Lecture _10 -- Part 2

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Modeling & Simulation

Lecture 10

Part 2

Generating Random

Variates

Instructor:

Eng. Ghada Al-MashaqbehThe Hashemite UniversityComputer Engineering Department

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The Hashemite University 2

Outline

Introduction.

Generating continuous randomvariates.

Generating empirical randomvariates.

Generating discrete random variates. Generating arrivals processes.

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Introduction

We have explored the general techniquesto generate random variates.

In this lecture we will explore the commonalgorithms used with the theoretical

distributions (both continuous anddiscrete) that we have dealt with before.

So, just pick the distribution and itsalgorithm and work with it.

In addition, we will deal with empiricalrandom variates generation and arrivalprocesses.

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Continuous Distributions

Uniform

Use the inverse transform algorithm.

Exponential Use the inverse transform algorithm.

Weibull Use the inverse transform algorithm.

UabaX )(

UX ln

/1)]1ln([ UX

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Continuous Distributions Gamma I

Pdf and cdf function

Where is the shape parameter and is thescale parameter.

otherwise0

0!

/1

)(

1

0

/x

j

xe

xFj

j

x

otherwise0

0)()(

/1

xex

xf

x

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Continuous Distributions Gamma II

No closed-form inverse cannot use theinverse transform method.

if

Gamma(1,1) is exponential with mean 1. For this purpose we will consider Gamma

distribution with = 1 and

),(~ gammaX)1,(~ gammaX

1,10

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Gamma(,1) Density

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Continuous Distributions Gamma(0

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Continuous Distributions Gamma(>1) I

Acceptance-rejection with the followingt(x) and use inverse transform togenerate Y from r(x):

)(

4

12

2

1

ec

x

xxt

otherwise

xx

xxR

,0

0,)(

10,1

)(

/1

1

u

u

uXuR

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Continuous Distributions Normal

Distribution function does not have closed form (soneither does the inverse) Can use numerical methods for inverse-transform Note that

If we can generate unit normal (or called thestandard normal distribution), then we can generateany normal.

Many algorithms were proposed to generate randomvariates from the normal distribution which differ intheir speed and accuracy.

We will discuss two algorithms: Box and Muller algorithm. Polar algorithm.

),(~

)1,0(~

NX

NX

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Continuous Distributions Normal:Box-Muller Method I

Algorithm

call.functionnextfor thesaveandReturn.3

2sinln2

,2cosln2Set2.

)1,0(~,tindependenGenerate.1

21

212

211

21

XX

UUX

UUX

UUU

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Continuous Distributions Normal:Box-Muller Method II

Technically, independent N(0,1), butserious problem if used with LCGs:

U1 and U2 if from the same stream could be

adjacent resulting in having X1 and X2 to bedependent.

Solution:

Generate U1 and U2 from different streams

(i.e. start with different seed values).

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Continuous Distributions Normal:Polar Method

An improved version of the Box and Muller algorithm Faster than Box and Muller algorithm.

Also generates random variates in pairs.

Algorithm

YVX

YVX

WWY

VVWUVUV

UUU

22

11

22

212211

21

/)ln2(

letOtherwise,1.steptogo1WIf2.

and,12and12Let

).1,0(~,tindependenGenerate.1

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Continuous Empirical Distributions

You have two cases: You have an equation for the empirical distribution. Or you use the actual observation.

For the first case you can use the inverse transformmethod.

For the second case you can use the following

algorithm exploits such fact which is an equivalentto the inverse transform method.

Algorithm (first you must rank the data):

n is the sample size (i.e. total number of empirical

Xs).

)()1()( )1(Return.2

1letand

,1Let).1,0(~Generate.1

III XXIPXX

PI

)U(n-PUU

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Empirical Distribution Function

For example, for this distribution n = 6

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Discrete Distributions

Can always use the inverse-transform methodbut it may not be the most efficient for alldistributions. Since it involve searching.

Another general method is the alias method,which works for every finite range discretedistribution Requires some initial steps and extra storage. Will not be discussed here.

We will explore how to generate random variates

from common discrete theoretical distributionswhich are mainly based on inverse transformmethod.

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Bernoulli

Mass function

Algorithm: Discrete Inverse Transform

0returnOtherwise.1returnIf.2

)1,0(~Generate.1

XXpU

UU

otherwise0

1

01

)( xp

xp

xp

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Binomial

Mass function

Use the fact that ifX~binomial(t,p) then

So use the convolution method which is based onthe discrete inverse transform of Bernoulli.

otherwise0

},...,1,0{)1()(

txppx

t

xpxtx

)(Bernoulli~

...,21

pY

YYYX

i

t

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Geometric

Mass function

Use inverse-transform

)1ln()1ln(

Return.2

)1,0(~Generate.1

pU

X

UU

otherwise0

},...,1,0{)1()(

txppxp

x

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The Hashemite University24

Negative Binomial

Mass function

Note thatX~negbin(s,p) iff

So use the convolution method which is based onthe discrete inverse transform of Geometric.

otherwise0

},...,1,0{)1(

1

1

)(txpp

s

x

xpxs

)(Geometric~...,21pYYYYX

i

t

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Poisson

Mass function

Algorithm

This algorithm is based on the relation betweenPoisson and exponential distributions.

Rather slow especially for large values of . Novery good algorithm for Poisson distribution.

otherwise0

},...,1,0{

!)(

tx

x

exp

x

1stepback togoand1Let.3

3.steptogoOtherwise.return,If

.byreplaceand)1,0(~Generate2.

0,1,Let.1

11

ii

iXab

bUbUU

ibea

ii

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Generating Arrival Processes

Remember that we model the arrivalprocess as a Poisson process and batchprocess based on the number of arrivalsper an arrival event.

For the Poisson process we have: Stationary Poisson Process (SPP): the arrival

rate is constant.

Non-Stationary Poisson Process (NSPP): the

arrival rate varies with time.

We will explore algorithms used togenerate arrival times from each type.

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Generating Arrival Times inSPP

Stationary with rate >0

Time between eventsAi=ti-ti-1 are IIDexponential

Algorithm

The above algorithm can be used with anyIID distribution (not only exponential) justfind the correct inverse of its cdf.

UttUU

ii ln1Return.2

)1,0(~Generate.1

1

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Generating Arrival Times in NSPP

Remember that a NSPP has an arrival rate that varies with

time. Also remember the methods that we have used in Chapter 6

to fit a NSPP to a stationary one. Thinning method. Inversion method.

Based on these two methods we will generate arrival timesfrom a NSPP.

(t)

it 1it

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Thinning Algorithm

1. Set t=ti-1

2. Generate U1, U

2IID U(0,1)

3. Replace tby

4. If return ti= t. Otherwise,

go back to step 2.

As you see the above algorithm is similar to the acceptancerejection method.

)(maxwhere,ln1 *1* tUtt

*

2 /)( tU

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Inversion Algorithm

1. Generate U~ U(0,1)

2. Set ti=t

i-1- ln U

3. Return

Where (see chapter 6):

As you see the above algorithm is similar to the inversetransform method.

)('1

ii tt

t

(s)ds(t) 0

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Additional Notes

The lecture covers the followingsections from the textbook:

Chapter 8

Sections:

8.3 (8.3.1, 8.3.2, 8.3.4, 8.3.5, 8.3.6, 8.3.16),

8.4 (8.4.1, 8.4.4, 8.4.5, 8.4.6, 8.4.7),

8.6 (8.6.1, 8.6.2)

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Lecture _10 -- Part 2