ReviewCh6 Random Processes

8/3/2019 ReviewCh6 Random Processes

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241-460 Introduction to Queueing

Netw orks : Engineering Approach

Assoc. Prof. Thossaporn KamolphiwongCentre for Network Research (CNR)

Department of Computer Engineering, Faculty of EngineeringPrince of Songkla University, Thailand

Stochastic ProcessesStochastic Processes

Email : [email protected]

Outline

Random Processes or Stochastic Processes

• Definitions

• Types of Stochastic Processes

• Random Sequences• Examples of Stochastic Processes

Bernoulli Process

ount ng rocess

Poisson Process

Stationary Process

Chapter 6 : Stochastic Processes



2

Stochastic Process

• Observation corresponds to function of time

OutcomesS


Random Variable

X ( )Random Variable

X (t ) = X (t , )

Stochastic Process

Definition : A stochastic process X ( t) or Random process is a rule for assigning to every a function

x(t, )

Definition : Sample Function

A sample function x(t, ) is the time function associated with outcome of an experiment

Definition : Ensemble

The ensemble of a stochastic process is the set of all possible time functions that can result from anexperiment




3

Example

x(t, 1)

1

2

x(t, 2)

x(t, 3)

Ensemble


3

Sample Space Sample Function

Stochastic Symbols

t : time dependent

x( t,) : sample functions, X ( t) : name of stochastic process




4

Example

Time instants T = 0, 1, 2,…, x(t,S 1)

T

where 1 N T 6

X (t ) = N T

X (t ) for T t < T + 1 , x(t,S 2)

S 11,2,6,3,…

t


t

24,2,6,5,…

Type of Stochast ic processes

• Based on the parameter space:

Discrete-time stochastic rocess:

Set I is countable ( t I )

Continuous-time stochastic process

Set I is continuous (t I )• Based on the state processes:

Discrete-state processes:

state space discrete

Continuous-state processes:

state space continuous




5

Stochastic processes Example

• Discrete-time, discrete-state processes

The number of occupied channels in a telephone

link at the arrival time of the k th customer,

k = 1,2,…

he number of ackets in the buffer of a

statistical multiplexer at the arrival time of the k th

customer, k = 1,2,…


(Continue)

• Continuous-time, discrete-state processes

The number of occupied channels in a telephone

link at time t > 0

The number of packets in the buffer of a




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Type of Stochastic Processes

Discrete-Time, Continuous-Value

t )

Continuous-Time, Continuous-Value

)

Continuous-Time, Discrete-Value

D ( t )

Discrete-Time, Discrete-Value

D ( t )

X D C

(

t X C C

(

t


X C

t

X D

t

Random Variables fromStochastic Processes

x(t,S 1)

x(t,S 2)

S 11,2,6,3,…

t

Random Variable : X (t )

x(0 ,1), x(1 ,2), x(2 ,6), x(3 ,3), …

x P PMF

x f PDF

t X

t X

:

:


t

S 24,2,6,5,…



7

Example

• Rolling a die, what is the PMF of X (3.5)?

T t < T + 1 x(t,S 1) 6

Solution

• The random variables X (3.5) is the value of the

die roll at time 3.

t

2

otherwise x x P 6,...,2,1

0

6 / 15.3


Random Sequence

Definition :

n

of X 0, X 1, X 2, …

Independent, Identically Distributed (iid ) Random

Sequences is a random sequence X n in which…, X -2, X -1, X 0, X 1, X 2,… are ii Ran om Varia es




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Theorem : Let X n is an iid random sequence.

IID Random Sequences

For a continuous-valued process, the joint PDF is

- ,

i X k X X x P x x x P k

,...,,21,...,1


k

i

i X k X X x f x x x f k

1

21,...,,...,,

1

Sum P rocess

• Many interesting random processes are obtainedas the sum of se uence of iid random variable X 1, X 2, …

S n = X 1 + X 2 + … + X n n = 1, 2, …




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Some Important Stochastic

Process• Bernoulli Process

• Poisson Process


Bernoulli P rocess

Definition : A Bernoulli ( p ) process X n is an iid

ran om sequence n w c eac n s a

Bernoulli ( p ) random variable

Example


,output X 1, X 2,… of a binary source is modeled as

a Bernoulli ( p = ½) process



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Counting Process

Definition Counting Process

s oc as c process s a coun ng process

if for every sample function, k (t, ) = 0

for t < 0 and k (t, ) is integer-valued and

nondecreasing with time


Sample path of counting process

N (t )

Arrival rate > 0

X 5

S 1 S 2 S 3 S 4 S 5t

X 4 X 3 X 2 X 1


X n : Bernoulli process

N (t ) = # of customers that arrive at a systemduring interval (0,t ]



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N (t )

Arrival rate > 0

(Continue)

• Sum of Process

S 1 = X 1

X 5

S 1 S 2 S 3 S 4 S 5t

X 4 X 3 X 2 X 1

S 2 = X 1 + X 2

S 3 = X 1 + X 2 + X 3

S 4 = X 1 + X 2 + X 3 + X 4

S 5 = X 1 + X 2 + X 3 + X 4 + X 5


N (t )

(Continue)

N (t ) = # of customers

X n : Bernoulli process

X 5

t S 1 S 2 S 3 S 4 S 5

Arrival rate > 0

X 4 X 3 X 2 X 1

T=m

T=m

t

S 1 S 2 S 3 S 4 S 5




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Counting Process

t

S 1 S 2 S 3 S 4 S 5

0, there is only 1 arrival ( X n = 1)

Choose << 1, success probability of T / m

5 10 15 m

nmn

N mT mT n

mn P

m

1


Prob. of N m arrival is

Binomial PMF

Counting Process

Binomial Process

nmn

N mT mT n

n P m

1

T n

m , 0 ,


otherwise

nnn P N m

,...,,

0

!

Poisson Process



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Poisson P rocess

• Any interval (t 0,t 1], # of arrivals is a Poisson PMF

1- 0

• # of arrivals in (t 0,t 1] dependents on the

independent Bernoulli trials

• 0, counting process in which # of arrivals in

any interval is Poisson process


Poisson P rocess

Definition : Poisson Process

a) # of arrivals in any interval (t 0 ,t 1] , N (t 1) –

N (t 0

),is a Poisson random variable withexpected value (t 1-t 0)

b) For any pair of nonoverlapping intervals (t 0 ,t 1]

0 , 1 ,

interval, N (t 1) – N (t 0) and N (t 1) – N (t 0)respectively, are independent random variables




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Poisson P rocess

t 0

N (t 0)

N (t 1)

1

N (t )


N (t ) = # of arrivals in the interval (t 0 ,t 1]

N (t 1) - N (t 0) = # of arrivals in the interval (t 0 ,t 1]

Poisson P rocess

• Process rate () = E [ N (t )]/ t

• Poisson random variable N t = N t – t

me

m

t t

m P t t

m

t N

,...,1,0!

0101

PMF is

0




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Example

Suppose that the number of calls that arrive at acom an call centre is a Poisson rocess with a rate of 120 per hour.

a) What is the probability of 3 calls in a minute?

b) What is the probability of at least two calls in a minute?


Solution

=120 calls/hour

t t 0 t 1

N (t )


1 - 0 = m nu e, = ca s = =

b) t 1 - t 0 = 1 minute, N (t ) > 2 calls P [ N (t ) > 2] = ?



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Solution

a) What is the probability of 3 calls in a minute?

P N t = 3 = ??

33

On average there are 120/60 =2 calls perminute. (= 2)

18.0!3!)3(01 01

eemt N P


Solution

b) What is the probability of at least two calls in a minute? P [ N (t ) > 2] = ??

101212 N P N P N P N P

P [ N > 2] = P [ N = 2] + P [ N = 3] + P [ N = 4] + …

P [ N = 0] + P [ N = 1] + P [ N = 2] + P [ N = 3] + … = 1

101 N P N P


594.0211

!1

2

!0

21

2

12

02

e

ee



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Theorem : For a Poisson process N (t ) of rate ,

Joint PMF

e o n o = 1 ,…, k , orordered time instances t 1 < ···< t k , is

otherwise

nnnn

e

nn

e

n

e

n P k

k k

nn

k

nnn

N

k k k

,...0

0

!!!1

112

2

1

1121211


1 = t 1 and i = (t i –t i-1), i = 2, 3,…

Example

t

t 0 t 4t 1 t 2 t 3α1


2

n1 = 2 n2 = 3 n3 = 4



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(Continue)

t t 0 t 4t 1 t 2 t 3

α1 α2 α3

n1 = 2 n2 = 3 n3 = 4

23 e


!32 12 t t N

!2

1

1

2

101

e

t t P N

!4

3

3

4

334

e

t t P N

(Continue)

t t 0 t 4t 1 t 2 t 3

α

1α

2α

3

n1 = 2 n2 = 3 n3 = 4


!4!3!2

321 4

3

3

2

2

1

eeet P N



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Example

Inquiries arrive at a recorded message deviceaccordin to a Poisson rocess of rate 15 inquiries per minute.

• Find the probability that in a 1-minute period, 3inquiries arrive during the first 10 seconds and 2inquiries arrive during the last 15 seconds.


(Continue)

3 = 15 inquiries/minute

t ( s)0 102

= ¼ /second

50 60


P [ N 1(10) = 3 and N 3(60)- N 2(45) = 2] = ??



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Solution

Arrival rate () =15/60 = ¼ inquiries per second,

!2

415

!3

41041524103

ee

P [ N 1(10) = 3, N 3(60)- N 2(45) = 2] = ??

= P [ N (10) = 3] P [ N (60 – 45) = 2]


Interarrival Time

Theorem: A counting process with independentex onential interarrival X X … is a Poisson

process of rate

N (t ) Arrival rate > 0


X 5

S 1 S 2 S 3 S 4 S 5 X 4 X 3 X 2 X 1

2th Interarrival time



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Theorem : For a Poisson process of rate ,

Interarrival time

the interarrival times X 1, X 2,… are an iid

random sequence with the exponential PDF

otherwise

xe x f

x

X

,0

0


Relationship between the Poissonand Exponential Distributions

Poisson distributionPoisson distributionrovides an a ro riate descri tionrovides an a ro riate descri tion

of the number of occurrencesof the number of occurrencesper intervalper interval

Ex onential distributionEx onential distribution


provides an appropriate descriptionprovides an appropriate description

of the length of the intervalof the length of the intervalbetween occurrencesbetween occurrences



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• Property 1 : Memoryless property

Properties of Poisson Process

h

n

nnnn e

t X P

t t t X ht X P

,|

h


If the arrival has not occurred by time t, the additional time until

the arrival, h + t, has the same exponential distribution as X n

t t+h time

(Continue)

t X ht X P

t X ht X P nnnn

,|

h

X n > t + h X n > t

n


+ met



23

Example

Connection requests arrive at a server according toa Poisson rocess with intensit = 5 re uests in a minute.

(a) What is the probability that exactly 2 newrequests arrive during the next 30 seconds?

at the server, what is the probability that ittakes more than 30 seconds before nextrequest arrives?


Solution

• N (t ) : # of requests arrive at a server at time t

requests arrive during the next 30 seconds?

30 s

t


t 0 1

P [ N (t 0 + 30) - N (t 0) = 2 ] = ??



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Solution

• # of new arrivals during a time interval followsPoisson distribution with the parameter

!2

5.052)(30

2

5.05

2

et N t N P

=(5/60)30 = 2.5

• N (t +30)- N (t ) ~ Poisson(2.5)

257.0

!2

.5.2

e


Solution

(b) If a new connection request has just arrived atthe server what is the robabilit that it takes more than 30 seconds before next requestarrives?

t

new connection next connection

P [t 1-t 0 > 30] = ??

Or P [T > 30+t | T > t ] = ??


t 0 1



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Solution

• Consider the process as a point process. Theinterarrival time follows ex onential distribution with parameter

P [t 1-t 0 > 30] = 1 – P [t 1-t 0 < 30]

= e-(5/60)30 = e-2.5 = 0.82

P [T > t+30 | T > t ] = e-(5)(30/60)

= e-2.5 = 0.82



• Property 2 : Let N 1(t ) and N 2(t ) be two

and 2. The counting process N (t ) = N 1(t ) + N 2(t )

is a Poisson process of rate 1 + 2.

1 N 1(t )


2

1 + 2

2

N (t )



26


• Property 3 : The counting processes N 1(t ) andt derived from a Bernoulli decom osition of

the Poisson process N (t ) are independentPoisson processes with rate p and (1- p).

N (t )= N 1(t )+ N 2(t )


1 p

N 2(t ) (1- p)

Example

A corporate Web server records hits (request forHTML document as a Poisson rocess at a rate of 10 hits per second. Each page is either aninternal request (with probability 0.7) from the

corporate intranet or an external request (withprobability 0.3) from the Internet.

• Over a 10-minute interval, what is the joint PMF of I , the number of internal requests, and X , the

number of external requests?




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Solution

Internal and external request arrival areindependent Poisson processes with rate of 7an its per secon

I = 7(600) = 4200 hits

X = 3(600) = 1800 hits

The joint PMF of I and X is

otherwise0

,...1,0,!

1800

!

4200

,

18004200

,

xi x

e

i

e xi

X I X I


Stationary Processes

• Stochastic process X (t ) ,

1 1 X (t 1) 1.

• Stationary process,

at t 1: X (t 1) with f X (t 1)( x) does not depend on t 1

Stationary process


• Same random variable at all time• The statistical properties of the processdo not change with time



• Definition : Stationary Process

Stationary Process

stoc astic process t is stationary i an on y

if for all sets of time instant t 1 ,…, t m , and any

time difference

mt X t X mt X t X x x f x x f mm

,...,,..., 1,...,1,..., 11


References

1. Alberto Leon-Garcia, Probability and RandomProcesses for Electrical En ineerin 3rd Ed., Addision-Wesley Publishing, 2008

2. Roy D. Yates, David J. Goodman, Probabilityand Stochastic Processes: A FriendlyIntroduction for Electrical and ComputerEngineering, 2nd, John Wiley & Sons, Inc, 2005

3. Jay L. Devore, Probability and Statistics forEngineering and the Sciences, 3rdedition, Brooks/Cole PublishingCompany, USA, 1991.


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ReviewCh6 Random Processes