wireless communication network lab. chapter 2 probability, statistics and traffic theories ee of...
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Wireless Communication Network Lab.
Chapter 2Probability, Statistics and Traffic
Theories
EE of NIU Chih-Cheng Tseng 1
Prof. Chih-Cheng Tseng
http://wcnlab.niu.edu.tw
Wireless Communication Network Lab.
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
EE of NIU Chih-Cheng Tseng 2
Wireless Communication Network Lab.
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.
EE of NIU Chih-Cheng Tseng 3
Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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)
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
Distributions of Discrete RVs (2)
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
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Wireless Communication Network Lab.
Distributions of Discrete RVs (3)
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
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Wireless Communication Network Lab.
Distributions of Discrete RVs (4)
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
Distributions of Continuous RVs (2)
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
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Wireless Communication Network Lab.
Distributions of Continuous RVs (3)
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
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Wireless Communication Network Lab.
Distributions of Continuous RVs (4)
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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.
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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
Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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 .
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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.
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Wireless Communication Network Lab.
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.
EE of NIU Chih-Cheng Tseng 40
Wireless Communication Network Lab. The state transition diagram of the continuous birth-
death process
Birth-Death Process (2)
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.
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Wireless Communication Network Lab.
Birth-Death Process (3)
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
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Wireless Communication Network Lab.
Birth-Death Process (4)
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
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Wireless Communication Network Lab.
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.
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Wireless Communication Network Lab.
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)
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System
Wireless Communication Network Lab.
M/M/1 Queuing System (3)
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
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Wireless Communication Network Lab.
M/M/1 Queuing System (4)
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
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Wireless Communication Network Lab.
M/M/1 Queuing System (5)
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)
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Wireless Communication Network Lab.
M/M/1 Queuing System (6)
By using the Little’s Law, the average dwell time (or system time) of customers is
1
(1 )
1
SS
LW
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Wireless Communication Network Lab.
M/M/1 Queuing System (7)
The average queue length
1
2
2
( 1) ( )
1
( )
qi
L i P i
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Wireless Communication Network Lab.
M/M/1 Queuing System (8)
The average waiting time of customers is
2
(1 )
( )
LW
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Wireless Communication Network Lab.
M/M/S/ Queuing Model
Queue
S
Servers
S
.
.
2
1
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Wireless Communication Network Lab.
State Transition Diagram
2 3 (S-1) S S S
0 1 2 S-1 S S+1……
…
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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)
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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.
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Wireless Communication Network Lab.
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,..}.
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
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
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Wireless Communication Network Lab.
Homework
P2.4P2.7P2.14P2.17P2.20
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