1 1 Slide Chapter 13 Simulation n Advantages and Disadvantages of Using Simulation n Modeling n Random Variables and Pseudo-Random Numbers n Time Increments

1 1 Slide Slide

Chapter 13Chapter 13SimulationSimulation

Advantages and Disadvantages of Using Advantages and Disadvantages of Using SimulationSimulation

ModelingModeling Random Variables and Pseudo-Random Random Variables and Pseudo-Random

NumbersNumbers Time Increments Time Increments Simulation LanguagesSimulation Languages Validation and Statistical ConsiderationsValidation and Statistical Considerations ExamplesExamples

2 2 Slide Slide

SimulationSimulation

SimulationSimulation is one of the most frequently is one of the most frequently employed management science techniques. employed management science techniques.

It is typically used to model It is typically used to model random processesrandom processes that are too complex to be solved by analytical that are too complex to be solved by analytical methods.methods.

3 3 Slide Slide

Advantages of SimulationAdvantages of Simulation

Among the advantages of simulation is the Among the advantages of simulation is the ability to gain insightsability to gain insights into the model solution into the model solution which may be impossible to attain through other which may be impossible to attain through other techniques. techniques.

Also, once the simulation has been developed, it Also, once the simulation has been developed, it provides a provides a convenient experimental laboratoryconvenient experimental laboratory to perform "what if" and sensitivity analysis.to perform "what if" and sensitivity analysis.

4 4 Slide Slide

Disadvantages of SimulationDisadvantages of Simulation

A large amount of time may be required to A large amount of time may be required to develop the simulation.develop the simulation.

There is no guarantee that the solution obtained There is no guarantee that the solution obtained will actually be optimal. will actually be optimal.

Simulation is, in effect, a Simulation is, in effect, a trial and error methodtrial and error method of comparing different policy inputs. of comparing different policy inputs.

It does not determine if some input which was It does not determine if some input which was not considered could have provided a better not considered could have provided a better solution for the model.solution for the model.

5 5 Slide Slide

Simulation ModelingSimulation Modeling

One begins a simulation by developing a One begins a simulation by developing a mathematical statement of the problem. mathematical statement of the problem.

The model should be realistic yet solvable The model should be realistic yet solvable within the speed and storage constraints of the within the speed and storage constraints of the computer system being used. computer system being used.

Input values for the model as well as probability Input values for the model as well as probability estimates for the random variables must then estimates for the random variables must then be determined.be determined.

6 6 Slide Slide

Random VariablesRandom Variables

Random variable values are utilized in the model Random variable values are utilized in the model through a technique known as through a technique known as Monte Carlo Monte Carlo simulationsimulation. .

Each random variable is mapped to a set of Each random variable is mapped to a set of numbers so that each time one number in that numbers so that each time one number in that set is generated, the corresponding value of the set is generated, the corresponding value of the random variable is given as an input to the random variable is given as an input to the model. model.

The mapping is done in such a way that the The mapping is done in such a way that the likelihood that a particular number is chosen is likelihood that a particular number is chosen is the same as the probability that the the same as the probability that the corresponding value of the random variable corresponding value of the random variable occurs.occurs.

7 7 Slide Slide

Pseudo-Random NumbersPseudo-Random Numbers

Because a computer program generates random Because a computer program generates random numbers for the mapping according to some numbers for the mapping according to some formula, the numbers are not truly generated in formula, the numbers are not truly generated in a random fashion. a random fashion.

However, using standard statistical tests, the However, using standard statistical tests, the numbers can be shown to appear to be drawn numbers can be shown to appear to be drawn from a random process. from a random process.

These numbers are called These numbers are called pseudo-random pseudo-random numbersnumbers. .

8 8 Slide Slide

Time IncrementsTime Increments

In a In a fixed-time simulation modelfixed-time simulation model, time periods , time periods are incremented by a fixed amount. For each are incremented by a fixed amount. For each time period a different set of data from the time period a different set of data from the input sequence is used to calculate the effects input sequence is used to calculate the effects on the model.on the model.

In a In a next-event simulation modelnext-event simulation model, time periods , time periods are not fixed but are determined by the data are not fixed but are determined by the data values from the input sequence.values from the input sequence.

9 9 Slide Slide

Simulation ProgramsSimulation Programs

The computer program that performs the The computer program that performs the simulation is called a simulation is called a simulatorsimulator. .

FlowchartsFlowcharts can be useful in writing such a can be useful in writing such a program. program.

While this program can be written in any general While this program can be written in any general purpose language (e.g. BASIC, FORTRAN, C++, purpose language (e.g. BASIC, FORTRAN, C++, etc.) special languages which reduce the amount etc.) special languages which reduce the amount of code which must be written to perform the of code which must be written to perform the simulation have been developed. simulation have been developed.

Special simulation languagesSpecial simulation languages include SIMSCRIPT, include SIMSCRIPT, SPSS, DYNAMO, and SLAM.SPSS, DYNAMO, and SLAM.

10 10 Slide Slide

Model Verification/ValidationModel Verification/Validation

Verification/validation of both the model and Verification/validation of both the model and the method used by the computer to carry out the method used by the computer to carry out the calculations is extremely important. the calculations is extremely important.

Models which do not reflect real world Models which do not reflect real world behavior cannot be expected to generate behavior cannot be expected to generate meaningful results. meaningful results.

Likewise, errors in programming can result in Likewise, errors in programming can result in nonsensical results. nonsensical results.

11 11 Slide Slide

Model Verification/ValidationModel Verification/Validation

Validation is generally done by having an expert Validation is generally done by having an expert review the model and the computer code for review the model and the computer code for errors.errors.

Ideally, the simulation should be run using actual Ideally, the simulation should be run using actual past data. past data.

Predictions from the simulation model should be Predictions from the simulation model should be compared with historical results.compared with historical results.

12 12 Slide Slide

Experimental DesignExperimental Design

Experimental designExperimental design is an important consideration is an important consideration in the simulation process. in the simulation process.

Issues such as the length of time of the simulation Issues such as the length of time of the simulation and the treatment of initial data outputs from the and the treatment of initial data outputs from the model must be addressed prior to collecting and model must be addressed prior to collecting and analyzing output data. analyzing output data.

Normally one is interested in results for the Normally one is interested in results for the steady steady state state (long run) operation of the system being (long run) operation of the system being modeled.modeled.

The initial data inputs to the simulation generally The initial data inputs to the simulation generally represent a represent a start-up periodstart-up period for the process and it for the process and it may be important that the data outputs for this may be important that the data outputs for this start-up period be neglected for predicting this long start-up period be neglected for predicting this long run behavior.run behavior.

13 13 Slide Slide

Experimental DesignExperimental Design

For each policy under consideration by the For each policy under consideration by the decision maker, the simulation is run by decision maker, the simulation is run by considering a long sequence of input data values considering a long sequence of input data values (given by a pseudo-random number generator). (given by a pseudo-random number generator).

Whenever possible, different policies should be Whenever possible, different policies should be compared by using the same sequence of input compared by using the same sequence of input data.data.

14 14 Slide Slide

Example: Dynogen, Inc.Example: Dynogen, Inc.

The price change of shares of Dynogen, The price change of shares of Dynogen, Inc. has been observed over the past 50 trades. Inc. has been observed over the past 50 trades. The frequency distribution is as follows: The frequency distribution is as follows:

Price ChangePrice Change Number of TradesNumber of Trades -3/8 -3/8 4 4 -1/4 -1/4 2 2 -1/8 -1/8 8 8 0 200 20 +1/8 +1/8 10 10 +1/4 +1/4 3 3 +3/8 +3/8 2 2 +1/2 +1/2 1 1

Total = 50Total = 50

15 15 Slide Slide


Relative Frequency Distribution andRelative Frequency Distribution andRandom Number MappingRandom Number Mapping

Price ChangePrice Change Relative FrequencyRelative Frequency Random NumbersRandom Numbers -3/8 -3/8 .08 .08 00 - 07 00 - 07 -1/4 -1/4 .04 .04 08 - 11 08 - 11

-1/8 -1/8 .16 .16 12 - 27 12 - 27 0 0 .40 .40 28 - 67 28 - 67 +1/8 +1/8 .20 .20 68 - 87 68 - 87 +1/4 +1/4 .06 .06 88 - 93 88 - 93 +3/8 +3/8 .04 .04 94 - 97 94 - 97 +1/2 +1/2 .02 .02 98 - 99 98 - 99 TOTAL 1.00TOTAL 1.00

16 16 Slide Slide


If the current price per share of Dynogen If the current price per share of Dynogen is 23, use random numbers to simulate the is 23, use random numbers to simulate the price per share over the next 10 trades. price per share over the next 10 trades.

Use the following stream of random Use the following stream of random numbers:numbers:

21, 84, 07, 30, 94, 57, 57, 19, 84, 8421, 84, 07, 30, 94, 57, 57, 19, 84, 84

17 17 Slide Slide


Simulation WorksheetSimulation Worksheet

Trade Random Price StockTrade Random Price Stock

NumberNumber NumberNumber ChangeChange PricePrice 1 21 -1/8 22 1 21 -1/8 22 7/87/8 2 84 +1/8 232 84 +1/8 23 3 07 -3/8 22 3 07 -3/8 22 5/85/8 4 30 0 22 4 30 0 22 5/85/8 5 94 +3/8 235 94 +3/8 23 6 57 0 236 57 0 23 7 57 0 237 57 0 23 8 19 -1/8 22 8 19 -1/8 22 7/87/8 9 84 +1/8 239 84 +1/8 23 10 84 +1/8 23 10 84 +1/8 23 1/81/8

18 18 Slide Slide


Spreadsheet for Stock Price SimulationSpreadsheet for Stock Price SimulationA B C D E F

1 Lower Upper Trade Price Stock2 Random Random Price Number Change Price3 Number Number Change 1 0.125 23.1254 0.00 0.08 -0.375 2 0.375 23.5005 0.08 0.12 -0.250 3 0.000 23.5006 0.12 0.28 -0.125 4 0.000 23.5007 0.28 0.68 0.000 5 0.000 23.5008 0.68 0.88 0.125 6 0.000 23.5009 0.88 0.94 0.250 7 0.125 23.625

10 0.94 0.98 0.375 8 0.125 23.75011 0.98 1.00 0.500 9 0.000 23.75012 10 0.125 23.875

A B C D E F1 Lower Upper Trade Price Stock2 Random Random Price Number Change Price3 Number Number Change 1 0.125 23.1254 0.00 0.08 -0.375 2 0.375 23.5005 0.08 0.12 -0.250 3 0.000 23.5006 0.12 0.28 -0.125 4 0.000 23.5007 0.28 0.68 0.000 5 0.000 23.5008 0.68 0.88 0.125 6 0.000 23.5009 0.88 0.94 0.250 7 0.125 23.625

10 0.94 0.98 0.375 8 0.125 23.75011 0.98 1.00 0.500 9 0.000 23.75012 10 0.125 23.875

19 19 Slide Slide


Theoretical Results and Observed ResultsTheoretical Results and Observed Results

Based on the probability distribution, the Based on the probability distribution, the expected price change per trade can be calculated expected price change per trade can be calculated by: by:

(.08)(-3/8) + (.04)(-1/4) + (.16)(-1/8) + (.40)(0)(.08)(-3/8) + (.04)(-1/4) + (.16)(-1/8) + (.40)(0)

+ (.20)(1/8) + (.06)(1/4) + (.04)(3/8) + (.02)(1/2) + (.20)(1/8) + (.06)(1/4) + (.04)(3/8) + (.02)(1/2) = +.005= +.005

The expected price change for 10 trades is The expected price change for 10 trades is

(10)(.005) = .05. Hence, the expected stock price (10)(.005) = .05. Hence, the expected stock price

after 10 trades is 23 + .05 = 23.05. after 10 trades is 23 + .05 = 23.05.

Compare this ending price with the Compare this ending price with the

spreadsheet simulation and “manual” simulation spreadsheet simulation and “manual” simulation

results on the previous slides.results on the previous slides.

20 20 Slide Slide

Example: Mark Off’s ProcessExample: Mark Off’s Process

Mark Off is a specialist at repairing large metal-Mark Off is a specialist at repairing large metal-cutting machines that use laser technology. His repair cutting machines that use laser technology. His repair territory consists of the cities of Austin, San Antonio, territory consists of the cities of Austin, San Antonio, and Houston. His day-to-day repair assignment and Houston. His day-to-day repair assignment locations can be modeled as a Markov process. The locations can be modeled as a Markov process. The transition matrix is: transition matrix is:

This Day’sThis Day’s Next Day's LocationNext Day's Location LocationLocation Austin San Antonio Houston Austin San Antonio Houston

Austin .60 .15 .25Austin .60 .15 .25

San Antonio .20 .75 .05San Antonio .20 .75 .05 Houston .15 .05 .80Houston .15 .05 .80

21 21 Slide Slide


Random Number MappingsRandom Number Mappings

Currently in Currently in Currently inCurrently in Currently in Currently in AustinAustin San AntonioSan Antonio HoustonHoustonNext-Day Random Next-Day Random Next-Day Next-Day Random Next-Day Random Next-Day

RandomRandom LocationLocation NumbersNumbers LocationLocation NumbersNumbers LocationLocation

NumbersNumbers

Austin 00 - 59 Austin 00 - 19 Austin 00 Austin 00 - 59 Austin 00 - 19 Austin 00 - 14- 14

San Ant. 60 - 74 San Ant. 20 - 94 San Ant. 15 San Ant. 60 - 74 San Ant. 20 - 94 San Ant. 15 - 19- 19

Houston 75 - 99 Houston 95 - 99 Houston 20 Houston 75 - 99 Houston 95 - 99 Houston 20 - 99- 99

22 22 Slide Slide


Assume Mark is currently in Houston. Assume Mark is currently in Houston. Simulate where Mark will be over the next 16 Simulate where Mark will be over the next 16 days. What percentage of time will Mark be in days. What percentage of time will Mark be in each of the three cities? each of the three cities?

Use the following random numbers:Use the following random numbers:

93, 63, 26, 16, 21, 26, 70, 55, 72, 89, 49, 64, 91, 93, 63, 26, 16, 21, 26, 70, 55, 72, 89, 49, 64, 91, 02, 52, 6902, 52, 69

23 23 Slide Slide


Simulation WorktableSimulation Worktable

Starting in HoustonStarting in Houston Random Day'sRandom Day's Random Day's Random Day's DayDay NumberNumber LocationLocation DayDay NumberNumber LocationLocation 1 93 Houston1 93 Houston 9 9 72 San Ant. 72 San Ant.

2 63 Houston2 63 Houston 10 10 89 89 San Ant. San Ant. 3 26 Houston3 26 Houston 11 11 49 49 San Ant. San Ant.

4 16 San Ant. 12 4 16 San Ant. 12 64 San Ant. 64 San Ant. 5 21 San Ant. 13 5 21 San Ant. 13 91 San Ant. 91 San Ant. 6 26 San Ant. 14 6 26 San Ant. 14 02 Austin 02 Austin 7 70 San Ant. 15 7 70 San Ant. 15 52 Austin 52 Austin 8 55 San Ant. 16 8 55 San Ant. 16 69 San Ant.69 San Ant.

24 24 Slide Slide


Repeat the simulation with Mark currently Repeat the simulation with Mark currently in Austin. Use the following random numbers:in Austin. Use the following random numbers:

13, 08, 60, 13, 68, 40, 40, 27, 23, 64, 36, 56, 25, 13, 08, 60, 13, 68, 40, 40, 27, 23, 64, 36, 56, 25, 88, 18, 7488, 18, 74

Compare the percentages with those found Compare the percentages with those found with Mark starting in Houston.with Mark starting in Houston.

25 25 Slide Slide


Simulation WorksheetSimulation Worksheet

Starting in AustinStarting in Austin Random Day's Random Day's Random Day's Random Day's

DayDay NumberNumber LocationLocation Day Day NumberNumber LocationLocation 1 13 Austin 1 13 Austin 9 23 San Ant. 9 23 San Ant. 22 08 Austin 08 Austin 10 10 64 San 64 San

Ant.Ant. 3 60 San Ant.3 60 San Ant. 11 36 San Ant. 11 36 San Ant. 4 4 13 Austin 13 Austin 12 56 San Ant. 12 56 San Ant. 5 68 San Ant.5 68 San Ant. 13 25 San Ant. 13 25 San Ant. 6 40 San Ant.6 40 San Ant. 14 88 San Ant. 14 88 San Ant. 7 40 San Ant.7 40 San Ant. 15 15 18 18

Austin Austin 8 8 27 San Ant. 27 San Ant. 16 74 San Ant. 16 74 San Ant.

26 26 Slide Slide


Simulation SummarySimulation Summary

Starting in HoustonStarting in Houston

Austin = 2/16 = 12.50%Austin = 2/16 = 12.50%

San Antonio = 11/16 = 68.75% San Antonio = 11/16 = 68.75%

Houston = 3/16 = 18.75%Houston = 3/16 = 18.75%

Starting in AustinStarting in Austin

Austin = 4/16 = 25%Austin = 4/16 = 25%

San Antonio = 12/16 = 75%San Antonio = 12/16 = 75%

Houston = 0/16 = 0%Houston = 0/16 = 0%

27 27 Slide Slide


Partial Spreadsheet with Variable Look-up TablePartial Spreadsheet with Variable Look-up Table

A B C D E F G H I123 LRN URN NDL LRN URN NDL LRN URN NDL6 0.00 0.60 Aus. 0.00 0.20 Aus. 0.00 0.15 Aus.7 0.60 0.75 S.A. 0.20 0.95 S.A. 0.15 0.20 S.A.8 0.75 1.00 Hou. 0.95 1.00 Hou. 0.20 1.00 Hou.9

Aus. S.A. Hou.Current Location

A B C D E F G H I123 LRN URN NDL LRN URN NDL LRN URN NDL6 0.00 0.60 Aus. 0.00 0.20 Aus. 0.00 0.15 Aus.7 0.60 0.75 S.A. 0.20 0.95 S.A. 0.15 0.20 S.A.8 0.75 1.00 Hou. 0.95 1.00 Hou. 0.20 1.00 Hou.9

Aus. S.A. Hou.Current Location

LRN = Lower Random Number URN = Upper Random NumberNDL = Next-Day Location

28 28 Slide Slide


Partial Spreadsheet with Simulation TablePartial Spreadsheet with Simulation Table

=IF(A13=$A$2,VLOOKUP(C13,$A$6:$C$8,3), IF(A13=$D$2,VLOOKUP(C13,$D$6:$F$8,3),

VLOOKUP(C13,$G$6:$I$8,3)))

A B C D E F111213141516171819

Next-Day

S.A.

S.A.

S.A. 0.61 S.A.

Number

0.33

CurrentLocation

Random

S.A.0.45

S.A.

LocationAus. 0.64 S.A.

S.A. 0.87 S.A.

S.A.S.A.S.A.

0.30

0.71S.A.

29 29 Slide Slide

Example: Wayne International AirportExample: Wayne International Airport

Wayne International Airport primarily serves Wayne International Airport primarily serves domestic air traffic. Occasionally, however, a domestic air traffic. Occasionally, however, a chartered plane from abroad will arrive with chartered plane from abroad will arrive with passengers bound for Wayne's two great passengers bound for Wayne's two great amusement parks, Algorithmland and Giffith's amusement parks, Algorithmland and Giffith's Cherry Preserve.Cherry Preserve.

Whenever an international plane arrives at the Whenever an international plane arrives at the airport the two customs inspectors on duty set up airport the two customs inspectors on duty set up operations to process the passengers.operations to process the passengers.

Incoming passengers must first have their Incoming passengers must first have their passports and visas checked. This is handled by one passports and visas checked. This is handled by one inspector. The time required to check a passenger's inspector. The time required to check a passenger's passports and visas can be described by the passports and visas can be described by the probability distribution on the next slide.probability distribution on the next slide.

30 30 Slide Slide


Time Required toTime Required to Check a Passenger'sCheck a Passenger's Passport and VisaPassport and Visa ProbabilityProbability

20 seconds 20 seconds .20.20




31 31 Slide Slide


After having their passports and visas After having their passports and visas checked, the passengers next proceed to the checked, the passengers next proceed to the second customs official who does baggage second customs official who does baggage inspections. Passengers form a single waiting inspections. Passengers form a single waiting line with the official inspecting baggage on a first line with the official inspecting baggage on a first come, first served basis. The time required for come, first served basis. The time required for baggage inspection has the following probability baggage inspection has the following probability distribution:distribution:

Time Required ForTime Required For

Baggage InspectionBaggage Inspection ProbabilityProbability No Time No Time .25 .25

1 minute 1 minute .60 .60 2 minutes 2 minutes .10 .10 3 minutes 3 minutes .05 .05

32 32 Slide Slide


Random Number MappingRandom Number Mapping

Time Required toTime Required to Check a Passenger's Check a Passenger's

RandomRandom Passport and VisaPassport and Visa ProbabilityProbability NumbersNumbers

20 seconds .20 00 - 20 seconds .20 00 - 1919

40 seconds .40 40 seconds .40 20 - 59 20 - 59

60 seconds .30 60 seconds .30 60 - 89 60 - 89

80 seconds .10 80 seconds .10 90 - 99 90 - 99

33 33 Slide Slide


Random Number MappingRandom Number Mapping

Time Required For Time Required For RandomRandom Baggage InspectionBaggage Inspection ProbabilityProbability NumbersNumbers

No Time .25 00 - No Time .25 00 - 2424

1 minute 1 minute .60 .60 25 - 8425 - 84

2 minutes .10 85 2 minutes .10 85 - 94- 94

3 minutes .05 95 3 minutes .05 95 - 99- 99

34 34 Slide Slide


Next-Event Simulation RecordsNext-Event Simulation Records

For each passenger the following For each passenger the following information must be recorded:information must be recorded:

•When his service begins at the passport When his service begins at the passport control inspectioncontrol inspection

•The length of time for this serviceThe length of time for this service

•When his service begins at the baggage When his service begins at the baggage inspectioninspection

•The length of time for this serviceThe length of time for this service

35 35 Slide Slide


Time RelationshipsTime Relationships

Time a passenger begins serviceTime a passenger begins service

by the passport inspectorby the passport inspector

= (Time the previous passenger started passport = (Time the previous passenger started passport service) service)

+ (Time of previous passenger's passport + (Time of previous passenger's passport service)service)

36 36 Slide Slide



Time a passenger begins serviceTime a passenger begins service

by the baggage inspectorby the baggage inspector

( If passenger does not wait in line for baggage ( If passenger does not wait in line for baggage inspection)inspection)

= (Time passenger completes service= (Time passenger completes service

with the passport control inspector) with the passport control inspector)

(If the passenger does wait in line for baggage (If the passenger does wait in line for baggage inspection)inspection)

= (Time previous passenger completes= (Time previous passenger completes

service with the baggage inspector)service with the baggage inspector)

37 37 Slide Slide



Time a customer completes serviceTime a customer completes service

at the baggage inspectorat the baggage inspector

= (Time customer begins service with baggage = (Time customer begins service with baggage inspector) inspector) + (Time required for baggage + (Time required for baggage inspection)inspection)

38 38 Slide Slide

A chartered plane from abroad lands at A chartered plane from abroad lands at Wayne Airport with 80 passengers. Simulate the Wayne Airport with 80 passengers. Simulate the processing of the first 10 passengers through processing of the first 10 passengers through customs. customs.

Use the following random numbers:Use the following random numbers:

For passport control:For passport control:

93, 63, 26, 16, 21, 26, 70, 55, 72, 8993, 63, 26, 16, 21, 26, 70, 55, 72, 89

For baggage inspection:For baggage inspection:

13, 08, 60, 13, 68, 40, 40, 27, 23, 6413, 08, 60, 13, 68, 40, 40, 27, 23, 64


39 39 Slide Slide


Simulation Worksheet (partial)Simulation Worksheet (partial)

Passport ControlPassport Control Baggage InspectionsBaggage InspectionsPass. Time Rand. Service Time Time Rand. Service Pass. Time Rand. Service Time Time Rand. Service

TimeTimeNum Begin Num. Time End Begin Num. Time EndNum Begin Num. Time End Begin Num. Time End

1 0:00 93 1:20 1:20 1:20 13 0:00 1 0:00 93 1:20 1:20 1:20 13 0:00 1:20 1:20

2 1:20 63 1:00 2:20 2:20 08 0:00 2 1:20 63 1:00 2:20 2:20 08 0:00 2:20 2:20

3 2:20 26 :40 3:00 3:00 60 1:00 3 2:20 26 :40 3:00 3:00 60 1:00 4:00 4:00

4 3:00 16 :20 3:20 4:00 13 0:00 4 3:00 16 :20 3:20 4:00 13 0:00 4:00 4:00

5 3:20 21 :40 4:00 4:00 68 1:00 5 3:20 21 :40 4:00 4:00 68 1:00 5:00 5:00

40 40 Slide Slide


Simulation Worksheet (continued)Simulation Worksheet (continued)

Passport ControlPassport Control Baggage InspectionsBaggage InspectionsPass. Time Rand. Service Time Time Rand. Service TimePass. Time Rand. Service Time Time Rand. Service TimeNum Begin Num. Time End Begin Num. Time EndNum Begin Num. Time End Begin Num. Time End

6 4:00 26 :40 4:40 5:00 40 1:00 6 4:00 26 :40 4:40 5:00 40 1:00 6:00 6:00

7 4:40 70 1:00 5:40 6:00 40 1:00 7 4:40 70 1:00 5:40 6:00 40 1:00 7:00 7:00

8 5:40 55 :40 6:20 7:00 27 1:00 8 5:40 55 :40 6:20 7:00 27 1:00 8:00 8:00

9 6:20 72 1:00 7:20 8:00 23 0:00 9 6:20 72 1:00 7:20 8:00 23 0:00 8:00 8:00

10 7:20 89 1:00 8:20 8:20 64 1:00 10 7:20 89 1:00 8:20 8:20 64 1:00 9:20 9:20

41 41 Slide Slide


ExplanationExplanation

For example, passenger 1 begins being For example, passenger 1 begins being served by the passport control inspector served by the passport control inspector immediately. His service time is 1:20 (80 immediately. His service time is 1:20 (80 seconds) at which time he goes immediately to seconds) at which time he goes immediately to the baggage inspector who waves him through the baggage inspector who waves him through without inspection. without inspection.

Passenger 2 begins service with passport Passenger 2 begins service with passport inspector 1:20 minutes (80 seconds) after inspector 1:20 minutes (80 seconds) after arriving there (as this is when passenger 1 is arriving there (as this is when passenger 1 is finished) and requires 1:00 minute (60 seconds) finished) and requires 1:00 minute (60 seconds) for passport inspection. He is waved through for passport inspection. He is waved through baggage inspection as well. baggage inspection as well.

This process continues in this manner.This process continues in this manner.

42 42 Slide Slide


QuestionQuestion

How long will it take for the first 10 How long will it take for the first 10 passengers to clear customs?passengers to clear customs?

AnswerAnswer

Passenger 10 clears customs after 9 Passenger 10 clears customs after 9 minutes and 20 seconds.minutes and 20 seconds.

43 43 Slide Slide


QuestionQuestion

What is the average length of time a What is the average length of time a customer waits before having his bags inspected customer waits before having his bags inspected after he clears passport control? How is this after he clears passport control? How is this estimate biased?estimate biased?

44 44 Slide Slide

AnswerAnswer

For each passenger calculate his waiting For each passenger calculate his waiting time:time:

(Baggage Inspection Begins) - (Passport Control (Baggage Inspection Begins) - (Passport Control Ends) Ends)

= 0+0+0+40+0+20+20+40+40+0 = 120 = 0+0+0+40+0+20+20+40+40+0 = 120 seconds. seconds.

120/10 = 12 seconds per passenger 120/10 = 12 seconds per passenger

This is a biased estimate because we This is a biased estimate because we assume that the simulation began with the system assume that the simulation began with the system empty. Thus, the results tend to underestimate empty. Thus, the results tend to underestimate the average waiting time.the average waiting time.


45 45 Slide Slide

End of Chapter 13End of Chapter 13

Documents

1 1 Slide Chapter 13 Simulation n Advantages and Disadvantages of Using Simulation n Modeling n Random Variables and Pseudo-Random Numbers n Time Increments