Question: Continuing the previous problem, perform a

> 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

> A laundromat has five washing machines. A typical machine breaks down once every five days. A repairman can repair a machine in an average of 2.5 days. Currently, three repairmen are on duty. The owner of the laundromat has the option of replacing them w

> 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

> 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

> You are considering a 10-year investment project. At present, the expected cash flow each year is $10,000. Suppose, however, that each year’s cash flow is normally distributed with mean equal to last year’s actual cash flow and standard deviation $1000.

> We commented on the 95th percentile on days required and the corresponding date to start production. If the company begins production on this date, then it is 95% sure to complete the order by the due date. We found this date to be August 2. Do you alway

> Mary Higgins is a freelance writer with enough spare time on her hands to play the stock market fairly seriously. Each morning she observes the change in stock price of a particular stock and decides whether to buy or sell, and if so, how many shares to

> Suppose you begin year 1 with $5000. At the beginning of each year, you put half of your money under a mattress and invest the other half in Whitewater stock. During each year, there is a 40% chance that the Whitewater stock will double, and there is a 6

> Consider an oil company that bids for the rights to drill in offshore areas. The value of the right to drill in a given offshore area is highly uncertain, as are the bids of the competitors. This problem demonstrates the “winner’s curse.” The winner’s cu

> A common decision is whether a company should buy equipment and produce a product in house or outsource production to another company. If sales volume is high enough, then by producing in house, the savings on unit costs will cover the fixed cost of the

> The DC Cisco office is trying to predict the revenue it will generate next week. Ten deals may close next week. The probability of each deal closing and data on the possible size of each deal (in millions of dollars) are listed in the file P11_55.xlsx. U

> A company is trying to determine the proper capacity level for its new electric car. A unit of capacity provides the potential to produce one car per year. It costs $15,000 to build a unit of capacity and the cost is charged equally over the next five ye

> During the next four months, a customer requires, respectively, 600, 800, 1200, and 900 units of a commodity, and no backlogging is allowed (that is, the customer’s requirements must be met on time). Production costs are $80, $100, $105, and $90 per unit

> The annual demand for Prizdol, a prescription drug manufactured and marketed by the NuFeel Company, is normally distributed with mean 50,000 and standard deviation 12,000. Assume that demand during each of the next 10 years is an independent random numbe

> Appliances Unlimited (AU) sells refrigerators. Any refrigerator that fails before it is three years old is replaced for free. Of all refrigerators, 2% fail during their first year of operation; 4% of all one-year-old refrigerators fail during their secon

> Consider a drill press containing three drill bits. The current policy (called individual replacement) is to replace a drill bit when it fails. The firm is considering changing to a block replacement policy in which all three drill bits are replaced when

> Consider a device that requires two batteries to function. If either of these batteries dies, the device will not work. Currently there are two new batteries in the device, and there are three extra new batteries. Each battery, once it is placed in the d

> The gamma distribution was used to model the skewness to the right of the lifetime distribution. Experiment to see whether the triangular distribution could have been used instead. Let its minimum value be 0, and choose its most likely and maximum values

> You have been asked to simulate the cash inflows to a toy company for the next year. Monthly sales are independent random variables. Mean sales for the months January through March and October through December are $80,000, and mean sales for the months A

> Based on Marcus (1990). The Balboa mutual fund has beaten the Standard and Poor’s 500 during 11 of the last 13 years. People use this as an argument that you can beat the market. Here is another way to look at it that shows that Balboa’s beating the mark

> A ticket from Indianapolis to Orlando on Deleast Airlines sells for $150. The plane can hold 100 people. It costs Deleast $8000 to fly an empty plane. Each person on the plane incurs variable costs of $30 (for food and fuel). If the flight is overbooked,

> Suppose you have invested 25% of your portfolio in four different stocks. The mean and standard deviation of the annual return on each stock are shown in the file P11_46.xlsx. The correlations between the annual returns on the four stocks are also shown

> You now have $10,000, all of which is invested in a sports team. Each year there is a 60% chance that the value of the team will increase by 60% and a 40% chance that the value of the team will decrease by 60%. Estimate the mean and median value of your

> A farmer owns 450 acres of land. He is going to plant each acre with wheat or corn. Each acre planted with wheat yields $2000 profit, requires three workers, and requires two tons of fertilizer. Each acre planted with corn yields $3000 profit, requires t

> You now have $5000. You will toss a fair coin four times. Before each toss you can bet any amount of your money (including none) on the outcome of the toss. If heads comes up, you win the amount you bet. If tails comes up, you lose the amount you bet. Yo

> You are playing Serena Williams in tennis, and you have a 42% chance of winning each point. (You are good!) a. Use simulation to estimate the probability you will win a particular game. Note that the first player to score at least four points and have at

> Based on Morrison and Wheat (1984). When his team is behind late in the game, a hockey coach usually waits until there is one minute left before pulling the goalie out of the game. Using simulation, it is possible to show that coaches should pull their g

> Consider the following card game. The player and dealer each receive a card from a 52-card deck. At the end of the game the player with the highest card wins; a tie goes to the dealer. (You can assume that Aces count 1, Jacks 11, Queens 12, and Kings 13.

> You are going to play the Wheel of Misfortune Game against the house. The wheel has 10 equally likely numbers: 5, 10, 15, 20, 25, 30, 35, 40, 45 ,and 50. The goal is to get a total as close as possible to 50 points without exceeding 50. You go first and

> For each part, make the change indicated, run the simulation, and comment on any differences between your outputs and the outputs in the example. a. The cost of a new camera is increased to $500. b. The warranty period is decreased to one year. c. The te

> Assume a very good NBA team has a 70% chance of winning in each game it plays. During an 82 game season what is the average length of the team’s longest winning streak? What is the probability that the team has a winning streak of at least 16 games? Use

> You have $5 and your opponent has $10. You flip a fair coin and if heads comes up, your opponent pays you $1. If tails comes up, you pay your opponent $1. The game is finished when one player has all the money or after 100 tosses, whichever comes first.

> The game of Chuck-a-Luck is played as follows: You pick a number between 1 and 6 and toss three dice. If your number does not appear, you lose $1. If your number appears x times, you win $x. On the average, use simulation to find the average amount of mo

> A martingale betting strategy works as follows. You begin with a certain amount of money and repeatedly play a game in which you have a 40% chance of winning any bet. In the first game, you bet $1. From then on, every time you win a bet, you bet $1 the n

> A furniture company manufactures desks and chairs. Each desk uses four units of wood, and each chair uses three units of wood. A desk contributes $250 to profit, and a chair contributes $145. Marketing restrictions require that the number of chairs produ

> The Mutron Company is thinking of marketing a new drug used to make pigs healthier. At the beginning of the current year, there are 1,000,000 pigs that could use the product. Each pig will use Mutron’s drug or a competitor’s drug once a year. The number

> Suppose that GLC earns a $2000 profit each time a person buys a car. We want to determine how the expected profit earned from a customer depends on the quality of GLC’s cars. We assume a typical customer will purchase 10 cars during her lifetime. She wil

> We are all aware of the fierce competition by mobile phone service companies to get our business. For example, AT&T is always trying to attract Verizon’s customers, and vice versa. Some even give away prizes to entice us to sign up for a guaranteed lengt

> The customer loyalty model in Example 11.9 assumes that once a customer leaves (becomes disloyal), that customer never becomes loyal again. Assume instead that there are two probabilities that drive the model, the retention rate and the rejoin rate, with

> Based on Babich (1992). Suppose that each week each of 300 families buys a gallon of orange juice from company A, B, or C. Let pA denote the probability that a gallon produced by company A is of unsatisfactory quality, and define pB and pC similarly for

> Seas Beginning sells clothing by mail order. An important question is when to strike a customer from the company’s mailing list. At present, the company strikes a customer from its mailing list if a customer fails to order from six consecutive catalogs.

> If the average bid for each competitor stays the same, but their bids exhibit less variability, does Miller’s optimal bid increase or decrease? To study this question, assume that each competitor’s bid, expressed as a multiple of Miller’s cost to complet