wireless communication network lab. chapter 2 probability, statistics and traffic theories ee of...

Wireless Communication Network Lab.

Chapter 2Probability, Statistics and Traffic

Theories

EE of NIU Chih-Cheng Tseng 1

Prof. Chih-Cheng Tseng

[email protected]

http://wcnlab.niu.edu.tw

mailto:[email protected]

http://wcnlab.niu.edu.tw/




Introduction

Several factors influence the performance of wireless systems• Density of mobile users

• Cell size

• Moving direction and speed of users (Mobility models)

• Call rate, call duration

• Interference, etc. Probability, statistics theory and traffic patterns, help

make these factors tractable



Random Variables (RV)

If S is the sample space of a random experiment, then a RV X is a function that assigns a real number X(s) to each outcome s that belongs to S.

RVs have two types• Discrete RVs: probability mass function, pmf.

• Continuous RVs: probability density function, pdf.



Discrete Random Variables (1)

A discrete RV is used to represent a finite or countable infinite number of possible values.• E.g., throw a 6-sided dice and calculate the probability of

a particular number appearing.

1 2 3 4 6

0.1

0.3

0.1 0.1

0.2 0.2

5

Probability

Number



For a discrete RV X, the pmf p(k) of X is the probability that the RV X is equal to k and is defined below:

p(k) = P(X = k), for k = 0, 1, 2, ... It must satisfy the following conditions

• 0 p(k) 1, for every k

• p(k) = 1, for all k

Discrete Random Variables (2)



The pdf fX(x) of a continuous RV X is a nonnegative valued function defined on the whole set of real numbers (-∞, ∞) such that for any subset S (-∞, ∞)

where x is simply a variable in the integral. It must satisfy following conditions

• fX(x) 0, for all x;

•

Continuous Random Variables

( ) ( )XSP X S f x dx

( ) 1.Xf x dx



Cumulative Distribution Function (CDF)

The CDF of a RV is represented by P(k) (or FX(x)), indicating the probability that the RV X is less than or equal to k (or x).• For discrete RV

• For continuous RV

( ) ( ) ( )x

X XF x P X x f x dx

;

( ) ( ) ( )k X k

P k P X k P X k

( ) ( )

( ) ( ) ( )

b

X Xa

X X

F a x b f x dx

F b F a P a X b



Probability Density Function (pdf)

The pdf fX(x) of a continuous RV X is the derivative of the CDF FX(x):

( )( ) X

X

dF xf x

dx

x

fX(x)

AreaCDF



Discrete RV --- Expected Value

The expected value or mean value of a discrete RV X

The expected value of the function g(X) of discrete RV X is the mean of another RV Y that assumes the values of g(X) according to the probability distribution of X

all

[ ] ( )k

E X kP X k

all

[ ( )] ( ) ( )k

E g X g k P X k



Discrete RV --- nth Moment

The n-th moment

The first moment of X is simply the expected value.

all

[ ] ( )n n

k

E X k P X k



Discrete RV --- nth Central Moment

The nth central moment is the moment about the mean value

The first central moment is equal to 0.

all

[( [ ]) ] ( [ ]) ( )n n

k

E X E X k E X P X k



Discrete RV --- Variance

The variance or the 2nd central moment

where s is called the standard deviation

2 2 2 2Var( ) [( [ ]) ] [ ] [ ]X E X E X E X E X



Continuous RV --- Expected Value

Expected value or mean value

The expected value of the function g(X) of a continuous RV X is the mean of another RV Y that assumes the values of g(X) according to the prob. distribution of X

[ ] ( )XE X xf x dx

[ ( )] ( ) ( )XE g X g x f x dx



Continuous RV --- nth Moment, nth Central Moment and Variance

The nth moment

The nth central moment

Variance or the 2nd central moment

22 2 2Var( ) [ ] [ ] [ ]X E X E X E X E X

[ ] ( )n nXE X x f x dx

[( [ ]) ] ( [ ]) ( )n nXE X E X x E X f x dx



Distributions of Discrete RVs (1)

Poisson distribution• A Poisson RV is a measure

of the number of events that occur in a certain time interval.

• The probability distribution of having k events is

k=0,1,2,…, and >0 • E[X]=l• Var(X)=l

( ) ,!

k eP X k

k

http://en.wikipedia.org/wiki/Poisson_distribution




Geometric distribution• A geometric RV indicate the

number of trials required to obtain the first success.

• The probability distribution of a geometric RV X is

• p is the probability of success• E[X]=1/p• Var(X)=(1-p)/p2

The only discrete RV with the memoryless property.

1( ) (1 ) , 1, 2,3,...kP X k p p k

http://en.wikipedia.org/wiki/Geometric_distribution




Binomial distribution• A binomial RV represents the

presence of k, and only k, out of n items and is the number of successes in a series of trials.

k=0, 1, 2, …, n, n=0, 1, 2,…• p is a success prob., and

• E[X]=np

• Var(X)=np(1-p)

( ) (1 )k n knP X k p p

k

!

!( )!

n n

k k n k

http://en.wikipedia.org/wiki/Binomial_distribution


http://en.wikipedia.org/wiki/File:Binomial_distribution_cdf.svg



When n is large and p is small, the binomial distribution approaches to the Poisson distribution with the parameter given by l = np

( 1) ( 1)( )!(1 ) (1 )

!( )!

( 1) ( 1)(1 ) (if is large)

!

(1 )!

( )(1 ) NOTE: lim (1 )

! !

k n k k n k

k n k

k

k n k

k kn x y

x

n n n n k n kp p p p

k k n k

n n n kp p n

k

n n np p

k

np ye e

k n k x



Distributions of Continuous RVs (1)

Normal distribution• The pdf of the normal RV X is

• The CDF can be obtained by

2

2

( )

21

( ) , for2

x

Xf x e x

2

2

( )

21

( )2

yx

XF x e dy




In general• X～ N(m,s2) is used

to represent the RV X as a normal RV with the mean and variance m and s2 respectively.

The case when m=0 and s = 1 is called the standard normal distribution.

http://en.wikipedia.org/wiki/Normal_distribution


http://en.wikipedia.org/wiki/File:Normal_Distribution_CDF.svg



Uniform Distribution• The values of a uniform RV are uniformly distributed over an

interval.• pdf of a uniform distributed RV X is

• CDF of a uniform distributed RV X is

• E[X]=(a+b)/2 and Var(X)=(b-a)2/12

1, for

( )0, otherwise.

X

a x bf x b a

0, for ,

( ) , for ,

1, for .

X

x a

x aF x a x b

b ax b

http://en.wikipedia.org/wiki/Uniform_distribution


http://en.wikipedia.org/wiki/File:Uniform_distribution_PDF.png



Exponential distribution• Generally used to describe the time interval between two consecutive

events

• pdf is

• CDF is

• l is the average rate.

• E[X]=1/l • Var(X)=1/l2

0, 0,( )

, for 0 .X x

xf x

e x

0, 0,( )

1 , for 0 .X x

xF x

e x

http://en.wikipedia.org/wiki/Exponential_distribution



Multiple RVs (1)

In some cases, the result of one random experiment is dictated by the values of several RVs, where these values may also affect each other.

A joint pmf of the discrete RVs X1, X2, …, Xn is

and represents the prob. that X1=x1, X2 = x2, …, Xn = xn.

1 2 1 1 2 2( , ,..., ) ( , ,..., )n n xp x x x P X x X x X x



Multiple RVs (2)

joint CDF

joint pdf

1 2, ,..., 1 2 1 1 2 2( , ,..., ) ( , ,..., )nX X X n n xF x x x P X x X x X x

1 2

1 2

, ,..., 1 2

, ,..., 1 21 2

( , ,..., )( , ,..., )

...n

n

nX X X n

X X X nn

F x x xf x x x

x x x



Conditional Probability

A conditional prob. is the prob. that X1=x1

given X2=x2, …, Xn=xn

For discrete RVs

For continuous RVs

1 1 2 21 1 2 2

2 2

( , ,..., )( | ,..., )

( ,..., )n n

n nn n

P X x X x X xP X x X x X x

P X x X x

1 1 2 21 1 2 2

2 2

( , ,..., )( | ,..., )

( ,..., )n n

n nn n

P X x X x X xP X x X x X x

P X x X x



Bayes’ Theorem

P(Ai│B) (read as: the prob. of B, given Ai) is

• P(Ai) and P(B) are the unconditional probabilities of Ai and B.


1

( , ) ( | ) ( )( | )

( ) ( )

( | ) ( )

( | ) ( )

i i ii

i in

k kk

P A B P B A P AP A B

P B P B

P B A P A

P B A P A


Stochastically Independence (or Independence)

Two events are independent if one may occur irrespective of the other.

A finite set of events is mutually independent if and only if (iff) every event is independent of any intersection of the other events.• If the RVs X1, X2,…, Xn are mutually independent

• Discrete RVs

• Continuous RVs

( , ) ( ) ( )( | ) ( ), when ( ) 0

( ) ( )

P X Y P X P YP Y X P Y P X

P X P X

1 2 1 21 2 1 2... ( , ,..., ) ( ) ( ) ( )n nX X X n X X X nF x x x F x F x F x

1 2 1 1 2 2( , ,..., ) ( ) ( ) ( )n n np x x x P X x P X x P X x



Important Properties (1)

Sum property of the expected value

• Expected value of the sum of RVs X1, X2, …, Xn

Product property of the expected value• Expected value of product of independent RVs

1 1

[ ]n n

i i i ii i

E a X a E X

1 1

[ ]n n

i ii i

E X E X



Important Properties (2)

Sum property of the variance

• Variance of the sum of RVs X1, X2,…, Xn is

• Cov[Xi,Xj] is the covariance of RVs Xi and Xj

• If Xi and Xj are indep.,

• Cov[Xi,Xj]=0 for i≠j

•

-12

1 1 1 1

Var Var( ) 2 Cov[ , ]n n n n

i i i i i j i ji i i j i

a X a X a a X X

Cov[ , ] [( [ ])( [ ])]

[ ] [ ] [ ]

i j i i j j

i j i j

X X E X E X X E X

E X X E X E X

2

1 1

Var Var( )n n

i i i ii i

a X a X



Central Limit Theorem

Whenever a random sample (X1, X2,…, Xn) of size n is taken from any distribution with expected value E[Xi] =m and variance Var(Xi)=s2 where i = 1, 2, …, n, then their arithmetic mean (or sample mean) is defined by

• The sample mean is approximated to a normal distribution with E[Sn] =m and Var(Sn)=s2/n

• The larger the value of the sample size n, the better the approximation to the normal

1

1 n

n ii

S Xn



Poisson Arrival Model

A Poisson process is a sequence of events randomly spaced in time.

For a time interval [0,t]， the probability of n arrivals in t units of time is

• The rate l of a Poisson process is the average number of events per unit of time (over a long time).

• The number of arrivals in any two disjoint intervals are independent.

( )( ) , for 0,1, 2,...

!

nt

n

tP t e n

n



Interarrival Times of Poisson Process

Interarrival times of a Poisson process

• We pick an arbitrary starting point t0 in time. Let T1 be the time until the next arrival. We have P(T1>t)=P0(t)=e-t.

• The CDF of T1 is (t)=P(T1≤ t)=1-e-t

• The pdf of T1 is (t)=e-t.

• Therefore, T1 has an exponential distribution with mean rate .

1TF

1Tf



Exponential Distribution

Similarly,

• T2 is the time between first and second arrivals

• T3 as the time between the second and third arrivals

• T4 as the time between the third and fourth arrivals and so on.

The random variables T1, T2, T3,… are called the interarrival times of the Poisson process.

T1, T2, T3,… are mutually independent and each has the exponential distribution with mean arrival rate .



Memoryless and Merging Properties

Memoryless property• A random variable X is said to be memoryless if

• The exponential/geometric distribution is the only continuous/discrete RV with the memoryless property.

Merging property• If we merge n Poisson processes with distributions for the

interarrival times where i = 1, 2,…, n into one single process, then the result is a Poisson process for which the interarrival times have the distribution 1-e-t with mean =1+2+…+n.

( | ) ( )P X t X P X t

1 ite



Basic Queuing Systems

What is queuing theory?• Queuing theory is the study of queues (sometimes called

waiting lines).

• It can be used to describe real world queues, or more abstract queues, found in many branches of computer science, such as operating systems.

Queuing theory can be divided into 3 sections• Traffic flow

• Scheduling

• Facility design and employee allocation



Kendall’s Notation (1)

D. G. Kendall in 1951 proposed a standard notation A/B/C/D/E for classifying queuing systems into different types.

A Distribution of inter arrival times of customers

B Distribution of service times

C Number of servers

D Maximum number of customers in the system

E Calling population size



Kendall’s Notation (2)

A and B can take any of the following distributions types

M Exponential distribution (Markovian)

D Degenerate (or deterministic) distribution

Ek Erlang distribution (k = shape parameter)

Hk Hyper exponential with parameter k



Little’s Law

Assuming a queuing environment to be operated in a steady state where all initial transients have vanished, the key parameters characterizing the system are• l ─ the mean steady-state customer arrival rate

• N ─ the average no. of customers in the system

• T ─ the mean time spent by each customer in the system (time spent in the queue plus the service time)

Little’s law: N = lT



Markov Process

A Markov process is one in which the next state of the process depends only on the present state, irrespective of any previous states taken by the process.

The knowledge of the current state and the transition probabilities from this state allows us to predict the next state.

A Markov chain is a discrete state Markov process.



Birth-Death Process (1)

Special type of Markov process If the population (or jobs) in the queue has n,

• birth of another entity (arrival of another job) causes the state to change to n+1.

• a death (a job removed from the queue for service) would cause the state to change to n-1.

Any state transitions can be made only to one of the two neighboring states.


Wireless Communication Network Lab. The state transition diagram of the continuous birth-

death process


1 2 3 n-1n n+1

n+2

n-10 1 2 n n+1……

0 1 2 n-2 n-1 n n+1

…

P(0) P(1) P(2) P(n-1) P(n) P(n+1)

P(i) is the steady state probability in state i.




In state n, we have

• P(i) is the steady state prob. of the state i.

• li (i=0, 1, 2, …) is the average arrival rate in the state i.

• mi (i=0, 1, 2, …) is the average service rate in the state i.

1 1( 1) ( 1) ( ) ( )n n n nP n P n P n




For state 0,

For state 1,

For state n,

00 1

1

(0) (1) (1) (0)P P P P

0 1 1

1 2

...( ) (0)

...n

n

P n P

0 2 1 1

02 1 1 0 2 1 1 0

1

0 1 1 0 1 0 1

1 2 1 2 1 2

(0) (2) ( ) (1)

(2) ( ) (1) (0) (2) ( ) (0) (0)

( )(2) (0) (2) (0)

P P P

P P P P P P

P P P P



M/M/1/∞ Queuing System (1)

M/M/1/∞ = M/M/1/∞/∞ = M/M/1 When a customer arrives in this system, it will be

served if the server is free, otherwise the customer is queued.

In this system, customers arrive according to a Poisson distribution and compete for the service in a FIFO (first-in-first-out) manner.

Service times are independent identically distributed (iid) random variables, the common distribution being exponential.



M/M/1 Queuing System (2)

The M/M/1 queuing model

The state transition diagram of the M/M/1 queuing system

Queue Server

0 1 2 i-1 i i+1…… …

P(0) P(1) P(2) P(i-1) P(i) P(i+1)


System



The equilibrium state equations are given by

So,

(0) (1), 0,

( ) ( ) ( 1) ( 1), 1

P P i

P i P i P i i

2

(1) (0) (0),

(2) (1) (0),

...

( ) ( 1) (0),

...

i

P P P

P P P

P i P i P

r =l/m is the flow intensity and r < 1




The normalized condition is given by

Since,

Therefore,

0

( ) 1i

P i

0 0

1( ) 1 (0) (0) 1 (0) 1

1i

i i

P i P P P

1

0

(1 ) 1 ( 1 )

1 1

ni

ni

a rS r

r

( ) (1 )iP i




The average number of customers in the system is given by

0 0

1

1

( ) (1 )

1(1 ) (1 )( )

1

1

iS

i i

i

i

L iP i i

i

Typo in Eq. (2.64)




By using the Little’s Law, the average dwell time (or system time) of customers is

1

(1 )

1

SS

LW




The average queue length

1

2

2

( 1) ( )

1

( )

qi

L i P i




The average waiting time of customers is

2

(1 )

( )

qq

LW



M/M/S/ Queuing Model

Queue

S

Servers

S

.

.

2

1



State Transition Diagram

2 3 (S-1) S S S

0 1 2 S-1 S S+1……

…



The Equilibrium State Equations

The equilibrium state equations

The steady state probabilities

• a=l/m

(0) (1), 0,

( ) ( ) ( 1) ( 1) ( 1), 1 ,

( ) ( ) ( 1) ( 1), ,

P P i

i P i P i i P i i S

S P i P i S P i i S

!

!

( ) (0), ,

( ) ( ) (0),

i

S

i

i SS S

P i P i S

P i P i S



Finding P(0)

Based on the normalized condition

We can obtain

If a<S and the utilization factor ρ =l/(Sm),

1

0 0 0

( ) (0) 1! !

ii SS

i i i

P i Pi S S

11

0 0

(0)! !

ii SS

i i

Pi S S

1

1

1

0

1

0

(0)! !

1

! ! 1

i SS

i

i SS

i

SP

i S S

i S

Typo in Eq. (2.73)



Queuing System Metrics (1)

The average number of customers in the system is

The average dwell time of a customer in the system is given by

02)1(!

)0()(

i

S

sS

PiiPL

2

1 (0)

!(1 )

SS

SL P

WS S



Queuing System Metrics (2)

The average queue length is

The average waiting time of customers is

1

2

(0)( ) ( )

( 1)!( )S

S

q

i

PL i S P i

S S

2

(0)

!(1 )

Sq

qL P

WS S



M/G/1 Queuing Model (Optional)

We consider a single server queuing system whose arrival process is Poisson with mean arrival rate .

Service times are independent and identically distributed with distribution function FB and pdf fb.

Jobs are scheduled for service as FIFO.



Basic Queuing Model (Optional)

Let N(t) denote the number of jobs in the system (those in queue plus in service) at time t.

Let tn (n= 1, 2,..) be the time of departure of the nth job and Xn be the number of jobs in the system at time tn, so that

Xn = N(tn), for n = 1, 2,… The stochastic process {Xn, n= 1, 2,…} can be modeled as a

discrete Markov chain, known as imbedded Markov chain of the continuous stochastic process N(t).

The imbedded Markov chain converts a non-Markovian problem, i.e. {N(t), t≥0}, into a Markovian one, i.e. {Xn, n= 1, 2,..}.



Queuing System Metrics (1) (Optional)

The average number of jobs in the system, in the steady state

is

The average dwell time of customers in the system is

The average waiting time of customers in the queue is

2 2[ ][ ]

2(1 )

E BE N

2[ ] 1 [ ]

2(1 )

E N E BWs

[ ] qE N W



Queuing System Metrics (2) (Optional)

Average waiting time of customers in the queue is

The average queue length is

2[ ]

2(1 )q

E BW

2 2[ ]

2(1 )q

E BL



Homework

P2.4P2.7P2.14P2.17P2.20


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