Set5 Simulation

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System Simulation

Dr. Dessouky


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Description

Simulation is a very powerful and widely used management

science technique for the analysis and study of complex systems.

Simulation may be defined as a technique that imitates theoperation of a real-world system as it evolves over time. This is

normally done by developing a simulation model. A simulation

model usually takes the form of a set of assumptions about the

operation of the system, expressed as mathematical or logical

relations between the objects of interest in the system.

Simulation has its advantages and disadvantages. We will focus

our attention on simulation models and the simulation technique.


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Simulation

What is simulation:

The process of designing a mathematical orlogical model of a real-system and then

conducting computer-based experiments

with the model to describe, explain, andpredict the behavior of the real system.


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Simulation

Where simulation fits in

SimulationProgramming

Analysis

Modeling

Probability &

Statistics


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Basic Terminology

In most simulation studies, we are concerned with

the simulation of some system.

Thus, in order to model a system, we must

understand the concept of a system. Definition: A system is a collection of entities that

act and interact toward the accomplishment of some

logical end.

Systems generally tend to be dynamictheir status

changes over time. To describe this status, we use

the concept of the state of a system.


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Example Simulation Model

Ford - # of Panels per day (throughput)

Emergency Room(beds, doctors, nurses), (minor, moderate, major,

critical)

TRWBallistic Missile Survivability against Soviet Threat

Paramount FarmsPistachio

Miami University Parking

HMTDisks Throughput

Christopher RanchGarlic Capacity

Power IntegrationSemiconductor Capacity, and random machinedown times


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Value of Simulation

Empirical Method verses mathematical

model

Allow you to calculate the extreme values

not just the expected value


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Simulation

What is simulation

Simulation is the actual running of themodel system to gain insight into its

performance.


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Simulation

Why use simulation

Simulation is used to better understand theexpected performance of the real system

and to test the effectiveness of the system

design.


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Simulation

Why use simulation

Without building them

experimental systemnew concepts

Without disturbing themcostly experimentation

unsafe experimentation.

Without destroying themDetermine limits of stress


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Queuing systems

Performance measures (output)

Data requirements (input)

Uses of model

Kendalls notation


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Queuing systems

System Performance measures (outputs)

Expected number of customers in system

Expected number of customers in queueExpected time in system

Expected time in queue

Server utilizationProbability of n customers in system

Throughput


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Queuing systems

Data requirements (Inputs)

Interarrival time distribution

Service time distributionNumber of servers

Queue discipline

System capacitySize of input population

Kendalls notation (M/M/s/FCFS/K/M)


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Alternative to simulation

Simulation

Analytic models

Physical experimentation Visit other sites


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Simulation vs. analytic modeling

Advantage:

various performance measures

greater realismeasier to understand

model the steady-state as well as the transit

behavior.

Disadvantage:

May not provide you with the optimal solution

time to construct model will be longer.


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Simulation vs. Physical

Advantage:

High Speed

Not disruptiveReplication easy

Control variations

Generally less costly

Disadvantage:

Realism

Validity


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Simulation vs. Alternatives

V A S P

Realism

V A

V A S PCost


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Representing system

System:

a collection of mutually interacting objects

designed to accomplish a goal (machines

repair system)

Entities:

denotes an element/object within boundary of

system (machines, operators, repairman)

Entitywork being performed on object

Resourceperforming the work


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Representing system

Attribute:

Characteristic or property or an entity

(machine ID, Type of breakdown, time that

machine went down)

Activity:

transforms the state of an object usually over

some time (repairman service time, machinerun time)


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Representing system

State of the system:

Numeric values that contain all the

information necessary to describe the system

at any time.

Delays:

Processes that take a conditional length of

time in the system


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Representing system

Events:

Change the state of the system(end of service

of machine,machine breaks down)

Queue:

it is set, used to model waiting


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Ex. Elevator systems

Entities

Elevators, people

SetsPeople waiting at each floor

Attributes

Elevatorscapacity, speed, destination,current location of each elevator

Peopleinter-arrival time at each floor,

destination of each people


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State of system:

# of people on each elevator

# of people in each floor Activities

Load/Unloading passenger

Travel to next floor (speed and distance)Persons travel to elevator


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Delays:

Persons waiting for elevator

Events:Elevator arrival

End unloading

End LoadingPerson Arrival


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Static Simulation vs. Dynamic

Simulation

There are two types of simulation models,

static and dynamic.

Definition: A static simulation model is arepresentation of a system at a particular

point in time.

We usually refer to a static simulation as aMonte Carlo simulation.


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Static Simulation vs. Dynamic

Simulation Definition: A dynamic simulation is a

representation of a system as it evolvesover time.

Within these two classifications, asimulation may be deterministic orstochastic.

A deterministic simulation model is onethat contains no random variables; astochastic simulation model contains oneor more random variables.


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Discrete Event vs. Continuous

Event Simulation

Discrete event:

state of system changes only at discrete points

in time(events)

ex. Machine repair problem

Programming

Look at system only when events occur; time isadvanced from event to event.


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Discrete Event vs. Continuous

Event Simulation

Continuous event:

state of system changes continuously over

time

Ex. Level of fluid in tank

Programming:

Advances time in small intervals. Use differentialequations to represent flows.


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An Example of a Discrete-Event

Simulation

To simulate a queuing system, we first have todescribe it.

We assume arrivals are drawn from an infinite

calling population. There is unlimited waiting room capacity, and

customers will be serve in the order of theirarrival (FCFS).

Arrivals occur one at a time in a random fashion. All arrivals are eventually served with the

distribution of service teams as shown in thebook.


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Service times are also assumed to be random.After service, all customers return to the callingpopulation.

For this example, we use the following variablesto define the state of the system: (1) the numberof customers in the system; (2) the status of theserverthat is, whether the server is busy or idle;

and (3)the time of the next arrival. An event is defined as a situation that causes the

state of the system to change instantaneously.


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All the information about them is maintained in alist called the event list.

Time in a simulation is maintained using a

variable called the clock time. We begin this simulation with an empty system

and arbitrarily assume that our first event, anarrival, takes place at clock time 0.

Next we schedule the departure time of the firstcustomer.Departure time = clock time now + generated service

time


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Also, we now schedule the next arrival into thesystem by randomly generating an interarrival timefrom the interarrival time distribution and setting

the arrival time asArrival time = clock time now + generated interarrival time

Both these events are their scheduled times aremaintained on the event list.

This approach of simulation is called the next-eventtime-advance mechanism, because of the way theclock time is updated. We advance the simulationclock to the time of the most imminent event.


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As we move from event to event, we carry out theappropriate actions for each event, including anyscheduling of future events.

The jump to the next event in the next-eventmechanism may be a large one or a small one; thatis, the jumps in this method are variable in size.

We contrast this approach with the fixed-incrementtime-advance method.

With this method, we advance the simulation clockin increments ofttime units, where tis someappropriate time unit, usually 1 time unit.


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For most models, however, the next eventmechanism tends to be more efficientcomputationally.

Consequently, we use only the next-eventapproach for the development of the models forthe rest of the chapter.

To demonstrate the simulation model, we need to

define several variables: TM= clock time of the simulation

AT= scheduled time of the next arrival


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DT= scheduled time of the next departure

SS= status of the server (1=busy, 0=idle)

WL = length of the waiting line

MX= length (in time units) of a simulation run

We now begin the simulation by initializingall the variables. This simple example

illustrates some of the basic concepts insimulation and the way in which simulationcan be used to analyze a particular problem.


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World ViewThe Structure concepts and views

under which the simulation is guided for the

development of the simulation model

Event Orientationdefines the changes in state that occur

at each event time

Process Orientationdescribes the process throughwhich the entities in the system flow

Activity Scanning Orientationdescribes the activities in

which the entities in the system engage


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Discrete Event Simulation

Event scheduling

Write modules that describe changes in the

state of the system at each event

Main program advances time

One subprogram for each event

General purpose programming language


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Discrete Event Simulation

Process interaction

Write modules that describe the progress of

entities through the system

As entities move the systems changes state

Entities are held to represent activities and

delays

Promodel programming language


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Event scheduling

Time is advanced from event to event

Future events listordered list of

upcoming eventsAs events are scheduled, they are added to the

list

As events occur they are removed from list

Activities in event ( one / event type)


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Event scheduling

List is required to keep track of entities in a

set

StatisticsTwo typesSample statisticsaverage of some values (W)

W = (W1 +W2+ +Wn)/n = Total Wait / # of wait

Time average statisticstime weighted (L)

L = (0(t1) + 1(t2-t1) + 2(t3-t2) + 1(t4-t3)) / t4


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Activity scanning

Activity scanning

Time is modeled in fixed time increments to

check if activity occurred

Small time increments is inefficient

Large time increments may miss activity

describes the activities in which the entities in

the system engage.


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Process Oriented

Process oriented:

Many simulation models include elementswhich occur in defined patterns

The logic associated with such a system orevents can be generalized and defined by asingle statement

A simulation language could then translate

such statement into the appropriate sequenceof events

describes the processes through which theentities in the system flow.


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Process Oriented

Process oriented:

These statements, define a sequence of events

which are automatically executed by the

simulation language as the entities move

through the process

Create arrival entities every t time units

However, since we are normally restricted to aset of standardized statement, provided by the

simulation language, our model flexibility is

not as great as with the event condition


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Feature provided by a language

Conceptual framework(entities, attributes,resource, queues)

Maintenance of event list

Random variable generation

Animation

Debugging function

Output analysis

Input analysis

Report generation


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Simulation Languages

One of the most important aspects of a simulationstudy is the computer programming.

Several special-purpose computer simulation

languages have been developed to simplifyprogramming.

The best known and most readily availablesimulation languages, including GPSS, GASP IV

and SLAM. Most simulation languages use one of two different

modeling approaches or orientations; eventscheduling or process interaction.


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GPSS uses the process-interaction approach.

SLAM allows the modeler to use either approach

or even a mixture of the two, whichever is the

most appropriate for the model being analyzed.

Of the general-purpose languages, FORTRAN is

the most commonly used in simulation.

In fact, several simulation languages, includingGASP IV and SLAM, use a FORTRAN base.


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To use GASP IV we must provide a mainprogram, an initialization routine, and the eventroutines.

For the rest of the program, we use the GASProutines.

Because of these prewritten routines, GASP IVprovides a great deal of programming flexibility.

GPSS, in contrast to GASP, is a highly structuredspecial-purpose language.

GPSS does not require writing a program in theusual sense.


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Building a GPSS model then consist of

combining these sets of blocks into a flow

diagram so that it represents the path an

entity takes as it passes through the system.

SLAM was developed by Pritsket and

Pegden (1979). It allows us to develop

simulation model as network models,

discrete-event models, continuous models, or

any combination of these.


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The decision of which language to use is one of

the most important that a modeler or an analyst

must make in performing a simulation study.

The simulation language offer severaladvantages.

The most important of these is that the special-

purpose languages provide a natural frameworkfor simulation modeling and most of the features

needed in programming a simulation model.


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The Simulation Modeling Steps

We now discuss the process for a completesimulation study and present a systematicapproach of carrying out a simulation.

A simulation study normally consists of severaldistinct stages. (See Figure in the book)

However, not all simulation studies consist of allthese stages or follow the order stated here.

On the other hand, there may even beconsiderable overlap between some of thesestages.


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Problem/Model Formulation

State the objective of the study.

Identify the Problem. Determine any underlying

causes if possible.

Determine the input variables.

Controllable Variables.

Uncontrollable Variables.

Make assumptions / boundaries that were used tosimplify the model.

Determine Performance measures used to

measure the objective. (Output)


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Data collection/acquisition

Determine the Data Collection System orEstimates to be used. Observe the system

Historical or Similar Systems Theoretical Estimates

Engineering Estimates

Operator Estimates

Vendor Estimates Identify the data collected.

How it was collected.

How it was represented in the model.


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Model Construction or

Development

Identify The Real System

Determine Conceptual Model -Activities

and Events Develop the Logical Model.

Identify the Programming Language used.

Computer Implementation (Promodel,Arena, Slam Systems).

d l C i


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Model Construction or

Development

Modeling Tips

Art vs. ScienceOver Simplification vs. Unnecessary Detail

Start Simple

Add stronger assumptions

M d l V ifi i d


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Model Verification and

Validation

Verification: Determining whether

simulation model works as intended.

Verifying the Model.

Structure: Walk Through of the Model

Debugger.

Trace = print or writing in process calculations.

Animation. Model testing

Analytical Model.

M d l V ifi i d


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Validation Verification.

Logical Model. Are events represented correctly?

Are mathematical formulas and relationshipscorrect?

Are statistical measures formulated correctly?

Computer Model/Simulation Model.

Does the code contain all aspects of the logicalmodel?

Are the statistics and formulas calculated correctly?

Does the model contain coding errors?

M d l V ifi i d


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Validation

Validation:Determine whether Simulation

of The Model is a credible representation

of a Real System.

Compare the model with the actual systems

by performing statistical tests. T-Test &

C.I.

Conceptual Model.

Does the model contain all relevant elements,

events and relationships?

Will the model answer the questions of concern?

M d l V ifi i d


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ValidationLogical Model.

Does the model contain all events included in theconceptual model?

Does the model contain all the relationships of theconceptual model?

Computer Model/Simulation Model. Is the computer model a valid representation of the

real system?

Can the computer model duplicate the performanceof the real system?

Does the computer model output have credibilitywith system experts and decision makers?


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Experimentation and Analysis of

Results

ExperimentationThe execution of the

simulation model to obtain output values

Analysis of ResultsThe process of analyzing

the simulation outputs to draw inferences and

make recommendations for problem resolution


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Implementation and Documentation

The process of implementing decisions

resulting from the simulation and

documenting the model and its use.


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Manual Simulation Example

Given the following arrival times for a singleserver system what will be the average numberin the queue, average number in the system,average time in system, average time in queue,the number of completed jobs, number in thequeue, number in the system, and serverutilization at time 15 if the service time is 3 timeunits for each entity.

1, 3, 5, 9,13,15,17


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Data Collection

Activities may be represented as

Constants

Random variables

Collection of data

Design a data collection form

Record more than single attribute in case you

need to use data in a different way.

Use several session to get representative data

Use control charts


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Data Collection

Machine Begin Repair End Repair Time

Elapsed


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Data Collection

Testing data

Independence

Randomness

Homogeneity


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Data Collection

Test of Independence

Ho: Measure A is independent of measure B

H1: Measure A is not independent of measure

B.

Inventory and day of week


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Data Collection

Test of Randomness

Ho: f(xi/xj) = f(xi) =Independent

Hi: f(xi/xj) f(xi) : Dependent

For example, when simulation a production

process in which the items can be defective or

good, it would be important to know if

successive items are randomly distributed withreputation good items followed by some of

defective items.


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Data Collection

Test of Homogeneity

Tests for whether multiple sets of data can be

considered as coming from statistical

population are generally referred to as tests ofhomogeneity distribution free.

Ho : G(x) =H(x)

H1 : G(x) H(x)Two different workers working on the same

machine.


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Random Variable

Two types

Discrete

Continuous


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Random Variable

Probability mass function

Discrete

P(X = xi) = p(xi)

p(xi) = 1


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Random Variable

Probability density function

Continuous

f(x) = ex

x > 0 P(X = a) = 0

-f(x) dx = 1

P(a < x < b) = ab f(x) dx


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Random Variable

Cumulative distribution function (CDF)

F(X) = P(X


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Random Variable

Expected value

= E(x) = xi p (xi)

= x f(x) dx


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Random Variable

Variance

))((

))((22

22

22

2

)(

]2[

])[()(

xpxx

xEx

iixip

E

xxE

xExV

i


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Random Variable

Standard deviation

Sums of R.V.

)()( XVXSD

)()()(

)()()(

2

2

21

2

1

2211

2211

xVaxVaYV

xEaxEayE

xaxaY


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Random Variable

n

XXSampleMean

i

11

)(222

2

n

n

n

XXanceSampleVari xxS

ii


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Poisson Probability Distribution

Consider a discrete r.v. which is often useful

when dealing with the number of occurrences

of an event over a specified interval of time.

Suppose we want to find the probability

distribution of the accidents at the intersection

of Rural and Apache during a one week

period.

The R.V. we are interested in is the number of

accidents.


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Poisson Probability Distribution

i. The Poisson Distribution provides a good model for the probability

distribution of the number of rare events that occur in space, time,

and volume where is the average at which events occur.

ii. Define: A r.v. is said to have a Poisson distribution if the p.m.f of

X is

P(x) = f(x) =!x

ex

, x = 0,1,

where is the rate per unit time or per unit areaiii.

)(

][

XV

XE


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Exponential Distribution

Previously, we discussed the Poisson random variable,

which was the number of events occurring in a given

interval. This number was a discrete r.v. and the

probabilities associated with it could be described by the

Poisson Probability Distribution.

Not only is the number of events a r.v., but the waiting

time between event is also a random variable. This r.v. is a

continuous r.v. for it can assume any positive value.

This r.v. is an exponential r.v. which can be described by

the exponential distribution.


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i. Pdf:

otherwise

xexf

x

0

0&0)(

where = rate at which events occur

ii. Correspondingly,

2

0

1)(

1][

0,1)()(

XV

XE

xedxexXPxF xx

x

iii. An important application of the exponential distribution is to

model the distribution of component lifetime. A reason for its

popularity is because of the memory-less property of the



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The Uniform Distribution

o The simplest distribution is the one in which a continuous r.v. can assume

any value within a interval [a, b]

Def:

A continuous r.v. X is said to have a uniform distribution on the

interval [a,b] if the probability distribution (pdf) of X is:

otherwise

bxaabxf

0

1

)(


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The cumulative distribution is

12

)()(

2)

1()(][

)(

)()()(

2ab

XV

abdx

abxdxxxfXE

abax

aba

abx

ax

abxdxxf

dxxfxXPXF

xx

x

x


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Note:

An important uniform distribution is

that for when a = 0 and b = 1, namelyU(0, 1)

A U(0,1) r.v. can be used to simulate

observation of other random variablesof the discrete and continuous type.


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The Triangular Distribution

Continuous Distribution

elsewhere

cxbacbc

xc

bxaacab

ax

xf

0

))((

)(2

))((

)(2

)(


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cx

cxbacbc

xc

bxaacab

axxF

axxF

1

))((

)(1

))((

)()(

0)(

2

2


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caxb

xxc

xxa

bcacabcbaxV

cbaxE

axxF

n

n

3

}max{

}min{

18)(

3

)(

0)(

1

1

222


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Normal Distribution

It is a fact that measurements on many random variables will follow a bell-

shaped distribution.

Random variable of this type are closely approximated by a Normal

Probability Distribution.

A continuous r.v. X is said to have a normal distribution if the pdf of X is

,,0,2

1)(

2

2

2

)(

xexf

x

The distribution contains 2 parameters ( and ). These are the expectedvalue and the variance and hence locate the center of the distribution and

measure its spread.


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Normal Distribution

The Standard Normal Distribution

To compute P(a x b) when X ~ N(, 2), we must evaluate

dxedxxf

b

a

xb

a

2

2

2

)(

2

1

)(

Note: None of the standard integration techniques can be used

to evaluate this pdf. Instead, for = 0, and 2 = 1, the pdf hasbeen evaluated and values have been computed. Using the

table, probabilities for any other values of and 2 can bedetermined


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Normal Distribution

The normal distribution for parameters values

= 0, and 2 = 1 is called the standard normaldistribution. A r.v. that has a standard

distribution is called a standard normal random

variable (denoted by Z). The pdf of Z is:

zezf

z

,2

1

)(2

2


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Normal Distribution

The cumulative distribution of Z is

(Z)bydenotedisand)()(

z

dyyfzZP

Note: The N(0,1) Table returns the cumulative

probability up to z or (z)


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Normal Distribution

Non-standard Normal Distribution

The table only provides probabilities for r.v.

following the N(0,1) distribution. Thus, when X~ N(, 2), (i.e. not = 0, 2 = 1), probabilitiesinvolving X are computed by standardizing

the r.v. to N(0,1) scale.


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Data Collection

Little variability model as a constant.

Variability model as a random variable.

Empirical vs. Theoretical, Select aDistribution, Estimate Parameter of

distribution, goodness or fit test.

2


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X2 goodness of fit test

Compare observed versus theoretical

density

A collection of data can be as a sample

from a specified p.d.f

H0: Xis are IID r.v. with density f(x)

H1: Xis are not IID r.v. with density f(x)

2


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Critical value

If H0 is true, TS ~ X2

k-1-(# of par estimated),

A large T.S.would cause rejection of H0

Reject Ho if T.S. > X2critical

k

i i

iiTS1

2

2


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Issuestest is an art

Number of intervals > 2

Size of intervals: Ei ~ same > 5 Requires relatively large amount of data


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K-S test

Compare observed with theoretical CDF

Limited to continuous distribution, known

parameters

H0: Xi are IID r.v. with CDF F(x)

H1: Xi are not IID r.v. with CDF F(x)

Test statisticFrom table

K S


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K-S test

Critical value

A large T.S would cause rejection

Critical value

))(max(),1

)(max(max^

XiFn

i

n

iXiFTS

n

nn

/22.110.0

/36.105.0/63.101.0

P i i


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Parameter estimation

Set of data x1, x2, xm

Methods of moments => equate E(X),

V(X) to x and S2

1

22

2

n

xnxs

n

xx

ii

P i i


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Parameter estimation

Maximum likelihood => find parameter

that max the likelihood of obtaining the

given sample

Produces efficient and consistent estimates

Not always unbiased

Superior properties to methods of moments

Common sense.

St ti ti l A l i f Si l ti


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Statistical Analysis of Simulations

As previously mentioned, output data fromsimulation always exhibit random variability,since random variables are input to the simulationmodel.

We must utilize statistical methods to analyzeoutput from simulations.

The overall measure of variability is generally

stated in the form of a confidence interval at agiven level of confidence.

Thus, the purpose of the statistical analysis is toestimate this confidence interval.

O l i


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Output analysis

Need multiple observations to estimate

variability

Y1, Y2, Y3, . Yn

Estimate a confidence interval for the

measure of performance

Estimate the number of observations

required to obtain the desired precision

O t t l i


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Output analysis

What is an observation?

Is observation a sample statistic or time

average statistic?

Is this a steady state simulation or

terminating simulation?

Are the observations independent or

correlated?

T i ti St d St t Si l ti


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Terminating vs Steady State Simulation

Often, the type of model determines whichtype of output analysis is appropriate for aparticular simulation.

However, the system or model may notalways be the best indicator of whichsimulation would be the most appropriate.

It is quite possible to use the terminatingsimulation approach for systems moresuited to steady-state simulations, and viceversa.

Ob ti Ti B d


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Observation vs Time Based

Observation (Sample) Average Time In System

Average Time In Queue

Time Based Average Number in System Average Number in Queue

Machine Utilization

T i ti i l ti


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Terminating simulation

Simulation in which the output measure of

performance is defined over a specific

interval of time with a specific starting

condition and a specific ending conditionRetail sales during a business day

Project network

Time to produce a batch of parts in a work cell

Military Simulations

T i ti i l ti


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Terminating simulation

Has a specified starting and ending

condition.

Each observation must have the same

starting and ending.

Observations are obtained by replication.

Use a different seed for random number

generation.

St d t t i l ti


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Steady state simulation

Simulation in which the output measure of

performance is defined over an infinite

interval of time independent of the initial

state of the system and stopping conditionAverage production from an assembly line of

well trained employees

Inventory simulation

Stead state sim lation


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Independent of starting and ending

condition.

Remove initial condition bias

Specify warm-up period (transient period) .

Set initial condition too steady state.

Have a very long run length



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1. Individual Yi average of individuals.

2. Replication Yi average of each one.

3. Batch means batch by time, by number.

Terminating vs Steady state simulation


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Terminating vs. Steady state simulation

Terminating

Observations are obtained by replication

Each observation must reflect the specified

starting and ending condition

Use a different seed for each replication

Y1, Y2, , Yr => one independent

observation per replication

Confidence interval for steady state


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y

simulation

Y1, Y2, . Yn

Trying to estimate a long run performance

measure independent of starting and

ending conditions

Two problems

Initial condition bias

Dependent observations

Confidence interval for steady state


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simulation

Outline

Removing initial condition bias

Creating independent observation

Replication/ deletion

Batch means

Confidence interval for replication


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Let Y1, Y2, and Y3YR be measures of

performance from R independent

replication.

Independent -> different seed for each run

1

)(

2,

22

2

2

1

R

R

RY YYsSti

r



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Approximate due to need for Yi ~ Normal

(1-) Confidence Interval => Probabilityof containing true mean

RS

RRS

YVarYVarR

YYVarYVar

R

RRR

2

2

2

12

1

1

1

1

))(...)((

)...()(

Number of replication needed


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Suppose we desire a confidence interval

Based on a preliminary run of R0replication, we have an estimate of S2 and

confidence interval

HalfLengthIY

0

2

211

0RStY

,R



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Find R such that

If R is large,

R

StI

,R

2

21

1

2

2

2

2

1

*

I

SR

RI

Z

ZsZ

tr

Test for comparing two means


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H0: 12 = 0 H1: 12 0Two approaches:

Form a (1) confident on 12 :)( 21,2/21 YYVtYY r

Reject H0 if confident does not contain 0.

Perform a t test )(0)(

21

21

YYV

YY

t

Reject if \t\ > tr,/2

Assumptions

Case 1: Y1, Y2 YR12

11 , sY

Case 2: Y1, Y2 YR22

22 , sY

Observations are independent

Observation are normally distributed

Variances are unknown/known.

Variances are equal/unequal

Observations are paired/unpaired.



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Equal Variance

1. Assumptions: independent, normal, unknown, unpaired, equalvariance.

2.2

)1()1(

2

)()(

21

2

22

2

11

21

2

2

2

12

RR

SRSR

RR

YYYYS ii

p

3.2

2

1

2

2121)()()(

R

S

R

SYVarYVarYYVar

pp

4.2

2

1

2

2,2/21 21:)1(

R

S

R

StYYconfident

pp

RR

5. t-test:

2

1

1

1

21)(

RRp

S

yyt

t-crit =2

,221

RRt

6. Note: Many simulations do not have equal variance.



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One sided test

Need to make hypothesis in advance

Use t test, adjust critical value


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Unequal Variance

1. Assumptions: independent, normal, unknown variance,

unpaired, unequal variance.

2.2

2

2

1

2

1

2121 )()()(

R

S

R

SYVarYVarYYVar

3.2

2

2

1

2

1

,2/21:)1(

R

S

R

StYYconfident

12

2

22

11

1

21

2

22

1

21

22

2

R

R

S

R

R

S

R

S

R

S



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Paired Test

Assumptions: independent, normal, unknown variance,equal # of replications

Case 1: Y1, Y2 YRCase 2: Y1, Y2 YR

Different: d1, d2 dR , where di = yi yi

1

)(2

2

R

ddS

R

dd id

i

H0: 12 = 0 d = 0H1: 12 0 d 0

R

SdVtdconfident dR

2

1,2/ )(:)1(

R

Sd

dt

2

Test for comparing two variances


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Test for comparing two variances

F-test for equal variance1.

2

2

2

11

2

2

2

10

H

H

2. Test statistics = F =2

2

2

1

S

S =

3. Critical Value =2

,1,1 21

RRF

4. Example

F =5.4/2.55 = 2.12

= .10, Fcritical = F9,9,. 05 = 3.18, can not reject Ho

Common Random Number


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Common Random Number

The process of comparing cases with thesame set of random numbers creating identical condition

Observation Confident Interval

Use the paired test

),(2)()()(

)()(

212121

212/,121

YYCovYVYVYYV

YYVtYY R

Random Numbers


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Random Numbers

Generation of U(0,1) random number

algorithm used by the RND function

Generation of random variates from

various distributionsalgorithm used by

EXPONENTIAL, UNIFORM, and so on

(these algorithms use U(0,1) randomnumbers.

Random Number Generation


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Desirable properties

Fast and efficient

Capable of repeating same sequence

Statistically equivalent to U(0,1)

Independent and dense

Large cycle length or period

Low storage requirements

Old methodtables



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Pseudo random number generators

A non random sequence of numbers each

completely determined by its predecessor, the

algorithm, and initially, the seed.

Linear Congruential Generator


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Zi = ( a * Zi-1 + C ) mod m

Z0 = seed

Ui= Z

i/ m (Random Number)

If we choose a, C, and m correctly, => then

we achieve a maximum period 0


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Rule For Full Period :

C is relatively prime to m.

other than 1, hence there is no integer that exactly

divides C and mEvery prime factor of M is also a prime factor

of A-1

If m is exactly dividable by 4, then A-1 must

be exactly dividable by 4



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A full period does NOT mean always a

good random number generator

Multiplicative Generators


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Zi = a * Zi-1 mod m

Z0 = seed

Saves an addition, more popular



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u t p cat ve Ge e ato s

C=0

M divides both m and c

Condition (a) is violated

Not full period

P = m1 is largest available period



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p

2b is not a good choice for m

only possible numbers

Let m = 2b - 1

Testing a random number generator


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g g

Testing the distribution

Generate 1000 or more observations

X2 test or K-S test for U(0, 1)

Use 100 intervals

Test for independence

Runs up

Tests designed to compare observed and

expected distribution

E(x) = .5 V(X) = 1/12, where a = 0, b=1

Random variate generation


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g

Assume a random number generator is

available to generate Ui ~ U(0, 1)

Goal: Generate Xi from a specified

distribution f(x) or p(x) of F(x)

Three methods

Inverse transformation method

Convolution method

Acceptance\Rejection method

Random variate generation


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g

Apply these methods to the five

distributions we are using in this class

Uniform

Triangular

Exponential

Normal

Poisson



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General ideause CDF

Select Ui

Find corresponding xi

That is xi = F-1(Ui)

Advantage of inverse transformation

method

One Ui per xi

Disadvantage

CDF may not always exist



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Exponential distribution

f(x) = e -x x 0

F(X) = 1 - e -x x 0

Ui = F(Xi) = 1 - e-xi

(1- Ui) = e-xi

ln(1- Ui) = - XiXi = - (1/) ln(1- Ui) = - (1/) ln(Ui)



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Triangular distribution

cxb

acbc

xcbxa

acab

axF x ,

))((

)(1,

))((

)(22

)(

acab

iu

ac

ab

iu

ac

ab

iu

ux ii acaba ))((

)1)()(( ux ii acbcc

ui

Yes

No

Convolution Method


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Applicable to situation where the random

variable of interest can be expressed as a

sum of other random variables that are IID

(independent identical distributed)

X=Y1+Y2+Y3. +Yn

Idea: Generate Y1. Yn and add these up

to calculate X

Convolution Method


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Normal distribution

Focus: Generating Zi ~ N(0, 1)

Generating Zi

Inverse transformation: F(x) does not exist

Acceptance\Rejection: Not bounded

),(~

NZx

x

Z iii

i

2

2

1

2

1)(

z

eZf


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Acceptance\Rejection Method


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p j

Applicable to distribution functions that

are hard to integrate

IdeaFind a majoring function t(x) where t(x) > f(x)

Sample values of x from t(x) call it x*

Sample Ui < f(x*) / t(x*), accept x*

Simplification for this classwe will

always use a rectangular majoring function

9.3 Random Numbers and Monte

Carol Simulation


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Carol Simulation

The procedure of generating these times from thegiven probability distributions is known assampling from probability distributions, orrandom variate generation, or Monte Carlo

sampling. We will discuss several different methods of

sampling from discrete distributions.

The principle of sampling from discretedistributions is based on the frequencyinterpretation of probability.


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In addition to obtaining the right frequencies, the

sampling procedure should be independent; that is,each generated service time should be independentof the service times that precede it and follow it.

This procedure of segmentation and using a roulette

wheel is equivalent to generating integer randomnumbers between 00 and 99.

This follows from the fact that each random numberin a sequence has an equal probability of showing

up, and each random number is independent of thenumbers that precede and follow it.


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Random Number Generators Since our interest in random numbers is for use


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Since our interest in random numbers is for usewithin simulations, we need to be able to generate

them on a computer. This is done by using mathematical functions called

random number generators.

Most random number generators use some form of acongruential relationships. Examples of suchgenerators include linear congruential generator, themultiplicative generator, and the mixed generator.

The lineal congruential generator is by far the mostwidely used.


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Each random number generated using thismethods will be a decimal number between 0and 1.

Random numbers generated using congruential

methods are called pseudorandom numbers.

Random number generators must have theseimportant characteristics:

1. The routine must be fast

2. The routine should not require a lot of core storage

3. The random numbers should be replicable; and

4. The routine should have a sufficiently long cycle


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Most programming languages have built-inlibrary functions that provide random (or

pseudorandom) numbers directly.

Computer Generation of Random

Numbers


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Numbers

We now take the method of Monte Carlosampling a stage further and develop a procedureusing random numbers generated on a computer.

The idea is to transform the U(0,1) random

numbers into integer random numbers between00 and 99 and then to use these integer randomnumbers to achieve the segmentation bynumbers.

We now formalize this procedure and use it togenerate random variates for a discrete randomvariable.


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The procedure consists of two steps:

1. We develop the cumulative probability

distribution (cdf) for the given random

variable, and2. We use the cdf to allocate the integer random

numbers directly to the various values of the

random variables.

9.4 An Example of Monte Carlo

Simulation


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Simulation

The book uses a Monte Carlo simulation tosimulate a news vendor problem.

The procedure in this simulation is different from

the queuing simulation, in that the presentsimulation does not evolve over time in the same

way.

Here, every day is an independent simulation.

Such simulations are commonly referred to asMonte Carlo simulations.

9.5 Simulations with Continuous

Random Variables


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Random Variables

In many simulations, it is more realistic andpractical to use continuous random variables.

We present and discuss several procedures forgenerating random variates from continuous

distributions. The basic principle is similar to the discrete case.

We first generate U(0,1) random number andthen transform it into a random variate from thespecified distribution.


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The selection of a particular algorithm willdepend on the distribution from which we want

to generate, taking into account such factors as

the exactness of the random variables, the

computations and storage efficiencies, and the

complexity of the algorithm.

The two most common used algorithms are the

inverse transformation method (ITM) and theacceptance-rejection method (ARM).

Inverse Transformation Method


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The inverse transformation method is generallyused for distribution whose cumulativedistribution function can be obtained in closedform.

Examples include the exponential, the uniform,the triangular, and the Weibull distributions.

For distributions whose cdf does not exist inclosed form, it may be possible to use some

numerical method, such as a power-seriesexpansion, within the algorithm to evaluate thecdf.


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The ITM is relatively easy to describe andexecute.

It consists of the following steps: Step1: Given the probability density formulaf(x) for a

random variable X, obtain the cumulative distributionfunction F(x) as

Step 2: Generate a random number r. Step 3: Set F(x) = rand solve forx.

x

dttfxF )()(

We consider the distribution given by the function


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We consider the distribution given by the function

A function of this type is called a ramp function. To obtain random variates from the distribution

using the inverse transformation method, we first

computer the cdf as

0

2)(

x

xf 0 x 2otherwise

4

2)(

2

0

x

dttxFx

In Step 2, we generate a random number r.


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Finally, in Step 3, we set F(x) =rand solve forx.

Since the service time are defined only for positive

values ofx, a service time of as thesolution forx. This equation is called a randomvariate generator or a process generator.

Thus, to obtain a service time, we first generate a

random number and then transform it using thepreceding equation.

rx

rx

2

4

2

rx 2


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As this example shows, the majoradvantage of the inverse transformation

method is its simplicity and ease of

application.

AcceptanceRejection Method


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There are several important distributions,including the Erlang (used in queuing models)and the beta (used in PERT), whose cumulativedistribution functions do not exist in closed form.

For these distributions, we must resort to othermethods of generating random variates, one ofwhich is the acceptancerejection method(ARM).

This method is generally used for distributionswhose domains are defined over finite intervals.


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Given a distribution whose pdf,f(x), is definedover the interval ax b, the algorithm consistsof the following steps: Step 1: Select a constantMsuch thatMis the largest

value off(x) over the interval [a, b]. Step 2: Generate two random numbers, r1 and r2.

Step 3: Computerx* = a + (ba)r1. (This ensuresthat each member of [a, b] has an equal chance to be

chosen asx

*.) Step 4: Evaluate the functionf(x) at the pointx*. Letthis be f(x*).


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There are several ways by which the method canbe made more efficient.

One of these is to use a function in Step 1 insteadof a constant.

We now give an intuitive justification of thevalidity of the ARM.

In particular, we want to show that the ARMdoes generate observations from the givenrandom variable X.

Direct and Convolution Methods for

the Normal Distribution


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the Normal Distribution

Both the inverse transformation method and theacceptancereject method are inappropriate forthe normal distribution, because (1) the cdf doesnot equal in closed form and (2) the distribution

is not defined over a finite interval. Other methods such asan algorithm based on

convolution techniques, and then a directtransformation algorithm that produces two

standard normal variates with mean 0 andvariance 1.


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The Direct Method


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The direct methods for the normal distributionwas developed by Box and Muller (1958).

Its not as efficient as some of the newer

techniques, it is easy to apply and execute. The algorithm generates two U(0,1) random

numbers, r1andr2, and then transforms them into

two normal variates, each with mean 0 and

variance 1, using the direct transformation.

1


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It is easy to transform these standardized normal

variates intro normal variates X1 and X2 fromthe distribution with mean and variance 2,using the equations

22

1

12

22

1

11

2cos)ln2(

2sin)ln2(

rr

rr

22

11

9.6 An Example of a Stochastic

Simulation


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Simulation

Cabot Inc. is a large mail order firm in Chicago. Orders arrive into the warehouse via telephones.

At present, Cabot maintains 10 operators on-line24 hours a day.

The operators take the orders and feed themdirectly into a central computer, using terminals.

Each operator has one terminal. At present, thecompany has a total of 11 terminals.

That is, if all terminals are working, there will be1 spare terminal.

Cabot managers believe that the terminal systemneeds evaluation because the downtime of


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needs evaluation, because the downtime of

operators due to broken terminals has beenexcessive.

They feel that the problem can be solved by thepurchase of some additional terminals for the spares

pool. It has been determined that a new terminal will cost

a total of $75 per week.

It has also been determined that the cost of terminal

downtime, in terms of delays, lost orders, and so onis $1000 per week.

Given this information, the Cabot managers would liketo determine how many additional terminals they


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to determine how many additional terminals they

should purchase. This model is a version of the machine repair problem.

It is easy to find an analytical solution to the problemusing the birth-death processes.

However, in analyzing the historical data for theterminals, it has been determined that although thebreakdown times can be represented by theexponential distribution, the repair times can be

adequately represented only by the exponentialdistribution.

This implies that analytical methods cannot be used


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p yand that we must use simulation.

To simulate this system, we first require theparameters of both the distributions.

The data show that the breakdown rate is

exponential and equal to 1 per week per terminal. In other words, the time breakdowns for a terminal

is exponential with a mean equal to 1 week.

Analysis for the repair times shows that this

distribution can be represented by the triangulardistribution which has a mean of 0.075 week.

0750025040010 xx


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The repair stuff on average can repair 13.33terminals per week.

To find the optimal number of terminals, wemust balance the cost of the additional terminalsagainst the increased revenues generated as aresult of the increase in the number of terminals.

In this simulation we increase the number ofterminals in the system, n, from the present totalof 11 in increments of 1.

125.0075.040050

075.0025.040010)(

xx

xxxf


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Once we have a value ofELn, we cancomputer the expected weekly downtime

costs, given by 1000(10-ELn).

Then the increase in revenue as a result ofincreasing the number of terminals from 11

to n is 1000(ELnEL11). Mathematically,

we computeELn T

A

T

dttN

EL

m

i

iT

n

10

)(

h


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where

T= length of simulation

N(t) = number of terminals on-line at time t (0tT)

Ai = area of rectangle under N(t) between ei-1 and ei

(where ei is the time of the ith event)m = number of events that occur in the interval [0,T]

Between time 0 and time e1, the time of the firstevent, the total on-line time for all the terminals is

given by 10ei, since each terminal is on-line for aperiod ofe1 time units.

If we now run this simulation over T time units and


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If we now run this simulation over Ttime units and

sum up the areasA1,A2, A3,, we can get anestimate forEL10 by dividing this sum by T. Thisstatistic is called a time-average statistic.

We would like to set up the process in such way that

it will be possible to collect the statistics tocomputer the areasA1,A2, A3,.

That is, as we move from event to event, we wouldlike to keep track of at least the number of terminals

on-line between the events and the time betweenevents.


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We first define the state of the system as thenumber of terminals in the repair facility.

The only time the state of the system will changeis when there is either a breakdown or a

completion of a repair. Therefore, there are two events in this simulation:

breakdown and completion of repairs.

To set up the simulation, our first task is todetermine the process generators for both thebreakdown and the repair times.

We use the ITM to develop the process generators.


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For the exponential distribution the processgenerator is simplyx = -log r

In case of the repair times, applying the ITM givesus

and

as the process generators.

)5.00(005.0025.0 rrx

)0.15.0()1(005.0125.0 rrx


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For each n, we start the simulation in the statewhere there are no terminals in the repair facility.

In this state, all 10 operators are on-line and anyremaining terminals are in the spares pool.

Our first action is the simulation is to schedulethe first series of events, the breakdown times forthe terminals presently on-line.

Having scheduled these events, we nextdetermine the first event, the first breakdown, bysearching through the current event list.


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We then move the simulation clock to the timeof this event and process this breakdown.

To process a breakdown, we take two separateseries of actions

1. Determine whether a spare is available.2. Determine whether the repair staff is idle.

These actions are summarized in the systemflow diagram showed in the book in Figure 17.

Otherwise, we process a completion of a repair.

To process the completion of a repair we also


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To process the completion of a repair, we also

undertake two series of actions.1. At the completion of a repair, we have an additional

working terminal, so we determine whether the terminalgoes directly to an operator or to the spares pool.

2. We check the repair queue to see whether any terminalsare waiting to be repaired.

We proceed with the simulation by moving fromevent to event until the termination time T.

At this time, we calculate all the relevant measuresof performance from the statistical counters.


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Our key measure is the net revenue for thecurrent value ofn.

If this revenue is greater than the revenue for asystem with n-1 terminals, we increase the value

ofn by 1 and repeat the simulation with n +1terminals in the system.

Otherwise, the net revenue has reached a peak.

The simulation outlined in this example can beused to analyze other policy options thatmanagement may have.


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Set5 Simulation

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