Chapter Six

Process Simulation

Using Arena

McGraw-Hill/Irwin © 2005 The McGraw-Hill Companies, Inc. All rights reserved.

Chapter Six Learning Objectives

Analyze queuing processes using Excel and Arena

Develop and utilize Arena simulation models for a simple queuing process

Use Arena to compare alternative queuing processes

Develop new Arena modeling techniques by using Arena Example and Smart models

Chapter Six Worksheet MenuChapter 6 Exhibits

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Process Analysis and Improvement: Tools and Techniques Copyright ©2005 by The McGraw-Hill Companies, Inc.

Filename Sheet / Page NameExhibit Number Application / Caption

Drawings.vsd MM1 1 M/M/1 Queueing Process Diagram

Worksheets.xls Exponential 2 Exponential Probability Density Function

Worksheets.xls MM1 3 M/M/1 Queue Calculations

Worksheets.xls MM1 Calculations 4 M/M/1 Queue Calculations

Worksheets.xls MM1 5 M/M/1 Queue Measures of Performance versus Traffic Intensity

Drawings.vsd MM3 6 M/M/3 Queueing Process Diagram

Worksheets.xls Manual Simulation 7 F/F/1 Queue Simulation Calculations

Worksheets.xls Manual Simulation 8 F/F/1 Queue Simulation Chart

Worksheets.xls Manual Simulation 9 M/F/1 Queue Simulation Calculations

Worksheets.xls Manual Simulation 10 M/F/1 Queue Simulation Chart

Worksheets.xls Manual Simulation 11 M/M/1 Queue Simulation Calculations

Worksheets.xls Manual Simulation 12 M/M/1 Queue Simulation Chart

Drawings.vsd Simulation Tasks 13 Process Simulation Tasks

Worksheets.xls Arena Components 15 Arena Simulation Model Development Environment Components

Drawings.vsd Arena File Structure 16 Arena File Structure

Worksheets.xls MM1 Comparison 20 M/M/1 Queue Comparison: Theorical versus Simulated

Worksheets.xls Arena Components 21 Arena Template Panel Modules

Worksheets.xls Arena Modules 22 Arena Basic Process Panel Modules

Worksheets.xls Arena Components 33 Arena Simulation Engine

Worksheets.xls Items Ordered Data 35 D&E Order Size

Drawings.vsd Comparison 46 Two Queueing Process Analysis Diagram

Worksheets.xls Arena Components 59 Arena Stand Alone Programs

Worksheets.xls Examples 60 Arena Examples

Worksheets.xls Smarts 61 Arena Smart Files

Worksheets.xls HSmart Arena Packaging Edition Smart File Examples

Worksheets.xls FSmart Arena Factory Analyzer Edition Smart File Examples

Related Files

Filename Source Application

Comparison.doe Text CD-ROM M/M/3 Queue Comparison Simulation Model

MM1 Statistics.doe Text CD-ROM M/M/1 Queue Simulation Model

MM1.doe Text CD-ROM M/M/1 Queue Simulation Model

MM3.doe Text CD-ROM M/M/3 Queue Simulation Model

MV1.doe Text CD-ROM M/Various/1 Queue Simulation Model

QueueStencil.vss Text CD-ROM Queue Visio Stencil

View Selected Worksheet or Drawing

Chapter Six Files

CD File Description

Chapter 6 Drawings.vsd Chapter 6 Drawings

Chapter 6 Presentation.ppt Chapter 6 PowerPoint Presentation

Chapter 6 Worksheets.xls Chapter 6 Worksheets

Comparison.doe M/M/3 Queue Comparison Simulation Model

MM1 Statistics.doe M/M/1 Queue Simulation Model

MM1.doe M/M/1 Queue Simulation Model

MM3.doe M/M/3 Queue Simulation Model

MV1.doe M/Various/1 Queue Simulation Model

QueueStencil.vss Queue Visio Stencil

Chapter Six Outline

Analyze queuing processes using Excel and Arena

Arena Simulation Environment Other Queueing Process Models Other Arena Programs and Examples

Analysis and Simulation of Queuing Processes

Queuing Process Analysis Using Excel M/M/1 Queue M/M/3 Queue

Manual Simulation Using Excel Process Simulation Tasks Why Use Arena for Process Analysis

and Improvement?

Queuing Process Analysis Using Excel

Process Boundary Defines process limits Arrival Source Supplies process customers Queue Allows customers to wait for service Server Provides customer service

Departure

Process Boundary

ArrivalSource

QueueSingle

Server

Exhibit 6-1 p. 155Page: MM1

M/M/1 Queue

M The time between arrivals follows the exponential distribution

M The time for service follows the exponential distribution

1 The process has 1 server

Exponential Probability Density Functions

f(x) = λe-λx, for 0 ≤ x < ∞

0.0

0.5

1.0

1.5

2.0

2.5

0.0 1.0 2.0 3.0 4.0 5.0 6.0

x

f(x

)

λ =1.0

λ =1.5

λ =2.0

Exhibit 6-2 p. 156Sheet: Exponential

Exponential Distributionf (x) = λe-λx, for 0 ≤ x < ∞E (X) = 1/λ V(X) = 1/λ2

F (x) = 1-e-λx, for 0 ≤ x < ∞

M/M/1 Queue Calculations

Column Customers / Minute Formula Units Source

B Mean Inter Arrival Time 1 / λ Seconds Input

C Mean Arrival Rate λ Customers / Minute 60 / Column B

D Mean Service Time 1 / μ Seconds Input

E Mean Service Rate μ Customers / Minute 60 / Column D

F Traffic Intensity ρ = λ / μ Dimensionless Column C / Column E

G Expected Number in Process Lp = ρ / (1 - ρ) Customers Column F / (1 - Column E)

H Expected Number in Queue Lq = Lp - ρ = ρ2 / (1 - ρ) Customers Column G - Column F

I Expected Time in Process Wp = Lp / λ =1 / (μ - λ) Minutes Column G / Column C

J Expected Time in Queue Wq = Lq / λ =ρ / (μ - λ) Minutes Column H / Column C

Exhibit 6-3 p. 156Sheet: MM1

Exhibit 6-4 p. 156Sheet: MM1 Calculations

M/M/1 Queue Calculations

Traffic Intensity = ρ = Mean Arrival Rate / Mean Service Rate =Mean Service Time / Mean Inter Arrival Time

λ = 1 / 60 = 0.0167 Customers per Seconds = 0.0167 * 60 = 1.00 Customers per Minute

μ = 1 / 55 = 0.0182 Customers per Seconds = 0.0182 * 60 = 1.09 Customers per Minute

ρ = 1.00 / 1.09 = 0.917 Expected Number in Process = Lp

= ρ / (1 - ρ) Expected Number in Queue = Lq

= Lp - ρ = ρ2 / (1 - ρ)

Exhibit 6-5 p. 167Sheet: MM1

M/M/3 Queue

ArrivalSource

Queue

Three

Servers

Process Boundary

Departure

Exhibit 6-6 p. 157Sheet: MM3

Manual Simulation Using Excel

Mean Inter Arrival Time in Seconds, 60 Mean Service Time in Seconds, 50 Arrival Option, 0 = Fixed or 1 = Exponential Service Option, 0 = Fixed or 1 =

Exponential F(60) / F(50) / 1 Exhibits 6-7 & 6-8 M(60) / F(50) / 1 Exhibits 6-9 & 6-10 M(60) / M(50) / 1 Exhibits 6-11 & 6-12

Manual Simulation F(60) / F(50) / 1

Exhibit 6-8 p. 158Sheet: Manual SimulationExhibit 6-7 p. 158

Sheet: Manual Simulation

Manual Simulation M(60) / F(50) / 1

Exhibit 6-10 p. 159 Sheet: Manual SimulationExhibit 6-9 p. 159

Sheet: Manual Simulation

Manual Simulation M(60) / M(50) / 1

Exhibit 6-12 p. 160Sheet: Manual Simulation

Exhibit 6-11 p. 160Sheet: Manual Simulation

Process Simulation Tasks

Exhibit 6-13 p. 160Page: Simulation Tasks

Why Use Arena for Process Analysis and Improvement?

Simulation History 1961 GPSS (Gordon – IBM) 1982 SIMAN (Pegden – Penn State) 1992 Arena (Systems Modeling Corp.) 2002 Arena Factory Analyzer (Rockwell Software)

Other Simulation Products AutoMod Flexsim ProModel Simul8 Witness

Arena Simulation Environment

Arena Modeling Environment Arena Template Panel Modules Arena Simulation Engine

Arena Modeling Environment

Modeling Environment Exhibit 6-14 Template Panel Modules Exhibit 6-21 Simulation Engine Exhibit 6-33 VBA Interface Chapter 8 Stand Alone Programs Exhibit 6-59

Arena Modeling Environment

Exhibit 6-14 p. 161Model: MM1.doe

Tool Bar

FlowchartWindow

SpreadsheetWindow

ProjectBar

StatusBar

Arena Modeling Environment

Exhibit 6-15 p. 162Sheet: Arena Components

Arena Modeling Environment

Exhibit 6-16 p. 163Page: Arena File Structure

Queueing Process Models

MM1.doeMM3.doeComparison.doeMV1.doe

MM1.doe @ 60 Minutes

Exhibit 6-17 p. 163TNOW = 60 Minutes

MM1.doe

Exhibit 6-18 p. 164Run Setup Dialog

Exhibit 6-19 p. 164Category Overview Report (Queue)

M/M/1 Queue Comparison: Theoretical versus Simulated

Exhibit 6-20 p. 165Sheet: MM1 Comparison

Arena Template Panel Modules

Create Module Process Module Dispose Module Entity Spreadsheet Queue Spreadsheet Resource Spreadsheet Variable Spreadsheet

Arena Template Panel Modules

Exhibit 6-21 p. 165Sheet: Arena Components

Arena Basic Process Template

Panel Module Type Module Reference

Basic Process Data Entity Entity

Basic Process Data Queue Queue

Basic Process Data Resource Resource

Basic Process Data Schedule Resource

Basic Process Data Set System

Basic Process Data Variable System

Basic Process Flowchart Assign Entity

Basic Process Flowchart Batch Entity

Basic Process Flowchart Create Entity

Basic Process Flowchart Decide Entity

Basic Process Flowchart Dispose Entity

Basic Process Flowchart Process Resource

Basic Process Flowchart Record Entity

Basic Process Flowchart Separate Entity

Exhibit 6-22 p. 166Sheet: Arena Modules

Exhibit 6-23 p. 166Arena Basic Process Panel Module Icons

Create Module

Exhibit 6-24 p. 166MM1.doe Create Module Dialog Box

Process Module

Exhibit 6-25 p. 167MM1.doe Process Module Dialog Box

Resource Detail

Exhibit 6-26 p. 168MM1.doe Process Module Resource Detail Dialog Box

Exhibit 6-27 p. 168 Arena Resource Picture Placement Window

Dispose Module

Exhibit 6-28 p. 168MM1.doe Dispose Module Dialog Box

Entity Spreadsheet

Exhibit 6-29 p. 169MM1.doe Entity Spreadsheet

Queue Spreadsheet

Exhibit 6-30 p. 169MM1.doe Queue Spreadsheet

Resource Spreadsheet

Exhibit 6-31 p. 169 MM1.doe Resource Spreadsheet

Variable Spreadsheet

Exhibit 6-32 p. 169 MM1.doe Variable Spreadsheet

Arena Simulation Engine

Entity Simulation Clock and Event Calendar Random Number Generator Statistic Collection Tools Crystal Reports Statistic Output

Arena Simulation Engine

Exhibit 6-33 p. 170Sheet: Arena Components

Entity

Entities are the dynamic elements in the Arena model Entities enter the model via a Create module and exit via a

Dispose module While a large arena model may contain thousands of entities at

any one point in time, or one entity is active. The active entity attempts to move as far as possible through the

logic modules of the model. In model MM1.doe, the only point at which an entity can be

delayed is the process module. When an entity enters the Service Process module, either the service

function is started immediately or the entity is held in the Service.Queue.

If Resource Server is available, the duration of the service is determined based on the random expression illustrated previously. The Arena simulation engine schedules an End-Of-Service event on the Event Calendar to take place at a future point in time.

If Resource Server is not available, the duration of the time in queue cannot be determined. The entity will be released from the queue only after the server has processed all previously waiting entities.

Simulation Clock and Event Calendar

The Simulation Clock control the advance of simulation time

When the active entity is halted The movement of another entity may

be started or The clock advances to the next

scheduled event, starting the associated entity

Simulation Clock and Event Calendar

Exhibit 6-34 p. 171MM1.doe Trace Report Extract

Random Number Generators

Statistical Distributions BETA (Alpha1, Alpha2) EXPOnential (Mean) GAMMa (Beta, Alpha) JOHNson (gamma, delta, lambda, xi) ERLAng (Mean, k) LOGNormal (LogMean, LogStd) NORMal (Mean, SD) POISson (Mean) TRIAngular (Min, Mode, Max) UNIForm (Min, Max) WEIBull (Beta, Alpha)

Empirical Data CONTinuous (Prob1, Value1, Prob2, Value2, ... ) DISCrete (Prob1, Value1, Prob2, Value2, ... )

Random Number Generator Example

Items Ordered FrequencyCumulative Frequency Cumulative %

1 11 11 11.0%2 17 28 28.0%3 18 46 46.0%4 23 69 69.0%5 21 90 90.0%6 10 100 100.0%

Discrete Distribution

0.0%

25.0%

50.0%

75.0%

100.0%

0 1 2 3 4 5 6

Item s OrderedC

um

ula

tive

%

Disc (0.11, 1, 0.28, 2, 0.46, 3, 0.69, 4, 0.90, 5, 1, 6)

Exhibit 6-35 p. 172Sheet: D&E Order Size

Statistic Collection Tools

CounterTally Time PersistentOutputFrequency

Counter

Exhibit 6-36 p. 173MM1.doe Record Number Served

Crystal Reports Statistic Output

Exhibit 6-37 p. 173MM1.doe Advanced Process Statistics Module

Exhibit 6-42 p. 176MM1.doe User Specified Report

Tally

Exhibit 6-38 p. 173MM1.doe Record Time Between

Time Persistent

Exhibit 6-39 p. 174Arena Expression Builder, Current Number in Queue

Output

Exhibit 6-40 p. 174Arena Expression Builder, Number In Entity

Frequency Report

Exhibit 6-42 p. 176MM1.doe Frequencies Report

MM3.doe

Exhibit 6-43 p. 177MM3.doe Network Logic Flow

MM3.doe @ 60 Minutes

Exhibit 6-44 p. 177MM3.doe at TNOW = 60 Minutes

MM3.doe Report

Exhibit 6-45 p. 177MM3.doe Category Overview Report (Queue)

Comparison.doe

Separate Module Assign Module

Variables Attributes

• Entity Type• Entity Picture

Other Entity Icon Simulation Results

Comparison.doe

Process Boundary

ArrivalSource

QueueSingle

Server

ArrivalSource

QueueSingle

Server

ArrivalSource

QueueSingle

Server

Departure

Departure

Queue

Triple

Servers

Process Boundary

Departure

Departure

Exhibit 6-46 p. 178Page: Comparison

Comparison.doe

Exhibit 6-47 p. 178Comparison.doe Network Logic Flow

Separate Module

Exhibit 6-48 p. 179Comparison.doe Module Separate 1

Assign Module

Exhibit 6-49 p. 179Comparison.doe Module Assign as Parallel

Exhibit 6-50 p. 179Comparison.doe Assignments Dialog

Assign Module Types

Variables (Available throughout model) Attributes (Available for each entity)

Entity Type Entity Picture User Defined

Other J (Only Arena Integer) M (Station)

Default Arena Entity Attributes

Entity.HoldCostRateEntity.JobStepEntity.PictureEntity.SequenceEntity.StationEntity.Type

Entity Icon or Picture

Exhibit 6-51 p. 180Entity Pictures Dialog Window

Exhibit 6-52 p. 180Arena Picture Editor

Comparison.doe @ 60 Minutes

Exhibit 6-53 p. 181Comparison.doe at TNOW = 60 Minutes

Comparison.doe Report

Exhibit 6-54 p. 181Comparison.doe Number in Queues Report

MV1.doe

Exhibit 6-55 p. 182MV1.doe Network Logic Flow

Branch Block

Exhibit 6-56 p. 182MV1.doe Branch Block

MV1.doe @ 60 Minutes

Exhibit 6-57 p. 182MV1.doe at TNOW = 60 Minutes

MV1.doe Report

Exhibit 6-58 p. 183MV1.doe Number in Queues Report

Arena Stand Alone Programs

Exhibit 6-59 p. 183Sheet: Arena Components

Arena Examples

Exhibit 6-60 p. 184Sheet: Examples

Model Name (*.DOE) DescriptionACME Medical Equipment

Example of how Arena can be used to model a complete business process. The following concepts are illustrated: Process module, Decide module, animation of several different resources, changing entity picture in mid-model to represent current status, generating a job number automatically in Arena, attaching the job number to an entity (Assign module), and permanent batching.

Bank Kiosk Analysis This study examines alternatives for utilizing automatic ("self-serve") kiosks in a bank lobby. The optimization problem is to minimize the total resource cost (of tellers and kiosks), while providing acceptable customer service. Options include how many tellers to assign to customer transactions, how many kiosks to place in the bank, and what proportion of customers will use tellers.

Banking Transactions Demonstrates the various transactions and activities at a bank, including ATM, teller, and drive-thru transactions.

Billing Department Complex model of a standard billing center. The following concepts are illustrated: Process module, assigning an attribute (Assign module), different processing by attribute (Decide module by condition), resource animation, modeling an error checking scenario, entity duplication for parallel processing, and modeling resource cost.

Arena Smart Files

Exhibit 6-61 p. 184Sheet: Smarts

\Program Files\Rockwell Software\Arena\Smarts 258 Number of Smart models

Category

Model Name Smarts*.DOE Application

VBA and Visual Basic® 001 VBA—VariableArray ValueModules 002 Batch and Separate ModulesBatching 002 Batch and Separate ModulesModules 003 Assign Module ExampleResources 004 Defining a Resource CapacityDecision Logic 005 Decide Module ExampleModules 005 Decide Module ExampleNamed Views 006 Using Named ViewsBasic Concepts 007 Basic ModelingHierarchical Submodels 008 Hierarchical ModelingHierarchical Submodels 009 Nested Submodel ExampleAnimation 010 Animating Resource Status

Arena SMART files, a collection of small example models that demonstrate a variety of modeling techniques and situations commonly encountered using Arena.Smart files have been specifically designed for use as a training or reference tool.

Each SMART File is listed in one or more of the categories below.

Summary

Process Simulation Using Arena Analysis and Simulation of Queuing

Processes Arena Simulation Environment Other Queueing Process Models Other Arena Programs and Examples

Next: Arena Applications