Simulation UNIT-I

SIMULATION

08/09/2016 1Dr. DEGA NAGARAJU, SMEC

08/09/2016 Dr. DEGA NAGARAJU, SMEC 2

08/09/2016 Dr. DEGA NAGARAJU, SMEC 3

War gaming: test strategies; training

Flight SimulatorTransportation systems: Improved operations; urban planning

Computer communicationnetwork: protocol design

Parallel computer systems: developing scalable software

Games

A few more applications …

12/22/2016 3Dr. DEGA NAGARAJU, SMBS

Areas of

Applications

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Business Process

Simulation


Applications:

COMPUTER SYSTEMS: hardware components, software

systems, networks, data base management, information

processing, etc..

MANUFACTURING: material handling systems, assembly

lines, automated production facilities, inventory control

systems, plant layout, etc..

BUSINESS: stock and commodity analysis, pricing policies,

marketing strategies, cash flow analysis, forecasting, etc..

GOVERNMENT: military weapons and their use, military

tactics, population forecasting, land use, health care

delivery, fire protection, criminal justice, traffic control, etc..

And the list goes on and on...08/09/2016 5Dr. DEGA NAGARAJU, SMEC

Examples of Applications at Disney World

Cruise Line Operation: Simulate the arrival and

check-in process at the dock.

Private Island Arrival: How to transport passengers

to the beach area? Drop-off point far from the

beach. Used simulation to determine whether

to invest in trams, how many trams to purchase,

average transport and waiting times, etc..


Why do we go for

simulation?

Is it possible to represent all real

life problems mathematically?

Method of last resort

Logical extension to

the analytical &

mathematical techniques

John Von Neumann & Stanislaw Ulam

Nuclear Shielding problem

1950-Digital Computers

Managerial decision makingAircraft-wind tunnel-aerodynamic characteristicsScale models of machines-plant layoutPilot training-flight simulatorCar Manufacturing SimulationTV games(chess playing game, snake and ladders)08/09/2016 7Dr. DEGA NAGARAJU, SMEC

Car Manufacturing Simulation.flv

Why do we go for simulati-

on?

Draw backs of scientific methods?

Draw backs of Analytical methods?

Draw backs of iterative

methods?

Certain processes : too costly or impossible

Difficult-mathematical equations

No straight forward analytical solution

Ex: Queuing problems, Job shop problems,

Multi-integral problems etc.

Difficulty in performing validating

experiments for mathematical models

Dynamic programming, queuing theory,

network models

Dynamic programming-optimal strategies-

uncertainties-analyze multi-planning

problems

DP-simple cases-less number of static

variables

LPP-data does not change over the entire

planning horizon

One time decision process-average values for

decision variables

Many real life situations-uncertainties


Webster’s Dictionary:

“ to assume the mere appearance of ,

without the reality”


Definition:

Simulation is the process of designing a

model of a real system and conducting

experiments with this model for the purpose

of either understanding the behavior of the

system and/or evaluating various strategies

for the operation of the system.


Allows us to:

Model complex systems in a detailed way

Describe the behavior of systems

Construct theories or hypotheses that account for the observed behavior

Use the model to predict future behavior, that is, the effects that will be produced by changes in the system

Analyze proposed systems


Brief HistoryNot a very old technique...

World War II

“Monte Carlo” simulation: originated with

the work on the atomic bomb. Used to

simulate bombing raids. Given the

security code name “Monte-Carlo”.

Still widely used today for certain problems

which are not analytically solvable (for

example: complex multiple integrals…)08/09/2016 12Dr. DEGA NAGARAJU, SMEC

Brief History (cont…..)

Late ‘50s, early ‘60s Computers improve

First languages introduced: SIMSCRIPT,

GPSS (General purpose simulation system) (IBM)

Simulation viewed at the tool of “last resort”

Late ‘60s, early ‘70s Primary computers were mainframes: accessibility

and interaction was limited

GASP IV introduced by Pritsker. Triggered a wave

of diverse applications. Significant in the evolution

of simulation.


Brief History (cont…….)

Late ‘70s, early ‘80s SLAM introduced in 1979 by Pritsker and Pegden.

Models more credible because of sophisticated tools.

SIMAN introduced in 1982 by Pegden. First language

to run on both a mainframe as well as a

microcomputer.

Late ‘80s through present Powerful PCs

Languages are very sophisticated (market almost

saturated)

Major advancement: graphics. Models can now be animated.


What can be simulated?

Almost anything can

and

almost everything has...


Introduction to Simulation

Simulation a) the imitation of the operation of a real-world process or system over time.

b) to develop a set of assumptions of mathematical, logical, and symbolic relationship between the entities of interest, of the system.

c) to estimate the measures of performance of the system with the simulation-generated data.

Simulation modeling can be useda) as an analysis tool for predicting the effect of changes to existing systems.

b) as a design tool to predict the performance of new systems .

Real-world process concerning the behavior of a system

A set of assumptionsModeling &

Analysis


Simula-

tion

Imitation of the operation of a real

world process

Whether done by hand or on a ComputerIt involves

1. The generation of an artificial history

of a system & 2. The observation of

that artificial history

Simulation model

Behavior of a system

Set of Assumptions:Mathematical, Logical,Symbolic relationships b/w entities, Objects of interest

Used as:an analysis tool,

a design tool08/09/2016 17Dr. DEGA NAGARAJU, SMEC

Simulation Models

Solved by:Differential Calculus,Probability Theory,Algebraic Methods

Solution Consists:One or more numerical parameters(Measures of

Performance)

Complex real world systems:

Numerical computer based simulation


Problem formulation -1

Policy maker/Analyst understand and agree with the formulation.

Setting of objectives and overall project plan -2

Model conceptualization -3

The art of modeling is enhanced by an ability to abstract the

essential features of a problem, to select and modify basic

assumptions that characterize the system, and then to enrich and

elaborate the model until a useful approximation results.

Data collection -4

As the complexity of the model changes, the required data

elements may also change.

Model translation -5

GPSS/HTM or special-purpose simulation software

Steps in a Simulation Study


Verified? -6

Is the computer program performing properly?

Debugging for correct input parameters and logical structure

Validated? -7

The determination that a model is an accurate representation of

the real system.

Validation is achieved through the calibration of the model

Experimental design -8

The decision on the length of the initialization period, the length

of simulation runs, and the number of replications to be made of

each run.

Production runs and analysis -9

To estimate measures of performances

Steps in a Simulation Study (Contd….)


More runs? -10

Documentation and reporting -11

Program documentation : for the relationships between input

parameters and output measures of performance, and for a

modification

Progress documentation : the history of a simulation, a

chronology of work done and decision made.

Implementation -12



Four phases according to Figure 1.3

First phase : a period of discovery or orientation

(step 1, step2)

Second phase : a model building and data collection

(step 3, step 4, step 5, step 6, step 7)

Third phase : running the model

(step 8, step 9, step 10)

Fourth phase : an implementation

(step 11, step 12)



The basic nature of Simulation

Two problems

Continuous

Discrete

State changes continuously with timeDeterministic in nature

Arrival & Sale of Merchandise occur in discrete stepsStochastic in nature

Common features essential to simulation

Common features:Mathematical model of the system under studyChange of the state in accordance with some equations (rules or laws) for a

long periodCollection of information about the system (solution to the problem)Programming the calculations for a digital computerSimulate or mimic the real system with the help of computerContinue the process until the desired analytic solution is obtained


Simulation as an analytic tool is useful only when done on a computer

The basic nature of Simulation (Cont….)


Simulation of Inventory control manually Pencil and paper

System which can be simulated on a digital computer

Can also be simulated manually

Each application of Simulation is adhoc to a great extent

Simulation is an art

The basic nature of Simulation (Cont….)


No unifying theory of computer simulation

No unified theory No fundamental theorems

No underlying principles

Experimental technique

To simulate is to experiment Fast and inexpensive method

Ex: Inventory control problem

When Simulation is the Appropriate Tool (1)

Simulation enables the study of, and experimentation with, the internal interactions

of a complex system, or of a subsystem within a complex system.

Informational, organizational, and environmental changes can be simulated, and the

effect of these alterations on the model’s behavior can be observed.

The knowledge gained in designing a simulation model may be of great value

toward suggesting improvement in the system under investigation.

By changing simulation inputs and observing the resulting outputs, valuable insight

may be obtained into which variables are most important and how variables interact.

Simulation can be used as a pedagogical device to reinforce analytic solution

methodologies.


Simulation can be used to experiment with new designs or policies prior to implementation, so as to prepare for what may happen.

Simulation can be used to verify analytic solutions.

By simulating different capabilities for a machine, requirements can be determined.

Simulation models designed for training allow learning without the cost and disruption of on-the-job learning.

Animation shows a system in simulated operation so that the plan can be visualized.

The modern system (factory, wafer fabrication plant, service organization, etc.) is so complex that the interactions can be treated only through simulation.

When Simulation is the Appropriate Tool (2)


When Simulation is the appropriate tool?

Study of the internal interactions of a complex System

Effect of Informational,

organizational and environmental

changes on model behavior

Used as a pedagogical device to reinforce analytic

solution methodologies

Experimentation with new design

To verify analytic solutions

Simulation models:For training without

the cost and disruption of on the

job learning

To study the interactions of a complex modern systems: factory, fabrication plant,

service organization


When Simulation is not Appropriate

When the problem can be solved using common sense.

When the problem can be solved analytically.

When it is easier to perform direct experiments.

When the simulation costs exceed the savings.

When the resources or time are not available.

When system behavior is too complex or can’t be defined.

When there isn’t the ability to verify and validate the model.


When

Simulation is

not

Appropriate?

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can

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are

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When there is n’t the ability to verify and validate the model

When the simulation costs exceed the savings


Advantages and Disadvantages of Simulation (1)

Advantages New polices, operating procedures, decision rules, information flows, organizational

procedures, and so on can be explored without disrupting ongoing operations of the real system.

New hardware designs, physical layouts, transportation systems, and so on, can be tested without committing resources for their acquisition.

Hypotheses about how or why certain phenomena occur can be tested for feasibility.

Insight can be obtained about the interaction of variables.

Insight can be obtained about the importance of variables to the performance of the system.

Bottleneck analysis can be performed indicating where work-in-process, information, materials, and so on are being excessively delayed.

A simulation study can help in understanding how the system operates rather than how individuals think the system operates.

“What-if” questions can be answered. This is particularly useful in the design of new system.


Advantages and Disadvantages of Simulation (2)

Disadvantages

Model building requires special training. It is an art that is learned over time and through experience. Furthermore, if two models are constructed by two competent individuals, they may have similarities, but it is highly unlikely that they will be the same.

Simulation results may be difficult to interpret. Since most simulation outputs are essentially random variables (they are usually based on random inputs), it may be hard to determine whether an observation is a result of system interrelationships or randomness.

Simulation modeling and analysis can be time consuming and expensive. Skimping on resources for modeling and analysis may result in a simulation model or analysis that is not sufficient for the task.

Simulation is used in some cases when an analytical solution is possible, or even preferable, as discussed in Section 1.2. This might be particularly true in the simulation of some waiting lines where closed-form queueing models are available.


1

• New polices, operating procedures, decision rules, information flows, organizational procedures, and so on can be explored without disrupting ongoing operations of the real system

2

• New hardware designs, physical layouts, transportation systems, and so on, can be tested without committing resources for their acquisition.

3• Hypotheses about how or why certain phenomena occur can be tested for feasibility.

4• Insight can be obtained about the interaction of variables.

5• Insight can be obtained about the importance of variables to the performance of the system.

6

• Bottleneck analysis can be performed indicating where work-in-process, information, materials, and so on are being excessively delayed

7

• A simulation study can help in understanding how the system operates rather than how individuals think the system operates.

8• “What-if” questions can be answered. This is particularly useful in the design of new system.

Advantages of Simulation


1

• Model building requires special training. It is an art that is learned over time and through experience. Furthermore, if two models are constructed by two competent individuals, they may have similarities, but it is highly unlikely that they will be the same.

2

• Simulation results may be difficult to interpret. Since most simulation outputs are essentially random variables (they are usually based on random inputs), it may be hard to determine whether an observation is a result of system interrelationships or randomness.

3

• Simulation modeling and analysis can be time consuming and expensive. Skimping on resources for modeling and analysis may result in a simulation model or analysis that is not sufficient for the task.

4

• Simulation is used in some cases when an analytical solution is possible, or even preferable. This might be particularly true in the simulation of some waiting lines where closed-form queueingmodels are available.

Disadvantages of Simulation


Areas of

Applications

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Business Process

Simulation


Why do we

study the

System?

To understand the relationships b/w its

components or to predict how the system will

operate under a new policy

Is it possible to

conduct experiment

with the system?

Yes, but not always

New system may not yet exist. It may

be in hypothetical form or at the

design stage.

Example: Developing & testing of

prototype models can be very

expensive and time consuming

Even System exists: No

experimentation.

Example:

Is it possible to double the

unemployment rate to determine the

effect of employment on inflation?

Is it possible to reduce the number of

tellers at the bank to study the effect

on the length of waiting lines?

Is it feasible to change the supply and

demand of goods arbitrarily to study

the economic systems?

How do we

define the

system

model?

Body of information gathered about

the system to study the system.

No unique model of the system

For same system-different models-by

different analysts

Establish the model

Structure: System boundary,

entities, attributes and

activities of the system.

Provide the data: Values of

the attributes, relationships

among the activities.

MODEL OF A SYSTEM

How the model

is derived for

the system?


Example for the Model of a SystemENTITY ATTRIBUTE ACTIVITY

SHOPPER NO OF ITEMSARRIVE

GET

BASKET AVAILABILITYSHOP

QUEUECHECK-OUT

COUNTER NUMBER OF OCCUPANCY

RETURNLEAVE

Elements of a Super Market


Types of models

Physical

Static Dynamic

Mathematical

Static

Numerical Analytical

Dynamic

Analytical Numerical

System Simulation

TYPES OF MODELS08/09/2016 39Dr. DEGA NAGARAJU, SMEC

Physical Models: Based on some analogy b/w such systems as mechanical & electrical, electrical & hydraulicSystem attributes represented by such measurements as a voltage or the position of a shaftSystem activities are reflected in the physical laws that drive the model:Example: amount of voltage applied – speed of the shaft of the motorVoltage applied – Velocity of the vehicleNumber of revolutions of the shaft – distance traveled by the vehicle

Mathematical

Models:

Used symbolic notations, mathematical equations

etc.

System attributes are represented by variables

System activities are represented by mathematical

functions


Dynamic Models:

Static Models: Also called as Monte Carlo SimulationRepresents the system at a particular point in time

Represents systems as they change over timeExample: Simulation of a bank from 9.00 am to 4.00 pm

Numerical Models:

Analytical Models: Only certain forms of equations are solvedExample: Linear differential equations are solved

Computational procedure is used to solve the models


Monte Carlo Simulation

Statistical distribution functions are created by using a

series of random numbers.

Data can be developed for many months or years in a

matter of few minutes on a digital computer.

Used to solve the problems which can’t be adequately

represented by mathematical models or where

the solution of the model is not possible by

analytical method.

The solution obtained is very close to the optimal but

not exact.


Steps in Monte

Carlo Simulation

Objectives,

factors affecting

the objectives

Variables, parameters, decision rules,

conditions to carry experimentation, type of

distribution used, the manner in which time

is changed, relationship b/w variables and

parameters

Starting conditions

for the simulation,

number of

simulation runs

Define a coding system that will

correlate the factors defined in step

1 with random numbers to be

generated;

Select the random number generator and create

the random numbers to be used;

Associate the generated random numbers with

the factors identified in step 1 and coded in step

4(i)

Select the best

course of action


Systems and System Environment

System

defined as a group of objects that are joined together in some regular interaction or interdependence toward the accomplishment of some purpose.

System Environment

changes occurring outside the system.

The decision on the boundary between the system and its environment may depend on the purpose of the study.


Components of a System Entity : an object of interest in the system.

Attribute : a property of an entity.

Activity : a time period of specified length.

State : the collection of variables necessary to describe the

system at any time, relative to the objectives of the

study.

Event : an instantaneous occurrence that may change the

state of the system.

Endogenous : to describe activities and events occurring

within a system.

Exogenous : to describe activities and events in an

environment that affect the system.


Components of a System


Discrete and Continuous Systems

Systems can be categorized as discrete or continuous.

Bank : a discrete system

The head of water behind a dam : a continuous system


DISCRETE SYSTEMS

State variables change only at a discrete set of points in

time

Example: Bank

State variable: Number of customers in the

bank

Note: The state variable changes only when a new

customer arrives or when the service provided to the

customer is completed08/09/2016 48Dr. DEGA NAGARAJU, SMEC

CONTINUOUS SYSTEMS

State variables change continuously over time

Example: Head of water behind a dam

State variable: Head of water behind a dam

Note: During and for some time after a rain storm,

water flows into the lake behind the dam. Water is

drawn from the dam for flood control and to make

electricity. Evaporation also decreases the water level.


A grocery store has one checkout counter. Customers arrive at this checkout counter at

random from 1 to 8 minutes apart and each interval time has the same probability of

occurrence. The service times vary from 1 to 6 minutes, with probability given below:

Simulate the arrival of 6 customers and calculate (i) Average waiting time for a customer,

(ii) Probability that a customer has to wait, (iii) Probability of a server being idle (iv)

Average service time, (v) Average time between arrival. Use the following sequence of

random numbers:

Assume the first customer arrives at time θ. Depict the simulation in a tabular form.

Service (minutes) 1 2 3 4 5 6

Probability 0.10 0.20 0.30 0.25 0.10 0.05

Random digit for

arrival

913 727 015 948 309 922

Random digit for

service time

84 10 74 53 17 79


Time between

arrivals

Probability Cumulative

Probability

Random digit

assignment

1 0.125 0.125 001-125

2 0.125 0.250 126-250

3 0.125 0.375 251-375

4 0.125 0.500 376-500

5 0.125 0.625 501-625

6 0.125 0.750 626-750

7 0.125 0.875 751-875

8 0.125 1.000 876-000

ARRIVAL TIME DISTRIBUTION

Arrival time varies from 1 to 8 minutes with equal probability, meaning that

the probability of each arrival = 1/8 = 0.125


Service Time ProbabilityCumulative

Probability

Random digit

assignment

1 0.10 0.10 01-10

2 0.20 0.30 11-30

3 0.30 0.60 31-60

4 0.25 0.85 61-85

5 0.10 0.95 86-95

6 0.05 1.00 96-00

SERVICE TIME DISTRIBUTION


Custom

er

Random

No. for

Arrival

Time

since

last

arrival

Arrival

time

Random

No. for

Service

Service

time

Time

service

begins

Time

custome

-r waits

in queue

Time

service

ends

Time

custome

r spends

in

system

Idle

time of

server

1 - - 0 84 4 0 0 4 4 0

2 913 8 8 10 1 8 0 9 1 4

3 727 6 14 74 4 14 0 18 4 5

4 015 1 15 53 3 18 3 21 6 0

5 948 8 23 17 2 23 0 25 2 2

6 309 3 26 79 4 26 0 30 4 1

18 3 21 12

SIMULATION TABLE


Total time customer waits in queue 3Average waiting0.5

time for customer Total no. of customers 6

No. of customers who wait 1Pr obability that a0.166

customer has to wait Total no. of customers 6

Total idle time of server 12Pr obability of server0.4

being idle Total run time of system 30

Total service time 18Average service3

time Total no. of customer 6

Sum of all times between arrivals(min utes) 26Average time betweenarrivals No. of arrivals 1 6 1


Total timecustomers wait in queue(min utes)Average waiting time of thosewho wait(min utes) Totalnumber of customers who wait

33min utes

1

Total timecustomer spends

in the system(min utes)Average time customer spendsin the system Total number of customers

213.5min utes

6


Demand

(daily)0 1 2 3 4

Probability 0.05 0.10 0.30 0.45 0.10

A book store wishes to carry ‘Ramayana’ in stock. Demand is probabilistic and

replenishment of stock takes 2 days (i.e., if an order is placed on March 1, it will be

delivered at the end of the day on March 3). The probabilities of demand are given

below:

Each time an order is placed, the store incurs an ordering cost of Rs. 10 per order.

The store also incurs a carrying cost of Rs. 0.50 per book per day. The inventory

carrying cost is calculated on the basis of stock at the time of each day. The manager

of the book store wishes to compare two options for his inventory decision.

A: Order 5 books when the inventory at the beginning of the day plus order

outstanding is less than 8 books.

B: Order 8 books when the inventory at the beginning of the day plus order

outstanding is less than 8.

Currently (beginning of the first day) the store has stock of 8 books plus 6 books

ordered 2 days ago and expected to arrive next day. Using Monte-Carlo Simulation

for 10 cycles, recommend which option the manager should choose. The two digit

random numbers are given below. 89, 34, 78, 63, 61, 81, 39, 16, 13, 73.


Demand Prob. Cum. Prob.Random

Nos.

0 0.05 0.05 01-05

1 0.10 0.15 06-15

2 0.30 0.45 16-45

3 0.45 0.90 46-90

4 0.10 1.00 91-00

Stock in hand = 8, and

stock on order = 6 (expected next day).


Demand Distribution

Random

No.

Demand

sales

Opt. stock

in handReceipt

Cl. stock

in hand

Opt. stock

on order

Order

Qty.

Cl. Stock

on order

89 3 8 - 5 6 - 6

34 2 5 6 9 - - -

78 3 9 - 6 - 5 5

63 3 6 - 3 5 - 5

61 3 3 - 0 5 5 10

81 3 0 5 2 5 5 10

39 2 2 - 0 10 - 10

16 2 0 5 3 5 - 5

13 1 3 5 7 0 5 5

73 3 7 - 4 5 - 5

No. of orders =4 Ordering cost = 4 x 10 = Rs. 40.

Closing stock of 10 days = 39, Carrying cost = 39 x 0.50 = 19.50

Cost for 10 days = 59.50

OPTION A


OPTION B

Random

No.

Demand

sales

Opt. stock

in handReceipt

Cl. stock

in hand

Opt. stock

on order

Order

Qty.

Cl. Stock

on order

89 3 8 - 5 6 - 6

34 2 5 6 9 - - -

78 3 9 - 6 - 8 8

63 3 6 - 3 8 - 8

61 3 3 - 0 8 - 8

81 3 0 8 5 - 8 8

39 2 5 - 3 8 - 8

16 2 3 - 1 8 - 8

13 1 1 8 8 - - -

73 3 8 - 5 - 8 8

No. of orders =3 Ordering cost = 3 x 10 = Rs. 30.

Closing stock of 10 days = 45, Carrying cost = 45 x 0.50 = 22.50

Cost for 10 days = 52.50

Since, option B has lower cost, manager should choose option B


Discrete-event simulation (General Principles)

The basic building blocks of all discrete-event simulation models

: entities and attributes, activities and events.

A system is modeled in terms of

its state at each point in time

the entities that pass through the system and the entities that represent system resources

the activities and events that cause system state to change.

Discrete-event models are appropriate for those systems for which changes in system state occur only at discrete points in time.

This chapter deals exclusively with dynamic, stochastic systems (i.e., involving time and containing random elements) which change in a discrete manner. 08/09/2016 60Dr. DEGA NAGARAJU, SMEC

System : A collection of entities (e.g., people and machines) that interact

together over time to accomplish one or more goals.

Model : An abstract representation of a system, usually containing

structural, logical, or mathematical relationships which describe a

system in terms of state, entities and their attributes, sets, processes,

events, activities, and delays.

System state : A collection of variables that contain all the information

necessary to describe the system at any time.

Entity : Any object or component in the system which requires explicit

representation in the model (e.g., a server, a customer, a machine).

Attributes : The properties of a given entity (e.g., the priority of a waiting

customer, the routing of a job through a job shop).


List : A collection of (permanently or temporarily) associated entities, ordered

in some logical fashion (such as all customers currently in a waiting line,

ordered by first come, first served, or by priority).

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

(such as an arrival of a new customer).

Event notice : A record of an event to occur at the current or some future

time, along with any associated data necessary to execute the

event; at a minimum, the record includes the event type and

the event time.

Event list : A list of event notices for future events, ordered by time of

occurrence also known as the future event list (FEL).

Activity : A duration of time of specified length (e.g., a service time or

inter arrival time), which is known when it begins (although it may be

defined in terms of a statistical distribution).


Delay : A duration of time of unspecified indefinite length, which is not

known until it ends (e.g., a customer's delay in a last-in, first-out

waiting line which, when it begins, depends on future arrivals).

Clock : A variable representing simulated time, called CLOCK in the

examples to follow.

An activity typically represents a service time, an inter arrival time, or any other

processing time whose duration has been characterized and defined by the modeler.

An activity's duration may be specified in a number of ways:

1. Deterministic-for example, always exactly 5 minutes;

2. Statistical-for example, as a random draw from among 2, 5, 7 with equal

probabilities;

3. A function depending on system variables and/or entity attributes-for example,

loading time for an iron ore ship as a function of the ship's allowed cargo

weight and the loading rate in tons per hour.


Simulation UNIT-I

Engineering

Transcript of Simulation UNIT-I