All in One simulation notes

7/21/2019 All in One simulation notes

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Chapter 1 Introduction toSimulation

Banks, Carson, Nelson & Nicol

Discrete-Event System Simulation


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Outline

When Simulation Is the Appropriate Tool

When Simulation Is Not Appropriate

Advantages and Disadvantages of Simulation

Areas of Application

Systems and System Environment

Components of a System Discrete and Continuous Systems

Model of a System

Types of Models Discrete-Event System Simulation

Steps in a Simulation Study


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Definition

A simulation is the imitation of the operation of real-world

process or system over time.

Generation of artificial history and observation of that

observation history

Amodelconstruct a conceptual framework that

describes a system The behavior of a system that evolves over time is

studied by developing a simulation model.

The model takes a set of expressed assumptions:

Mathematical, logical

Symbolic relationship between the entities


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Goal of modeling and simulation

A model can be used to investigate a wide verity of what

if questions about real-world system.

Potential changes to the system can be simulated and predicate

their impact on the system.

Find adequate parameters before implementation

So simulation can be used as Analysis tool for predicating the effect of changes

Design tool to predicate the performance of new system

It is better to do simulation before Implementation.


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How a model can be developed?

Mathematical Methods

Probability theory, algebraic method , Their results are accurate

They have a few Number of parameters

It is impossible for complex systems Numerical computer-based simulation

It is simple

It is useful for complex system


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When Simulation Is the Appropriate Tool

Simulation enable the study of internal interaction of a subsystemwith complex system

Informational, organizational and environmental changes can besimulated and find their effects

A simulation model help us to gain knowledge about improvement ofsystem

Finding important input parameters with changing simulation inputs

Simulation can be used with new design and policies beforeimplementation

Simulating different capabilities for a machine can help determinethe requirement

Simulation models designed for training make learning possiblewithout the cost disruption

A plan can be visualized with animated simulation

The modern system (factory, wafer fabrication plant, serviceorganization) is too complex that its internal interaction can betreated only by simulation


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When Simulation Is Not Appropriate

When the problem can be solved by common

sense. When the problem can be solved analytically.

If it is easier to perform direct experiments.

If cost exceed savings. If resource or time are not available.

If system behavior is too complex.

Like human behavior


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Advantages and disadvantages of simulation

In contrast to optimization models, simulation

models are run rather than solved.Given as a set of inputs and model characteristics the

model is run and the simulated behavior is observed


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Advantages of simulation New policies, operating procedures, information flows and son on

can be explored without disrupting ongoing operation of the realsystem.

New hardware designs, physical layouts, transportation systemsand can be tested without committing resources for theiracquisition.

Time can be compressed or expanded to allow for a speed-up orslow-down of the phenomenon( clock is self-control).

Insight can be obtained about interaction of variables and importantvariables to the performance.

Bottleneck analysis can be performed to discover where work inprocess, the system is delayed.

A simulation study can help in understanding how the systemoperates.

What if questions can be answered.


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Disadvantages of simulation

Model building requires special training.

Vendors of simulation software have been activelydeveloping packages that contain models that only

need input (templates).

Simulation results can be difficult to interpret.

Simulation modeling and analysis can be time

consuming and expensive.

Many simulation software have output-analysis.


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Areas of application Manufacturing Applications

Semiconductor Manufacturing

Construction Engineering and project management

Military application

Logistics, Supply chain and distribution application

Transportation modes and Traffic

Business Process Simulation

Health Care

Automated Material Handling System (AMHS) Test beds for functional testing of control-system software

Risk analysis Insurance, portfolio,...

Computer Simulation CPU, Memory,

Network simulation Internet backbone, LAN (Switch/Router), Wireless, PSTN (call center),...


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Systems and System Environment

A system is defined as a groups of objects thatare joined together in some regular interactiontoward the accomplishment of some purpose.An automobile factory: Machines, components parts

and workers operate jointly along assembly line

A system is often affected by changes occurringoutside the system: system environment. Factory : Arrival orders

Effect of supply on demand : relationship between factory

output and arrival (activity of system)

Banks : arrival of customers


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Components of system Entity

An object of interest in the system : Machines in factory

Attribute The property of an entity : speed, capacity

Activity A time period of specified length :welding, stamping

State A collection of variables that describe the system in any time : status of machine

(busy, idle, down,) Event

A instantaneous occurrence that might change the state of the system:breakdown

Endogenous Activities and events occurring with the system

Exogenous Activities and events occurring with the environment


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Discrete and Continues Systems

A discrete system is one in which the state variables

change only at a discrete set of points in time : Bank

example


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Discrete and Continues Systems (cont.)

A continues system is one in which the state variables

change continuously over time: Head of water behind the

dam


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Model of a System

To study the system

it is sometimes possible to experiments with system This is not always possible (bank, factory,)

A new system may not yet exist

Model: construct a conceptual framework that

describes a system

It is necessary to consider those accepts of systems

that affect the problem under investigation

(unnecessary details must remove)


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Types of Models


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Characterizing a Simulation ModelCharacterizing a Simulation Model

Deterministic or Stochastic Does the model contain stochastic components?

Randomness is easy to add to a DES

Static or Dynamic Is time a significant variable?

Continuous or Discrete Does the system state evolve continuously or only at

discrete points in time?

Continuous: classical mechanics Discrete: queuing, inventory, machine shop models


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Discrete-Event Simulation Model

Stochastic: some state variables are random

Dynamic: time evolution is importantDiscrete-Event: significant changes occur at

discrete time instances


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Model TaxonomyModel Taxonomy


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DES Model DevelopmentDES Model Development

How to develop a model:

1) Determine the goals and objectives2) Build a conceptualmodel

3) Convert into a specification model

4) Convert into a computational model

5) Verify

6) Validate

Typically an iterative process


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Three Model LevelsThree Model Levels

Conceptual Very high level

How comprehensive should the model be? What are the state variables, which are dynamic, and which are

important?

Specification

On paper May involve equations, pseudocode, etc.

How will the model receive input?

Computational

A computer program General-purpose PL or simulation language?


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Verification vs. ValidationVerification vs. Validation

Verification Computational model should be consistent with

specification model

Did we build the model right?

Validation

Computational model should be consistent with thesystem being analyzed

Did we build the right model?

Can an expert distinguish simulation output fromsystem output?

Interactive graphics can prove valuable


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Steps in Simulation

Study


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Computer Science, Informatik 4

Communication and Distributed Systems

Simulation


Dr. Mesut Gne


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Chapter 9

Verification and Validation of Simulation Models


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3hapter 9. Verification and Validation of Simulation Models

Purpose Overview

The goal of the validation process is:

To produce a model that represents true

behavior closely enough for decision-making

purposes

To increase the models credibility to an

acceptable level

Validation is an integral part of model

development:

Verification: building the model correctly,

correctly implemented with good input and

structure

Validation: building the correct model, anaccurate representation of the real system

Most methods are informal subjective

comparisons while a few are formal

statistical procedures


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Modeling-Building, Verification Validation

Steps in Model-Building

Observing the real system

and the interactions amongtheir various components

and of collecting data on

their behavior.

Construction of aconceptual model

Implementation of an

operational model


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Verification

Purpose: ensure the conceptual model is reflected accurately in the

computerized representation.

Many common-sense suggestions, for example:

Have someone else check the model.

Make a flow diagram that includes each logically possible action a

system can take when an event occurs.

Closely examine the model output for reasonableness under a variety of

input parameter settings.

Print the input parameters at the end of the simulation, make sure they

have not been changed inadvertently.

Make the operational model as self-documenting as possible.

If the operational model is animated, verify that what is seen in theanimation imitates the actual system.

Use the debugger.

If possible use a graphical representation of the model.


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Examination of Model Output for Reasonableness

Two statistics that give a quick indication of model

reasonableness are current contents and total counts

Current content: The number of items in each component of thesystem at a given time.

Total counts: Total number of items that have entered each

component of the system by a given time.

Compute certain long-run measures of performance, e.g.

compute the long-run server utilization and compare to

simulation results.


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Examination of Model Output for Reasonableness

A model of a complex network of queues consisting of

many service centers.

If the current content grows in a more or less linear fashion asthe simulation run time increases, it is likely that a queue is

unstable

If the total count for some subsystem is zero, indicates no items

entered that subsystem, a highly suspect occurrence If the total and current count are equal to one, can indicate that

an entity has captured a resource but never freed that resource.


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Other Important Tools

Documentation

A means of clarifying the logic of a model and verifying its

completeness. Comment the operational model, definition of all variables and

parameters.

Use of a trace

A detailed printout of the state of the simulation model over time.- Can be very labor intensive if the programming language does not

support statistic collection.

- Labor can be reduced by a centralized tracing mechanism

- In object-oriented simulation framework, trace support can beintegrated into class hierarchy. New classes need only to add little for

the trace support.


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Trace - Example

Simple queue from Chapter 2

Trace over a time interval [0, 16]

Allows the test of the results by pen-and-paper method

Definition of Variables:

CLOCK = Simulation clock

EVTYP = Event type (Start, Arrival, Departure, Stop)NCUST = Number of customers in system at time CLOCK

STATUS = Status of server (1=busy, 0=idle)

State of System Just After the Named Event Occurs:

CLOCK = 0 EVTYP = Start NCUST=0 STATUS = 0

CLOCK = 3 EVTYP = Arrival NCUST=1 STATUS = 0CLOCK = 5 EVTYP = Depart NCUST=0 STATUS = 0

CLOCK = 11 EVTYP = Arrival NCUST=1 STATUS = 0

CLOCK = 12 EVTYP = Arrival NCUST=2 STATUS = 1

CLOCK = 16 EVTYP = Depart NCUST=1 STATUS = 1

...

There is a customer,

but the status is 0


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Calibration and Validation

Validation: the overall process of comparing the model and its behavior to the

real system.

Calibration: the iterative process of comparing the model to the real system

and making adjustments.

Comparison of the model to

real system

Subjective tests

- People who areknowledgeable about

the system

Objective tests

- Requires data on the

real systems behavior

and the output of the

model



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Calibration and Validation

Danger during the calibration phase

Typically few data sets are available, in the worst case only one, and the

model is only validated for these.

Solution: If possible collect new data sets

No model is ever a perfect representation of the system

The modeler must weigh the possible, but not guaranteed, increase inmodel accuracy versus the cost of increased validation effort.

Three-step approach for validation:

1. Build a model that has high face validity.

2. Validate model assumptions.

3. Compare the model input-output transformations with the real systems

data.



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High Face Validity

Ensure a high degree of realism:

Potential users should be involved in model construction from its

conceptualization to its implementation. Sensitivity analysis can also be used to check a models face

validity.

Example: In most queueing systems, if the arrival rate of

customers were to increase, it would be expected that serverutilization, queue length and delays would tend to increase.

For large-scale simulation models, there are many input

variables and thus possibly many sensitity tests.

- Sometimes not possible to perform all of theses tests, select themost critical ones.



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Validate Model Assumptions

General classes of model assumptions:

Structural assumptions: how the system operates.

Data assumptions: reliability of data and its statistical analysis.

Bank example: customer queueing and service facility in a bank.

Structural assumptions

- Customer waiting in one line versus many lines

- Customers are served according FCFS versus priority

Data assumptions, e.g., interarrival time of customers, service times for

commercial accounts.

- Verify data reliability with bank managers

- Test correlation and goodness of fit for data



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Validate Input-Output Transformation

Goal: Validate the models ability to predict future behavior

The only objective test of the model.

The structure of the model should be accurate enough to make goodpredictions for the range of input data sets of interest.

One possible approach: use historical data that have been reserved

for validation purposes only.

Criteria: use the main responses of interest.

SystemInput Output

ModelOutputInput

Model is viewed as an

input-outputtransformation



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Bank Example

Example: One drive-in window serviced by one teller, only one or two

transactions are allowed.

Data collection: 90 customers during 11 am to 1 pm.

- Observed service times {Si, i = 1,2, , 90}.

- Observed interarrival times {Ai, i = 1,2, , 90}.

Data analysis let to the conclusion that:

- Interarrival times: exponentially distributed with rate = 45

- Service times:N(1.1, 0.22)Input variables



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Bank Example The Black Box

A model was developed in close consultation with bank management and

employees

Model assumptions were validated

Resulting model is now viewed as a black box:

Input Variables

Possion arrivals

= 45/hr: X11, X12,

Services times,

N(D2, 0.22): X21, X22,

D1 = 1 (one teller)

D2 = 1.1 min

(mean service time)

D3 = 1 (one line)

Uncontrolled

variables, X

Controlled

Decision

variables, D

Model Output Variables, Y

Primary interest:Y1 = tellers utilization

Y2 = average delay

Y3 = maximum line length

Secondary interest:

Y4 = observed arrival rateY5 = average service time

Y6 = sample std. dev. of

service times

Y7 = average length of time

Model

black box

f(X,D) = Y



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Comparison with Real System Data

Real system data are necessary for validation.

System responses should have been collected during the same time

period (from 11am to 1pm on the same day.)

Compare the average delay from the model Y2

with the actual delay

Z2:

Average delay observed,Z2 = 4.3 minutes, consider this to be the true

mean value 0 = 4.3.

When the model is run with generated random variatesX1n andX2n, Y2should be close toZ2.


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Hypothesis Testing

Compare the average delay from the model

Y2 with the actual delayZ2

Null hypothesis testing: evaluate whether the simulation and the realsystem are the same(w.r.t. output measures):

- IfH0 is not rejected, then, there is no reason to consider the model

invalid

- IfH0 is rejected, the current version of the model is rejected, and the

modeler needs to improve the model

minutes3.4

minutes34

21

20

=

): E(YH

.): E(YH




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Hypothesis Testing

Conduct the

ttest:

Chose level of significance (= 0.5) and sample size (n = 6).

Compute the same mean and sample standard deviation overthe n replications:

Compute test statistics:

Hence, rejectH0.- Conclude that the model is inadequate.

Check: the assumptions justifying a ttest, that the observations(Y2i) are normally and independently distributed.

minutes51.21

1

22 == =

n

i

iYn

Yminutes82.0

1

)(1

2

22

=

==

n

YY

S

n

i

i

test)sided-2a(for571.25.346/82.0

3.451.2

/

020 =>=

=

= criticalt

nS

Yt




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Hypothesis Testing

Similarly, compare the model output with the observed output for

other measures:

Y4

Z4,

Y5

Z5,

and Y6

Z6




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Type II Error

For validation, the power of the test is:

Probability[ detecting an invalid model ] = 1

= P(Type II error) = P(failing to rejectH0|H

1is true)

Consider failure to rejectH0 as a strong conclusion, the modeler would

want to be small.

Value of depends on:

- Sample size, n

- The true difference, , betweenE(Y) and :

In general, the best approach to control error is:

Specify the critical difference, .

Choose a sample size, n, by making use of the operating characteristics

curve (OC curve).

=

)(YE




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Type II Error

Operating characteristics curve

(OC curve).

Graphs of the probability of aType II Error () versus for agiven sample size n

For the same error

probability with smaller

difference the required

sample size increses!




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Type I and II Error

Type I error (

):

Error of rejecting a valid model.

Controlled by specifying a small level of significance .

Type II error (

):

Error of accepting a model as valid when it is invalid.

Controlled by specifying critical difference and find the n.

For a fixed sample size

n, increasing

will decrease

.

Associated RiskModeling TerminologyStatistical Terminology

Failure to reject an invalid modelType II: failure to reject H0 when H1 is true

Rejecting a valid modelType I: rejecting H0 when H0 is true




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Confidence Interval Testing

Confidence interval testing: evaluate whether the simulation and the

real system performance measures are close enough

If

Yis the simulation output, and

=

E(Y)

The confidence interval (CI) for is:

n

S

tY n 1,2




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Validating the model:

Suppose the CI does not contain 0:

- If the best-case error is > , model

needs to be refined.

- If the worst-case error is , acceptthe model.

- If best-case error is , additionalreplications are necessary.

Suppose the CI contains 0:

- If either the best-case or worst-case

error is > , additional replications

are necessary.

- If the worst-case error is , acceptthe model.

is a dif ference value chosen

by the analyst, that is small

enough to allow valid

decisions to be based on

simulations!




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Bank example:

= 4

, and close enough is

= 1 minute of

expected customer delay.

A 95% confidence interval, based on the 6 replications is

[1.65, 3.37] because:

0= 4.3 falls outside the confidence interval,

- the best case |3.37 4.3| = 0.93 < 1, but

- the worst case |1.65 4.3| = 2.65 > 1

Additional replications are needed to reach a decision.

6

82.0

571.251.2

5,025.0

n

StY




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Using Historical Output Data

An alternative to generating input data:

Use the actual historical record.

Drive the simulation model with the historical record and then compare

model output to system data.

In the bank example, use the recorded interarrival and service times for

the customers {An, Sn, n = 1,2,}.

Procedure and validation process: similar to the approach used for

system generated input data.




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Using a Turing Test

Use in addition to statistical test, or when no statistical test is readily

applicable.

Utilize persons knowledge about the system.

For example:

Present 10 system performance reports to a manager of the system.

Five of them are from the real system and the rest are fake reports

based on simulation output data.

If the person identifies a substantial number of the fake reports, interview

the person to get information for model improvement.

If the person cannot distinguish between fake and real reports with

consistency, conclude that the test gives no evidence of model

inadequacy.

Turing TestDescribed by Alan Turing in 1950. A human jugde is involved in a natural languageconversation with a human and a machine. If the judge cannot reliably tell which of the

partners is the machine, then the machine has passed the test.




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Summary

Model validation is essential:

Model verification

Calibration and validation

Conceptual validation

Best to compare system data to model data, and make comparison

using a wide variety of techniques.

Some techniques that we covered: Insure high face validity by consulting knowledgeable persons.

Conduct simple statistical tests on assumed distributional forms.

Conduct a Turing test.

Compare model output to system output by statistical tests.




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Simulation


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Chapter 10

Output Analysis for a Single Model




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3hapter 10. Output Analysis for a Single Model

Purpose

Objective: Estimate system performance via simulation

If is the system performance, the precision of the estimator can

be measured by:

The standard error of .

The width of a confidence interval (CI) for .

Purpose of statistical analysis:

To estimate the standard error or CI .

To figure out the number of observations required to achieve desirederror or CI.

Potential issues to overcome:

Autocorrelation, e.g. inventory cost for subsequent weeks lack statisticalindependence.

Initial conditions, e.g. inventory on hand and number of backorders attime 0 would most likely influence the performance of week 1.




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Outline

Distinguish the two types of simulation:

transient vs.

steady state

Illustrate the inherent variability in a stochastic discrete-event

simulation.

Cover the statistical estimation of performance measures.

Discusses the analysis of transient simulations.

Discusses the analysis of steady-state simulations.




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Type of Simulations

Terminating versus non-terminating simulations

Terminating simulation:

Runs for some duration of time TE, whereEis a specified event that

stops the simulation.

Starts at time 0 under well-specified initial conditions.

Ends at the stopping time TE

.

Bank example: Opens at 8:30 am (time 0) with no customers present

and 8 of the 11 teller working (initial conditions), and closes at 4:30 pm

(Time TE= 480 minutes).

The simulation analyst chooses to consider it a terminating system

because the object of interest is one days operation.




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Type of Simulations

Non-terminating simulation:

Runs continuously, or at least over a very long period of time.

Examples: assembly lines that shut down infrequently, hospital

emergency rooms, telephone systems, network of routers, Internet.

Initial conditions defined by the analyst.

Runs for some analyst-specified period of time TE.

Study the steady-state (long-run) properties of the system, properties

that are not influenced by the initial conditions of the model.

Whether a simulation is considered to be terminating or non-

terminating depends on both

The objectives of the simulation study and

The nature of the system


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Stochastic Nature of Output Data

M/G/1 queueing example (cont.):

Batched average queue length for 3 independent replications:

Inherent variability in stochastic simulation both within a single replication

and across different replications.

The average across 3 replications, can be regarded as

independent observations, but averages within a replication, Y11, , Y15,

are not.

1, Y1j 2, Y2j 3, Y3j[0, 1000) 1 3.61 2.91 7.67

[1000, 2000) 2 3.21 9.00 19.53

[2000, 3000) 3 2.18 16.15 20.36

[3000, 4000) 4 6.92 24.53 8.11

[4000, 5000) 5 2.82 25.19 12.62

[0, 5000) 3.75 15.56 13.66

ReplicationBatching Interval

(minutes) Batch, j

,,, .3.2.1 YYY




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Measures of performance

Consider the estimation of a performance parameter,

(or

), of a

simulated system.

Discrete time data: [Y1, Y2, , Yn], with ordinary mean:

Continuous-time data: {Y(t), 0 t TE} with time-weighted mean:

Point estimation for discrete time data.

The point estimator:

- Is unbiased if its expected value is , that is if:

- Is biased if: and is called bias of

=)(E Desired

)(E

=

=n

i

iYn 1

1

)(E




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Point Estimator

Point estimation for continuous-time data.

The point estimator:

- Is biased in general where: .

- An unbiased or low-bias estimator is desired.

Usually, system performance measures can be put into the common

framework of or :

The proportion of days on which sales are lost through an out-of-stocksituation, let:

)(E

= ET

E

dttYT 0

)(1

=otherwise,0

dayonstockofoutif,1)(

iiY




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Point Estimator

Performance measure that does not fit:

quantile or percentile:

Estimating quantiles: the inverse of the problem of estimating a proportion

or probability.

Consider a histogram of the observed values Y:

- Find such that 100p% of the histogram is to the left of (smaller than) .

A widely used used performance meausure is the median, which is the

0.5 quantile or 50-th percentile.

pY = }Pr{




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Confidence-Interval Estimation

To understand confidence intervals fully, it is important to distinguish

between measures of error, and measures of risk, e.g., confidence

interval versus prediction interval.

Suppose the model is the normal distribution with mean , variance 2

(both unknown).

Let Yi. be the average cycle time for parts produced on the i-threplication

of the simulation (its mathematical expectation is ).

Average cycle time will vary from day to day, but over the long-run the

average of the averages will be close to .

Sample variance acrossR replications:

=

=R

i

i YYR

S1

2

...

2 )(1

1




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Confidence Interval (CI):

A measure of error.

Where Yi are normally distributed.

We cannot know for certain how far is from but CI attempts to bound

that error.

A CI, such as 95%, tells us how much we can trust the interval to actually

bound the error between and . The more replications we make, the less error there is in (converging

to 0 asR goes to infinity).

R

StY

R 1,..2

..Y

..Y

..Y

Quantile of the tdistribution with R-1

degrees of freedom.




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Prediction Interval (PI):

A measure of risk.

A good guess for the average cycle time on a particular day is our

estimator but it is unlikely to be exactly right.

PI is designed to be wide enough to contain the actualaverage cycle time

on any particular day with high probability.

Normal-theory prediction interval:

The length of PI will not go to 0 asR increases because we can never

simulate away risk. PIs limit is:

RStY R

111,2/.. +

2/z




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A terminating simulation: runs over a simulated time interval [0

, TE].

A common goal is to estimate:

In general, independent replications are used, each run using a

different random number stream and independently chosen initial

conditions.

E

E

n

i

i

TttYdttY

T

E

Yn

E

=

=

=

0),(outputcontinuousfor,)(1

outputdiscretefor,1

ET

0

1



Statistical Background


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Important to distinguish within-replication data from across-replication

data.

For example, simulation of a manufacturing system

Two performance measures of that system: cycle time for parts and work

in process (WIP).

Let Yij be the cycle time for thej-thpart produced in the i-threplication.

Across-replication data are formed by summarizing within-replication

data ..iY

R

2

RRRnRR1

2

2

222n2221

1

2

111n1211

H,S,YYYY

H,S,YYYYH,S,YYYY

R

2

1

L

MMMM

L

L

2

Across-Replication

Data

Within-Replication Data





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Across Replication:

For example: the daily cycle time averages (discrete time data)

- The average:

- The sample variance:

- The confidence-interval half-width:

Within replication:

For example: the WIP (a continuous time data)

- The average:

- The sample variance:

==R

i

iYR

Y1

... 1

=

=R

i

i YYR

S1

2

...

2 )(1

1

R

StH R 1,2/ =

= EiT

i

Ei

i dttYT

Y0

. )(1

( ) = EiT

ii

Ei

i dtYtY

T

S0

2

.

2 )(1





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Overall sample average, , and the interval replication sample

averages, , are always unbiased estimators of the expected daily

average cycle time or daily average WIP.

Across-replication data are independent (different random numbers)

and identically distributed (same model), but within-replication data

do not have these properties.

..Y

.iY


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Confidence Intervals with Specified Precision


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Assume that an initial sample of size

R0 (independent) replications has

been observed.

Obtain an initial estimate

S0

2of the population variance

2

.

Then, choose sample sizeR such thatR

R0:

Since t/2, R-1 z/2, an initial estimate ofR:

R is the smallest integer satisfyingRR0 and

Collect

R - R0additional observations.

The 100(1- ) CI for :

on.distributinormalstandardtheis,2/

2

02/

z

SzR

2

01,2/

StR

R

R

StY R 1,2/..



Confidence Intervals with Specified Precision


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Call Center Example: estimate the agents utilization over the first 2 hours of

the workday.

Initial sample of size R0 = 4 is taken and an initial estimate of the population varianceis S

02 = (0.072)2 = 0.00518.

The error criterion is = 0.04 and confidence coefficient is 1-= 0.95, hence, the finalsample size must be at least:

For the final sample size:

R = 15 is the smallest integer satisfying the error criterion, soR - R0 = 11 additionalreplications are needed.

After obtaining additional outputs, half-width should be checked.

14.1204.0

00518.0*96.12

22

0025.0 ==

Sz

R 13 14 15

t 0.025, R-1 2.18 2.16 2.14

15.39 15.1 14.83( )201,2/ / St R



Quantiles


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Here, a proportion or probability is treated as a special case of a

mean.

When the number of independent replications

Y1, ,YR is large

enough that

t

/2,n-1

= z

/2

, the confidence interval for a probability

pis

often written as:

A quantile is the inverse of the probability to the probability

estimation problem:

1

)1( 2/

R

ppzp

Find such that Pr(Y ) = p

p is given

The sample proportion



Quantiles


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The best way is to sort the outputs and use the (R*p)-

th smallest value,

i.e., find such that100p of the data in a histogram of Yis to the left of

.

Example: If we haveR=10 replications and we want thep = 0.8 quantile,first sort, then estimate by the (10)(0.8) = 8-thsmallest value (round ifnecessary).

5.6

sorted data

7.1

8.8

8.9

9.5

9.7

10.1

12.2 this is our point estimate

12.5

12.9



Quantiles


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1

)1(

1

)1(where

2/

2/

+=

=

R

ppzpp

R

ppzpp

u

l

Confidence Interval of Quantiles: An approximate (1

-

)100

confidence interval for can be obtained by finding two values

l

and

u

.

lcuts off 100pl%of the histogram (theRpl smallest value of the sorteddata).

u cuts off 100pu%of the histogram (theRpu smallest value of the sorted

data).



Quantiles


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Example: Suppose

R = 1000reps, to estimate the

p = 0.8quantile

with a 95

confidence interval.

First, sort the data from smallest to largest.

Then estimate of by the (1000)(0.8) = 800-thsmallest value, and thepoint estimate is 212.03.

And find the confidence interval:

The point estimate is

The 95% c.i. is [188.96, 256.79]

aluessmallest v820and780theisc.i.The

82.011000

)8.1(8.96.18.0

78.011000)8.1(8.96.18.0

thth

=

+=

=

=

up

pl

A portion of the 1000

sorted values:

Output Rank

180.92 779

188.96 780

190.55 781

208.58 799

212.03 800

216.99 801

250.32 819

256.79 820

256.99 821



Output Analysis for Steady-State Simulation


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Consider a single run of a simulation model to estimate a steady-

state or long-run characteristics of the system.

The single run produces observations Y1, Y2, ... (generally the samples of

an autocorrelated time series). Performance measure:

- Independent of the initial conditions.

measurediscretefor,1

1lim

=

=n

i

i

n

Yn

(with probability 1)

measurecontinuousfor,)(10lim

= E

E

T

ET

dttYT

(with probability 1)


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Initialization Bias


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Methods to reduce the point-estimator bias caused by using artificial

and unrealistic initial conditions:

Intelligent initialization.

Divide simulation into an initialization phase and data-collection phase.

Intelligent initialization

Initialize the simulation in a state that is more representative of long-runconditions.

If the system exists, collect data on it and use these data to specify morenearly typical initial conditions.

If the system can be simplified enough to make it mathematically solvable,e.g. queueing models, solve the simplified model to find long-run expectedor most likely conditions, use that to initialize the simulation.



Initialization Bias


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Divide each simulation into two phases:

An initialization phase, from time 0 to time T0.

A data-collection phase, from T0 to the stopping time T0+TE.

The choice of T0 is important:- After T0 , system should be more nearly representative of steady-state behavior.

System has reached steady state: the probability distribution of the systemstate is close to the steady-state probability distribution (bias of responsevariable is negligible).



Initialization Bias


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M/G/1 queueing example: A total of 10 independent replications were

made.

Each replication beginning in the empty and idle state.

Simulation run length on each replication was T0+TE= 15000 minutes. Response variable: queue length,LQ(t,r) (at time tof ther-th replication).

Batching intervals of 1000 minutes, batch means

Ensemble averages:

To identify trend in the data due to initialization bias

The average corresponding batch means acrossreplications:

The preferred method to determine deletion point.

=

=R

r

rjj YR

Y1

.

1Rreplications



Initialization Bias


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A plot of the ensemble averages, , versus 1000

j,

forj = 1,2, ,15.

),..( dnY



Initialization Bias


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Cumulative average sample mean (after deletingdobservations):

Not recommended to determine the initialization phase.

It is apparent that downward bias is present and this bias can be reducedby deletion of one or more observations.

+=

=n

dj

jYdn

dnY

1

...

1),(



Initialization Bias


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No widely accepted, objective and proven technique to guide how

much data to delete to reduce initialization bias to a negligible level.

Plots can, at times, be misleading but they are still recommended.

Ensemble averages reveal a smoother and more precise trend as the # ofreplications,R, increases.

Ensemble averages can be smoothed further by plotting a movingaverage.

Cumulative average becomes less variable as more data are averaged.

The more correlation present, the longer it takes for to approachsteady state.

Different performance measures could approach steady state at differentrates.

jY.



Error Estimation


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If {Y1, , Yn} are not statistically independent, then S2/n is a biased

estimator of the true variance.

Almost always the case when {Y1, , Yn} is a sequence of outputobservations from within a single replication (autocorrelated sequence,time-series).

Suppose the point estimator is the sample mean

Variance of is very hard to estimate.

For systems with steady state, produce an output process that isapproximately covariance stationary (after passing the transient phase).

- The covariance between two random variables in the time series dependsonly on the lag, i.e. the number of observations between them.

Y

== n

i iY

nY

11



Error Estimation


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For a covariance stationary time series, {

Y1, , Yn}:

Lag-kautocovariance is:

Lag-kautocorrelation is:

If a time series is covariance stationary, then the variance of is:

The expected value of the variance estimator is:

Y

),cov(),cov( 11 kiikk YYYY ++ ==

2

kk=

+=

=

1

1

2

121)(n

k

kn

k

nYV

1

1/e wher,)(

2

==

n

cnBYVB

n

SE

c

11 k



Error Estimation


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a)

Stationary time series Yi exhibiting

positive autocorrelation.

Serie slowly drifts above and then

below the mean.

b)

Stationary time series Yi exhibiting

negative autocorrelation.

c) Nonstationary time series with an

upward trend

kk mostfor0>

kk mostfor0


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The expected value of the variance estimator is:

If Yi are independent, then S2/n is an unbiased estimator of

If the autocorrelationkare primarily positive, then S2/n is biased low as

an estimator of .

If the autocorrelationkare primarily negative, then S2/n is biased high as

an estimator of .

)(YV

ofvariancetheis)(and

1

1/e wher,)(

2

YYV

n

cnBYVB

n

SE

==

)(YV

)(YV



Replication Method


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Use to estimate point-estimator variability and to construct a confidence

interval.

Approach: makeR replications, initializing and deleting from each one the

same way.

Important to do a thorough job of investigating the initial-condition bias:

Bias is not affected by the number of replications, instead, it is affected only by

deleting more data (i.e., increasing T0) or extending the length of each run (i.e.

increasing TE).

Basic raw output data {

Yrj, r = 1, ..., R; j = 1, , n} is derived by:

Individual observation from within replication r.

Batch mean from within replication rof some number of discrete-time

observations.

Batch mean of a continuous-time process over time intervalj.


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Replication Method


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To estimate the standard error of , the sample variance and

standard error:

..Y

R

SYesYRY

RYY

RS

R

r

r

R

r

r =

=

=

==

).(.and1

1)(

1

1..

1

2

..

2

.

1

2

...

2

Mean of the

undeleted

observations

from the r-th

replication.

Mean of

),(,),,(1 dnYdnY R K

Standard error



Replication Method


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Length of each replication (

n) beyond deletion point (

d):

(n d ) > 10d or TE > 10T0

Number of replications (

R) should be as many as time permits, up to

about 25 replications.

For a fixed total sample size (n), as fewer data are deleted (d):

CI shifts: greater bias.

Standard error of decreases: decrease variance.),(.. dnY

Reducing

bias

Increasing

varianceTrade off



Replication Method


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The point estimator and standard error are:

The 95%CI for long-run mean queue length is:

A high degree of confidence that the long-run mean queue length is between 4.84and 12.02 (if dand n are large enough).

M/G/1 queueing example:

SupposeR = 10, each of length TE= 15000

minutes, starting at time 0 in the empty and

idle state, initialized for T0 = 2000 minutes

before data collection begins.

Each batch means is the average number of

customers in queue for a 1000-minute interval.

The 1-st two batch means are deleted (d= 2).

( ) 59.1)2,15(..and43.8)2,15( .... == YesY

)59.1(26.242.8)59.1(26.243.8

// 1,2/..1,2/..

+

+

Q

RR

L

RStYRStY



Sample Size


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To estimate a long-run performance measure, , within

with confidence 100(1

-

)

.

M/G/1 queueing example (cont.):

We know:R0 = 10, d= 2 andS02

= 25.30. To estimate the long-run mean queue length, LQ, within = 2 customers

with 90%confidence (= 10%).

Initial estimate:

Hence, at least 18 replications are needed, next tryR = 18,19, using . We found that:

Additional replications needed isR R0 = 19-10 = 9.

1.172

)30.25(645.12

22

005.0

==

Sz

R

( ) 93.18)4/3.25*73.1(/19 220119,05.0 === StR

( )201,05.0 / StR R



Sample Size


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An alternative to increasing

Ris to increase total run length

T0+TEwithin each replication.

Approach:

- Increase run length from (T0+TE) to (R/R0)(T0+TE), and- Delete additional amount of data, from time 0 to time (R/R0)T0.

Advantage: any residual bias in the point estimator should be further

reduced.

However, it is necessary to have saved the state of the model at time

T0+TEand to be able to restart the model.



Batch Means for Interval Estimation


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Using a single, long replication:

Problem: data are dependent so the usual estimator is biased.

Solution: batch means.

Batch means:

divide the output data from 1 replication (after appropriate

deletion) into a few large batches and then treat the means of these batches

as if they were independent.

A continuous-time process, {Y(t), T0 t T0+TE}:

kbatches of size m = TE/ k, batch means:

A discrete-time process

, {

Y

i

, i = d+1,d+2, , n

}:

kbatches of size m = (n d)/k, batch means:

+=

+=jm

mji

dij Ym

Y1)1(

1

+= jm

mjj dtTtY

mY

)1(0 )(

1



Batch Means for Interval Estimation

YYYYYYYY


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Starting either with continuous-time or discrete-time data, the

variance of the sample mean is estimated by:

If the batch size is sufficiently large, successive batch means will be

approximately independent, and the variance estimator will be

approximately unbiased.

No widely accepted and relatively simple method for choosing an

acceptable batch size

m. Some simulation software does it

automatically.

( )

==

=

=

k

j

jk

j

j

kk

YkY

k

YY

kk

S

1

22

1

22

)1(1

1

kmdmkdmdmdmddd YYYYYYYY ++++++++ ...,,,...,...,,,...,,,...,, 1)1(2111

deleted1Y 2Y kY



Summary


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Stochastic discrete-event simulation is a statistical experiment.

Purpose of statistical experiment: obtain estimates of the performance measures

of the system.

Purpose of statistical analysis: acquire some assurance that these estimates are

sufficiently precise.

Distinguish: terminating simulations and steady-state simulations.

Steady-state output data are more difficult to analyze

Decisions: initial conditions and run length

Possible solutions to bias: deletion of data and increasing run length

Statistical precision of point estimators are estimated by standard-error or

confidence interval

Method of independent replications was emphasized.


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Chapter 11

Comparison and Evaluation of Alternative System Designs



Purpose


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3hapter 11. Comparison and Evaluation of Alternative Designs

Purpose: comparison of alternative system designs.

Approach: discuss a few of many statistical methods that can be

used to compare two or more system designs.

Statistical analysis is needed to discover whether observed

differences are due to:

Differences in design or

The random fluctuation inherent in the models



Outline


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For two-system comparisons:

Independent sampling.

Correlated sampling (common random numbers).

For multiple system comparisons:

Bonferroni approach: confidence-interval estimation, screening, and

selecting the best.

Metamodels



Comparison of Two System Designs


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Goal: compare two possible configurations of a system

Two possible ordering policies in a supply-chain system, two possible

scheduling rules in a job shop.

Approach: the method of replications is used to analyze the output

data.

The mean performance measure for system i

Denoted by i , i = 1,2.

To obtain point and interval estimates for the difference in mean

performance, namely

1

2

.


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From replication r of system i, the simulation analyst obtains an estimate


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From replication rof system i, the simulation analyst obtains an estimateYirof the mean performance measure i .

Assuming that the estimators Yirare (at least approximately) unbiased:

1 =E(Y1r), r = 1, , R1; 2 =E(Y2r), r = 1, , R2 Goal: compute a confidence interval for 1 2 to compare the two

system designs

Confidence interval for 1 2:

- If CI is totally to the left of 0, strong evidence for the hypothesis that1 2 < 0 (1 < 2 ).

- If CI is totally to the right of 0, strong evidence for the hypothesis that1 2 > 0 (1 > 2 ).

- If CI contains 0, no strong statistical evidence that one system is better thanthe other

If enough additional data were collected (i.e.,Ri increased), the CI would

most likely shift, and definitely shrink in length, until conclusion of1 < 2 or 1 > 2 would be drawn.





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In this chapter:

A two-sided 100(1-)% CI for 1 2 always takes the form of:

All three techniques assume that the basic data Yir are approximately

normally distributed.

).(. 2.1.,2/2.1. YYestYY

Sample

mean for

system i

Degree

of

freedom

Standard error

of the estimator


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Independent Sampling with Equal Variances


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Different and independent random number streams are used

to simulate the two systems

All observations of simulated system 1 are statistically

independent of all the observations of simulated system 2.

The variance of the sample mean, , is:

For independent samples:

iY.

( ) ( )

21,

2.

. ,iRR

YV

YVi

i

i

i

i ===

( ) ( ) ( )2

22

1

21

2.1.2.1.RR

YVYVYYV

+=+=


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Independent Sampling with Unequal Variances

If the assumption of equal variances cannot safely be made,


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an approximate 100(1-) CI can be computed as:

With degrees of freedom:

In this case, the minimum number of replicationsR1 > 7 andR2 > 7 is recommended.

( )( ) ( ) ( ) ( )

intergerantoround,

1//1//

//

2

2

2221

2

121

2

2221

21

+

+=RRSRRS

RSRS

( ) 2

2

2

1

2

12.1...R

S

R

SYYes +=



Common Random Numbers (CRN)

For each replication, the same random numbers are used to simulate


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both systems R1=R

2=R.

For each replication r, the two estimates, Yr1 and Yr2, are correlated.

However, independent streams of random numbers are used on differentreplications, so the pairs (Yr1 ,Ys2 ) are mutually independent for r s.

Purpose: induce positive correlation between (for eachr) to

reduce variance in the point estimator of .

2.1. ,YY

RRR

YYYVYVYYV

211222

21

2.1.2.1.2.1.

2

,cov2

+=

+=

2.1. YY

Correlation:

12

is positive




Compare variance from independent sampling with variance


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from CRN:

Variance of arising from CRN is less than that of

independent sampling (withR1=R2).2.1. YY

RVV INDCRN21122 =


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It is never enough to simply use the same seed for the random-


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number generator(s):

The random numbers must be synchronized: each random number used

in one model for some purpose should be used for the same purpose inthe other model.

Example: if the i-thrandom number is used to generate a service time at

work station 2 for the 5-tharrival in model 1, the i-thrandom number

should be used for the very same purpose in model 2.


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Vehicle inspection example (cont.)


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Dr. Mesut Gne


Each replication run of 16 hours

Model 1

Model 2 with independen random numbers

Model 2 with common random numbers

without synchronisation

Model 2 with common random numberswith synchronisation


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CRN with Specified Precision

Goal: The error in our estimate of

1

2

to be less than .


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Dr. Mesut Gne


Approach: determine the of replicationsR such that the half-width of

CI:

Vehicle inspection example (cont.):

R0 = 10, complete synchronization of random numbers

yield 95% CI:

Suppose = 0.5 minutes for practical significance, we knowR is thesmallest integer satisfyingRR0 and:

Since , a conservation estimate of Ris:

Hence, 35 replications are needed (25 additional).

= 2.1.,2/ .. YYestH

minutes9.04.0

2

1,2/

DR StR

2

1,2/ 0

DR StR1,2/1,2/ 0 RR tt


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xipxijxi2xi1

yi

i xipxijxi2xi1

yi

i


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p xij j=1,2,,p

yi i=1,2,,n

i p xij

yi


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ServerWaiting line

Calling population


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ServerWaiting line

Calling populationti

ti+1

Arrivals


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The interarrival and servicetimes are taken from

distributions!


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The simulation run isbuild by meshingclock, arrival and

service times!

Chronological ordering of events


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Number of customers in the system


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min74.1100

174

customersofNumber

queueintimeWaiting===

w

46046waithocustomer wofNumber

)wait( ===p


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46.0100customersofNunber

)wait( p

24.0418

101

run timeSimulation

serveroftimeIdle

server)idle( ===

p

min17.3100

317

customersofNumber

timeService===

s

min2.3605.0202.0101.0)()(0

=+++==

=

L

s

spssE

min19.499

415

1-arrivalsofNumber

arrivalsbetweenTimes===

min5.42

81

2)( =

+=

+=

baE

min22.354

174

that waitcustomersofNumber

queueintimeWaiting===

waitedw

min91.4100

491

customersofNumber

systeminspendcustomersTime===

t

min91.417.374.1 =+=+= swt

Average of 50 Trials


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Histogram for the Average Customer Waiting Time

0

2

15

17

7

5

2

1

0 0

1

0

2

4

6

8

10

12

14

16

18

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 >4.5

Average customer waiting time [min]

Occurrences(No.ofTrials)


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Computer Science, Informatik 4Communication and Distributed Systems


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Simulation


Dr. Mesut Gne



136/447

Chapter 3

General Principles


General Principles Introduction

Framework for modeling systems by discrete-event

simulation


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Dr. Mesut Gne3Chapter 3. General Principles

A system is modeled in terms of its state at each point in time

This is appropriate for systems where changes occur only atdiscrete points in time


Concepts in Discrete-Event Simulation

Concepts of dynamic, stochastic systems that change in a discretemanner


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A record of an event to occur at the current or some future time, along with any

associated data necessary to execute the event.

Event notice

An instantaneous occurrence that changes the state of a system.Event

A collection of associated entities ordered in some logical fashion in a waiting line.

Holds entities and event notices

Entities on a list are always ordered by some rule, e.g. FIFO, LIFO, or ranked by

some attribute, e.g. priority, due date

List, Set

The properties of a given entity.Attributes

An object in the system that requires explicit representation in the model, e.g., people,

machines, nodes, packets, server, customer.

Entity

A collection of variables that contain all the information necessary to describe the

system at any time.

System state

An abstract representation of a system, usually containing structural, logical, or

mathematical relationships that describe the system.

Model

A collection of entities that interact together over time to accomplish one or more

goals.

System



A duration of time of specified length which is known when it beginsActivity

A list of event notices for future events, ordered by time of occurrence; known as the

future event list (FEL).

Always ranked by the event time

Event list


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A variable representing the simulated time.Clock

A duration of time of unspecified indefinite length, which is not known until it ends.

Customers delay in waiting line depends on the number and service times of othercustomers.

Typically a desired output of the simulation run.

Delay

A duration of time of specified length, which is known when it begins.

Represents a service time, interarrival time, or any other processing time whose

duration has been characterized by the modeler. The duration of an activity can bespecified as:

Deterministic Always 5 time units

Statistical Random draw from {2, 5, 7}

A function depending on system variables and entities

The duration of an activity is computable when it begins

The duration is not affected by other events

To track activities, an event notice is created for the completion time, e.g., let

clock=100 and service with duration 5 time units is starting

Schedule an end of service-event for clock + 5 = 105

Activity


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Activity vs. Delay


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A1 A2 A3D1 D2

Activity1 Activity2

Delay

Delay

t



A model consists of static description of the model and

the dynamic relationships and interactions between the components


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Some questions that need to be answered for the dynamic behavior Events

- How does each event affect system state, entity attributes, and set contents?

Activities- How are activities defined?

- What event marks the beginning or end of each activity?

- Can the activity begin regardless of system state, or is its beginning conditioned on the

system being in a certain state? Delays

- Which events trigger the beginning (and end) of each delay?

- Under what condition does a delay begin or end?

System state initialization- What is the system state at time 0?

- What events should be generated at time 0 to prime the model that is, to get thesimulation started?


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Event-scheduling/Time-advance algorithm

Future event list (FEL)

All event notices are chronologically ordered in the FEL

At current time t the FEL contains all scheduled events


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At current time t, the FEL contains all scheduled events

The event times satisfy: t< t1 t2 t3 ... tn The event associated with t1 is the imminent event, i.e., the next

event to occur

Scheduling of an event

- At the beginning of an activity the duration is computed and an end-of-activity event is placed on the future event list

The content of the FEL is changing during simulation run

- Efficient management of the FEL has a major impact on the

performance of a simulation run- Class: Data structures and algorithms



(1,t2) Type 1 event to occur at t2

(3,t1) Type 3 event to occur at t1(5,1,6)t

Future event listStateClockOld system snapshot at time t


Step 1: Remove the event notice for the


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(2,tn) Type 2 event to occur at tn

...


(2,tn) Type 2 event to occur at tn

...


(4,t*) Type 4 event to occur at t*

(1,t2) Type 1 event to occur at t2(5,1,5)t1

Future event listStateClock

New system snapshot at time t1

Step 1: Remove the event notice for the

imminent event from FEL

event (3, t1) in the example

Step 2: Advance Clock to imminent event time

Set clock = t1

Step 3: Execute imminent event

update system state

change entity attributes

set membership as needed

Step 4: Generate future events and place their

event notices on FEL

Event (4, t*)

Step 5: Update statistics and counters



System snapshot at time 0

Initial conditions

Generation of exogenous events


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Generation of exogenous events

- Exogenous event, is an event which happens outside the system,but impinges on the system, e.g., arrival of a customer



Generation of events

Arrival of a customer

- At t=0 first arrival is generated and scheduled


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At t 0 first arrival is generated and scheduled

- When the clock is advanced to the time of thefirst arrival, a second arrival is generated

- Generate an interarrival time a*

- Calculate t* = clock+ a*

- Place event notice at t* on the FEL

Bootstrapping




Service completion of a customer

- A customer completes service at t


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p

- If the next customer is present a new service time s* is generated- Calculate t* = clock + s*

- Schedule next service completion at t*

- Additionally: Service completion event will scheduled at the arrival

time, when there is an idle server

- Service time is an activity

- Beginning service is a conditional event

Conditions: Customer is present and server is idle

- Service completion is a primary event




Alternate generation of runtimes and downtimes

- At time 0, the first runtime will be generated and an end-of-runtime


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, g

event will be scheduled

- Whenever an end-of-runtime event occurs, a downtime will be

generated, and a end-of-downtime event will be scheduled

- At the end-of-downtime event, a runtime is generated and

Documents

All in One simulation notes