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Performance Evaluation of Computer Systems

By

Behzad Akbari

Tarbiat Modares University

Spring 2009

Introduction to Probabilities: Discrete Random Variables

These slides are based on the slides of Prof. K.S. Trivedi (Duke University)

In the Name of the Most High

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Random Variables

Sample space is often too large to deal with directly Recall that in the sequence of Bernoulli trials, if we

don’t need the detailed information about the actual pattern of 0’s and 1’s but only the number of 0’s and 1’s, we are able to reduce the sample space from size 2n

to size (n+1). Such abstractions lead to the notion of a random

variable

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Discrete Random Variables

A random variable (rv) X is a mapping (function) from the sample space S to the set of real numbers

If image(X ) finite or countable infinite, X is a discrete rv Inverse image of a real number x is the set of all sample points that

are mapped by X into x:

It is easy to see that

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Probability Mass Function (pmf)

Ax : set of all sample points such that,

pmf

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pmf Properties

Since a discrete rv X takes a finite or a countable infinite set

values,

the last property above can be restated as,

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Distribution Function

pmf: defined for a specific rv value, i.e., Probability of a set

Cumulative Distribution Function (CDF)

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Distribution Function properties

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Equivalence: Probability mass function Discrete density function(consider integer valued random variable)

cdf:

pmf:

)( kXPpk

x

kkpxF

0)(

Discrete Random Variables

)1()( kFkFpk

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Common discrete random variables

Constant Uniform Bernoulli Binomial Geometric Poisson

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Constant Random Variable

pmf

CDF

c

1.0

1.0

c

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Discrete Uniform Distribution

Discrete rv X that assumes n discrete value with equal probability 1/n

Discrete uniform pmf

Discrete uniform distribution function:

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Bernoulli Random Variable

RV generated by a single Bernoulli trial that has a binary valued outcome {0,1} Such a binary valued Random variable X is called the indicator or Bernoulli random variable so that

Probability mass function:

)0(1

)1(

XPpq

XPp

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Bernoulli Distribution

CDF

x0.0 1.0

q

p+q=1

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Binomial Random Variable

Binomial rv a fixed no. n of Bernoulli trials (BTs) RV Yn: no. of successes in n BTs Binomial pmf b(k;n,p)

Binomial CDF

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Binomial Random Variable

In fact, the number of successes in n Bernoulli trials can be seen as

the sum of the number of successes in each trial:

where Xi ’s are independent identically distributed Bernoulli random

variables. nn XXXY ...21

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Binomial Random Variable: pmf

pk

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0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4 5 6 7 8 9 10

x

CD

FBinomial Random Variable: CDF

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Reliability of a k out of n system

Series system:

Parallel system:

Applications of the binomial

njnjn

j

njparallel

njnjn

nj

njseries

jnjn

kj

nj

n

kjkofn

RRRRnbR

RRRRnnbR

RRRnjbRnkBR

]1[1]1[])[(),;0(1

][]1[])[(),;(

]1[])[(),;(),;1(1

1

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Transmitting an LLC frame using MAC blocks p is the prob. of correctly transmitting one block Let pK(k) be the pmf of the rv K that is the number of

LLC transmissions required to transmit n MAC blocks correctly; then

Applications of the binomial

nknkK

nnK

nK

ppkp

and

ppp

ppnnbp

])1(1[])1(1[)(

])1(1[)2(

),;()1(

1

2

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Geometric Distribution

Number of trials upto and including the 1st success.

In general, S may have countably infinite size

Z has image {1,2,3,….}. Because of independence,

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Geometric Distribution (contd.)

Geometric distribution is the only discrete distribution that exhibits MEMORYLESS property.

Future outcomes are independent of the past events. n trials completed with all failures. Y additional trials are performed

before success, i.e. Z = n+Y or Y=Z-n

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Geometric Distribution (contd.)

Z rv: total no. of trials upto and including the 1st success. Modified geometric pmf: does not include the successful

trial, i.e. Z=X+1. Then X is a modified geometric random variable.

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The number of times the following statement is executed:

repeat S until B

is geometrically distributed assuming …. The number of times the following statement is

executed:

while B do S

is modified geometrically distributed assuming ….

Applications of the geometric

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Negative Binomial Distribution

RV Tr: no. of trials until rth success.

Image of Tr = {r, r+1, r+2, …}. Define events: A: Tr = n B: Exactly r-1 successes in n-1 trials. C: The nth trial is a success.

Clearly, since B and C are mutually independent,

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Poisson Random Variable

RV such as “no. of arrivals in an interval (0,t)” In a small interval, Δt, prob. of new arrival= λΔt. pmf b(k;n, λt/n); CDF B(k;n, λt/n)=

What happens when

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Poisson Random Variable (contd.)

Poisson Random Variable often occurs in situations, such as, “no. of packets (or calls) arriving in t sec.” or “no. of components failing in t hours” etc.

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Poisson Failure Model

Let N(t) be the number of (failure) events that occur in the time interval (0,t). Then a (homogeneous) Poisson model for N(t) assumes:

1.The probability mass function (pmf) of N(t) is:

Where > 0 is the expected number of event occurrences per unit time

2.The number of events in two non-overlapping intervals are mutually independent

/ !k

tP N t k kt e

,2,1,0k

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Note:

For a fixed t, N(t) is a random variable (in this case a discrete random variable known as the Poisson random variable). The family {N(t), t 0} is a stochastic process, in this case, the homogeneous Poisson process.

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Poisson Failure Model (cont.)

The successive interevent times X1, X2, … in a homogenous Poisson model, are mutually independent, and have a common exponential distribution given by:

To show this:

Thus, the discrete random variable, N(t), with the Poisson distribution, is related to the continuous random variable X1, which has an exponential distributionThe mean interevent time is 1/, which in this case is the mean time to failure

eXttP 11 0t

tetNPtXP )0)(()( 1

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Probability mass function (pmf) (or discrete density function):

Distribution function (CDF):

k!

)( )(

ktektNPp t

k

k!)(

0

k

x

k

t texF

Poisson Distribution

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pk

t=1.0

Poisson pmf

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t1 2 3 4 5 6 7 8 9 10

0.5

0.1

CDF1

t=1.0

Poisson CDF

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t=4.0

pk

t=4.0

Poisson pmf

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t

CDF

1 2 3 4 5 6 7 8 9 10

0.5

0.1

1

t=4.0

Poisson CDF

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Probability Generating Function (PGF)

Helps in dealing with operations (e.g. sum) on rv’s Letting, P(X=k)=pk , PGF of X is defined by,

One-to-one mapping: pmf (or CDF) PGF See page 98 for PGF of some common pmfs

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Discrete Random Vectors

Examples: Z=X+Y, (X and Y are random execution times) Z = min(X, Y) or Z = max(X1, X2,…,Xk)

X:(X1, X2,…,Xk) is a k-dimensional rv defined on S

For each sample point s in S,

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Discrete Random Vectors (properties)

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Independent Discrete RVs

X and Y are independent iff the joint pmf satisfies:

Mutual independence also implies:

Pair wise independence vs. set-wide independence

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Discrete Convolution

Let Z=X+Y . Then, if X and Y are independent,

In general, then,