Handout simulasi computer

Department of Industrial Engineering
Faculty of Science & Technology The Sunan Kalijaga State Islamic University
2 0 0 9

Program Studi Teknik Industri
Fakultas Sains dan Teknologi Universitas Islam Negeri Sunan Kalijaga
Tahun 2009

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Chapter 1Chapter 1Chapter 1Chapter 1
Introduction to SimulationIntroduction to Simulation
By : Arya WirabhuanaBy : Arya Wirabhuana
The Opportunity GameThe Opportunity Game
15 1
2
2
3
33
3
4
4
5 400
400
500
500
500500
600
600
600
700 200
200
300
300
300400
400
400
400
500
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Cost to Play: $1000
Payoff ($): (A Spinner) x (B Spinner) – (C Spinner)
Return ($): Payoff – Cost-to-Play
Spinner A Spinner B Spinner C
Arya Wirabhuana - Industrial Computer Simulation
UIN Sunan Kalijaga Yogyakarta
May 02, 2009 - Page (1)

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ProblemsProblems
If il bl ti (f l i th ) i blIf available time (for playing the game) is no problem,
and if there is no constraint on available working
capital, would a prudent person choose to play this
game (repeatedly)?
(In other words, what is the expected (that is, the
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long-run average) Return?
Alternative SolutionAlternative Solution
ApproachesApproaches
S l h bl h i llSolve the problem mathematically
Perform experiments with real system
Perform experiments with a model (representation) of
the real system
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Mathematical ModelMathematical Model
E t d lt f i di id l Expected B Spinner ResultExpected results of individual
Spinners
(long-run individual spinner results)
Expected – A-Spinner Result
Outcome Probability Outcome x Probability
1 1 / 10 0 1
Expected – B-Spinner Result
400 2 / 10 80
500 4 / 10 200
600 3 / 10 180
700 1 / 10 70
Sum (Expected Outcome) : 530
5
1 1 / 10 0.1
2 2 / 10 0.4
3 4 / 10 1.2
4 2 / 10 0.8
5 1 / 10 0.5
Sum (Expected Outcome) : 3.0
Expected – C-Spinner Result
200 2 / 10 40
300 3 / 10 90
400 4 / 10 160
500 1 / 10 50
Sum (Expected Outcome) : 340
Mathematical ModelMathematical Model
(cont’)(cont’)
What is the Expected Return in the OpportunityWhat is the Expected Return in the Opportunity
Game?
Payoff = (A-Spinner) x (B-Spinner) – (C-Spinner)
Return = Payoff – (Cost-to-Play)
substitute expected spinner results to get expected Payoff and Return
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substitute expected spinner results to get expected Payoff and Return
Expected Payoff = (3.0) x (530) – (340) = $1,250
Expected Return = $1,250 - $1,000 = $250
May 02, 2009 - Page (3)

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Complicated QuestionComplicated Question
Wh t th h th t ill lWhat are the chances that a person will lose money
in a single play of the game?
The answer to this question can be developed
mathematically, but doing so requires:
computing the relative frequency with which each of the possible
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returns occurs,
then using these relative frequencies to determine the cumulative
frequencies for the returns ordered from lowest to highest,
and finally looking up the cumulative frequency for negative
returns
Complicated QuestionComplicated Question
(cont’)(cont’)
Thi i d b ti ll f th iThis is done by enumerating all of the various
possible spinner combinations and using the law of
multiplication to compute the probability associated
with each combination
(and then using the law of addition to add
probabilities for identical outcomes to determine the
ll b bilit f th t t )
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overall probability of that outcome)
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Solution bySolution by
EnumerationEnumeration
Spinner A
1
2
3
Spinner B
400
500
600
Spinner C
200
300
400
Probability
0.032
0.048
0.064
Start Return
$300
$200
$100
4 / 10
4 / 10
2 / 10
3 / 10
4 / 10
1 / 10
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4
5
700 500 0.016 $0
1 / 10
and so on, for all other
combinations of Spinner A,
Spinner B and Spinner C
Opportunity GameOpportunity Game
OutcomesOutcomes
Return Relative Frequency Cumulative Frequencyq y q y
-$1100 0.002 0.002
-$1000 0.012 0.014
-$900 0.025 0.039
-$800 0.029 0.068
… … …
-$200 0.072 0.290
-$100 0.044 0.334
34 distinct returns,
ranging from
-$1100 to $2300
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$0 0.052 0.386
$100 0.074 0.460
$200 0.069 0.528
… … …
$2100 0.004 0.995
$2200 0.003 0.998
$2300 0.002 1.000
-$1100 to $2300
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Opportunity GameOpportunity Game
HistogramHistogram
A hi t h i l ti f i fA histogram showing relative frequencies of
various “Return” ranges in the Opportunity Game
20
25
30
35
40
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0
5
10
15
-1000 -500 0 500 1000 1500 2000 2500
5.2%
31%
27.2%
13.2%
15%
6.8% 0.2%1.4%
More ComplicatedMore Complicated
QuestionsQuestions
If h d $2000 i ki it l dIf a person had $2000 in working capital, and
enough time to play the game up to 25 times, what
are chances that the person would:
go bankrupt?
lose money, but not go bankrupt?
break even?
make money?
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make money?
exceed the expected gain of $6250?
($6250 = 25 x $250)
May 02, 2009 - Page (6)

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More ComplicatedMore Complicated
Questions (cont’)Questions (cont’)
A th ti l h l b t k tA mathematical approach can also be taken to
answer each of these questions, but the calculations,
although straightforward, are quite tedious!
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S l th bl th ti llSolve the problem mathematically
Perform experiments with a model (representation) of
the real system
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Real SystemReal System
It ld b t t t th “ t it ”It would be easy to construct the “opportunity game”
spinners and play the game repeatedly (without dollar
consequences), say 1,000 times, then use the
average result as an estimate of the expected result
More generally, experimentation on “the real system”
can be done in concept, but often cannot be done in
ti
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practice
Experimenting on the real system requires of course
that the system exists, and it might not (the goal
might in fact be to design a system)
Real System (cont’)Real System (cont’)
If th t d i t it i ht t b f ibl tIf the system does exists, it might not be feasible to
experiment with it, for reasons such as these:
Economic reasons
(it might be prohibitively expensive to interrupt the ongoing use
of the real system)
Political reasons
(it might be difficult to get permission from the system’s
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(it might be difficult to get permission from the system s
“owners” to experiment with the system)
Real-system experiments might take too long
(days, weeks, or months of experimentation might be required,
and so the findings might not be available in time to do any good)
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S l th bl th ti llSolve the problem mathematically
Perform experiments with a model
(representation) of the real system
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Model of theModel of the
Real SystemReal System
F i l ti i i l t h iFor our purposes, simulation is a numerical technique
for conducting experiments with a model that
describes or mimics the behaviour of a system
A model is a representation of a system that behaves
like the system itself behaves
(the model may not behave like the system in all respects, but the
d l t b h lik th t t l t i th t th t
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model must behave like the system at least in those respects that
are important for the purpose at hand)
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Model of theModel of the
Real System (cont’)Real System (cont’)
I l d l ti h i lIn general, models sometimes are physical, e.g.,
blueprints of a house
a three dimensional model of a shopping mall
a mock-up of the control panels in a jetliner
Models sometimes are logical abstractions based on
the rules that govern the operation of a system, for
l
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example,
a computer program that plays the “opportunity game” by
determining spinner results at random and combining the results
to determine the payoff and return.
Spreadsheet OutputSpreadsheet Output
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Model Classification andModel Classification and
Steps in a Simulation StudySteps in a Simulation Study
Definition of SimulationDefinition of Simulation
Simulation is the imitation of an operation of a realSimulation is the imitation of an operation of a real-
world process or system over time.
Simulation is a method of understanding,
representing and solving complex interdependent
system.
Simulation is the process of designing a model of a
real system and conducting experiments with this
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model for the purpose either of understanding the
behavior of the system or of evaluating various
strategies (with the limits imposed by a criterion or a
set of criteria) for the operation of the system.
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Definition of SimulationDefinition of Simulation
(cont’)(cont’)
Si l ti i l i t t d th t d l ithSimulation in general is to pretend that one deals with
a real thing while really working with an imitation.
A flight simulator on a PC is computer model of some
aspects of the flight: it shows on the screen the
controls and what the “pilot” (the youngster who
operates it) is supposed to see from the “cockpit” (his
h i )
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armchair).
When to use ModelWhen to use Model
T fl i l t i f d h th th lTo fly a simulator is safer and cheaper than the real
airplane.
For precisely this reason, models are used in
industry, commerce and military: it is very costly,
dangerous and often impossible to make experiments
with real systems.
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Provided that models are adequate descriptions of
reality (they are valid), experimenting with them can
save money, suffering and even time.
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When to useWhen to use
SimulationsSimulations
Systems which change with time such as a gasSystems which change with time such as a gas
station where cars come and go (called dynamic
systems) and involve randomness (nobody can
guess at exactly which time and next cars should
arrive at the station) are good candidates for
simulation.
Modeling complex dynamic systems theoretically
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need too many simplifications and the emerging
models may not be therefore valid.
Simulation does not require that many simplifying
assumptions, making it the only tool even in absence
of randomness.
How to simulate?How to simulate?
Suppose we are interested in a gas station We maySuppose we are interested in a gas station. We may
describe the behaviour of this system graphically by
plotting the number of cars in the station; the state of
the system.
Every time a car arrives the graph increases by one
unit while a departing car causes the graph to drop
one unit.
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This graph (called sample path), could be obtained
from observation of a real station, but could also be
artificially constructed.
Such artificial construction and the analysis of the
resulting sample path consists of the simulation.
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Types of ModelsTypes of Models
M d l b l ifi d b i th ti lModels can be classified as being mathematical or
physical.
A mathematical model uses symbolic notation and
mathematical equations to represent a system.
A simulation model is particular type of mathematical
model of a system.
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Type of SimulationType of Simulation
Si l ti d l b f th l ifi d b iSimulation models may be further classified as being:
Static model or Dynamic model
Deterministic model or Stochastic model
Discrete model or Continuous model
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Static vs DynamicStatic vs Dynamic
St ti d l d d i d l l ifi tiStatic models and dynamic models are classification
by the dependency on time
A static simulation model, sometimes called a Monte
Carlo simulation, represents a system at a particular
point in time.
For example, Mark Six, inventory level
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Dynamic simulation models represent systems in
which state of the variables change over time. The
simulation of a bank from 9:00am to 4:00pm is an
example of a dynamic simulation.
For example, service time, waiting time.
Deterministic vsDeterministic vs
StochasticStochastic
Cl ifi ti b th t f th i blClassification by the nature of the variables
Simulation models that contain no random variables
are classified as deterministic.
For example, deterministic arrivals would occur at a dentist’s
office if all arrived at the scheduled appointment time.
A stochastic simulation model has one or more
d i bl i t
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random variables as input.
Random inputs lead to random outputs.
For example, random arrival, random product demand, random
incoming calls.
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Deterministic vsDeterministic vs
Stochastic (cont’)Stochastic (cont’)
Si th t t d th bSince the outputs are random, they can be
considered only as estimates of the true
characteristics of a model.
For example, the simulation of a bank would usually involve
random interarrival times and random service times.
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Discrete vs ContinuousDiscrete vs Continuous
Di t d ti d l d fi d iDiscrete and continuous models are defined in an
analogous manner, classification by system nature.
A discrete model is one in which the state variable(s)
change only at a discrete set of points in time.
The bank is an example of a discrete system, since
the state variable, the number of customers in the
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bank, changes only when a customer arrives or when
the service provided a customer is complete.
Other examples, busy/idle counter, occupied/free
machine.
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Discrete vs ContinuousDiscrete vs Continuous
(cont’)(cont’)
A continuous model is one in which the stateA continuous model is one in which the state
variable(s) change continuously over time.
An example is the head of water behind a dam.
During and for some time after a rain storm, water
flows into the lake behind the dam.
Water is drawn from the dam for flood control and to
make electricity.
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y
Evaporation also decreases the water level.
But, continuous system can be approximated by a
discrete-event system, depending on the expected
preciseness and the objective of the study.
ApplicationsApplications
-- Service ApplicationsService Applications
St ffiStaffing
A bank manager might determine that three tellers on
duty results in a tolerable wait for service during most
of the day, but that her customers’ “time in queue” is
too long during the busy lunch hour and in the late
afternoon.
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She could then assess the impacts of adding
additional part-time help during the peak hours.
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ApplicationsApplications
-- Service ApplicationsService Applications
(cont’)(cont’)
P d I tProcedure Improvement
Many organizations have learned that internal
consumers are customers.
In an effort to improve the responsiveness of their
administrative and support functions many of these
companies are using simulation to model revised
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procedures designed to streamline processing of
paperwork, telephone calls and other daily
transactions.
Advantages ofAdvantages of
SimulationSimulation
New policies operating procedures decision rulesNew policies, operating procedures, decision rules,
information flows, organizational procedures, and so
on can be explored without disrupting ongoing
operations of the real system.
New hardware designs, physical layouts,
transportation systems, and so on, can be tested
without committing resources for their acquisition.
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Hypotheses about how or why certain phenomena
occur can be tested for feasibility.
Time can be compressed or expanded allowing for a
speedup or slowdown of the phenomena under
investigation.
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Advantages ofAdvantages of
Simulation (cont’)Simulation (cont’)
Insight can be obtained about the interaction ofInsight can be obtained about the interaction of
variables.
Insight can be obtained about the importance of
variables to the performance of the system.
Bottleneck analysis can be performed indicating
where work-in-process, information, materials, and so
on are being excessively delayed.
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g y y
A simulation study can help in understanding how the
system operates rather than how individuals think the
system operates.
“What-if” questions can be answered.
Disadvantages ofDisadvantages of
Model building requires special trainingModel building requires special training.
Simulation results may be difficult to interpret.
Simulation modeling and analysis can be time
consuming and expensive. Skimping on resources
for modeling and analysis may result in a simulation
model or analysis that is not sufficient for the task.
Simulation is used in some cases when an analytical
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Simulation is used in some cases when an analytical
solution is possible, or even preferable. This might
be particularly true in the simulation of some waiting
lines where closed-form queueing models are
available.
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Defense of SimulationDefense of Simulation
V d f i l ti ft h b ti lVendors of simulation software have been actively
developing packages that contain all or part of
models that need only input data for their operation.
Many simulation software vendors have developed
output analysis capabilities within their packages for
performing very thorough analysis.
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Simulation can be performed faster today than
yesterday, and even faster tomorrow. This is
attributable to the advances in hardware that permit
rapid running of scenarios.
Defense of SimulationDefense of Simulation
(cont’)(cont’)
Cl d f d l t bl t l t fClosed-form models are not able to analyze most of
the complex systems that are encountered in
practice.
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Steps in aSteps in a
Simulation StudySimulation Study
Problem formulationProblem formulation
Setting of objectives and overall project plan
Model Conceptualization Data Collection
Model translation
Experimental design
Production runs and analysis
More runs?
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Model translation
Verified?
Validated?
Documentation
and reporting
Implementation
No
Yes
NoNo
Yes
(cont’)(cont’)
P bl f l tiProblem formulation
If the statement is provided by the policy makers, or those that
have the problem, the analyst must ensure that the problem being
described is clearly understood. If a problem statement is being
developed by the analyst, it is important that the policy makers
understand and agree with the formulation.
Setting of objectives and overall project plan
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The objectives indicate the questions to be answered by simulation.
The overall project plan should include a statement of the
alternative systems to be considered, and a method for evaluating
the effectiveness of these alternatives.
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(cont’)(cont’)
Model conceptualizationModel conceptualization
This is another important and difficult subject. The basic steps are
to consider all the related factors first, then evaluate each one
(keep or ignore) and reach the final model.
Data collection
The more data you have the more complete information you
have the more precise model you can build the better
solution you would get
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solution you would get.
Model translation
Program the model into a computer language. Simulation
languages are powerful and flexible. In most cases, some
computer software packages are involved. The model development
time is greatly reduce. Furthermore, software packages have
added features that enhance their flexibility.
(cont’)(cont’)
Verified?Verified?
Verification pertains to the computer program prepared for the
simulation model. Is the computer program performing properly?
If the input parameters and logical structure or the model are
correctly represented in the computer, verification has been
complete.
Validated?
Validation is the determination that a model is an accurate
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Validation is the determination that a model is an accurate
representation of the real system. Validation is usually achieved
through the calibration of the model, an iterative process of
comparing the model to actual system behaviour and using the
discrepancies between the two, and the insights gained, to improve
the model.
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(cont’)(cont’)
Experimental designExperimental design
The alternatives that are to be simulated must be determined. For
each system design that is simulated, decisions need to be made
concerning the length of the initialization period, the length of
simulation runs, and the number of replications to be made of each
run.
Production runs and analysis
Production runs and their subsequent analysis are used to
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Production runs, and their subsequent analysis, are used to
estimate measures of performance for the system designs that are
being simulated.
More runs?
The analyst determines of additional runs are needed and what
design those additional experiments should follow.
(cont’)(cont’)
Documentation and reportingDocumentation and reporting
Program documentation:
If the program is going to be used again by the same or
different analysts, it may be necessary to understand
how the program operates.
The model users can change parameters at will in an
effort to determine the relationships between input
parameters and output measures of performance, or to
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p p p ,
determine the input parameters that “optimize” some
output measure of performance.
Progress report:
It provides the important written history of a simulation
project.
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(cont’)(cont’)
I l t tiImplementation
The success of the implementation phase depends on how well the
previous eleven steps have been performed.
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Concepts of DiscreteConcepts of Discrete--EventEvent
Si l tiSi l tiSimulationSimulation
Discrete Event ModelDiscrete Event Model
In the discrete approach to system simulation stateIn the discrete approach to system simulation, state
changes in the physical system are represented by a
series of discrete changes or events at specific
instants of time and such models are known as
discrete event models.
The time and state are the two important coordinates
used in describing simulation models.
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Between events, the states of the entities remain
constant.
The change in state is brought about by events which
from the driving force behind every discrete event
simulation model.
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System TerminologySystem Terminology
S tSystem:
A collection of entities (e.g., people and machines) that interact
together over time to accomplish one or more goals.
Model:
An abstract representation of a system, usually containing
structural logical, or mathematical relationships which describe a
system in terms of state, entities and their attributes, sets,
3
system in te ms of state, entities and thei att ibutes, sets,
processes, events, activities, and delays.
System state:
A collection of variables that contain all the information necessary
to describe the system at any time.
(cont’)(cont’)
E titiEntities:
Any object or component in the system which requires explicit
representation in the model (e.g., a server, a customer, a machine).
Attributes:
The properties of a given entity (e.g., the priority of a waiting
customer, the routing of a job through a job shop).
List (Set Queue):
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List (Set, Queue):
A collection of (permanently or temporarily) associated entities,
ordered in some logical fashion (such as all customers currently in
a waiting line, ordered by first come, first served, or by priority).
May 02, 2009 - Page (26)

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(cont’)(cont’)
E tEvent:
An instantaneous occurrence that changes the state of a system
(such as an arrival of a new customer).
Event notice:
A record of an event to occur at the current or some future time,
along with any associated data necessary to execute the event; at a
minimum, the record includes the event type and the event time.
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minimum, the eco d includes the event type and the event time.
Event list:
A list of event notices for future events, ordered by time of
occurrence; also known as the future event list (FEL).
(cont’)(cont’)
A ti itActivity:
A duration of time of specified length (e.g., a service time or inter-
arrival time), which is known when it begins (although it may be
defined in terms of a statistical distribution).
Delay:
A duration of time of unspecified indefinite length, which is not
known until it ends (e.g., a customer’s delay in a last-in, first-out
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known until it ends (e.g., a custome s delay in a last in, fi st out
waiting line which, when it begins, depends on future arrivals).
Clock:
A variable representing simulated time.
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(Example 2.1)(Example 2.1)
S t t tSystem state:
LQ(t), the number of customer waiting to be served at time t
LC(t), 0 or 1 indicate counter being idle or busy at time t
Entities
Neither the customers nor the servers need to be explicitly
represented, unless certain customer averages are desired
E t
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Events:
Arrival event
Service completion
(Example 2.1)(Example 2.1)
A ti itiActivities:
Interarrival time
Service time
Unconditional wait
Delay:
A customer’s wait in queue until counter becomes free
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Conditional wait
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Main ApproachesMain Approaches
Event scheduling approachEvent-scheduling approach
concentrate on the events and how they affect the system state.
The simulation evolves over time by executing events in increasing
order of their times of occurrence.
Examples: FORTRAN, GASP IV, C++
Process-interaction approach
concentrate on a single entity (e.g. a customer) and the sequence
of events and activities it undergoes as it PASSES THROUGH
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of events and activities it undergoes as it PASSES THROUGH
THE SYSTEM. At any given time, the system may contain many
processes (e.g. customers) interacting with each other while
competing for a set of resources.
Example: GPSS
Main ApproachesMain Approaches
(cont’)(cont’)
I t ti d i i t d hInteractive, menu-driven, animated approach
Recently available on PCs.
Examples: PROMODEL, SIGMA
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Future Event ListFuture Event List
(FEL)(FEL)
B th th t h d li d thBoth the event-scheduling and the process-
interaction approaches use a variable time advance;
that is, when all events and system state changes
have occurred at one instant of simulated time, the
simulation clock is advanced to the time of the next
imminent event on the FEL.
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EventEvent--SchedulingScheduling
ApproachApproach
Thi li t t i ll t ti f t th tThis list contains all event notices for events that
have been scheduled to occur at a future time.
Scheduling a future event means that at the instant
an activity begins, its duration is computed or drawn
as a sample from a statistical distribution and the
end-activity event, together with its event time, is
l d th f t t li t
12
placed on the future event list.
In the real world, most future events are not
scheduled but merely happen – such as random
breakdowns or random arrivals.
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ApproachApproach
(Example)(Example)
List of all activities’ SCHEDULED TIME OFList of all activities’ SCHEDULED TIME OF
COMPLETION (EVENTS)
The FEL is a SET ordered in completion times
t1 < t2 < … < tn
Example
Consider a single server queue with the following arrival times for the
first 10 customers:
13
f
0 4 8 10 13 14 17 20 27 29
and service times for these customers
5 5 1 3 2 1 4 7 3 1
Assume that completions are given priority over arrivals
ApproachApproach
(Example)(Example)
Time = 0
FEL
4 ARRIVAL
5 COMPLETION
.
.
QUEUE
0 φ
14
.
.
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ApproachApproach
(Example)(Example)
Time = 4
FEL
5 COMPLETION
8 ARRIVAL
10 COMPLETION
13 ARRIVAL
QUEUE
4 #1
arrival time customer
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14 ARRIVAL
17 ARRIVAL
20 ARRIVAL
27 ARRIVAL
29 ARRIVAL
ProcessProcess--InteractionInteraction
ApproachApproach
A i th lif l f titA process is the life cycle of one entity.
This life cycle consists of various events and
activities.
Some activities may require the use of one or more
resources whose capacities are limited.
These and other constraints cause processes to
16
p
interact, the simplest example being an entity forced
to wait in a queue (on a list) because the resource it
needs is busy with another entity.
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Approach (cont’)Approach (cont’)
I i t i tiIn more precise terms, a process is a time-
sequenced list of events, activities, and delays,
including demands for resources, that define the life
cycle of one entity as it moves through a system.
We see the interaction between two customer
processes as customer n + 1 is delayed until the
i t ’ “ d i t”
17
previous customer’s “end-service event” occurs.
ApproachApproach
(Example)(Example)
Customer n
TimeTime
Arrival
event
Begin
service
End-
service-
event
Interaction
ActivityDelay
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TimeTime
Arrival
event
Begin
service
End-
service-
eventActivityDelay
Customer n + 1
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1
Mathematical and StatisticalMathematical and Statistical
M d l i Si l tiM d l i Si l tiModels in SimulationModels in Simulation
Queueing ModelsQueueing Models
Si l ti i ft d i th l i f iSimulation is often used in the analysis of queueing
models.
Typical measures of system performance include
server utilization (percentage of time server is busy),
length of waiting lines, and delays of customers.
Decision maker is involved in trade-offs between
2
server utilization and customer satisfaction in terms
of line lengths and delays.
Calling population Waiting line Server
May 02, 2009 - Page (34)

2
I i l h l i t th l
Flow DiagramFlow Diagram
In a single-channel queueing system there are only
two possible events that can affect the state of the
system.
The entry of a unit into the system or the completion of service on a
unit
The server has only two possible states:
it i ith b idl
3
it is either busy or idle
Departure event
Being server
idle time
Remove the waiting unit
from the queue
Being servicing the unit
Another unit
waiting ?
NO YES
Example 2.1Example 2.1
Si l h l t fi t iSingle-channel queue serves customers on a first-in,
first-out (FIFO) basis
Customer
Number
Arrival Time
(Clock)
Time Service
Begins
(Clock)
Service Time
(Duration)
Time Service
Ends (Clock)
1 0 0 2 2
2 2 2 1 3
Table 2.4. Simulation Table Emphasizing Clock Times
4
2 2 2 1 3
3 6 6 3 9
4 7 9 2 11
5 9 11 1 12
6 15 15 4 19
May 02, 2009 - Page (35)

3
Chronological OrderChronological Order
Th f th t t f tThe occurrence of the two types of events
Event Type Customer Number Clock Time
Arrival 1 0
Departure 1 2
Arrival 2 2
Departure 2 3
Arrival 3 6
Table 2.5. Chronological Ordering of Events
5
Arrival 4 7
Departure 3 9
Arrival 5 9
Departure 4 11
Departure 5 12
Arrival 6 15
Departure 6 19
Chronological OrderingChronological Ordering
(cont’)(cont’)
N b i t t ti tNumber in system at time t
ofcustomersinthesystem
1
2
4 5
6
Numbero
0 4 8 12 16 20
1 2 3 4 5 6
May 02, 2009 - Page (36)

4
TerminologyTerminology
λ i l t ( b f lli it it f ti )λ mean arrival rate (number of calling units per unit of time)
μ mean service rate of one server (number of calling units served
per unit of time)
1/ μ mean service time for a calling unit
s number of parallel service facilities in the system
Lq mean length of the queue
L mean number in the system (those in queue + being served)
7
L mean number in the system (those in queue + being served)
Wq mean time spent waiting in the queue
W mean time spent in the system (Wq + 1/ μ)
ρ server utilization factor
Statistical ModelsStatistical Models
in Simulationin Simulation
Di t Di t ib ti P i (λ)Discrete Distribution – Poisson (λ)
estimate “number of arrivals per unit time”
where
P(x) = the probability of X successes given a knowledge of λ
!
)(
x
e
xP
x
λλ−
=
8
λ = expected number of successes
e = mathematical constant approximated by 2.71828
x = number of successes per unit
λ=)(xE λ=)(xV
May 02, 2009 - Page (37)

5
Poisson DistributionPoisson Distribution
D f N(t) i P i ifDef: N(t) is a Possion process if
Arrivals occurs individually (at rate λ)
N(t) has stationary increments: The distribution of the numbers of
arrivals between t and t+s depends on the length of the interval s
and not on the starting point t.
N(t) has independent increments: The numbers of arrivals during
nonoverlapping time intervals (t, t+s) and (t’, t’+s’) are
9
independent random variables.
Uniform DistributionUniform Distribution
C ti Di t ib ti U if di t ib tiContinuous Distribution – Uniform distribution
A random variable x is uniformly distributed on the interval (a, b):
ab
xf
−
=
1
)( bxa ≤≤
)( bU
10
),(~ baUx
2
)(
ba
xE
+
=
12
)(
)(
2
ab
xV
−
=
May 02, 2009 - Page (38)

6
(cont’)(cont’)
Th if di t ib ti l it l l iThe uniform distribution plays a vital role in
simulation. Random numbers, uniformly distribution
between zero to 1, provide the means to generate
random events.
11
ExponentialExponential
DistributionDistribution
C ti Di t ib ti E ti l di t ib ti hContinuous Distribution – Exponential distribution has
been used to model interarrival times when arrivals
are completely random and to model service times
which are highly variable.
A random variable x is exponentially distributed with parameter λ>0:
12
x
exf λ
λ −
=)( 0≥x
λ
1
)( =xE 2
1
)(
λ
=xV
May 02, 2009 - Page (39)

7
MemorylessMemoryless
M lMemoryless
)(
)(
)|(
sXP
sXandtsXP
sXtsXP
>
>+>
=>+>
)(
)(
sXP
tsXP
>
+>
=
13
t
s
ts
e
e
e λ
λ
λ
−
−
+−
==
)(
)( tXP >=
Example ofExample of
MemorylessMemoryless
S th t th lif f i d t i l l iSuppose that the life of an industrial lamp, in
thousands of hours, is exponentially distributed with
failure rate λ=1/3 (one failure every 3000 hours, on
the average). Find the probability that the industrial
lamp will last for another 1000 hours, given that it is
operating after 2500 hours.
14
)1()5.2|5.3( >=>> XPXXP
3/1−
= e
717.0=
May 02, 2009 - Page (40)

1
Properties of Random NumbersProperties of Random Numbers
Random NumberRandom Number
GenerationGeneration
A i l ti f t i hi h thA simulation of any system or process in which there
are inherently random components requires a
method of generating or obtaining numbers that are
random, in some sense.
The earliest methods were carried out by hands such
as throwing dice.
2
As computers (and simulation) became more widely
used, increasing attention was paid to methods of
random number generation compatible with the
computers work.
May 02, 2009 - Page (41)

2
Random NumberRandom Number
Generation (cont’)Generation (cont’)
Th f th h i th 1940’ d 1950’Therefore, the research in the 1940’s and 1950’s
turned to numerical or arithmetic ways to generate
“random” numbers.
These method are sequential, with each new number
being determined by one or several of its
predecessors according to a fixed mathematical
f l
3
formula.
The first such arithmetic generate generator,
proposed by von Neumann and Metropolic in the
1940’s is the famous midsquare method.
Midsquare MethodMidsquare Method
A l f id th d f tiAn example of midsquare method for generating a
uniform [0, 1] random numbers:
i Zi Ui Zi
2
0 7182 - 51581124
1 5811 0.5811 33767721
2 7677 0.7677 58936329
3 9363 0.9363 87665769
4
4 6657 0.6657 44315649
5 3156 0.3156 09960336
. . . .
. . . .
. . . .
May 02, 2009 - Page (42)

3
Midsquare MethodMidsquare Method
(cont’)(cont’)
Drawbacks for midsquare methodDrawbacks for midsquare method
If Zi = 0 for some i, then Uj = Zj = 0 for all j > i.
Ui+1 is determined by Ui, i.e. Ui+1 is a function of Ui. Therefore,
Ui and Ui+1 are not independent.
Nowadays, the random numbers generated from
computers are more complicated and appear to be
independent, in that they pass a series of statistical
t t
5
test.
But after all, the random numbers generated from
computers are still not purely random
(pseudorandom, but too awkward to use this term).
Good Random NumberGood Random Number
A “good” arithmetic random number generator shouldA “good” arithmetic random number generator should
posses several properties:
Above all, the numbers produced should appear to be distributed
uniformly on [0, 1] and should not exhibit any correlation with
each other; otherwise, the simulation’s results may be completely
invalid.
From a practical standpoint, we would naturally like the generator
to be fast and avoid the need for a lot of storage.
6
f f f g
We would like to be able to reproduce a given stream of random
numbers exactly, for at least two reasons. First, this can
sometimes make debugging or verification f the computer program
easier. More important, we might want to use identical random
numbers in simulating different systems in order to obtain a more
precise comparison.
May 02, 2009 - Page (43)

4
Good Random NumberGood Random Number
Generation (cont’)Generation (cont’)
Th h ld b i i i th t f d i lThere should be provision in the generator for producing several
separate “stream” of random numbers. As we shall see, a stream
is simply a subsegment of the numbers produced by the generator,
with one stream beginning where the previous stream ends.
7
Methods forMethods for
I T f T h i (ITT)Inverse Transform Technique (ITT)
Exponential distribution
Uniform distribution
Triangular distribution
Empirical discrete distribution
Empirical continuous distribution
A t R j ti T h i
8
Acceptance-Rejection Technique
May 02, 2009 - Page (44)

5
Inverse TransformInverse Transform
TechniqueTechnique
Th i t f t h i b d tThe inverse transform technique can be used to
sample from the exponential, the Weibull and the
uniform distributions, and empirical distribution.
Additionally, it is the underlying principle for sampling
from a wide variety of discrete distributions.
A step by step procedure for the inverse transform
9
techniques, illustrated by the exponential distribution,
is as follows:
Step 1 Compute the cdf of the desired random variable.
For the exponential distribution, the cdf is
x
exF λ−
−=1)( 0≥x
Technique (cont’)Technique (cont’)
St 2 S t F( ) R th f XStep 2 Set F(x) = R on the range of X.
For the exponential distribution, it becomes
on the range
Since X is a random variable (with the exponential
distribution in this case), it follows that
is also a random variable, here called R. As will be shown
Re x
=− −λ
1 0≥x
Re x
=− −λ
1
10
,
later, R has a uniform distribution over the interval (0, 1).
May 02, 2009 - Page (45)

6
St 3 S l th ti F(X) R f X i t f R F thStep 3 Solve the equation F(X) = R for X in terms of R. For the
exponential distribution, the solution proceeds as follows:
1
)1ln(
1
1
RX
Re
Re
X
X
−
−=−
−=
=−
−
−
λ
λ
λ
11
)1ln(
1
RX −=
λ
Step 4 Generate (as needed) uniform random number R1 R2 R3Step 4 Generate (as needed) uniform random number R1, R2, R3,…,
and compute the desired random variable by
where
One simplification that is usually employed is to replace
by to yield
)(1
ii RFX −
=
)1ln(
1
)(1
ii RRF −
−
=−
λ
R1 R
12
by to yield
which is justified since both and are uniformly
distributed on (0, 1).
iR−1 iR
ii RX ln
1
λ
−
=
iR−1iR
May 02, 2009 - Page (46)

7
C id d i bl X th t i if l di t ib t d thConsider a random variable X that is uniformly distributed on the
interval [a, b].
The pdf of X is given by
Step 1 The cdf is given by
⎪⎩
⎪
⎨
⎧
≤≤
−=
otherwise,0
,
1
)(
bxa
abxf
0 ba
)(
1
ab −
13
p f g y
Step 2 Set F(X) = (X – a) / (b – a) = R
Step 3 Solving for X in terms of R yields X = a + (b – a) R
⎪
⎩
⎪
⎨
⎧
>
≤≤
−
−
<
=
bx
bxa
ab
ax
ax
xF
,1
,
,0
)(
Triangular DistributionTriangular Distribution
C id d i bl X hi h h dfConsider a random variable X which has pdf
This distribution is called a triangular distribution with
endpoints (0, 2) and mode at 1.
Step 1 The cdf is given by
⎪
⎩
⎪
⎨
⎧
≤<−
≤≤
=
otherwise,0
21,2
10,
)( xx
xx
xf
14
p f g y
⎪
⎪
⎪
⎩
⎪
⎪
⎪
⎨
⎧
>
≤<
−
−
≤<
≤
=
2,1
21,
2
)2(
1
10,
2
0,0
)( 2
2
x
x
x
x
x
x
xF
1 20
1
f(x)
May 02, 2009 - Page (47)

8
Empirical DiscreteEmpirical Discrete
DistributionsDistributions
Service Time Probability Cumulative Probability Random Number
1 0.10 0.10 0.01 – 0.10
2 0.20 0.30 0.11 – 0.30
3 0.30 0.60 0.31 – 0.60
4 0.25 0.85 0.61 – 0.85
5 0.10 0.95 0.86- 0.95
6 0.05 1.00 0.96 – 000
p(x)
15
0 1
1
1
2
Draw a sample
from a 0-1 uniform
distribution
Convert the 0-1 sample to an
equivalent sample from the
target population
0 1 2 3 4
0.3
0.2
0.1
5 6
Empirical ContinuousEmpirical Continuous
DistributionDistribution
If th d l h b bl t fi d th ti lIf the modeler has been unable to find a theoretical
distribution that provides a good model for the input
data, then it may be necessary to use the empirical
distribution of the data.
Suppose that 100 broken-widget repair times have been collected.
The data are summarized in the following table in terms of the
number of observations in various interval. For example, there
16
f p ,
were 31 observations between 0 and 0.5 hour, 10 between 0.5 and
1 hour, and so on.
Interval (Hours) Frequency Relative Frequency Cumulative Frequency
0.0 ≤ x ≤ 0.5 31 0.31 0.31
0.5 < x ≤ 1.0 10 0.10 0.41
1.0 < x ≤ 1.5 25 0.25 0.66
1.5 < x ≤ 2.0 24 0.24 1.00
May 02, 2009 - Page (48)

9
Empirical ContinuousEmpirical Continuous
Distribution (cont’)Distribution (cont’)
F(x)
(1.5, 0.66)
(2.0, 1.0)
1.0
0.8
0.6
R1=0.8
3
F(x)
eprobability
)( 1
1
1 = −
RFX
17
(1.0, 0.41)
(0.5, 0.31)
0 0.5 1.0 1.5 2.0
X1=1.7
5
0.4
0.2
x
Repair times
Cumulative
( )
75.1
5.10.2
66.000.1
66.0
5.1
)(
1
1
11
=
−⎥
⎦
⎤
⎢
⎣
⎡
−
−
+=
=
R
X
RFX
AcceptanceAcceptance--RejectionRejection
TechniqueTechnique
S th t d t d i th d fSuppose that we need to devise a method for
generating random variates, X, uniformly distributed
between ¼ and 1.
Step 1: Generate a random number u ~ U (0, 1)
Step 2a: If , accept X = u, then go to step 3.
Step 2b: If , reject u, and return to step 1.
Step 3: If another uniform random variate on [1/4 1] is needed
25.0≥u
25.0<u
18
Step 3: If another uniform random variate on [1/4, 1] is needed,
repeat the procedure beginning at step 1. If not, stop.
May 02, 2009 - Page (49)

10
Tests for RandomTests for Random
NumbersNumbers
Th d i bl ti f d bThe desirable properties of random numbers
uniformity and independence
To insure that these desirable properties are
achieved, a number of tests can be performed.
The tests can be placed in two categories according
to the properties of interest.
19
Test for uniformity
Test for independence
Tests for RandomTests for Random
Numbers (cont’)Numbers (cont’)
F t t U th hi t t tFrequency test: Uses the chi-square test to
compare the distribution of the set of numbers
generated to a uniform distribution.
Runs test: Tests the runs up and down or the runs
above and below the mean by comparing the actual
values to expected values. The statistic for
i i th hi
20
comparison is the chi-square.
May 02, 2009 - Page (50)

11
Frequency TestFrequency Test
FREQUENCY TESTFREQUENCY TEST
Random numbers about from the uniform distribution
and several tests have been developed to test for this
condition. We will consider the χ2 goodness-of-fit
test.
The goodness-of-fit test requires that:
21
50 observations in total
Expected frequency of at least five in each class
The following table shows the results of placing a total of 100
observations in 10 evenly spaced classes
Frequency Test (cont’)Frequency Test (cont’)
F T tFrequency Test
Classes Observed Frequency Expected Frequency (fo – fe)2 / fe
0.00 – 0.10 9 10 0.10
0.10 – 0.20 12 10 0.40
0.20 – 0.30 10 10 0.00
0.30 – 0.40 11 10 0.10
0.40 – 0.50 8 10 0.40
0.50 – 0.60 10 10 0.00
22
0.60 – 0.70 10 10 0.00
0.70 – 0.80 7 10 0.90
0.80 – 0.90 12 10 0.40
0.90 – 1.00 11 10 0.10
100 100 2.40
May 02, 2009 - Page (51)

12
Frequency Test (cont’)Frequency Test (cont’)
The question isThe question is,
Do these numbers come from the uniform distribution?
Calculating the χ2 statistic from the data using the equation
Gives a value of χ2 = 2.40. In testing the null hypothesis that the
random numbers come from the uniform distribution,
H R U [0 1]
∑
−
=
e
eo
f
ff 2
2 )(
χ
23
H0 : Ri ~ U [0, 1]
one compares the calculated χ2 to the value obtained from the
table based on (10-1) = 9 degree of freedom and a α = 0.05.
This χ2 value is found to be 16.919, which is larger than the
calculated χ2 value.
Therefore, we accepted the null hypothesis, and find our random
number generation acceptable.
Run Up and Down TestRun Up and Down Test
RUNS UP AND DOWN TESTRUNS UP AND DOWN TEST
Numbers can pass a uniformity test and still not be
random.
For example, the numbers 0.00, 0.10, 0.20, 0.30, 0.40, …obviously
are not random.
The numbers also must be sequentially random to be
j d d t l d
24
judged truly random.
A variety of runs test can be used for this purpose.
We will consider a run up and down test.
May 02, 2009 - Page (52)

13
(cont’)(cont’)
I f b if b i f ll d bIn a sequence of numbers, if a number is followed by
a larger number, this is an upward run.
Likewise, a number followed by a smaller number is a
downstream run.
If the numbers are truly random, one would expect to
find a certain numbers of runs up and down.
25
In a sequence of N numbers, one should expect to
find runs equal to the following equation:
90
2916
3
12 2 −
=
−
=
NN
δμ
(cont’)(cont’)
A l th t th f ll i 40As an example, assume that the following 40
numbers have been generated.
0.43, 0.32, 0.48, 0.23, 0.90, 0.72, 0.94, 0.11, 0.14, 0.67,
0.61, 0.25, 0.45, 0.56, 0.87, 0.54, 0.01, 0.64, 0.65, 0.32, 0.03,
– + – + – + – +
– – –+ +
26
0.93, 0.08, 0.58, 0.41, 0.32, 0.03, 0.18, 0.90, 0.74, 0.32,
0.75, 0.42, 0.71, 0.66, 0.03, 0.44, 0.99, 0.40, 0.51
– – –
– – –
+ + +
+ + + +
May 02, 2009 - Page (53)

14
(cont’)(cont’)
One should expect to find 26 33 runsOne should expect to find 26.33 runs
There were 26 runs in the sequence of numbers.
612
79.6
90
294016
33.26
3
1402
2
=
=
−×
=
=
−×
=
δ
δ
μ
961310
33.2626
96.1
33.26:
33.26:
025.0
1
0
−>−=
−
=
−
=
±=
≠
=
μ
μ
μ
X
Z
Z
H
H
27
We consider to accept the generated numbers are
random.
61.2=δ96.131.0
61.2
−>−===
δ
Z
Table of Random DigitsTable of Random Digits
10097 32533 76520 13586 34673 54876 80959 09117 39292 7494510097 32533 76520 13586 34673 54876 80959 09117 39292 74945
37542 04805 64894 74296 24805 24037 20636 10402 00822 91655
08422 68953 19645 09303 23209 02560 15953 34764 35080 33606
99019 02529 09376 70715 38311 31165 88676 74397 04436 27659
12807 99970 80157 36147 64032 36653 98951 16877 12171 76833
66065 74717 34072 76850 36697 36170 65813 39885 11190 29170
31060 10805 45571 82406 35303 42614 86799 07439 23403 09732
85269 77602 02051 65692 68665 74818 73053 85247 18623 88579
28
63573 32135 05325 47048 90553 57548 28468 28709 83491 25624
73796 45753 03529 64778 35808 34282 60935 20344 35273 88435
98520 17767 14905 68607 22109 40558 60970 93433 50500 73998
11805 05431 39808 27732 50725 68248 29405 24201 52775 67851
…
May 02, 2009 - Page (54)

1
Data Collection and ParameterData Collection and Parameter
E ti tiE ti tiEstimationEstimation
Input ModelingInput Modeling
I l ld i l ti li ti d t i iIn real-world simulation applications, determining
appropriate distributions for input data is a major task
from the standpoint of time and resource
requirements.
Faulty models of the inputs will lead to outputs whose
interpretation may give rise to misleading
d ti
2
recommendations.
Steps to develop a useful model for input data
Collect data from the real system of interest
Identify a probability distribution to represent the input process
May 02, 2009 - Page (55)

2
Input ModelingInput Modeling
(cont’)(cont’)
Ch t th t d t i ifi i t f thChoose parameters that determine a specific instance of the
distribution family
Evaluate the chosen distribution and the associated parameters for
goodness-of-fit
3
Data CollectionData Collection
Pl d t ll tiPlan your data collection process
Always try to find ways that can help you collect data
efficiently and accurately (equipment, barcoding,
receipts, personnel, video, etc)
Collect only data that is useful for your project
4
May 02, 2009 - Page (56)

3
IdentifyingIdentifying
the Distributionthe Distribution
HISTOGRAMSHISTOGRAMS
Divide the range of the data into intervals
Label the horizontal axis to conform to the intervals
selected
Determine the frequency of occurrences within each
interval
5
Label the vertical axis so that the total occurrences
can be plotted for each interval
Plot the frequencies on the vertical axis
(cont’)(cont’)
SELECTING THE FAMILY OF DISTRIBUTOINSSELECTING THE FAMILY OF DISTRIBUTOINS
Recall if the histogram drawn from your resembles
any kind of statistical distribution
Use physical basis (e.g. usage, discrete or
continuous) of the distribution as a guide
Use software
6
The exponential, normal, and Poisson distributions
are frequently encountered and are not difficult to
analyze from a computational standpoint
May 02, 2009 - Page (57)

4
(cont’)(cont’)
QUANTILE QUANTILE PLOTSQUANTILE-QUANTILE PLOTS
Evaluate the fit of the chosen distribution(s)
Compare the actual values with the values derived
from the chosen distribution
The nearer to become a straight line, the better the
accuracy
7
y
99.79 99.56 100.17 100.33
100.26 100.41 99.98 99.83
100.23 100.27 100.02 100.47
99.55 99.62 99.65 99.82
99.96 99.90 100.06 99.85
(cont’)(cont’)
Observed Valueq-q plot
99 40
99.60
99.80
100.00
100.20
100.40
100.60
100.80
Estimated
Observed Value
j Value j Value j Value j Value
1 99.55 6 99.82 11 99.98 16 100.26
2 99.56 7 99.83 12 100.02 17 100.27
3 99.62 8 99.85 13 100.06 18 100.33
4 99.65 9 99.90 14 100.17 19 100.41
5 99.79 10 99.96 15 100.23 20 100.47
8
99.20
99.40
99.40 99.60 99.80 100.00 100.20 100.40 100.60
Observed
⎟
⎠
⎞
⎜
⎝
⎛ −−
20
211 j
F
Estimated Value
j Value j Value j Value j Value
1 99.43 6 99.82 11 100.01 16 100.20
2 99.58 7 99.86 12 100.04 17 100.25
3 99.66 8 99.90 13 100.08 18 100.32
4 99.73 9 99.94 14 100.12 19 100.40
5 99.78 10 99.97 15 100.16 20 100.55
May 02, 2009 - Page (58)

5
Parameter EstimationParameter Estimation
S l M d S l V iSample Mean and Sample Variance
Calculate sample mean ( ) and variance ( ) from
the collected data
Based on the distribution chosen, convert the
parameters from the sample mean and variance
which is (are) used for the distribution
X 2
S
9
Distribution Parameter(s) Suggested Estimator(s)
Poisson α
Exponential λ
Normal μ, σ 2
X=αˆ
X/1ˆ =λ
22
ˆ
ˆ
S
X
=
=
σ
μ
GoodnessGoodness--ofof--Fit TestsFit Tests
P id h l f l ( tit ti ) id fProvides helpful (quantitative) guidance for
evaluating the suitability of a potential input model
Used in large samples size data
Use tables to determine accept or reject
10
May 02, 2009 - Page (59)

6
(cont’)(cont’)
Chi S T tChi-Square Test
This test is applied to for testing the hypothesis that a random
sample of size n of the random variable X follows a specific
distributional form
The test is valid for large sample sizes, for both discrete and
continuous distributional assumptions
∑
−
k
EO 2
2 )(
11
Oi is the observed frequency in the ith class interval
Ei is the expected frequency in that class interval
∑=
=
i i
ii
E
EO
1
2
0
)(
χ
(cont’)(cont’)
Example 9 13 (Poisson Assumption)Example 9.13 (Poisson Assumption)
H0 : the random variable is Poisson distributed
H1 : the random variable is not Poisson distributed
For α = 3.64, the probabilities associated with various values of x:
⎪⎩
⎪
⎨
⎧
==
−
otherwise,0
,...2,1,
!)( x
x
e
xp
x
αα
12
It is significantly to reject H0 at the 0.05 level of significance.
P(0) = 0.026 P(4) = 0.192 P(8) = 0.020
P(1) = 0.096 P(5) = 0.140 P(9) = 0.008
P(2) = 0.174 P(6) = 0.085 P(10) = 0.003
P(3) = 0.211 P(7) = 0.044 P(11) = 0.001
2.6 19.2 2.0
9.6 14.0 0.8
17.4 8.5 0.3
21.1 4.4 0.1
E(x)=np
68.271.112
117,05.0 <=−−χ
May 02, 2009 - Page (60)

7
(cont’)(cont’)
xi Observed frequency, Oi Expected Frequency, Ei
0 12 2.6
1 10 9.6
2 19 17.4 0.15
3 17 21.1 0.80
4 10 19.2 4.41
5 8 14.0 2.57
i
ii
E
EO 2
)( −
22 12.2 7.87
13
6 7 8.5 0.26
7 5 4.4
8 5 2.0
9 3 0.8
10 3 0.3
11 1 0.1
100 100.0 27.68
17 7.6 11.62
(cont’)(cont’)
E l 9 14 (E ti l A ti )Example 9.14 (Exponential Assumption)
H0 : the random variable is Exponential distributed
H1 : the random variable is not Exponential distributed
Let k = 8, then each interval will have probability p = 0.125
( ) ia
i eaF λ−
−=1
084.0/1ˆ == Xλ
1
14
ia
eip λ−
−=1
590.1)125.01ln(
084.0
1
1 =−−=a
)1ln(
1
ipai −−=⇒
λ
,677.11,252.8,595.5,425.3,1590.0 54321 ===== aaaaa
755.24,503.16 76 == aa
May 02, 2009 - Page (61)

8
(cont’)(cont’)
Class Interval Observed
frequency, Oi
Percentage Factor Expected
Frequency, Ei
[0, 1.590) 19 P(X≤0.159) – P(X ≤0) = 0.125 6.25 26.01
[1.590, 3.425) 10 P(X≤3.425) – P(X ≤1.590) = 0.125 6.25 2.25
[3.425, 5.595) 3 P(X≤5.595) – P(X ≤3.425) = 0.125 6.25 0.81
[5.595, 8.252) 6 P(X≤8.252) – P(X ≤5.595) = 0.125 6.25 0.01
[8.252, 11.677) 1 P(X≤11.677) – P(X ≤8.252) = 0.125 6.25 4.41
[11.677, 16.503) 1 P(X≤16.503) – P(X ≤11.677) = 0.125 6.25 4.41
i
ii
E
EO 2
)( −
x
exXP λ−
−=≤ 1)(
15
It is significantly to reject H0 at the 0.05 level of significance.
[11.677, 16.503) 1 P(X≤16.503) P(X ≤11.677) 0.125 6.25 4.41
[16.503, 24.755) 4 P(X≤24.755) – P(X ≤16.503) = 0.125 6.25 0.81
[24.755, ∞) 6 P(X≤ ∞) – P(X ≤24.755) = 0.125 6.25 0.01
50 1.000 50 39.6
6.126.39 2
118,05.0
2
0 =>= −−χχ
Selecting Input ModelsSelecting Input Models
without Datawithout Data
Engineering dataEngineering data
A product or process has performance ratings provided by the
manufacturer (for example, a laser printer fan produce 4
pages/minute)
Expert option
Talk to people who are experienced with the process or similar
processes.
Ph i l ti l li it ti
16
Physical or conventional limitations
Most real processes have physical limits on performance (for
example, computer data entry cannot be faster than a person can
type)
The nature of the process
Select the family of distribution
May 02, 2009 - Page (62)

1
Model Development and ModelModel Development and Model
V ifi tiV ifi tiVerificationVerification
Model BuildingModel Building
One of the most important and difficult tasks facing aOne of the most important and difficult tasks facing a
model developer is the verification and validation of
the simulation model.
To reduce the degree of skeptic about model’s validity
To increase the model’s credibility
Verification is concerned with building the model
right. It is utilized in the comparison of the
2
conceptual model to computer representation that
implements that conception.
Is the model implemented correctly in the computer?
Are the input parameters and logical structure of the model
correctly represented?
May 02, 2009 - Page (63)

2
(cont’)(cont’)
V lid ti i d ith b ildi th i ht d lValidation is concerned with building the right model.
It is utilized to determine that a model is an accurate
representation of the real system.
Validation is usually achieved through the calibration
of the model, an iterative process of comparing the
model to actual system behavior and using the
di i b t th t d th i i ht
3
discrepancies between the two, and the insights
gained, to improve the model.
This process is repeated until model accuracy is
judged to be acceptable.
(cont’)(cont’)
Th fi t t i d l b ildi i t f b iThe first step in model building consists of observing
the real system and the interactions among its
various components and collecting data on its
behavior.
Ask person who are familiar with the system.
New questions may arise.
Model developers will return to this step of learning true system
4
Model developers will return to this step of learning true system
structure and behavior.
May 02, 2009 - Page (64)

3
(cont’)(cont’)
Th d t i d l b ildi i th t tiThe second step in model building is the construction
of a conceptual model.
A collection of assumptions on the components and the structure of
the system, plus hypotheses on the values of model input
parameters.
The third step is the translation of the operational
model into a computer-recognizable form – the
5
model into a computer recognizable form the
computerized model.
ModelModel--buildingbuilding
ProcessProcess
Real system
Conceptual model
1. Assumptions on system components
2. Structural assumptions, which define the interactions
between system components
Conceptual validation
Calibration
And
validation
6
between system components
3. Input parameters and data assumptions
Operational model
(Computerized
representation)
Model verification
May 02, 2009 - Page (65)

4
VerificationVerification
Th f d l ifi ti i t th tThe purpose of model verification is to assure that
the conceptual model is reflected accurately in the
computerized representation.
The conceptual model quite often involves some
degree of abstraction about system operations, or
some amount of simplification of actual operations.
7
Verification asks the question:
Is the conceptual model (assumptions on system components and
system structure, parameter values, abstractions and
simplifications) accurately represented by the operational model?
Three Classes ofThree Classes of
TechniqueTechnique
C t h iCommon-sense techniques
Thorough documentation
Traces
8
May 02, 2009 - Page (66)

5
CommonCommon--sensesense
TechniquesTechniques
Ch k d b th th it d lChecked by someone other than its developer.
Make a flow diagram and follow each event type.
Examine the output for reasonableness under a
variety of settings of the input parameters.
Print the input parameters at the end of simulation to
ensure that these parameters values have not been
9
p
changed inadvertently.
Make the model as self-documenting as possible.
CommonCommon--sensesense
Techniques (cont’)Techniques (cont’)
If the operational model is animated verify that whatIf the operational model is animated, verify that what
is seen in the animation imitates the actual system.
Use the debugger provided by the simulation
software.
Use a variety of graphics to represent different model
states.
For example (reasonableness)
10
For example, (reasonableness)
Current contents and total count
Current content refers to the number of items in
each component of the system at a given time.
Total count refers to the total number of items that
have entered each component of the system by a give
time.
May 02, 2009 - Page (67)

6
OftOft--neglectedneglected
DocumentationDocumentation
TechniqueTechnique
D t ti i l i t t fDocumentation is also important as a means of
clarifying the logic of a model and verifying its
completeness.
If a model builder writes brief comments in the
computerized model, plus definitions of all variables
and parameters, and descriptions of each major
ti f th t i d d l it b h
11
section of the computerized model, it becomes much
simpler for someone else, or the model builder at a
later date, to verify the model logic.
Trace TechniqueTrace Technique
A more sophisticated techniqueA more sophisticated technique.
A trace is a detailed computer printout which gives
the value of every variable in a computer program,
every time that one of these variables changes in
value.
The purpose of the trace is to verify the correctness
of the computer program by making detailed paper-
12
p p g y g p p
and-paper calculations.
Some software allows a selective trace.
Whenever the queue before a certain resource reaches five or
more, turn on the trace.
May 02, 2009 - Page (68)

7
RecommendationsRecommendations
It i d d th t th fi t t l b i dIt is recommended that the first two always be carried
out.
Close examination of model output for
reasonableness is especially valuable and
informative.
A trace can also provide information if it is selective.
13
The generalized trace can be extremely time
consuming.
May 02, 2009 - Page (69)

1
Model Calibration and ModelModel Calibration and Model
V lid tiV lid tiValidationValidation
Calibration of ModelsCalibration of Models
Verification and validation although conceptuallyVerification and validation, although conceptually
distinct, usually are conducted simultaneously by the
modeler.
Validation is the overall process of comparing the
model and its behavior to the real system and its
behavior.
Calibration is the iterative process of comparing the
2
p p g
model to the real system, making adjustments (or
even major changes) to the model, comparing the
revised model to reality, making additional
adjustments, comparing again, and so on.
May 02, 2009 - Page (70)

2
Comparison of theComparison of the
ModelsModels
The comparison of the model to reality is carried outThe comparison of the model to reality is carried out
by a variety of tests
Subjective and Objectives
Subjective tests usually involve people, who are
knowledgeable about one or more aspects of the
system, making judgments about the model and its
output.
3
Objective tests always require data on the system’s
behavior plus the corresponding data produced by
the model. Then one or more statistical tests are
performed to compare some aspect of the system
data set to the same aspect of the model data set.
Iterative Process ofIterative Process of
Calibration a ModelCalibration a Model
Compare model
RealReal
systemsystem
Initial modelInitial model
First revisionFirst revision
of modelof model
Revise
R i
Compare model
to reality
Compare revised
model to reality
4
Second revisionSecond revision
of modelof model
Revise
Revise
Compare second
Revision to reality
May 02, 2009 - Page (71)

3
ThreeThree--step Validationstep Validation
ApproachApproach
B ild d l th t h hi h f liditBuild a model that has high face validity
Validate model assumptions
Compare the model input-output transformations to
corresponding input-output transformations for the
real system
5
Face ValidityFace Validity
The first goal of the simulation modelers is toThe first goal of the simulation modelers is to
construct a model that appears reasonable on its
face to model users and others who are
knowledgeable about the real system being
simulated.
The potential users should be involved in model
construction. (Conceptual Implementation)
6
Sensitive analysis can be used to check a model’s
face validity.
E.g., if the arrival rate of customer were to increase, it would be
expected that …
utilization of servers, lengths of lines, and delays would tend to
increase.
May 02, 2009 - Page (72)

4
Validation of ModelValidation of Model
AssumptionsAssumptions
Model assumptions fall into two general classes:Model assumptions fall into two general classes:
Structural assumptions and data assumptions
Structural assumptions involve questions of how the
system operates and usually involve simplifications
and abstractions of reality.
E.g., number of tellers may be fixed or variable
Verified by actual observation, discussion with managers, etc.
7
Data assumptions should be based on the collection
of reliable data and correct statistical analysis of the
data.
E.g., Interarrival time, service times, etc.
Input data analysis
Validating InputValidating Input--OutputOutput
TransformationsTransformations
DecisionDecision
RandomRandom
variablesvariables
MODELMODEL
“Black box”“Black box”
OutputOutput
variablesvariables
8
DecisionDecision
variablesvariables
Input variables Model Output variables
May 02, 2009 - Page (73)

5
InputInput--Output ValidationOutput Validation
–– Artificial Input DataArtificial Input Data
ARTIFICIAL INPUT DATAARTIFICIAL INPUT DATA
When the model is run using generated random
variates, it is expected that observed values should
be close to collected values.
Hypothesis test – average customer delay: (p.380)
( ) minutes34: 20 =YEH 512
1
== ∑YY
n
9
( )
( ) minutes3.4:
minutes3.4:
21
20
≠
=
YEH
YEH
( )
( )
82.0
1
51.2
1
2
22
2
1
22
=
−
−
=
==
∑
∑
=
=
n
YY
YS
Y
n
Y
n
i ii
i
i
nS
Y
t 02
0
μ−
=
(Example)(Example)
T bl 10 2 ( 383)Table 10.2 (p.383)
Replication Y4, Observed
arrival rate
Y5, Average service
time
Y2, Average
Delay
1 51 1.07 2.79
2 40 1.12 1.12
3 45.5 1.06 2.24
4 50 5 1 10 3 45
10
4 50.5 1.10 3.45
5 53 1.09 3.13
6 49 1.07 2.38
Sample mean 2.51
Standard deviation 0.82
May 02, 2009 - Page (74)

6
(cont’)(cont’)
Hypothesis test (cont’):Hypothesis test (cont’):
Degree of freedom = n – 1 = 5
Since ,
( )
34.5
682.0
3.451.2
3.4
0
20
−=
−
=
==
t
YEμ
571.234.5 5,025.00 =>= tt
11
reject H0 and conclude that the model is inadequate
in its prediction of average customer delay at α=0.05.
AreAre thesethese assumptionsassumptions metmet inin thethe presentpresent case?case?
(p(p..385385))
–– Historical Input DataHistorical Input Data
HISTORICAL INPUT DATAHISTORICAL INPUT DATA
An alternative to generating input data is to use the
actual historical record to drive the simulation model
and then to compare model output to system data.
12
May 02, 2009 - Page (75)

7
(cont’)(cont’)
Input Data
Set
System
Output, Zij
Model
Output, Wij
Observed
Difference, dj
Squared Deviation from
Mean,
1 Zi1 Wi1
2 Zi2 Wi2
3 Zi3 Wi3
… … …
( )2
dd j −
( )2
2 dd −
( )2
3 dd −
111 ii WZd −=
222 ii WZd −=
333 ii WZd −=
( )2
1 dd −
13
K ZiK WiK
( )2
ddK −
∑=
=
K
j
jd
K
d
1
1
( )∑=
−
−
=
K
j
jd dd
K
S
1
22
1
1
444 ii WZd −=
(Example)(Example)
E l 10 4 ( 392)Example 10.4 (p.392)
0:
0:
1
0
≠
=
d
d
H
H
μ
μ
37.1
585.8705
2.5343
0 ==
−
=
KS
d
t
d
dμ
72
10580.7
2.5343
5
×=
=
=
dS
d
K
14
Since ,
the null hypothesis cannot be rejected at α = 0.05.
278.237.1 4,025.00 =<= tt
May 02, 2009 - Page (76)

8
I dditi t t ti ti l t t h t ti ti l
–– Turing TestTuring Test
In addition to statistical test, or when no statistical
test is readily applicable, persons knowledgeable
about system behavior can be used to compare
model output to system output.
SystemSystem
ff
15
performanceperformance
OutputOutput
ReportsReports ?
ConclusionConclusion
Th l f th lid ti i t f ldThe goal of the validation process is twofold:
to produce a model that represents true system behavior closely
enough for the model to be used as a substitute for the actual
system for the purpose of experimenting with the system;
to increase to an acceptable level the credibility of the model, so
that the model will be used by managers and other decision
makers.
16
May 02, 2009 - Page (77)

1
Output Analysis for a Single ModelOutput Analysis for a Single Model
Output Analysis for aOutput Analysis for a
Single ModelSingle Model
O t t l i i th i ti f d t t dOutput analysis is the examination of data generated
by a simulation.
Its purpose is to predict the performance of a system
or to compare the performance of two or more
alternative system designs.
This lecture deals with the analysis of a single
2
system, while next lecture deals with the comparison
of two or more systems.
May 02, 2009 - Page (78)

2
Type of Simulation withType of Simulation with
respect to Outputrespect to Output
AnalysisAnalysis
Wh l i i l ti t t d t di ti tiWhen analyzing simulation output data, a distinction
is made between terminating or transient simulation
and steady-state simulation.
A terminating simulation is one that runs for some
duration of time TE, where E is a specified event (or
set of events) which stops the simulation.
3
Example 11.1: Shady Grove Bank operates 8:30 – 16:30, then
TE = 480min.
Example 11.3: A communication system consists of several
components. Consider the system over a period of time, TE , until
the system fails. E = {A fails, or D fails, or (B and C both fail)}
Terminating SimulationTerminating Simulation
When simulating a terminating system the initialWhen simulating a terminating system, the initial
conditions of the system at time 0 must be specified,
and the stopping time TE, or alternatively, the
stopping event E, must be well defined.
Whether a simulation is considered to be terminating
or not depends on both the objectives of the
simulation study and the nature of the system.
4
Examples 11.1 and 11.3 are considered the terminating systems
because:
Ex. 11.1: the objective of interest is one day’s operation;
Ex. 11.3: short-run behavior, from time 0 until the first system
failure.
May 02, 2009 - Page (79)

3
SteadySteady--statestate
A t i ti t i t th tA nonterminating system is a system that runs
continuously, or at least over a very long period of time.
For example, assembly lines which shut down infrequently,
continuous production systems of many different types, telephone
systems and other communications systems such as the Internet,
hospital emergency rooms, fire departments, etc.
A steady-state simulation is a simulation whose
5
objective is to study long-run, or steady-state,
behavior of a nonterminating system.
The stopping time, TE, is determined not by the nature of
the problem but rather by the simulation analyst, either
arbitrarily or with a certain statistical precision in mind.
Stochastic Nature ofStochastic Nature of
Output DataOutput Data
Consider one run of a simulation model over a periodConsider one run of a simulation model over a period
of time [ 0, T ]. Since the model is an input-output
transformation, and since some of the model input
variables are random variable, it follows that the
model output variables are random variables.
The stochastic (or probability) nature of output
variables will be observed.
6
Example 2.2 (Able-Baker carhop problem)
Input: randomness of arrival time and service time
Output: randomness of utilization and time spent in the system per
customer.
May 02, 2009 - Page (80)

4
Output Analysis forOutput Analysis for
TerminatingTerminating
C id th ti ti f f tConsider the estimation of a performance parameter,
θ (or φ), of a simulated system.
The simulation output data is of the form {Y1, Y2, … ,
Yn} (discrete-time data) for estimating θ.
E.g. the delay of customer i, total cost in week i.
The simulation output data is of the form {Y(t), 0 ≤ t ≤
7
TE} (continuous-time data) for estimating φ.
E.g. the queue length at time t, the number of backlogged orders at
time t.
Point Estimation:
∑=
=
n
i
iY
n 1
1ˆθ dttY
T
ET
E
∫=
0
)(
1ˆφ
Simulations (cont’)Simulations (cont’)
B th C t l Li it d Th (CLT) f 30By the Central Limited Theorem (CLT), for n ≥ 30,
where
Interval Estimation:
An approximate 100(1 α)% confidence interval for θ
nS
t
/)ˆ(
ˆ
)ˆ(ˆ
ˆ
θ
θθ
θσ
θθ −
=
−
=
1
)ˆ(
)ˆ( 1
2
2
−
−
=
∑=
n
Y
S
n
i
i θ
θ
8
An approximate 100(1 - α)% confidence interval for θ
is given by:
n
S
t
n
S
t nn
)ˆ(ˆ)ˆ(ˆ
1,2/1,2/
θ
θθ
θ
θ αα −− +≤≤−
May 02, 2009 - Page (81)

5
Simulations (Example)Simulations (Example)
E l 11 10 (Abl B k C h P bl )Example 11.10 (Able Baker Carhop Problem)
Run, r Utilization, Average System Time,
1 0.808 3.74
2 0.875 4.53
3 0.708 3.84
4 0.842 3.98
rρˆ rwˆ
8420708087508080 +++
9
922.0694.0
)036.0)(18.3(808.0
)ˆ(ˆˆ
)036.0(
)4(3
)808.0842.0()808.0808.0(
)ˆ(ˆ
808.0
4
842.0708.0875.0808.0
ˆ
3,025.0
2
22
2
≤≤
±
±
=
−++−
=
=
+++
=
ρ
ρσρ
ρσ
ρ
t
L
Number of ReplicationsNumber of Replications
PRECISION LEVELPRECISION LEVEL
Suppose that an error criterion ε is specified; in other
words, it is desired to estimate θ by to within
with high probability, say at least 1 – α.
θˆ ε±
1,2/
)ˆ(
≤− ε
θ
α
R
S
t R
10
2
2/
2
1,2/
)ˆ()ˆ(
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≥
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
≥ −
ε
θ
ε
θ
αα
S
z
S
tR
R
R
May 02, 2009 - Page (82)

6
Number of ReplicationsNumber of Replications
(Example)(Example)
E l 11 12 (Abl B k C h P bl )Example 11.12 (Able Baker Carhop Problem)
Suppose that it is desired to estimate Able’s
utilization in Example 11.7 to within with
probability 0.95. An initial sample size R0=4 is taken.
Step 1:
04.0±
1344.12
)040(
)00518.0()96.1(
2
22
2
2
0
2
025.0
≈==
Sz
11
Step 2:
)04.0( 22
ε
R 13 14 15
t0.025, R-1 2.18 2.16 2.14
15.39 15.10 14.83
2
2
0
2
1,025.0
ε
St R−
R = 15
Additional replications:
R – R0 = 15 – 4 = 11
SteadySteady--StateState
Prior to beginning analysis of output data thePrior to beginning analysis of output data, the
modeler must take every effort to ensure that the
output represents an accurate estimate of the true
system values.
One useful technique for improving the reliability of
output results from steady-state simulation is to
provide an initialization period for which statistics are
12
not kept.
A steady-state condition implies that a simulation has
reached a point in time where the state of the model
is independent of the initial start-up conditions.
May 02, 2009 - Page (83)

7
Simulations (cont’)Simulations (cont’)
Th t f ti i d t hi t d t tThe amount of time required to achieve steady-state
conditions is referred to as a warm-up period.
Data collection begins after a warm-up period is
completed.
Determining the length of this period can be
accomplished by utilizing moving averages calculated
13
from the output produced by multiple model
replications.
=)(wYi
∑−=
+ +
w
ws
si wY )12/(
∑
−
−−=
+ −
1
)1(
)12/(
i
is
si iY
wnwifor −+= ,,1 K
wifor ,,1 K=
WarmWarm--up Periodup Period
Determine A Warm-up Period in a Steady-state Simulation
Period Average Cost w = 5 w = 10 w = 19
1 422.00 422.00 422.00 422.00
2 468.16 522.20 522.20 522.20
3 676.45 502.72 502.72 502.72
4 572.88 568.92 568.92 568.92
5 374.10 571.26 571.26 571.26
6 842.90 560.94 560.94 560.94
7 625.92 587.72 563.86 563.86
8 4 3 08 8 46 4 3 4 3
14
8 473.08 585.46 574.53 574.53
9 685.88 568.25 569.68 569.68
10 528.79 588.95 578.06 578.06
11 500.22 611.93 565.67 565.67
12 716.52 575.28 568.46 558.81
13 443.33 546.57 569.65 561.05
14 487.13 593.43 564.58 563.82
May 02, 2009 - Page (84)

8
Moving AverageMoving Average
Average Monthly Cost
1200
Moving Average for w = 5
1200
0
200
400
600
800
1000
1
5
9
13
17
21
25
29
33
37
0
200
400
600
800
1000
1
5
9
13
17
21
25
29
33
37
Moving Average for w = 10 Moving Average for w = 19
15
0
200
400
600
800
1000
1200
1
5
9
13
17
21
25
29
33
37
0
200
400
600
800
1000
1200
1
5
9
13
17
21
25
29
33
37
d = 12
Observed cost during i-th period and j-th replication
Period Rep 1 Rep 2 Rep 3 Rep 4 Rep 5
13 376.81 500.97 192.96 509.00 636.92
14 352.05 329.30 587.45 336.11 530.74
15 518.96 634.81 716.81 533.05 1899.13
16 673.88 853.97 563.86 179.72 864.17
17 376.99 1098.67 290.92 205.43 276.93
18 139.26 339.08 563.49 319.10 189.20
19 199.54 4032.35 355.94 138.88 215.99
20 542 79 908 48 633 90 349 55 727 93
Period Rep 1 Rep 2 Rep 3 Rep 4 Rep 5
27 589.21 649.52 544.64 296.36 289.96
28 103.42 936.04 393.21 771.45 151.18
29 219.14 1338.29 163.15 169.59 938.19
30 169.36 841.89 651.41 492.09 232.72
31 791.25 137.11 734.38 807.81 401.16
32 1360.99 274.57 457.19 148.87 231.46
33 530.04 1259.50 497.51 1300.90 990.27
34 198 98 275 20 177 60 723 29 414 02
16
20 542.79 908.48 633.90 349.55 727.93
21 383.47 317.29 165.11 345.11 106.20
22 276.26 387.05 366.19 789.89 613.58
23 336.39 388.63 605.87 315.20 818.90
24 562.06 323.83 1311.94 339.76 312.95
25 931.46 236.80 706.43 484.30 658.11
26 182.82 352.79 991.44 271.73 1815.32
34 198.98 275.20 177.60 723.29 414.02
35 523.28 1012.45 904.77 212.75 523.06
36 633.30 723.07 431.72 245.69 158.22
37 631.47 455.12 1256.28 287.57 351.78
38 807.24 1627.84 994.93 215.50 603.36
39 271.41 138.54 352.43 441.73 352.38
Avg. 469.70 752.74 565.96 427.05 567.14
May 02, 2009 - Page (85)

9
C fid t I t l f St d St t Si l tiConfident Interval for a Steady-State Simulation
dn
Y
dnY
n
dj
rj
r
−
=
∑+= 1
),(
5
14.56705.42796.56574.75270.469
),(
1
)(
1
++++
== ∑=
R
r
r dnY
R
RY
17
49.15752)(
1
1
)( 2
1
2
=−
−
= ∑=
R
r
r YY
R
RS
5/)51.125)(78.2(57.556/)(4,025.0 ±=± RRStY
04.15657.556 ±=
May 02, 2009 - Page (86)

1
Comparison and Evaluation ofComparison and Evaluation of
Alt ti S t D iAlt ti S t D iAlternative System DesignsAlternative System Designs
Basic Concept ofBasic Concept of
Confidence IntervalConfidence Interval
C fid I t l id f l b dConfidence Interval provides range of values based
on observations from 1 sample, .
A probability that the population parameter falls
somewhere within the interval.
Confidence Interval
Sample Statistic
(Point Estimate)
( )SX ,
2
Confidence Limit
(Lower)
Confidence Limit
(Upper)
( )XestX n ..1,2/ −± α
X
C.I. for a mean:
n
S
tX
n
S
tX nn ⋅+≤≤⋅− −− 1,2/1,2/ αα μ
May 02, 2009 - Page (87)

2
Two SystemsTwo Systems
O f th t i t t f i l ti i thOne of the most important uses of simulation is the
comparison of alternative system designs.
A two-sided 100(1 – α)% C.I. for θ1 – θ2 will always be
Parameter Estimator
System 1 θ1
System 2 θ2
1Y
2Y
3
A two sided 100(1 α)% C.I. for θ1 θ2 will always be
of the form:
( ) ( )21,2/21 .. YYestYY v −±− α
( ) ( ) ( ) ( )21,2/212121,2/21 .... YYestYYYYestYY vv −+−≤−≤−−− αα θθ
Comparison ofComparison of
AlternativesAlternatives
C I b d th t diff θ θ ithi thC.I. bounds the true difference θ1 – θ2 within the range
with probability 1 – α.
( x )
0
21 YY −
0
( x )
2121 0 θθθθ <⇔<−
0 θθθθ >⇔>−
4
0
21 YY −
0
( x )
21 YY −
2121 0 θθθθ >⇔>
2121 0 θθθθ =⇔=−
May 02, 2009 - Page (88)

3
Independent SamplingIndependent Sampling
with Equal Varianceswith Equal Variances
I d d t li th t diff t dIndependent sampling means that different and
independent random-number streams will be used to
simulate the two systems.
where( ) ( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+⋅
−−−
=−+
21
2
2121
2,2
11
21
RR
S
YY
t
p
RR
μμ
α
( ) ( )
2
11
21
2
22
2
112
−+
−−−
=
RR
SRSR
Sp
5
⎠⎝ 21 21
( )
21
21
11
..
RR
SYYes p +=−
( ) ( )212,2/21 ..21
YYestYY RR −±− −+α
Independent SamplingIndependent Sampling
with Unequal Varianceswith Unequal Variances
If th ti f l i t f l bIf the assumption of equal variances cannot safely be
made, an approximate 100(1 – α)% C.I. for θ1 – θ2 can
be computed as follows.
where( ) ( )
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
−−−
=
2
2
2
1
2
1
2121
,2
R
S
R
S
YY
t v
μμ
α
( )
( ) ( )2222
2
2
2
21
2
1 +
=
RSRS
RSRS
v
6
⎠⎝ 21 ( ) ( )
11 2
22
1
11
−
+
− R
RS
R
RS
( ) ⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+=−
2
2
2
1
2
1
21..
R
S
R
S
YYes
( ) ( )21,2/21 .. YYestYY v −±− α
May 02, 2009 - Page (89)

4
Correlated SamplingCorrelated Sampling
C l t d li th t f h li tiCorrelated sampling means that, for each replication,
the same random numbers are used to simulate both
system. Therefore, R1 and R2 must be equal, say
R1 = R2 = R.
where21 rrr YYD −= ∑=
=
R
r
rD
R
D
1
1
R
1D μ−
7
( )∑=
−
−
=
R
r
rD DD
R
S
1
22
1
1
R
S
D
t
D
D
R
μ
α =−1,2
( ) ( )
R
S
YYesDes D
=−= 21.... ( )DestD R ..1,2 −± α
Experimental DesignExperimental Design
E i t l d i id f d idiExperimental design provides a way of deciding
before the runs are made which particular
configurations to simulate so that the desired
information can be obtained with the least amount of
simulating.
The input parameters and structural assumptions
i d l ll d f t d th t t
8
composing a model are called factors, and the output
performance measures are called responses.
May 02, 2009 - Page (90)

5
Experimental DesignExperimental Design
(cont’)(cont’)
E h ibl l f f t i ll d l l f thEach possible value of a factor is called a level of the
factor.
A combination of factors all at a specified level is
called a treatment.
Factors can be either quantitative or qualitative.
These factors are collectively called decision
9
y
variables, or policy variables.
E.g., queue discipline (policy variable), number of physicians
(decision variable).
Factorial DesignFactorial Design
S ti i l ti l d tSometimes simulation analyses are used to
determine the effects that various factors exert on
selected performance criteria.
Factorial designed experiments are one means of
providing this type of information.
The results produced from these experiments can be
10
statistically analyzed to measure the 1) main effects,
and 2) interactive effects that selected factors exert
on performance indices (system responses).
May 02, 2009 - Page (91)

6
Factorial Design (cont’)Factorial Design (cont’)
A i ff t (d t E ) i th h iA main effect (denote Ei) is the average change in a
response resulting from raising the ith factor from a
specified “low level” to a specified “high level”.
Suppose we perform a simulation to investigate three
factors (lot size, quantity of machines, and set-up
time) regarding their individual effects on a product’s
k
11
makespan.
Factor Low Level (–) High Level (+)
Lot Size, E1 5 10
Machine Quantity, E2 1 2
Rework Rate, E3 6% 12%
Main Effect FactorMain Effect Factor
A design matrix with 2x2x2 design points is
t t dconstructed.
Design Point Factor 1 Level
(Lot Size)
Factor 2 Level
(Machine Qty)
Factor 3 Level
(Rework Rate)
Response
(Makespan)
1 + + + R1 = 5.7
2 – + + R2 = 5.0
3 + – + R3 = 12.1
4 – – + R4 = 11.1
5 + + – R5 = 5.7
12
5 R5 5.7
6 – + – R6 = 5.0
7 + – – R7 = 12.1
8 – – – R8 = 11.1
+ R1 – R2 + R3 – R4 + R5 – R6 + R7 – R8
2 k-1
Main effect factor #1 E1 =
May 02, 2009 - Page (92)

7
Main Effect Factor andMain Effect Factor and
Interactive EffectInteractive Effect
FactorsFactors
+ R1 + R2 – R3 – R4 + R5 + R6 – R7 – R8
2 k-1
+ R1 + R2 + R3 + R4 – R5 – R6 – R7 – R8
2 k-1
Interactive effect factor #1 and #2 E12 =
+ R1 – R2 – R3 + R4 + R5 – R6 – R7 + R8
13
12
2 k-1
+ R1 – R2 + R3 – R4 – R5 + R6 – R7 + R8
2 k-1
+ R1 + R2 – R3 – R4 – R5 – R6 + R7 + R8
2 k-1
Analyzing FactorialAnalyzing Factorial
Designed ExperimentsDesigned Experiments
A i t ti ff t t ll if th ff t f iAn interactive effect tells us if the effect of a given
factor is influenced by the level of another factor.
If there is a significant interactive effect, then we
cannot be certain that a main effect is due solely to
the raising or lowering of a factor level.
Main Effects Raising a lot size from 5 to 10 is
t i d t k
14
Lot Size, E1 0.8
Machine Quantity, E2 -6.3
Rework Rate, E3 0.0
Interactive Effects
Lot Size & Machine Qty, E12 -0.2
Lot Size & Rework Rate, E13 0.0
Machine Qty & Rework Rate, E23 0.0
to increase product makespan an
average of 0.8 days per part.
Adding an additional machine
decreased the makespan by an
average of 6.3 days.
May 02, 2009 - Page (93)

8
Documentation andDocumentation and
ConclusionsConclusions
D t ti b di id d i t fiDocumentation can be divided into five areas:
Objectives and Assumptions
Model Input Parameters
Experimental Design
Results
Conclusions
Obj ti d A ti
15
Objectives and Assumptions
All objectives and assumptions should be recorded at the onset of
any simulation project. Any changes or modifications made
during the course of building a model need to be included in the
final report.
Conclusions (cont’)Conclusions (cont’)
M d l I t P tModel Input Parameters
This section contains a recap of the data used with a simulation.
System flow charts, mathematical calculations, performance
criteria, solution constraints, solution restrictions, and any cost
related information should be included.
Experimental Design
The information summarized in this category is comprised of
16
he info mation summa i ed in this catego y is comp ised of
descriptions regarding the alternatives investigated, the
experiments designed for comparing alternatives, starting
conditions, stopping conditions, a history of the random number
streams employed with each experiment, and an account for the
number of model replications performed for each alternative.
May 02, 2009 - Page (94)

9
Conclusions (cont’)Conclusions (cont’)
R ltResults
This section is composed of the output data produced by a
simulation. It also provides an overview of the statistical analyses
performed on the data. Tables and graphical charts which
illustrate the findings are very beneficial.
Conclusions
One of the final steps in any decision-making process is to make
17
One of the final steps in any decision making p ocess is to make
conclusions and recommendations. This demands that benefit-to-
cost ratios be investigated for each alternative. What are the total
costs (tangible and intangible) needed to implement an alternative,
and what are the total benefits anticipated from doing it?
Risk and UncertaintiesRisk and Uncertainties
Si d i i ki i b d th t fSince decision-making is based on the precept of
prediction, risk and uncertainties are almost always
involved. The potential labor requirements
forecasted with a given alternative may fall within a
range. This can be classified as an uncertainty. The
potential outcome for an alternative may also vary.
This can be designated as a risk Any uncertainties
18
This can be designated as a risk. Any uncertainties
and risks associated with an alternative should be
discussed in the final documentation.
May 02, 2009 - Page (95)

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