This document discusses methods of simulation and Monte Carlo simulation. It is authored by a group including Roy Thomas, Sam Scaria, Sonu Sebastian, and others. The document defines simulation as using a model of a real system to conduct experiments on a computer in order to describe, explain, and predict the behavior of the real system. Monte Carlo simulation is described as using probability and sampling to solve complicated equations. Key steps of Monte Carlo simulation include drawing a flow diagram, determining variable distributions, selecting random numbers, and applying mathematical functions to obtain solutions. Examples of applications include queuing problems, inventory problems, and risk analysis.

1. Methods of SimulationMethods of Simulation

2. GROUP MEMBERSGROUP MEMBERS
 ROY THOMASROY THOMAS
 SAM SCARIASAM SCARIA
 SONU SEBASTIANSONU SEBASTIAN
 SILPA MATHEWSILPA MATHEW
 AMMU VIJAYANAMMU VIJAYAN
 SIJU JOSESIJU JOSE
 SAJITH P SSAJITH P S
 SCARIA JOSEPHSCARIA JOSEPH

3. What is simulation:What is simulation:
The process of designing aThe process of designing a
mathematical or logical model of amathematical or logical model of a
real-system and then conductingreal-system and then conducting
computer-based experiments withcomputer-based experiments with
the model to describe, explain, andthe model to describe, explain, and
predict the behavior of the realpredict the behavior of the real
system.system.

4. Monte carlo method &Monte carlo method &
system simulationsystem simulation
methodmethod

5. What is a Monte Carlo simulation?What is a Monte Carlo simulation?
• Monte carlo method is a substitution forMonte carlo method is a substitution for
the mathematical evaluation of a model.the mathematical evaluation of a model.
• Darker and Kac define monte carloDarker and Kac define monte carlo
method as combination of probabilitymethod as combination of probability
methods & sampling techniques providingmethods & sampling techniques providing
solution to complicated partial or integralsolution to complicated partial or integral
differential equation.differential equation.
• In short, monte carlo technique isIn short, monte carlo technique is
concerned with experiments on randomconcerned with experiments on random
numbers & it provides solutions tonumbers & it provides solutions to
complicated OR problems.complicated OR problems.

6. Uses of monte carlo techniqueUses of monte carlo technique
 Where one is dealing with a problemWhere one is dealing with a problem
which has not yet arisen.which has not yet arisen.
 Where the mathematical andWhere the mathematical and
statistical problems are toostatistical problems are too
complicated and some alternativecomplicated and some alternative
methods are needed.methods are needed.
 To estimate parameters to a model.To estimate parameters to a model.

7. Steps of Monte Carlo methodSteps of Monte Carlo method
 A Flow diagram is drawn.A Flow diagram is drawn.
 Probability distribution for theProbability distribution for the
variables of our interest isvariables of our interest is
determined.determined.
 Probability distribution is convertedProbability distribution is converted
to cumulative distribution function.to cumulative distribution function.

8.  Sequence of random numbers isSequence of random numbers is
selected .selected .
 Sequence of values of the variablesSequence of values of the variables
of our interest is determined with theof our interest is determined with the
sequence of random numberssequence of random numbers
obtained.obtained.
 Some standard mathematicalSome standard mathematical
functions is applied to the sequencefunctions is applied to the sequence
of values obtainedof values obtained

9. AdvantageAdvantage
 Find solution of complicatedFind solution of complicated
mathematical expressions.mathematical expressions.
 Difficulties of trial and errorDifficulties of trial and error
experimentation are avoided byexperimentation are avoided by
these method.these method.

10. DisadvantagesDisadvantages
 These are costly way of getting aThese are costly way of getting a
solution of any problem.solution of any problem.
 These method do not provide optimalThese method do not provide optimal
answer to the problems. Theanswer to the problems. The
answers are good only when the sizeanswers are good only when the size
of the sample is sufficiently large.of the sample is sufficiently large.

11. ApplicationsApplications
 It is applied to a wide diversity ofIt is applied to a wide diversity of
problems such as queuing problems,problems such as queuing problems,
inventory problems, risk analysisinventory problems, risk analysis
concerning a major capitalconcerning a major capital
investment.investment.
 It is very useful in budgeting.It is very useful in budgeting.

12. System Simulation MethodSystem Simulation Method
 Under this method operatingUnder this method operating
environment is produced andenvironment is produced and
systems allows for analysing thesystems allows for analysing the
response from the environment toresponse from the environment to
alternative management actions.alternative management actions.
 The method is complicated andThe method is complicated and
costly.costly.

13. Generation of random numbersGeneration of random numbers
 Random numbersRandom numbers
It is a number in a sequence ofIt is a number in a sequence of
numbers whose probability ofnumbers whose probability of
occurrence is same as that of anyoccurrence is same as that of any
other number in that sequence.other number in that sequence.

14.  Pseudo-random Numbers:Pseudo-random Numbers:
Random numbers are called pseudoRandom numbers are called pseudo
random numbers when they arerandom numbers when they are
generated by some deterministicgenerated by some deterministic
process. But they qualify the preprocess. But they qualify the pre
determined statistical test fordetermined statistical test for
randomness.randomness.

15. Generating of random numbers:Generating of random numbers:
 For solving simulation problems,For solving simulation problems,
there is the need of generating athere is the need of generating a
sequence of random numbers.sequence of random numbers.
 Random numbers may be found byRandom numbers may be found by
computer ,by random tables,computer ,by random tables,
manually etc.manually etc.

16.  Most common method to obtainMost common method to obtain
random numbers is to generate themrandom numbers is to generate them
by a computer programme.by a computer programme.
 These numbers lie between 0 andThese numbers lie between 0 and
1,in conjunction with the cumulative1,in conjunction with the cumulative
probability distribution of a randomprobability distribution of a random
variable including 0 but not 1.variable including 0 but not 1.

17. Waiting Line simulation modelWaiting Line simulation model
 In this type problems the simulationIn this type problems the simulation
technique can be applied to solvetechnique can be applied to solve
problems of complex nature.problems of complex nature.
 The uncertain characteristics of thisThe uncertain characteristics of this
model are the arrival behaviour ofmodel are the arrival behaviour of
the customer in the system and thethe customer in the system and the
service time distribution.service time distribution.

18. Inventory Simulation ModelInventory Simulation Model