ME or Vxii Simulation

7/30/2019 ME or Vxii Simulation

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Parul Institute of Engineering & Technology

Subject Code : 171901

Name Of Subject :Operation Research

Name of Unit : Simulation

Topic : Monte Carlo Simulation& Generation of random

Numbers

Name of Faculty : R.N Barot

Name of Students: (i) Shah Devanshu[49]

(ii) Mehta Hardik[50]


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Simulation:A type of model where the computer is used to

imitate the behavior of the system

Monte Carlo simulation: Simulation that makes use of

internally generated (pseudo) random numbers.It is a technique which involves conducting experiments on

the model of the system under study, with some probability

distribution to draw random samples using random numbers.

Monte Carlo Simulation:


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3

Ways to Study System

Focus of class

System

Experiment w/

actualsystemExperiment w/

modelof

system

Physical

ModelMathematical

Model

Analytical

Model

Simulation

Model


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In a Monte Carlo simulation we attempt to follow the `time

dependence of a model for which change, or growth, does

not proceed in some rigorously predefined fashion (e.g.

according to Newtons equations of motion) but rather in a

stochastic manner which depends on a sequence of randomnumbers which is generated during the simulation.

MC method is often referred to as the method of last

resort, as it is apt to consume large computing resources;


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1) Calculate Probability distribution for variables to beanalysed .

2) Calculate a cumulative probability distribution for each

random variable.

3) Generate random numbers.4) Assign appropriate set of random numbers

5) Conduct simulation experiment using random numbers.

6) Repeat the steps until satisfactory runs are not completed.

7) Design and implement the course of outcome ofsimulation and maintain it.

Steps of Monte Carlo simulation:


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consuming vast computing resourceshave historically had to be executed upon the fastestcomputers available at the timeand employ the most advanced algorithms implemented with substantial programming acumen.

The name "Monte Carlo" comes from the city of MonteCarlo in the principality of Monaco, famous for itsgambling house

Characteristics:


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Monte Carlo methods can help in design and prediction of

behavior of systems in nuclear applications and radiationphysics

The use of MC in the area of nuclear power has undergone

an important evolution. Notable are the extensions to

compute burnup in reactor cores, and full core neutronic

simulations. Random numbers generated by the computer

are used to simulate naturally random processes

many previously intractable thermodynamic and quantummechanics problems have been solved using Monte Carlo

techniques.

Applications:


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Random number generators produce a flow of numbers that

Are observations from a continuous uniform distribution

between 0 and 1

Are independent of each other

Remember the discussion on pseudorandom numbers

Need of RNs in simulation

Generation of random variates to recapture prob. distribution

of stochastic process parameters

Random number generation


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The Midsquare method (von Neumann and Metropolis1940s)

EX: Start with an arbitrary 3-digit #

Z0 = 123 Z02 = (123)2 = 15129

Z1 = 512 U1 = 0.512 Z12 = (512)2 = 262144Z2 = 621 U2 = 0.621 Z22 = (621)2 = 385641

EX: A 4-digit example (see L&K Table 7.1)

ProblemsZero problem 002 004 zero will stay

Cycle length

The one trap 001 001 001

Methods of number generation


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Random-Number Generators

The Midproduct Method

Variation of Midsquare method and improved

EX:Start with two arbitrary 3-digit #s

Z0 = 123 Z1 = 456

Z0Z1 = (123)(456) = 56088 U1 = 0.608 Z2 = 608

Z1Z2 = (456)(608) = 277248 U2 = 0.772 Z3 = 772

Z2Z3 = (608)(772) = 469376 U3 = 0.693 Z4 = 693


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Linear Congruential Generators (LCG)

Problems and Concerns Zis are not really random at all, i.e., Zi can be

determined by m, a, c, and Z0

However, by careful choice of these four parameters

we try to make the corresponding Uis appear to beIID U(0,1) random variates subjected to some tests

Uis are 0, 1/m, 2/m, 3/m, (m-1)/m, impossible toget fractions, say 0.3/m or 1.5/m

i.e., m needs be very large to have desired density

Ex: m109, there are at least a billion possiblevalues, a good approximation to the truecontinuous U(0,1) distribution


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Finding the ith RN Directly from Z0

Also -

Zi can be obtained directly from Z0 withoutgoing through Z1, Z2, , Zi-1,

For i = 1, 2, ...Zi = [a

iZ0 + c(ai - 1)/(a- 1)] (mod m)

This shows Zis are not really random at all.

===> Predictablity

However, Zis are predictable does not meanthey are correlated.


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Linear Congruential Generators (LCG)

Problems and Concerns

Looping (or Repeating cycle):

LCG: m = 16, a = 5, c = 3, Z0= 7 for i = 1, 2, , 19

Zi = (5 Zi-1 + 3) (mod 16) Z0 = 7

Z1 = Z17 = 6, Z2 = Z18 = 1, Z3 = Z19 = 8

(see L&K Table 7.2)

Length of cycle mIf length of cycle = m, then LCG has a full period

If m and loop m, then LCG is adequate


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ME or Vxii Simulation