution of any important output such as net profit to report, which two wou

> You are an avid basketball fan, and you would like to build a simulation model of an entire game so that you could compare two different strategies, such as man to-man versus zone defense. Is this possible? What might make this simulation model difficult

> Suppose you are an HR (human resources) manager at a big university, and you sense that the university is becoming too top-heavy with full professors. That is, there do not seem to be as many younger professors at the assistant and associate levels as th

> Maggie Stewart loves desserts, but due to weight and cholesterol concerns, she has decided that she must plan her desserts carefully. There are two possible desserts she is considering: snack bars and ice cream. After reading the nutrition labels on the

> We have separated the examples in this chapter into operations, finance, marketing, and sports categories. List at least one other problem in each of these categories that could be attacked with simulation. For each, identify the random inputs, possible

> If a batch fails to pass inspection, the entire batch is unusable. Change the model so that if a batch fails to pass inspection, it is reworked, and at the end of the rework, its entire yield (the same yield determined in column C) is usable. However, th

> Suppose you are using an underwater probe to search for a sunken ship. At any time in the search, your probe is located at some point (x, y) in a grid, where the distance between lines in the grid is some convenient unit such as +00 meters. The sunken sh

> Nucleon is trying to determine whether to produce a new drug that makes pigs healthier. The product will be sold in years 1 to 5. The following information is relevant: ■ A fixed cost is incurred on January 1 of year 0 and will be between $1 billion and

> It is January 1 of year 0, and Merck is trying to determine whether to continue development of a new drug. The following information is relevant. You can assume that all cash flows occur at the ends of the respective years. ■ Clinical trials (the trials

> It is January 1 of year 0, and Lilly is considering developing a new drug called Dialis. We are given the following information ■ On March 15 of year 0, Lilly incurs a fixed cost that is assumed to follow a triangular distribution with best case $10 mill

> It is now May 1 of year 0, and GM is deciding whether to produce a new car. The following information is relevant. ■ The fixed cost of developing the car is incurred on January 1 of year 1 and is assumed to follow a triangular distribution with smallest

> In this version of “dice blackjack,” you toss a single die repeatedly and add up the sum of your dice tosses. Your goal is to come as close as possible to a total of 7 without going over. You may stop at any time. If your total is 8 or more, you lose. If

> Based on Bukiet et al. (1997). Many Major League teams (including Oakland, Boston, LA Dodgers, and Toronto) use mathematical models to evaluate baseball players. A common measure of a player’s offensive effectiveness is the number of runs generated per i

> The Ryder Cup is a three-day golf tournament played every other year with 12 of the best U.S. golfers against 12 of the best European golfers. They play 16 team matches (each match has two U.S. golfers against two European golfers) on Friday and Saturday

> Continuing the previous problem, perform a sensitivity analysis on the selling price of VXPs. Let this price vary from $500 to $650 in increments of $10, and keep track of the values in the decision variable cells and the objective cell. Discuss your fin

> A popular restaurant in Indianapolis does a brisk business, filling virtually all of its seats from 6 p.m. until 9 p.m. Tuesday through Sunday. Its current annual revenue is $2.34 million. However, it does not currently accept credit cards, and it is thi

> You are unemployed, 21 years old, and searching for a job. Until you accept a job offer, the following situation occurs. At the beginning of each year, you receive a job offer. The annual salary associated with the job offer is equally likely to be any n

> Suppose you want to run five simulations, where the probability of passing inspection is varied from 0.6 to 1.0 in increments of 0.1. Use the RISKSIMTABLE function appropriately to do this. Comment on the effect of this parameter on the key outputs. In p

> The Tinkan Company produces one-pound cans for the Canadian salmon industry. Each year the salmon spawn during a 24-hour period and must be canned immediately. Tinkan has the following agreement with the salmon industry. The company can deliver as many

> Chemcon has taken over the production of Nasacure from a rival drug company. Chemcon must build a plant to produce Nasacure by the beginning of 2010. Once the plant is built, the plant’s capacity cannot be changed. Each unit sold brings in $10 in revenue

> Based on Hoppensteadt and Peskin (1992). The following model (the Reed–Frost model) is often used to model the spread of an infectious disease. Suppose that at the beginning of period 1, the population consists of five diseased people (called infectives)

> Rework the previous problem for a case in which the one-year warranty requires you to pay for the new device even if failure occurs during the warranty period. Specifically, if the device fails at time t, measured relative to the time it went into use, y

> Suppose you buy an electronic device that you operate continuously. The device costs you $300 and carries a one-year warranty. The warranty states that if the device fails during its first year of use, you get a new device for no cost, and this new devic

> A truck manufacturer produces the Off Road truck. The company wants to gain information about the discounted profits earned during the next three years. During a given year, the total number of trucks sold in the United States is 500,000 1 50,000G – 40,0

> It costs a pharmaceutical company $75,000 to produce a 1000-pound batch of a drug. The average yield from a batch is unknown but the best case is 90% yield (that is, 900 pounds of good drug will be produced), the most likely case is 85% yield, and the wo

> A company faces the following demands during the next three weeks: week 1, 2000 units; week 2, 1000 units; week 3, 1500 units. The unit production costs during each week are as follows: week 1, $130; week 2, $140; week 3, $150. A holding cost of $20 per

> An automobile manufacturer is considering whether to introduce a new model called the Racer. The profitability of the Racer depends on the following factors: ■ The fixed cost of developing the Racer is triangularly distributed with parameters $3, $4, and

> Play Things is developing a new Lady Gaga doll. The company has made the following assumptions: ■ The doll will sell for a random number of years from 1 to 10. Each of these 10 possibilities is equally likely. ■ At the beginning of year 1, the potential