1. Chapter 2 Probability, Statistics and Traffic Theories
Prof. Chih-Cheng Tseng
EE of NIU
Chih-Cheng Tseng

2. 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

3. 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

4. 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
0.2 5
Probability
Number
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Chih-Cheng Tseng

5. Discrete Random Variables (2)
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
EE of NIU
Chih-Cheng Tseng

6. Continuous Random Variables
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;
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Chih-Cheng Tseng

7. 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
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Chih-Cheng Tseng

8. Probability Density Function (pdf)
The pdf fX(x) of a continuous RV X is the derivative of the CDF FX(x): x
fX(x)
Area
CDF
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Chih-Cheng Tseng

9. 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
EE of NIU
Chih-Cheng Tseng

10. Discrete RV --- nth Moment
The n-th moment
The first moment of X is simply the expected value.
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Chih-Cheng Tseng

11. Discrete RV --- nth Central Moment
The nth central moment is the moment about the mean value
The first central moment is equal to 0.
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12. Discrete RV --- Variance
The variance or the 2nd central moment
where s is called the standard deviation
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13. 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
EE of NIU
Chih-Cheng Tseng

14. Continuous RV --- nth Moment, nth Central Moment and Variance
The nth moment
The nth central moment
Variance or the 2nd central moment
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15. 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
EE of NIU
Chih-Cheng Tseng

16. 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.
EE of NIU
Chih-Cheng Tseng

17. 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)
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Chih-Cheng Tseng

18. 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
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19. Distributions of Continuous RVs (1)
Normal distribution
The pdf of the normal RV X is
The CDF can be obtained by
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Chih-Cheng Tseng

20. 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.
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Chih-Cheng Tseng

21. 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
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22. 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
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23. A joint pmf of the discrete RVs X1, X2, …, Xn is
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.
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Chih-Cheng Tseng

24. Multiple RVs (2)
joint CDF
joint pdf
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25. Conditional Probability
A conditional prob. is the prob. that X1=x1 given X2=x2, …, Xn=xn
For discrete RVs
For continuous RVs
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Chih-Cheng Tseng

26. P(Ai│B) (read as: the prob. of B, given Ai) is
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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Chih-Cheng Tseng

27. 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
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Chih-Cheng Tseng

28. 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
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29. 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
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Chih-Cheng Tseng

30. 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
EE of NIU
Chih-Cheng Tseng

31. 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.
EE of NIU
Chih-Cheng Tseng

32. 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 .
EE of NIU
Chih-Cheng Tseng

33. 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 .
EE of NIU
Chih-Cheng Tseng

34. 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.
EE of NIU
Chih-Cheng Tseng

35. 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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Chih-Cheng Tseng

36. 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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Chih-Cheng Tseng

37. A and B can take any of the following distributions types
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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38. 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
EE of NIU
Chih-Cheng Tseng

39. 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.
EE of NIU
Chih-Cheng Tseng

40. 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

41. Birth-Death Process (2)
The state transition diagram of the continuous birth-death process 1 2 3
n-1 n
n+1
n+2
n-1 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.
EE of NIU
Chih-Cheng Tseng

42. 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.
EE of NIU
Chih-Cheng Tseng

43. Birth-Death Process (4)
For state 0,
For state 1,
For state n,
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44. 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.
EE of NIU
Chih-Cheng Tseng

45. 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 
System  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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Chih-Cheng Tseng

46. M/M/1 Queuing System (3) The equilibrium state equations are given by
So,
r =l/m is the flow intensity and r < 1
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47. The normalized condition is given by
M/M/1 Queuing System (4)
The normalized condition is given by
Since,
Therefore,
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48. The average number of customers in the system is given by
M/M/1 Queuing System (5)
The average number of customers in the system is given by
Typo in Eq. (2.64)
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Chih-Cheng Tseng

49. M/M/1 Queuing System (6)
By using the Little’s Law, the average dwell time (or system time) of customers is
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50. The average queue length
M/M/1 Queuing System (7)
The average queue length
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51. The average waiting time of customers is
M/M/1 Queuing System (8)
The average waiting time of customers is
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52. M/M/S/ Queuing Model Servers S . 2 1 S  Queue EE of NIU
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53. State Transition Diagram1 2
S-1 S
S+1 ……  …  2 3
(S-1) S
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54. The Equilibrium State Equations
The steady state probabilities
a=l/m
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55. Finding P(0) Based on the normalized condition We can obtain
If a<S and the utilization factor ρ =l/(Sm),
Typo in Eq. (2.73)
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Chih-Cheng Tseng

56. 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
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57. Queuing System Metrics (2)
The average queue length is
The average waiting time of customers is
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58. 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.
EE of NIU
Chih-Cheng Tseng

59. 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,..}.
EE of NIU
Chih-Cheng Tseng

60. 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
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Chih-Cheng Tseng

61. Queuing System Metrics (2) (Optional)
Average waiting time of customers in the queue is
The average queue length is
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62. Homework
P2.4
P2.7
P2.14
P2.17
P2.20
EE of NIU
Chih-Cheng Tseng