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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
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Stochastic Process
• Observation corresponds to function of time
OutcomesS
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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Example
x(t, 1)
1
2
x(t, 2)
x(t, 3)
Ensemble
Chapter 6 : Stochastic Processes
3
Sample Space Sample Function
Stochastic Symbols
t : time dependent
x( t,) : sample functions, X ( t) : name of stochastic process
Chapter 6 : Stochastic Processes
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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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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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,…
Chapter 6 : Stochastic Processes
(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
Chapter 6 : Stochastic Processes
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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
Chapter 6 : Stochastic Processes
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
:
:
Chapter 6 : Stochastic Processes
t
S 24,2,6,5,…
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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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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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
Chapter 6 : Stochastic Processes
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, …
Chapter 6 : Stochastic Processes
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Some Important Stochastic
Process• Bernoulli Process
• Poisson Process
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
,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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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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
Chapter 6 : Stochastic Processes
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 ,
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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Poisson P rocess
t 0
N (t 0)
N (t 1)
1
N (t )
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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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?
Chapter 6 : Stochastic Processes
Solution
=120 calls/hour
t t 0 t 1
N (t )
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
1 = t 1 and i = (t i –t i-1), i = 2, 3,…
Example
t
t 0 t 4t 1 t 2 t 3α1
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
!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
Chapter 6 : Stochastic Processes
!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.
Chapter 6 : Stochastic Processes
(Continue)
3 = 15 inquiries/minute
t ( s)0 102
= ¼ /second
50 60
Chapter 6 : Stochastic Processes
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]
Chapter 6 : Stochastic Processes
Interarrival Time
Theorem: A counting process with independentex onential interarrival X X … is a Poisson
process of rate
N (t ) Arrival rate > 0
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
+ met
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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?
Chapter 6 : Stochastic Processes
Solution
• N (t ) : # of requests arrive at a server at time t
requests arrive during the next 30 seconds?
30 s
t
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
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 ] = ??
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
Properties of Poisson Process
• 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 )
Chapter 6 : Stochastic Processes
2
1 + 2
2
N (t )
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Properties of Poisson Process
• 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 )
Chapter 6 : Stochastic Processes
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?
Chapter 6 : Stochastic Processes
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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
Chapter 6 : Stochastic Processes
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
Chapter 6 : Stochastic Processes
• Same random variable at all time• The statistical properties of the processdo not change with time
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• 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
Chapter 6 : Stochastic Processes
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.
Chapter 6 : Stochastic Processes