TKK · 2004-01-12 · Lect01.ppt S-38.145 - Introduction to Teletraffic Theory – Spring 2004 1 1....

1Lect01.ppt S-38.145 - Introduction to Teletraffic Theory – Spring 2004

1. Introduction

2

1. Introduction

Contents

• Purpose of Teletraffic Theory

• Teletraffic models

• Classical model for telephone traffic • Classical model for data traffic

3

1. Introduction

• Telecommunication system from the traffic point of view:

• Ideas: – the system serves the incoming traffic– the traffic is generated by the users of the system

Traffic point of view

system

incomingtraffic

outgoingtrafficusers

4

1. Introduction

Interesting questions

• Given the system and incoming traffic, what is the quality of service experienced by the user?

• Given the incoming traffic and required quality of service, how should the system be dimensioned?

• Given the system and required quality of service, what is the maximum traffic load?

system

incomingtraffic

outgoingtrafficusers

5

1. Introduction

General purpose (1)

• Determine relationships between the following three factors:– quality of service– system capacity– traffic load

quality of service

traffic loadsystem capacity

6

1. Introduction

General purpose (2)

• System can be– a single device (e.g. a link between two telephone exchanges, a processor

routing packets in a data network, a statistical multiplexer in ATM network) or

– a whole communications network (e.g. telephone or data network) or a part of it

• Typically a system consists of– a device (hardware) and

– principles controlling it (software)

• Traffic, on the other hand, consists of (depending on the case)– calls, packets, bursts, cells, etc.

7

1. Introduction

General purpose (3)

• Quality of service can be described– from the users point of view

• e.g. call blocking probability, distribution of delay experienced by a packet stream

– from the system point of view (= performance of the system)

• e.g. utilization

• It is also possible to think quality of service– from the offered traffic point of view

• e.g. blocking experienced by ATM-connection requests, or other connection level property; grade of service

– from the carried traffic point of view

• e.g. cell loss during an accepted ATM connection, or other criteria during an accepted connection; quality of service

8

1. Introduction

Example

• Telephone traffic– traffic = telephone calls

– system = telephone network

– quality of service = probability that “the line is not busy”

PRRRR!!!1234567

9

1. Introduction

Relationships between the three factors

• Qualitatively, the relationships are as follows:

• To describe the relationships quantitatively,mathematical models are needed

system capacity quality of service quality of service

traffic load traffic load system capacity

with givenquality of service

with givensystem capacity

with giventraffic load

10

1. Introduction

Teletraffic models

• Teletraffic models are usually stochastic (= probabilistic)– systems themselves are usually deterministic

but traffic is typically stochastic

• The underlying reason for it is thus the stochastic nature of traffic:– “you never know, who calls you and when”

• It follows that the variables, which are needed to describe the quality of service, are also stochastic by nature, i.e. they are random variables:

– number of ongoing calls– number of packets in a buffer

• Random variable is described by its distribution, e.g.– probability that there are n ongoing calls

– probability that there are n packets in the buffer

• A stochastic process describes the temporal development of a random variable

11

1. Introduction

Related fields

• Probability Theory

• Stochastic Processes

• Queueing Theory• Statistical Analysis (of traffic measurements)

• Operations Research

• Optimization Theory• Decision Theory (Markov decision processes)

• Simulation Techniques (object oriented programming)

12

1. Introduction

Difference between the real system and the model

• It is good to keep in mind the difference between the real system and the model

– (Ideally), a model is a description of only a certain part or property of the real system

– Often, due to various reasons, the model is not very accurate but rather approximative

caution is needed when conclusions are drawn

13

1. Introduction

Practical goals

• Network planning– dimensioning

– optimization

– performance analysis

• Network management and control– efficient operation

– fault recovery– traffic management

– routing

– accounting

14

1. Introduction

Literature

• Teletraffic Theory– V. B. Iversen, Chapter 1 of “Teletraffic Engineering Handbook”, available from

http://www.tele.dtu.dk/teletraffic/– Teletronikk (1995) Vol. 91, Nr. 2/3, Special Issue on “Teletraffic”

– COST 242, Final report (1996) “Broadband Network Teletraffic”, Eds. J. Roberts, U. Mocci, J. Virtamo, Springer

– J.M. Pitts and J.A. Schormans (1996) “Introduction to ATM Design and Performance”, Wiley

– J. Roberts, “Traffic Theory and the Internet”, available from http://www.comsoc.org/ci/public/preview/roberts.html

• Queueing Theory– L. Kleinrock (1975) “Queueing Systems, Volume I: Theory”, Wiley

– L. Kleinrock (1976) “Queueing Systems, Volume II: Computer Applications”, Wiley– D. Bertsekas and R. Gallager (1992) “Data Networks”, 2nd ed., Prentice-Hall

– P.G. Harrison and N.M. Patel (1993) “Performance Modelling of Communication Networks and Computer Architectures”, Addison-Wesley

15

1. Introduction

Contents

• Purpose of Teletraffic Theory



16

1. Introduction

Teletraffic models

• Two phases of teletraffic modelling:

– modelling of the incoming traffic traffic model

– modelling of the system itself system model

• Roughly, teletraffic models can be divided into two categories by the system model

– loss systems (loss models)

– waiting/queueing systems (queuing models)

• During this course, we present simple teletraffic models describing a single resource

• These models can be combined to create models for whole telecommunication networks

– loss networks– queueing networks

17

1. Introduction

Simple teletraffic model

• Customers arrive at rate λ (customers per time unit on average)

– 1/λ = average inter-arrival time

• Customers are served by n parallel servers

• When busy, a server serves at rate µ (customers per time unit)

– 1/µ = average service time of a customer

• There are m waiting places in the system

• It is assumed that blocked customers (arriving in a full system) are lost

1

n

mλ

µ

18

1. Introduction

Exercise

• Consider the simple teletraffic model presented on the previous slide– What is the traffic model?– What is the system model?

1

n

mλ

µ

19

1. Introduction

Pure loss system

• No waiting places (m = 0)

– If the system is full (with all n servers occupied) when a customer arrives, she is not served at all but lost. System is thus lossy.

• From the customer’s point of view, it is interesting to know e.g. – What is the probability that the system is full when she arrives?

• From the system’s point of view, it is interesting to know e.g. – What is the utilization factor of the servers?

1

n

µ

λ

20

1. Introduction

Pure waiting system

• Infinite number of waiting places (m = ∞)

– If all n servers are occupied when a customer arrives, she occupies one of the waiting places. No customers are lost but some of them have to wait before getting served; system is thus lossless.

• From the customer’s point of view, it is interesting to know e.g. – what is the probability that she has to wait “too long”?

• From the system’s point of view, it is interesting to know e.g. – what is the utilization factor of the servers?

1

n

∞λ

µ

21

1. Introduction

Mixed system

• Finite number of waiting places (0 < m < ∞)

– If all n servers are occupied but there are free waiting places when a customer arrives, she occupies one of the waiting places

– If all n servers and all m waiting places are occupied when a customer arrives, she is not served at all but lost

– Some customers are lost and some customers have to wait before getting served. Also this system is thus lossy.

1

n

mλ

µ

22

1. Introduction

Infinite system

• Infinite number of servers (n = ∞)– No customers are lost or even have to wait before getting served; it is a

lossless system.

– This (hypothetical) system is often much more simple to analyze than the corresponding real system with finite capacity.

– Sometimes, this can be the only way to get even some approximate results for the real system

1

∞

µ

λ•••

23

1. Introduction

Little’s formula

• Consider a system where

– new customers arrive at rate λ• Stability assumption:

– Customers do not accumulate in the system, occasionally system is empty

• Consequence:

– Customers depart from the system at rate λ• Denote

• Little’s formula:

λ λ

systemin thespendscustomer a timeaverage

systemin thecustomersofnumber average

==

T

N

TN λ=

24

1. Introduction

Contents

• Purpose of the Teletraffic Theory



25

1. Introduction

Classical model for telephone traffic (1)

• Loss models have traditionally been used to describe (circuit-switched) telephone networks

– Pioneering work made by Danish mathematician A.K. Erlang (1878-1929)

• Consider a link between two telephone exchanges (classical teletraffic problem)

– traffic consists of the ongoing telephone calls on the link

26

1. Introduction

Classical model for telephone traffic (2)

• Erlang modelled this as a pure loss system (m = 0)– customer = call

• λ = call arrival rate

– service time = (call) holding time

• h = 1/µ = average holding time

– server = channel on the link

• n = number of channels on the link

1

n

µ

λ

27

1. Introduction

Traffic process

654321

6543210

time

time

call arrival timesblocked call

channel-by-channeloccupation

nr of channelsoccupied

traffic volume

call holdingtime

chan

nels

nr o

f cha

nnel

s

28

1. Introduction

Traffic intensity

• In telephone networks:

• The amount of traffic is described by the traffic intensity a• Definition: the traffic intensity a is

the product of the arrival rate λ and the mean holding time h:

• Note that the traffic intensity is a dimensionless quantity. However, to emphasize the context, the “unit” of the traffic intensity a is called erlang (erl)

• By Little’s formula: the traffic intensity tells the number of ongoing calls in the corresponding (hypothetical) infinite system.

ha λ=

Traffic ↔ Calls

29

1. Introduction

Example

• Consider a local exchange. Assume that, – on the average, there are 1800 new calls in an hour, and

– the mean holding time is 3 minutes

• It follows that the traffic intensity is

• If the mean holding time increases from 3 minutes to 10 minutes, then

erlang9060/31800 =∗=a

erlang30060/101800 =∗=a

30

1. Introduction

Characteristic traffic

• Here are typical characteristic traffics for some subscriber categories (of ordinary telephone users):

– private subscriber: 0.01 - 0.04 erlang

– business subscriber: 0.03 - 0.06 erlang

– private branch exchange (PBX): 0.10 - 0.60 erlang– pay phone: 0.07 erlang

• This means that, for example, – a typical private subscriber uses from 1% to 4% of her time in the telephone

(during a “busy hour”)

• Referring to the previous example, note that – it takes between 2250 - 9000 private subscribers to generate 90 erlang

traffic

31

1. Introduction

Blocking

• In a loss system some calls are lost– a call is lost if all n channels are occupied when the call arrives

– the term blocking refers to this event

• There are (at least) two different types of blocking quantities:– Call blocking Bc = probability that an arriving call finds all n channels

occupied = the fraction of calls that are lost

– Time blocking Bt = probability that all n channels are occupied at an arbitrary time = the fraction of time that all n channels are occupied

• The two blocking quantities are not necessarily equal– If calls arrive according to a Poisson process, then Bc = Bt

• Call blocking is a better measure for the quality of service experienced by the subscribers

• Typically, time blocking is easier to calculate

32

1. Introduction

Call rates

• In a loss system, there are three types of call rates:

– λoffered = arrival rate of all call attempts

– λcarried = arrival rate of carried calls

– λlost = arrival rate of lost calls

• Note:

clost

ccarried

lostcarriedoffered

)1(

B

B

λλλλ

λλλλ

=−=

=+=

λoffered λcarried

λlost

33

1. Introduction

Traffic streams

• The three call rates lead to the following three traffic concepts:

– Traffic offered aoffered = λofferedh

– Traffic carried acarried = λcarriedh

– Traffic lost alost = λlosth

• Note:

• Traffic offered and traffic lost are hypothetical quantities, but traffic carried is measurable; by Little’s formula, it is the average number of occupied channels on the link

clost

ccarried

lostcarriedoffered

)1(

aBa

Baa

aaaa

=−=

=+=

34

1. Introduction

Teletraffic analysis

• System capacity– n = number of channels on the link

• Traffic load– a = (offered) traffic intensity

• Quality of service (from the subscribers’ point of view)– Bc = probability that an arriving call finds all n channels occupied

• If we assume an M/G/n/n pure loss system, that is

– calls arrive according to a Poisson process (with rate λ)

– call holding times are independently and identically distributed according to any distribution with mean h

35

1. Introduction

Erlang’s formula

=

==n

iia

na

i

n

anB

0!

!c ),Erl(

• Then the quantitive relation between the three factors (system capacity, traffic load and quality of service) is given by the Erlang’s formula

• Note:

• Other names:– Erlang’s blocking formula

– Erlang’s B-formula

– Erlang’s loss formula– Erlang’s first formula

1!0,12)1(! =⋅⋅⋅−⋅= nnn

36

1. Introduction

Example

• Assume that there are n = 4 channels on a link and the offered traffic is a = 2.0 erlang. Then the call blocking probability Bc is

• If the link capacity is raised to n = 6 channels, then Bc reduces to

%5.9212

2121)2,4Erl(

2416

68

242416

!42

!32

!22

!42

c 432

4

≈=++++

=++++

==B

%2.121

)2,6Erl(

!62

!52

!42

!32

!22

!62

c 65432

6

≈++++++

==B

37

1. Introduction

Required capacity vs. traffic

• Given the quality of service requirement that Bc < 20%,the required capacity n depends on the traffic intensity a as follows:

2.0),Erl(|,2,1min)( <== aNNan

capacity n

traffic a

10 20 30 40 50

10

20

30

40

50

38

1. Introduction

Required quality of service vs. traffic

• Given the capacity n = 10 channels, the required quality of service 1 − Bc depends on the traffic intensity a as follows:

),10Erl(1)(1 c aaB −=−

10 20 30 40 500

0.2

0.4

0.6

0.8

1

traffic a

quality of service 1 − Bc

39

1. Introduction

Required quality of service vs. capacity

• Given the traffic intensity a = 10.0 erlang, the required quality of service 1 − Bc depends on the capacity n as follows:

)0.10,Erl(1)(1 c nnB −=−

quality of service 1 − Bc

capacity n

10 20 30 40 500

0.2

0.4

0.6

0.8

1

40

1. Introduction

Contents

• Purpose of the Teletraffic Theory



41

1. Introduction

Classical model for data traffic (1)

• Queueing models are suitable for describing (packet-switched) data networks

– Pioneering work made by many people in 60’s and 70’s (ARPANET), especially L. Kleinrock (http://www.lk.cs.ucla.edu/)

• Consider a link between two packet routers– traffic consists of data packets transmitted on the link

R

R

R

R

42

1. Introduction

Classical model for data traffic (2)

• This can be modelled as a pure waiting system with a single server (n = 1) and an infinite buffer (m = ∞)

– customer = packet

• λ = packet arrival rate (packets per time unit)

• L = average packet length (data units)

– server = link, waiting places = buffer

• C = link’s speed (data units per time unit)

– service time = packet transmission time

• 1/µ = L/C = average packet transmission time

1

n

∞λ

µ

43

1. Introduction

Traffic process

packet arrival times

link utilizationtime

state of packets in the system (waiting/being transmitted)

number of packets in the system

waitingtime

transmissiontime

time

time

4

3

2

1

0

1

0

44

1. Introduction

Traffic load

• In packet-switched data networks:

• The amount of traffic is described by the traffic load ρ• Definition: the traffic load ρ is

the quotient between the arrival rate λ and the service rate µ = C/L:

– Note that the traffic load is a dimensionless quantity (as traffic intensity in loss systems)

– By Little’s formula:traffic load tells the number of customers in service = probability of “server busy” = the utilization factor of the server

C

Lλµλρ ==

Traffic ↔ Packets

45

1. Introduction

Example

• Consider a link between two packet routers. Assume that, – on the average, 10 new packets arrive in a second,

– the mean packet length is 400 bytes, and

– the link speed is 64 kbps.

• It follows that the traffic load is

• If the link speed is increased to 150 Mbps, then the load is just

• Note:– 1 byte = 8 bits

– 1 kbps = 1 kbit/s = 1 kbit per second = 1,000 bits per second– 1 Mbps = 1 Mbit/s = 1 Mbit per second = 1,000,000 bits per second

%505.0000,64/840010 ==∗∗=ρ

%02.00002.0000,000,150/840010 ==∗∗=ρ

46

1. Introduction

Teletraffic analysis

• System capacity– C = link speed in kbps

• Traffic load

– λ = packet arrival rate in packet/s (considered here as a variable)

– L = average packet length in kbits (assumed here to be constant 1 kbit)

• Quality of service (from the users’ point of view)– Pz = probability that a packet has to wait “too long”, that is longer than a

given reference value z (assumed here to be constant 0.1 s)

• If we assume an M/M/1 queueing system, that is

– packets arrive according to a Poisson process (with rate λ)

– packet lengths are independent and identically distributed according to exponential distribution with mean L

47

1. Introduction

• Then the quantitive relation between the three factors is given by the following waiting time formula

• Note:

– The system is stable only in the former case (ρ < 1), otherwise the number of packets in the queue will grow to infinity eventually

Waiting time formula for an M/M/1 queue

≥≥<<−−

==)1(if,1

)1(if),)(exp(),;,Wait(

ρλρλλ

λλ

CL

CLzzLCP L

CCL

z

48

1. Introduction

Example

• Assume that packets arrive at rate λ = 50 packet/s and the link speed is C = 64 kbps. Then the probability Pz that an arriving packet has to wait too long (i.e. longer than z = 0.1 s) is

• Note that the system is stable, since

%19))4.1exp()1.0,1;50,64Wait(6450 ≈−==zP

16450 <== C

Lλρ

49

1. Introduction

Required link speed vs. arrival rate

• Given the quality of service requirement that Pz < 20%,the required link speed C depends on the arrival rate λ as follows:

2.0)1.0,1;,Wait(|min)( <>= λλλ cLcC

arrival rate λ10 20 30 40 50

0

10

20

30

40

50

60

70

link speed C

50

1. Introduction

Required quality of service vs. arrival rate

• Given the link speed C = 50 kbps, the required quality of service 1 − Pz depends on the arrival rate λ as follows:

)1.0,1;,50Wait(1)(1 λλ −=− zP

10 20 30 40 500

0.2

0.4

0.6

0.8

1

arrival rate λ

quality of service 1 − Pz

51

1. Introduction

Required quality of service vs. link speed

• Given the arrival rate λ = 50 packet/s, the required quality of service 1 − Pz depends on the link speed C as follows:

)1.0,1;50,Wait(1)(1 CCPz −=−

link speed C

60 70 80 90 1000

0.2

0.4

0.6

0.8

1

quality of service 1 − Pz


2. Modelling of telecommunication systems(part 1)

2

2. Modelling of telecommunication systems (part 1)

Contents

• Telecommunication networks

• Network level: switching and routing

• Link level: multiplexing and concentration• Shared media: multiple access

3


Resource Number Max. Util.

N(N-1) 1%

N(N-1)/2 1%

0 --

A B

DC

Example: Why networks? (1)

• Assume that – there are N = 100 persons who

want to be connected with each other

• Solution 1:Separate networks

– point-to-point links with separate terminals for each user pair

– no switches

• Comments:– no resource sharing

– very low utilization

4


A B

DC


• Simultaneous connections A-B and C-D in solution 1

5


A B

C D


N 100%

N(N-1)/2 1%

N 100%


• Still assume that – there are N = 100 persons who


• Solution 2:Fully meshed network

– one terminal per user– one 1x(N-1) switch per user

– point-to-point links for each switch pair

• Comments:– partial resource sharing– better utilization

6


A B

C D



7


D

BA

C


• Still assume that – there are N = 100 persons who


• Solution 3:Star network

– one terminal per user– just one NxN switch for the

whole network

• Comments:– complete resource sharing

– best utilization

• In reality, network topologies are much more complex. Why?


N 100%

N 100%

1 100%

8


A B

C D



9


• A simple model of a telecommunication network consists of

– nodes• terminals

• network nodes

– links between nodes

• Access network– connects the terminals to the

network nodes

• Trunk network– connects the network nodes to

each other

Telecommunication network

10


Shared medium as an access network

• In the previous model,– connections between terminals

and network nodes are point-to-point type ( no resource sharing within the access netw.)

• In some cases, such as – mobile telephone network– local area network (LAN)

connecting computers

the access network consists of shared medium:

– users have to compete for the resources of this shared medium

– multiple access (MA)techniques are needed

11


Network topologies

star tree

meshed fullymeshed

ring

bus

12


Network hierarchy

• Networks typically constructed level-by-level

• Relations to network topology– flat topologies (topology within one level)

– hierarchical topologies

• One natural hierarchy: – access vs. trunk network

• Traditionally: – many hierarchical levels (5 in AT&T)

• Current tendency: – to reduce the number of levels in hierarchy– “We see future large national networks with only three levels.”

13


Contents




14


Switching modes

• Circuit switching– telephone networks

– mobile telephone networks

– even applied to data networks

• Packet switching– data networks

– two possibilities• connection oriented: e.g. X.25, Frame Relay

• connectionless: e.g. Internet (IP), SS7 (MTP)

• Cell switching– special case of packet switching; fixed length packets (cells)

– integration of different traffic types (voice, data, video) multiservice networks (e.g. ATM-networks)

15


Circuit switching (1)

• Connection oriented:– connections set up end-to-end

before information transfer– resources reserved for the

whole duration of connection

• Information transfer as continuous stream

A

B

16


Circuit switching (2)

• Before information transfer– delay (to set up the connection)

• During information transfer– no overhead– no extra delays

A

B

17


Connectionless packet switching (1)

• Connectionless:– no connection set-up

– no resource reservation

• Information transfer as discrete packets

– varying length

– contains a header, with e.g. global address (of the destination)

A

B

BB

B

B

18


Connectionless packet switching (2)

• Before information transfer– no delays

• During information transfer– overhead (header bytes)– packet processing delays

– queueing delays (since packets compete for joint resources) A

B

BB

B

B

19


Connection oriented packet switching (1)

• Connection oriented:– end-to-end virtual connections

set up before information transfer

– no resource reservation

• Information transfer as discrete packets

– varying length

– local address (logical channel index)

• essentially shorter than global address

21

4

1

A

B

20


Connection oriented packet switching (2)

• Before information transfer– delay (to set up the virtual

connection)

• During information transfer– overhead (however, less than in

connectionless mode)

– packet processing delays (less, due to the shorter address)

– queueing delays (since packets compete for joint resources)

21

4

1

A

B

21


Cell switching (1)

• Connection oriented:– end-to-end virtual connections

set up before information transfer

– resource reservation possible but not mandatory

• Information transfer as discrete packets (cells)

– fixed (small) length– local address (VPI/VCI)

21

4

1111

4

A

B

22


Cell switching (2)

• Before information transfer– delay (to set up the connection)

• During information transfer– overhead (per packet even more

than in connectionless mode)

– packet processing delays (less, due to the shorter address and the fixed length of cells)

– queueing delays (if resources not reserved beforehand)

21

4

1111

4

A

B

23


Switching modes: summary

• Circuit switching– suitable for real-time traffic

(voice, RT-video, …)– inefficient for VBR traffic and

data– transparent but inflexible

• Cell switching– quite flexible– efficient use of network

resources

– seq. integrity guaranteed– real-time guarantees possible

– possible to integrate different traffic types

• Connection oriented packet switching

– quite flexible

– efficient use of network resources

– seq. integrity guaranteed

– no real-time guarantees

• Connectionless packet switching– flexible and fault tolerant

– efficient use of network resources

– seq. integrity not guaranteed

– no real-time guarantees

24


Contents




25


Analogue vs. digital systems (1)

• Originally telecommunication networks (i.e. telephone networks) were purely analogical

– First: digital trunks between exchanges

– Then: digital exchanges

– In the current telephone network, the telephone itself and the access line are still based on the analogue technology

– ISDN and GSM are the first completely digital telephone networks(including the terminals and the access part)

• Packet switched networks have always been completely digital– e.g. LANs

• Cell switched networks (ATM) are also completely digital

26


Analogue vs. digital systems (2)

• Analogical circuit switched system:– one connection occupies a single one or a multiple of channels

– link capacity n expressed in number of channels

• Digital circuit switched system:– one connection occupies a single one or a multiple of channels

– channel capacity expressed in bits per second (bps, kbps, Mbps, ...)

• typically: 64 kbps– link capacity expressed either in number of channels or in bits per second

(being then a multiple of the channel capacity)

• Digital packet/cell switched system:– link capacity occupied dynamically on-demand

– capacity demand (of a connection) expressed in bits per second

– link capacity C expressed in bits per second

27


multiplexer1

n

1

Transmission: multiplexing (1)

• Originally, – each connection in a telephone network required its own physical link

• By multiplexing, the capacity of a single physical link is divided into multiple channels

– each connection typically occupies one channel– thus, multiple connections can be conveyed by a single link

• The device implementing this is called a multiplexer

28


Transmission: multiplexing (2)

• In circuit switched networks, there are two different multiplexing techniques:

– frequency division multiplexing (FDM)

– time division multiplexing (TDM)

• Multiplexing principle used in packet switched networks is – statistical multiplexing

multiplexer1

n

1

29


1 channelFDM multiplexer1

n

1n channels

Frequency division multiplexing (FDM)

• FDM– oldest multiplexing technique, used in analogue circuit switched systems

– fixed portion (frequency band) of the link bandwidth reserved for each channel

– reserved bandwidth identifies the connection

• FDM multiplexer is lossless– input: n 1-channel physical connections

– output: 1 n-channel physical connection

30


Time division multiplexing (TDM)

• TDM– used in digital circuit switched systems

– information conveyed on a link transferred in time frames of fixed length

– fixed portion (time slot) of each frame reserved for each channel– location of reserved slot in frame identifies the connection

• TDM multiplexer is lossless– input: n 1-channel physical connections

– output: 1 n-channel physical connection

TDM multiplexer

1

slot

frame

1

n31


Statistical multiplexing

• Statistical multiplexing– used in digital packet/cell switched systems (e.g. Internet, ATM)

– information transferred in packets of varying or fixed length

– each packet belongs to exactly one connection• packet header includes the connection identifier

– link capacity reserved dynamically and asynchronously as packets arrive need for buffering

packet

Statistical multiplexer1

n

1

32


Statistical multiplexer

• Statistical multiplexer is (typically) lossy

– input: n physical connections with link speeds Ci (i = 1,…,n)

– output: 1 physical connection with link speed C ≤ C1 + ... + Cn

• However, the loss probability can be decreased by enlarging the buffer– with an “infinite” buffer, it is enough that C exceeds

the average aggregated input rate

Statistical multiplexer

1C

C1

Cn

1

n33


C/L

Teletraffic model for a statistical multiplexer

• Multiplexer can be modelled as – a pure waiting system (as below) if the buffer is large

– a mixed system if the buffer is small

• Traffic consists of packets– each packet is transmitted with the full link speed C

– let L denote the average packet length

– packet service rate µ will be µ = C/L

– stability requirement: packet arrival rate λ < µ

λ

34


Transmission: concentration

• Concentration– used in circuit switched systems (analogue/digital)

• typically in the access network part for economical reasons

• however, switches are also (implicitly) concentrators

• In concentration, – traffic (= connections) from n 1-channel links is concentrated on

a single m-channel link, where m < n

– Idea: the probability that all n connections are active is typically very small

concentrator

1

m

1

n35


Concentrator

• Concentrator is lossy– input: n 1-channel physical connections

– output: 1 m-channel physical connection with (m < n)

• Outgoing link should be dimensioned (i.e. m should be chosen) so that

– the call blocking probability (that all m channels are occupied when a new call arrives) is small enough

– In other words: the quality of service requirement should be fulfilled

concentrator

1

m

1

n

36


1

m

1/h

λ(n,x)

Teletraffic model for a concentrator

• Concentrator can be modelled as – a pure loss system (as below) with m parallel servers

• Traffic consists of connections– traffic generated by a finite number (n) of sources

arrival rate λ is not constant but depends on the total number of sources (n) and the number of active sources (x): λ = λ(n,x)

– let h denote the average connection holding time

– service rate µ will be µ = 1/h

37


Contents




38


Multiple access techniques used in mobile telephone networks

• Mobile telephone networks are geographically divided into cells– Each cell has its own base station

• The radio frequency band reserved for the network access (within a cell) is divided into channels

– The users (located in that cell) compete for these channels– Dynamic channel assignment is made centralized by the base station

• Various multiple access methods: – frequency division multiple access (FDMA)

– time division multiple access (TDMA)– code division multiple access (CDMA)

39


FDMA and TDMA

• Frequency division multiple access (FDMA)– used in analogue mobile networks, e.g. NMT– radio frequency band reserved for the network divided into subbands

(channels)– each connection occupies one channel– thus, simultaneous connections use separate frequency (sub)bands– cf. frequency division multiplexing

• Time division multiple access (TDMA)– used in digital mobile networks, e.g. GSM– information transferred in frames of fixed length,– fixed portion (time slot) of each frame reserved for each channel– each connection occupies one channel– thus, simultaneous connections use the same frequency band, but different

time slots– utilization of the frequency band better than in FDMA– cf. time division multiplexing

40


CDMA

• Code division multiple access (CDMA)– used in digital mobile networks, e.g. IS-95 (USA)

– information coded before transfer in such a way that signals transmitted to a certain receiver can be received only at this receiver (others see the signals as white noise)

– each code corresponds to a channel

– each connection reserves one channel

– thus, simultaneous connections use the same frequency band but different codes

– in general, the utilization of the frequency band is better than in TDMA

– however, the notion “system capacity” in CDMA is elastic (contrary to FDMA and TDMA):

• the more codes (channels), the greater the interference!

41


Teletraffic modelling of various multiple access techniques

• All multiple access techniques mentioned above (FDMA, TDMA and CDMA) can be modelled as a pure loss system

• Traffic consists of calls– calls either fresh or handovers– fresh calls may arrive according to a Poisson process,

but is it so with the handovers?

– due to handovers, call holding time is now different from conversation holding time (from the base station point of view, call ends as soon as the phone has moved out of the coverage area of this station)

– one more new feature: mobility modelling

• System capacity (that is, the number of parallel channels) depends on – the width of the frequency band reserved for the network– the multiple access technique used

42


Multiple access techniques in computer LANs

• A computer LAN (local area network) transfers packets between any stations connected to this LAN

– the stations compete for this joint transmission medium

– the medium can be used by (only) one station at a time

– dynamic channel assignment is made in a fully distributed manner by the stations themselves

• Various multiple access methods:– Random Access: ALOHA, Slotted ALOHA (originally in satellite links)– Carrier Sense Multiple Access / Collision Detection (CSMA/CD):

Ethernet, IEEE 802.3– Token Bus: IEEE 802.4

– Token Ring: IEEE 802.5

– Carrier Sense Multiple Access / Collision Avoidance (CSMA/CA): IEEE 802.11 WLAN

43


Random Access

• Stations transmit packets totally independently of each other as soon as new packets arrive

– no prior actions to avoid collisions

– theoretical maximum for the throughput is less than 20% of the LAN speed

– e.g. ALOHA

• Assuming a fixed packet length, – a slotted system (slot = transmission time of a packet) doubles the

theoretical maximum throughput

– e.g. Slotted ALOHA

44


Analysis of Random Access (1)

• Assume that the stations generate fixed length packets according to a Poisson process with intensity ν

• Let T denote the time needed to transmit a packet

– stability requirement: ν < 1/T• In a non-synchronized system (ALOHA), two packets collide with each

other if and only if their interarrival time is < T– collided packets are retransmitted after a random interval (again and again

until the transmission is successful without collisions)

• Assume further that the aggregate packet stream (including all the transmitted and retransmitted packets) still obeys a Poisson process (which is certainly not true) but with a higher intensity λ such that λ <1/T. This assumption is called the Poisson approximation.

45


Analysis of Random Access (2)

• Consider a station where a new packet arrives at time 0– No collisions during the transmission (0,+T) time if and only if

no other packet arrivals (to any station) between time interval (−T,+T)

– Due to the Poisson approximation, this happens with probability exp(−2λT)

• Thus,

– the throughput ν is ν = λ⋅exp(−2λT)

• This is maximized by λmax = 1/(2T) corresponding to

– an offered load of λmaxT = 1/2 = 50%

• The maximum throughput νmax is

– νmax = λmax⋅exp(−2λmaxT) = 1/(2eT) ≈ 0.184/T ≈ 20% (1/T)

46


Analysis of Random Access

• Let’s consider a slotted system (Slotted ALOHA) where the packets are transmitted in slots of length T

– In this system, two packets collide with each other if and only if they arrived during the same slot

– Due to the Poisson approximation, no collisions with probability exp(−λT),(there are no collisions if during (-T,0) there are no other packet arrivals)

• Thus, the throughput ν is ν = λ⋅exp(−λT)– this is maximized by λmax = 1/T corresponding to

– an offered load of λmaxT = 1 = 100%

• The maximum throughput νmax is

– νmax = λmax⋅exp(−λmaxT) = 1/(eT) ≈ 0.368/T ≈ 40% (1/T)

• Note that this is twice the throughput of a pure ALOHA system


3. Modelling of telecommunication systems(part 2)

2


Contents

• Circuit switched network modelled as a loss network

• Packet switched network modelled as a queueing network

3


Teletraffic model of a circuit switched network (1)

• Consider a circuit switched network

– e.g. a telephone network

• Traffic:– telephone calls– each (carried) call occupies one

channel on each link among its route

• System:– telephone machines (terminals)– exchanges (network nodes)

– access links (from terminals to exchanges)

– trunks (between exchanges)

A

B

4


Teletraffic model of a circuit switched network (2)

• Quality of service:– described by the end-to-end

call blocking probability(prob. that a desired connection cannot be set up due to congestion along the route of the connection)

• In our model we assume that – the network nodes and the

whole access network are non-blocking

• Thus, a call is blocked – if and only if all channels are

occupied in any trunk network link along the route of that call

A

B

5


Links j = 1,…,J

• In our model,– all links are bidirectional (why?)

• We index the links in the trunk network by

– j = 1,…,J

– In the example on the right J = 6

• Let

– nj = nr of channels in link j(that is: the link capacity)

– n = (n1,…,nJ)

• Each link is modelled as a pureloss system

A

B

1

2

3

45

6

6


Routes r = 1,…,R

• We define a route as a – set of (bidirectional) links

connecting two network nodes

• Denote R as the number of different routes indexed by – r = 1,…,R– in the example on the right

R = 12 + 10 + 7 + 3 = 32– for example, there are three

routes between nodes a and b: 1,2, 6,3, 5,4,3

• Let djr = 1 if link j belongs to route r (djr = 0 otherwise)– D = (djr | j = 1,…,J; r = 1,…,R)

A

B

1

2

3

45

6a

b

7


Traffic classes and state of the system

• Assume that– connections in the network are routed always in the same way (fixed

routing)– In the prev. slide, the route between users A and B is selected to be 6,3

• Thus, end-to-end call blocking prob. is equal for all the connections following the same route

• Thus, the (traffic) class of a connection is determined by the route that the connection follows

• Let xr be the number of active connections following route r– x = (x1,…,xR)

• Vector x is called the state of the system

8


State space S

• The number of active connections xr for any traffic class r is limited by the link capacities nj along the corresponding route r :

• The same in vector form:

• Thus, the state space S (that is: the set of admissible states) is

• Note that, the state space S is R-dimensional and finite (why?)

jnxd j

R

rrjr allfor

1≤

=

nxD ≤⋅

|0 nxDx ≤⋅≥=S

9


Example

• 3 links with capacities:– link a-c: 3 channels

– link b-c: 3 channels

– link c-d: 4 channels

• 2 routes:– route a-c-d

– route b-c-d– The other 4 routes (which?) are

ignored in this model

• State space:

– S = (0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3),(2,0),(2,1),(2,2),(3,0),(3,1)

a

b

c d

0

4

3

2

1

10 2 3 4

S

x1

x2

02 ≥x01 ≥x

3

34

10


)(|0 nexDx ≤+⋅≥= rrS

Set Sr of non-blocking states for class r

• Consider – an arriving call belonging to class r (that is: following route r)

• It will not be blocked by link j belonging to route r– if there is at least one free channel on link j:

• The same in vector form (er being here the unit vector in direction r):

• The set Sr of non-blocking states is thus

rjnxd j

R

rrjr ∈≤

=allfor1-

1'''

nexD ≤+⋅ )( r

11


Set SrB of blocking states for class r

• The set SrB of blocking states

for class r is clearly:

• Summary: – an arriving call of class r is

blocked (and lost) if and only if the state x of the system belongs to set Sr

B

• Example (continued):– The blocking states S1

B for connections of class 1 (using route a-c-d) are marked in the figure

– S1B = (1,3),(2,2),(3,0),(3,1)

rBr SSS \=

a

b

c d

0

4

3

2

1

10 2 3 4x1

x2

12


Stationary state probabilities (1)

• Assume that – new connection requests belonging to traffic class r arrive (independently)

according to a Poisson process with intensity λr

– call holding times independently and identically distributed with mean h

• Denote

– ar = λrh (traffic intensity for class r)

13


• Then it is possible to show that the so called “stationary state probability” π(x) for any state x ∈ S is as follows:

where G is a normalizing constant:

and the functions fr(xr) are defined as follows:

∏=

− ⋅=R

rrr xfG

1

1 )()(xπ

!)(

r

xr

rr x

axf

r=


∏∈ =

=S

R

rrr xfG

x 1)(

14



• Probability π(x) is said to be of product-form– However, the number of active connections of different classes are not

independent (since the normalizing constant G depends on each xr)

– Only if all the links had infinite capacities, all the traffic classes would be independent of each other

– Thus, it is the limited resources shared by the traffic classes that makes them dependent on each other

15


PASTA

• Consider, for a while, – any simple teletraffic model (as defined in slide 15 of lecture 1)

with Poisson arrivals

• According to so called PASTA (Poisson Arrivals See Time Averages) property,

– arriving calls (obeying a Poisson process) see the system in the stationary state

• This is an important observation – applicable in many problems

• For example, – it allows us to calculate the end-to-end blocking probabilities in our circuit

switched network model (since we assumed that new calls arrive according to a Poisson process)

16


End-to-end call blocking: exact formula

• The probability that the system is (at an arbitrary time) in such a state that it cannot accept any more connections of type r (that is: the end-to-end time blocking probability for class r) is clearly given by the sum

• But, due to the PASTA property, – the end-to-end call blocking probability Br equals the corresponding end-to-

end time blocking probability:

• In this case that end-to-end time and call blocking are the same it is sufficient to refer to end-to-end blocking.

∈ BrSx

x)(π

∈=

BrS

rBx

x)(π

17


Example

• Consider the example presented in slide 9 and in slide 11

• The end-to-end blocking probability B1 for class 1 will be

( )( ) ( ) ( ) ( )!1!3!2!1!2!3!2!1!1!3!2!1

!1!3!2!2!3!1

1

12

31

22

12

21

32

22

12

11

32

22

12

12

31

22

21

32

11

1111

1

)1,3()0,3()2,2()3,1(

aaaaaaaaaaaa

aaaaaa

B

++++++++++++

+++

=+++= ππππ

18


Approximative methods

• In practice, – it is extremely hard (even impossible) to apply the exact formula– this is due to the so called state space explosion:

there are as many dimensions in the state spaces as there are routes in our model exponential growth of the state space

• Thus, approximative methods are needed– such as the simple product bound method– more complex reduced load approximation (Erlang fixed point

approximation)

• In both methods we– estimate first blocking probabilities in each separate link

(common to all traffic classes)– calculate then the end-to-end blocking probabilities for each class

based on the hypothesis that “blocking occurs independently in each link”

19


• Consider first the blocking probability B(j) in an arbitrary link j– Let R(j) denote the set of routes that use link j

• If the capacities of all the other links (but j) were infinite,

– link j could be modelled as a loss system where new calls arrive according to a Poisson process with intensity λ(j),

– In this case, the blocking probability could be calculated from formula

– Note that this is really an approximation, since the traffic offered to link j is smaller due to blockings in other links (and not even of Poisson type).

∈=

)()(

jRrrj λλ

),Erl()()(

∈≈

jRrrj anjB

Product Bound (1)

20


• Consider then the end-to-end call blocking probability Br for class r– Let J(r) denote the set of the links that belong to route r

– Note that an arriving call of class r will not be blocked, if it is not blocked in any link j ∈ J(r)

• If blocking occurred independently in each link, – an arriving call of class r would be blocked with probability

– Note that for (very) small values of B(j)’s, we can use the following approximation:

∏ ∈ −−≈ )( ))(1(1 rJjr jBB

∈≈ )( )(rJjr jBB

Product Bound (2)

21


Contents

• Circuit switched network modelled as a loss network

• Packet switched network modelled as a queueing network

22


Teletraffic model of a connectionless packet switched network (1)

• Consider a connectionless packet switched network

– e.g. an Internet subnetwork

• Traffic:– data packets which compete the

resources by queuing– identified by their source (A) and

destination (B)

• System:– workstations & servers

(terminals)– routers (network nodes)– access links

(from terminals to routers)– trunks (between routers)

A

B

BB

B

B

23


Teletraffic model of a connectionless packet switched network (2)

• Quality of service:– described e.g. by the average

end-to-end delay (the mean time to get from the source (A) to the destination (B))

• However, in our model – we restrict ourselves to the

average trunk network delay(the mean time to get from the source router (a) to the destination router (b))

– implicitly, we assume that the delay due to access network is negligible (or, at least, almost deterministic)

A

B

BB

B

B

a

b

24


Delay components

• Trunk network delay consists of – propagation delays (in links)

– transmission delays (in links)

– processing delays (in nodes)– queueing delays (before transmission and before processing)

• Note that – propagation and transmission delays are deterministic, – processing delays might be random, and

– queueing delays are surely random

• In our model, – we will take into account the transmission and the related queueing delays

– but we will ignore the propagation delays in links and the delays in nodes (the processing and the related queueing delays, which would require an extended model; think how to do it)

25


Links j = 1,…,J

• In this case we separate the directions so that

– all links are unidirectional (why?)

• We index the links in the trunk network by

– j = 1,…,J– in the example on the right

J = 12

• Let

– Cj = capacity of link j (in bps)

1

3

5

712

9

A

B

BB

B

B

2

46

11

810

a

b

26


Routes r = 1,…,R

• We define here a route as an – ordered set of (unidirectional)

links connecting two network nodes (called origin and destination)

• We index the routes by – r = 1,…,R– in the example on the right:

R = 2∗(12+10+7+3) = 64– there are three routes

from node a to node b: (1,3), (11,6), (10,8,6)

– for these routes, node a is the origin and node b is the destination

1

3

5

712

9

A

B

BB

B

B

2

46

11

810

a

b

27


Individual link model

• Each link is modelled as a – pure waiting system, I.e. with a single server (n=1) and an infinite buffer

(m= ∞)

• Let

– λj = arrival rate of packets to be transmitted on link j (in packets/s)

– L = mean packet length (in bits)

– 1/µj = L/Cj = average packet transmission time on link j (in seconds)

• Stability requirement: λj < µj

Cj/Lλj

28


• Let

– λ(r) = arrival rate of packets following route r

– R(j) = the set of routes that use link j• these routes are found from the routing tables of the routers in a form

which usually relates the destination address to the corresponding next link on the path

• It follows that

∈=

)()(

jRrj rλλ

Packet arrival rates in links

29


Queueing network model

1

3

5

712

9

A

B

2

4

6

11

810

• The delay experienced by a packet on a route r is (in this model) the sum of the queuing delays along the route

• Note: – Average end-to-end delay is

equal for all the packets following the same route

• Thus, – the traffic class of a packet is

determined by the route r that the connection follows

a

b

30


State space S

• Let xj be the number of packets in queue j(including the packet being transmitted (if any))

– x = (x1,…,xJ)

• Vector x is called the state of the system– A more detailed state description (including the position and traffic class of

each packet in the whole system) is not needed under the assumptions that we will make later!

• In this case, as xj can have any non-negative value, the state space Sis

• Note that the state space S is now infinite

0 ≥= xS

31


02 ≥x01 ≥x

0

4

3

2

1

10 2 3 4

S

x1

x2

a b c1 2

Example

• 2 links:– link a-b

– link b-c

• 3 routes:– route a-b

– route b-c

– route a-b-c

• State space:

– S = (0,0),(1,0),(0,1), (2,0),(1,1),(0,2), (3,0),(2,1),(1,2),(0,3), ...

32


Stationary link state probabilities (1)

• Assume that – new packets following route r arrive (independently)

according to a Poisson process with intensity λ(r)– packet lengths are independently and exponentially distributed

with mean L

• It follows that – new packets to be transmitted on link j arrive (independently)

according to a Poisson process with intensity λj, where

– packet transmission times are independently and exponentially distributed with mean 1/µj = L/Cj

∈=

)()(

jRrj rλλ

33


• Assume further that

– the system is stable: λj < µj for all j– packet length is independently redrawn (from the same distribution)

every time the packet moves from one link to another

• This is so called Kleinrock’s independence assumption

• Under these assumptions, it is possible to show that

– the stationary state probability π(x) for any state x ∈ S is as follows:

where ρj denotes the traffic load of link j:

∏=

−=J

j

xjj

j

1)1()( ρρπ x

1<==j

j

j

jj C

Lλµλ

ρ


34


j

jjX

ρρ−

=1


• Probability π(x) is again said to be of product-form– Now, the number of packets in different queues are independent (why?)

• Each individual queue j behaves as an M/M/1 queue

– Number of packets in queue j follows a geometric distribution with mean

35


∈−

∈==

)(

1

)()(

rJjrJjj

jjTrT λµ

Average trunk network delay

• Consider then the average trunk network delay for class r– Let J(r) denote the set of the links that belong to route r

• In our model, the average trunk network delay will be – the sum of average delays experienced in the links along the route

(including both the transmission delay and the queueing delay)

• By Little’s formula, the average link delay is

• Thus, the average trunk network delay for class r is

jjj

j

jj

jj

XT

λµρρ

λλ −=

−⋅== 11

1

1Lect04.ppt S-38.145 - Introduction to Teletraffic Theory - Spring 2004

4. Traffic measurements and modelling

2


Contents

• Traffic measurements

• Traffic variations

• Traditional modelling of telephone traffic• Traditional modelling of data traffic

• Novel models for data traffic

3


Traffic measurements (1)

• Traffic measurements are needed for – network design and traffic management

• a basis for dimensioning• traffic modelling• traffic predictions

– implementation of dynamic traffic control• measurement based connection admission control• dynamic routing• congestion detection and control

– performance evaluation of a network or its components– evaluation of quality of service– but also for getting accounting information

• More and more information about traffic is needed because of – new services and networks– new users, new ways to use, new tariffs (as for existing services and

networks)– tougher competition 4



• Traffic measurements in telephone networks– traffic on different links

• traffic process (carried traffic intensity = number of occupied channels)

• call arrival process (interarrival times)• call holding times

– traffic on different trunk network nodes

• distribution of incoming traffic from different directions• distribution of outgoing traffic in different directions

– traffic on different access network nodes

• distribution according to the type of traffic source

– e.g. residential vs. business subscribers• use of different services

5



• Traffic measurements in data networks (Internet/LAN)– traffic on different links (or network nodes)

• traffic process (carried traffic intensity in bits per second)• packet level (e.g. IP)

– packet arrival process (interarrival times)– packet lengths– packet loss ratio

• connection level (e.g. TCP applications: ftp / http / email / telnet etc.)– connection arrival process– connection holding times – total amount of information transferred

– end-to-end traffic measurements• delay and its variation (jitter) experienced by packets• average throughput of a connection

6


Analysis of traffic measurements

• Traditional statistical methods:– parameter estimation

• traffic intensity

• traffic variability (short term variance, coefficient of variation)• traffic peakedness

– estimation of traffic auto-correlation

• New approach:– scalability analysis

• self-similarity

• multifractal characterization

7


Contents





8


Traffic variations in different time scales (1)

• Predictive variations– Trend (years)

• traffic growth: due to

– existing services (new users, new ways to use, new tariffs)– new services

– Regular year profile (months)

– Regular week profile (days)– Regular day profile (hours)

• including “busy hour”

– Variations caused by predictive (regular and irregular) external events• regular: e.g. Christmas day• irregular: e.g. World Championships, televoting

– Note: different profiles for different types of user groups

9


Traffic variations in different time scales (2)

• Non-predictive variations– Short term stochastic variations (seconds - minutes)

• related to the independence of users

– random call arrivals– random call holding times

– Long term stochastic variations (hours - ...)

• random deviations around the profiles• each day, week, month, etc. is different

– Variations caused by non-predictive external events• e.g. earthquakes, hurricanes

10


Busy hour (1)

• An estimate of the traffic load is needed for dimensioning– however, it is not meaningful to use “the highest possible peak value”

• In telephone networks, standard way is to use so called busy hourtraffic for dimensioning

– This is unambiguous only for a single day (let’s call it daily peak hour)– For dimensioning, however, we have to look at not only a single day but

many more (why?)

– At least three different definitions for busy hour (covering several days) have been proposed:

• Average Daily Peak Hour (ADPH)

• Time Consistent Busy Hour (TCBH)• Fixed Daily Measurement Hour (FDMH)

Busy hour ≈ the continuous 1-hour period for which the traffic volume is greatest

11


• Let

– N = number of days during which measurements are done (e.g. N = 10)

– an(∆) = measured avg. traffic volume during an interval ∆ of day n

– max∆ an(∆) = daily peak hour traffic volume of day n

• Busy hour traffic a using different methods:

• Obviously (why?),

Busy hour (2)

= ∆ ∆= N

n nN aa1

1ADPH )(max

=∆ ∆= N

n nN aa1

1TCBH )(max

=∆= N

n nN aa1 fixed

1FDMH )(

11ADPHTCBHFDMH aaa ≤≤ 12


Contents





13


Traffic ↔ Calls

Modelling of telephone traffic

• In telephone networks:

• Traffic model (for a single link) should specify – the type of the call arrival process (at what times a new call attempts to

enter the system)– the distribution of call holding times (how long does a single call last)

• These together specify – the traffic process that tells the number of ongoing calls

= number of occupied channels= instantaneous intensity of the traffic carried (in erlangs)

• Note:– Traffic volume refers to

the amount of carried traffic during some time interval = integral of the instantaneous traffic intensity over this interval

14


Traffic process

654321

6543210

time

time




traffic volume

call holdingtime

chan

nels

num

ber

of c

hann

els

15


Call arrival process (1)

• Aggregated traffic (in trunk network)

– Traditional model: Poisson process (with some intensity λ > 0)

• In a short time interval of length ∆, there are two possibilities:either a new call arrives (with probability λ∆)or nothing happens (with probability 1 − λ∆)

• Disjoint intervals are independent of each other

• As a result: call interarrival times are independently and exponentially distributed with mean 1/λ

– This is found to be a good model when user population is large (“infinite”) and users make independent decisions

– Corresponding teletraffic models are loss models: • Erlang model (finite link capacity)

• Poisson model (infinite link capapcity)

16



• Overflow traffic in trunk network due to hierarchical alternative routing

– Model:e.g. Interrupted Poisson process (IPP) or more generally Markov modulated Poisson Process (MMPP)

– In addition to the traffic process itself, there is a modulating process that tells whether the arrivals of an ordinary Poisson process will be realized or not

– Traffic stream consists of the calls blocked in the direct link

b

ca

direct route: a - calternative route: a - b - c

17



• Traffic generated by an individual user (subscriber)– Traditional model: exponential on-off process

• The user alternates between on and off states– When on, a call is going on– When off, the user is “idle”

• The times spent in different states are assumed to be independent and exponentially distributed (with state-dependent mean)

• Traffic generated by a superposition of users in access network – Finite number of individual users

• modelled separately as above• making independent decision

– Corresponding teletraffic models are loss models: • Engset model (insufficient link capacity)• Binomial model (sufficient link capacity)

18


Call holding time (1)

• Basic assumption:– call holding times are independent and identically distributed (IID)

• Distribution of call holding times– traditional model: exponential distribution

• one parameter simple!

• memoryless property: given that the holding time is at least (any) t,the probability that the call will end in a short time interval (t, t+∆)depends just on ∆ (but not on t)

• exponential tail– more complicated models:

• normal distribution (two parameters: mean and variance)• log-normal distribution (two parameters)• hyper-exponential (with two parameters)• Weibull distribution (with two/three parameters)

19


Call holding time (2)

• Call holding time distribution is typically different for – business and residential calls (daytime vs. evening calls)

– ordinary and “data” calls (fax, Internet access, etc.)

20


Contents





21


Traditional modelling of data traffic on packet level

• Traditional traffic model on packet level– new packets arrive according to a Poisson process

packet interarrival times independent and exponentially distributed– packet lengths are independent and exponentially distributed

packet transmission times (in links) independent and exponentially distributed

– system model is a single server queueing model (M/M/1-FIFO)

22


Traffic process at the packet level (1)

time

time

Packet arrival times

Buffer occupation (number of packets)43210

C

Link occupation

23


Traffic process at the packet level (2)

time

time

C

Link occupation (averaged)

Link occupation (continuous)

C

24


Modelling data traffic on flow level (1)

• A flow is a kind of an approximation of connection level traffic observed on the packet level

• Flow is a sequence of subsequent packets, which have the same source and the same destination, and belong together on some criteria

– in the simplest form, classifying flows would take into account only source and destination information

– on a finer level of granularity, higher layer protocols could be used as criteria in classifying (e.g. TCP, UDP, HTTP, FTP, …)

– subsequent flows are separated by a timer; subsequent packets can belong to same flow only if their mutual distance in time is small enough

– Note that the definition of flow is very flexible. How to classify flows in practice is a completely another story

• selection of granularity• selection of timer timeouts

25



• Classifying flows based on the ”nature” of traffic– Streaming flows, which are characterized by reserved “bandwidth” and

lifetime• e.g. audio and video streams using UDP

– Elastic flows, which are characterized by the size

• E.g. document transfer (HTTP, FTP, …) using TCP• Note that the used bandwidth and flow lifetime are determined

dynamically by state of the network

26



• Flow level model of elastic traffic– new flows are generated according to a Poisson process– distribution of flow sizes may be selected freely, e.g. based on

measurements– If we are interested in a single link only, we could use a single server

processor sharing queuing system, where all the clients are simultaneously receiving service, but sharing the service capacity (M/G/1-PS queuing model)

– PS queuing discipline is an idealization of the fact TCP-flows reduce their speed when experiencing congestion (adaptive behaviour)

• Flow level model of streaming traffic using UDP– new flows are generated according to a Poisson process– distribution of lifetimes may be selected freely, e.g. based on

measurements– Reserved “bandwidth” fixed or random (independent of the state of the

network)– If modelling only UDP-traffic, simple blocking models could be used? (cf.

ATM traffic models on connection level)

27


Modelling of ATM traffic (1)

• Three different time scales:

Call level

Burst level

Cell level

28


Modelling of ATM traffic (2)

• Call level – “traffic unit” = connection

– loss model (for CBR and VBR connections)

• Burst level– “traffic unit” = burst of varying length (and possibly of varying rate)

– (traditional) fluid buffer models:

• superposition of exponential ON-OFF sources (Anick-Mitra-Sondhi model, (A-M-S))

• burst arrivals according to Poisson process (Kosten model)

• Cell level– “traffic unit” = fixed length cell

– queueing models:

• superposition of periodic sources (N*D/D/1)• cell arrivals according to Poisson process (M/D/1)

29


Contents





30


Bellcore Ethernet measurements

• Ethernet (LAN) measurements by Leland, Willinger, … (‘89-’92)– high-accuracy recording of hundreds of millions Ethernet packets

• including both the arrival time and the length– see: IEEE/ACM Trans. Networking, vol. 2, nr. 1, pp. 1-15, February 1994

• Conclusions:– Ethernet traffic seems to be extremely varying

• presence of “burstiness” across an extremely wide range of time scales (from microseconds to milliseconds, seconds, minutes, hours, …)

• bad from the performance point of view– Ethernet traffic is statistically self-similar (fractal-like)

• it looks the same in all time scales when zooming• a single parameter (the Hurst parameter) describes the fractal nature• good from the modelling point of view (parsimony!)

– Traditional data traffic models (Poisson process, exponential distribution) do not capture these properties!

31


Internet measurements

• Internet (WAN) measurements by Paxson and Floyd (‘93-’95)– both the connection and the packet level concerned

– see: IEEE/ACM Trans. Networking vol. 3, nr. 3, pp. 226-244, June 1995

• Connection level conclusions:– For interactive TELNET traffic (and other user-initiated sessions),

• connection arrivals are well-modelled by a Poisson process (with hourly fixed rates)

– But for connections within user-initiated sessions (FTP data, HTTP) and machine-generated connections

• connection arrivals are more bursty than in a Poisson process suggests (and even correlated)

• Packet level conclusions – empirical distribution of TELNET packet interarrival times is

• heavy-tailed (not exponential as traditionally modelled)

32


New models for data traffic (1)

• “New” distributions:– Subexponential distributions

• “worse than exponential tail”• e.g. log-normal, Weibull and Pareto distributions

– Heavy-tailed distributions• “power-law tail”

• e.g. Pareto distribution (with location parameter a and shape parameter β)

• In the new models for data traffic, these distributions are used to describe

– packet lengths and interarrival times– connection holding times and interarrival times

0,0,)/( >>≥=> ββ axxaxXP

33


New models for data traffic (2)

• “New” models for traffic processes:– processes exhibiting long range dependence (LRD)

– self-similar processes

• If a stochastic process is (asymptotically) self-similar with positive correlations, then it exhibits long range dependence (LRD)

– e.g. fractional Brownian motion (FBM)

• suitable for describing aggregated traffic (in trunk network)

• just three parameters (thus, parsimonious!)

• one of them, so called Hurst parameter H, describes the grade of long range dependence (when in the interval (½,1))

• Self-similarity and long range dependence can be caused by heavy-tailed distributions (see the following Example)

34


1

∞

µ

λ•••

Example

• Consider an infinite system (M/G/∞)– new customers arrive according to a Poisson process

– service times independent and identically distributed• service time distribution heavy-tailed

• service time has an infinite variance

• e.g. Pareto distribution with shape parameter β < 2

• Then the traffic volume process is asymptotically self-similar and containing long range dependence


5. Basic probability theory

2


Contents

• Basic concepts

• Discrete random variables

• Discrete distributions (nbr distributions)• Continuous random variables

• Continuous distributions (time distributions)

• Other random variables

3


Sample space, sample points, events

• Sample space Ω is the set of all possible sample points ω ∈ Ω– Example 0. Tossing a coin: Ω = H,T

– Example 1. Casting a die: Ω = 1,2,3,4,5,6

– Example 2. Number of customers in a queue: Ω = 0,1,2,...

– Example 3. Call holding time (e.g. in minutes): Ω = x ∈ ℜ | x > 0

• Events A,B,C,... ⊂ Ω are measurable subsets of the sample space Ω– Example 1. “Even numbers of a die”: A = 2,4,6

– Example 2. “No customers in a queue”: A = 0

– Example 3. “Call holding time greater than 3.0 (min)”: A = x ∈ ℜ | x > 3.0

• Denote by the set of all events A ∈ – Sure event: The sample space Ω ∈ itself

– Impossible event: The empty set ∅ ∈

4


Combination of events

• Union “A or B”: A ∪ B = ω ∈ Ω | ω ∈ A or ω ∈ B

• Intersection “A and B”: A ∩ B = ω ∈ Ω | ω ∈ A and ω ∈ B

• Complement “not A”: Ac = ω ∈ Ω | ω ∉ A

• Events A and B are disjoint if

– A ∩ B = ∅• A set of events B1, B2, … is a partition of event A if

– (i) Bi ∩ Bj = ∅ for all i ≠ j

– (ii) ∪i Bi = AB1

B2

B3

A

5


Probability

• Probability of event A is denoted by P(A), P(A) ∈ [0,1]– Probability measure P is thus

a real-valued set function defined on the set of events , P: → [0,1]

• Properties:

– (i) 0 ≤ P(A) ≤ 1

– (ii) P(∅) = 0

– (iii) P(Ω) = 1

– (iv) P(Ac) = 1 − P(A)

– (v) P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

– (vi) A ∩ B = ∅ P(A ∪ B) = P(A) + P(B)

– (vii) Bi is a partition of A P(A) = Σi P(Bi)

– (viii) A ⊂ B P(A) ≤ P(B)

A

B

6


Conditional probability

• Assume that P(B) > 0• Definition: The conditional probability of event A

given that event B occurred is defined as

• It follows that

)()()|(

BPBAPBAP ∩=

)|()()|()()( ABPAPBAPBPBAP ==∩

7


Theorem of total probability

• Let Bi be a partition of the sample space Ω• It follows that A ∩ Bi is a partition of event A. Thus (by slide 5)

• Assume further that P(Bi) > 0 for all i. Then (by slide 6)

• This is the theorem of total probability

B1

B2

B3

B4

A

∩= i i

viiBAPAP )()(

)(

Ω

= i ii BAPBPAP )|()()(

8


Bayes’ theorem

• Let Bi be a partition of the sample space Ω• Assume that P(A) > 0 and P(Bi) > 0 for all i. Then (by slide 6)

• Furthermore, by the theorem of total probability (slide 7), we get

• This is Bayes’ theorem

– Probabilities P(Bi) are called a priori probabilities of events Bi

– Probabilities P(Bi | A) are called a posteriori probabilities of events Bi(given that the event A occured)

)()|()(

)()(

)|(AP

BAPBPAP

BAPi

iiiABP == ∩

=j jj

iiBAPBP

BAPBPi ABP

)|()()|()(

)|(

9


Statistical independence of events

• Definition: Events A and B are independent if

• It follows that

• Correspondingly:

)()()( BPAPBAP =∩

)()|()(

)()()(

)( APBAPBP

BPAPBP

BAP === ∩

)()|()(

)()()(

)( BPABPAP

BPAPAP

BAP === ∩

10


• Definition: Real-valued random variable X is a real-valued and measurable function defined on the sample space Ω, X: Ω → ℜ

– Each sample point ω ∈ Ω is associated with a real number X(ω)

• Measurability means that all sets of type

belong to the set of events , that is

X ≤ x ∈

• The probability of such an event is denoted by PX ≤ x

Random variables

Ω⊂≤Ω∈=≤ )(|: xXxX ωω

11


Example

• A coin is tossed three times

• Sample space:

• Let X be the random variable that tells the total number of tails in these three experiments:

3,2,1,T,H|),,( 321 =∈=Ω iiωωωω

ω HHH HHT HTH THH HTT THT TTH TTT

X(ω) 0 1 1 1 2 2 2 3

12


Indicators of events

• Let A ∈ be an arbitrary event

• Definition: The indicator of event A is a random variable defined as follows:

• Clearly:

∉∈

=A

AA ω

ωω

,0

,1)(1

)(1)(01

)(11

APAPP

APP

cA

A

−===

==

13


• Definition: The cumulative distribution function (cdf) of a random variable X is a function FX: ℜ → [0,1] defined as follows:

• Cdf determines the distribution of the random variable, – that is: the probabilities PX ∈ B, where B ⊂ ℜ and X ∈ B ∈

• Properties:– (i) FX is non-decreasing

– (ii) FX is continuous from the right

– (iii) FX (−∞) = 0

– (iv) FX (∞) = 1

Cumulative distribution function

)( xXPxFX ≤=

FX(x)

x0

1

14


Statistical independence of random variables

• Definition: Random variables X and Y are independent iffor all x and y

• Definition: Random variables X1,…, Xn are totally independent iffor all i and xi

, yYPxXPyYxXP ≤≤=≤≤

,..., 1111 nnnn xXPxXPxXxXP ≤≤=≤≤

15


Maximum and minimum of independent random variables

• Let the random variables X1,…, Xn be totally independent

• Denote: Xmax := maxX1,…, Xn. Then

• Denote: Xmin := minX1,…, Xn. Then

,, 1max xXxXPxXP n ≤≤=≤

1 xXPxXP n ≤≤=

,, 1min xXxXPxXP n >>=>

1 xXPxXP n >>=

16


Contents

• Basic concepts





17


Discrete random variables

• Definition: Set A ⊂ ℜ is called discrete if it is

– finite, A = x1,…, xn, or

– countably infinite, A = x1, x2,…

• Definition: Random variable X is discrete ifthere is a discrete set SX ⊂ ℜ such that

• It follows that

– PX = x ≥ 0 for all x ∈ SX

– PX = x = 0 for all x ∉ SX

• The set SX is called the value set

1 =∈ XSXP

18


Point probabilities

• Let X be a discrete random variable

• The distribution of X is determined by the point probabilities pi,

• Definition: The probability mass function (pmf) of X is a function pX: ℜ → [0,1] defined as follows:

• Cdf is in this case a step function:

Xiii SxxXPp ∈== ,:

∉∈=

===,0

,:)( i

X

XiX Sx

SxxpxXPxp

≤=≤=

xxiiX

i

pxXPxF:

)(

19


Example

x

pX(x)

probability mass function (pmf)

x

FX(x)

cumulative distribution function (cdf)

x1 x2 x3 x4 x1 x2 x3 x4

1 1

SX = x1, x2, x3, x4

20


Independence of discrete random variables

• Discrete random variables X and Y are independent if and only if for all xi ∈ SX and yj ∈ SY

, jiji yYPxXPyYxXP =====

21


Expectation

• Definition: The expectation (mean value) of X is defined by

– Note 1: The expectation exists only if Σi pi|xi| < ∞– Note 2: If Σi pi xi = ∞, then we may denote E[X] = ∞

• Properties:

– (i) c ∈ ℜ E[cX] = cE[X]

– (ii) E[X + Y] = E[X] + E[Y]

– (iii) X and Y independent E[XY] = E[X]E[Y]

==⋅===∈∈ i

iiSx

XSx

X xpxxpxxXPXEXX

)(:][:µ

22


Variance

• Definition: The variance of X is defined by

• Useful formula (prove!):

• Properties:

– (i) c ∈ ℜ D2[cX] = c2D2[X]

– (ii) X and Y independent D2[X + Y] = D2[X] + D2[Y]

]])[[(:]Var[:][: 222 XEXEXXDX −===σ

222 ][][][ XEXEXD −=

23


Covariance

• Definition: The covariance between X and Y is defined by

• Useful formula (prove!):

• Properties:

– (i) Cov[X,X] = Var[X]

– (ii) Cov[X,Y] = Cov[Y,X]

– (iii) Cov[X+Y,Z] = Cov[X,Z] + Cov[Y,Z]

– (iv) X and Y independent Cov[X,Y] = 0

])][])([[(:],[Cov:2 YEYXEXEYXXY −−==σ

][][][],Cov[ YEXEXYEYX −=

24


Other distribution related parameters

• Definition: The standard deviation of X is defined by

• Definition: The coefficient of variation of X is defined by

• Definition: The kth moment, k=1,2,…, of X is defined by

][][:][: 2 XVarXDXDX ===σ

][][:][:

XEXD

X XCc ==

][:)( kkX XE=µ

25


Average of IID random variables

• Let X1,…, Xn be independent and identically distributed (IID)with mean µ and variance σ2

• Denote the average (sample mean) as follows:

• Then (prove!)

==

n

iinn XX

1

1:

nn

nn

n

XD

XD

XE

σ

σ

µ

=

=

=

][

][

][22

26


Law of large numbers (LLN)

• Let X1,…, Xn be independent and identically distributed (IID)with mean µ and variance σ2

• Weak law of large numbers: for all ε > 0

• Strong law of large numbers: with probability 1

0|| →>− εµnXP

µ→nX

27


Contents

• Basic concepts





28


Bernoulli distribution

– describes a simple random experiment with two possible outcomes:success (1) and failure (0); cf. coin tossing

– success with probability p (and failure with probability 1 − p)

• Value set: SX = 0,1• Point probabilities:

• Mean value: E[X] = (1 − p)⋅0 + p⋅1 = p

• Second moment: E[X2] = (1 − p)⋅02 + p⋅12 = p

• Variance: D2[X] = E[X2] − E[X]2 = p − p2 = p(1 − p)

)1,0(),(Bernoulli ∈∼ ppX

pXPpXP ==−== 1,10

29


Binomial distribution

– number of successes in an independent series of simple random experiments (of Bernoulli type); X = X1 + … + Xn (with Xi ∼ Bernoulli(p))

– n = total number of experiments

– p = probability of success in any single experiment

• Value set: SX = 0,1,…,n• Point probabilities:

• Mean value: E[X] = E[X1] + … + E[Xn] = np

• Variance: D2[X] = D2[X1] + … + D2[Xn] = np(1 − p) (independence!)

)1,0(,...,2,1),,(Bin ∈∈∼ pnpnX

( ) inini ppiXP −−== )1(

( )12)1(!

)!(!!

⋅−⋅=−=

nnn

ininn

i

30


Geometric distribution

– number of successes until the first failure in an independent series of simple random experiments (of Bernoulli type)

– p = probability of success in any single experiment

• Value set: SX = 0,1,…• Point probabilities:

• Mean value: E[X] = i ipi(1 − p) = p/(1 − p)

• Second moment: E[X2] = i i2pi(1 − p) = 2(p/(1 − p))2 + p/(1 − p)

• Variance: D2[X] = E[X2] − E[X]2 = p/(1 − p)2

)1,0(),(Geom ∈∼ ppX

)1( ppiXP i −==

31


Memoryless property of geometric distribution

• Geometric distribution has so called memoryless property:for all i,j ∈ 0,1,...

• Prove!

– Tip: Prove first that PX ≥ i = pi

| jXPiXjiXP ≥=≥+≥

32


Minimum of geometric random variables

• Let X1 ∼ Geom(p1) and X2 ∼ Geom(p2) be independent. Then

and

• Prove! – Tip: See slide 15

)(Geom,min: 2121min ppXXX ∼=

2,1,211

1min ∈== −−

iXXPpp

pi

i

33


Poisson distribution

– limit of binomial distribution as n → ∞ and p → 0 in such a way that np → a

• Value set: SX = 0,1,…• Point probabilities:

• Mean value: E[X] = a

• Second moment: E[X(X −1)] = a2 E[X2] = a2 + a

• Variance: D2[X] = E[X2] − E[X]2 = a

0),(Poisson >∼ aaX

aia eiXP

i −==!

34


Example

• Assume that – 200 subscribers are connected to a local exchange

– each subscriber’s characteristic traffic is 0.01 erlang

– subscribers behave independently

• Then the number of active calls X ∼ Bin(200,0.01)

• Corresponding Poisson-approximation X ≈ Poisson(2.0)• Point probabilities:

0 1 2 3 4 5

Bin(200,0.01) .1326 .2679 .2693 .1795 .0893 .0354

Poisson(2.0) .1353 .2701 .2701 .1804 .0902 .0361

35


Properties

• (i) Sum: Let X1 ∼ Poisson(a1) and X2 ∼ Poisson(a2) be independent. Then

• (ii) Random sample: Let X ∼ Poisson(a) denote the number of elements in a set, and Y denote the size of a random sample of this set (each element taken independently with probability p). Then

• (iii) Random sorting: Let X and Y be as in (ii), and Z = X − Y. Then Y and Z are independent (given that X is unknown) and

)(Poisson 2121 aaXX +∼+

)(Poisson paY ∼

))1((Poisson apZ −∼36


Contents

• Basic concepts





37


Continuous random variables

• Definition: Random variable X is continuous ifthere is an integrable function fX: ℜ → ℜ+ such that for all x ∈ ℜ

• The function fX is called the probability density function (pdf)

– The set SX, where fX > 0, is called the value set

• Properties:

– (i) PX = x = 0 for all x ∈ ℜ– (ii) Pa < X < b = Pa ≤ X ≤ b = a

b fX(x) dx

– (iii) PX ∈ A = A fX(x) dx

– (iv) PX ∈ ℜ = -∞∞ fX(x) dx = SX

fX(x) dx = 1

∞−=≤=

x

XX dyyfxXPxF )(:)(

38


Example

x

fX(x)

probability density function (pdf)

x

FX(x)

cumulative distribution function (cdf)

x1 x2 x3 x1 x2 x3

1

SX = [x1, x3]

39


Expectation and other distribution related parameters

• Definition: The expectation (mean value) of X is defined by

– Note 1: The expectation exists only if -∞∞ fX(x)|x| dx < ∞

– Note 2: If -∞∞ fX(x)x = ∞, then we may denote E[X] = ∞

– The expectation has the same properties as in the discrete case (see slide 21)

• The other distribution parameters (variance, covariance,...) are defined just as in the discrete case

– These parameters have the same properties as in the discrete case (see slides 22-24)

∞

∞−== dxxxfXE XX )(:][:µ

40


Contents

• Basic concepts





41


Uniform distribution

– continuous counterpart of “casting a die”

• Value set: SX = (a,b)• Probability density function (pdf):

• Cumulative distribution function (cdf):

• Mean value: E[X] = ab x/(b − a) dx = (a + b)/2

• Second moment: E[X2] = ab x2/(b − a) dx = (a2 + ab + b2)/3

• Variance: D2[X] = E[X2] − E[X]2 = (b − a)2/12

babaX <∼ ),,(U

),(,1

)( baxab

xf X ∈−

=

),(,:)( baxxXPxFabax

X ∈=≤= −−

42


Exponential distribution

– continuous counterpart of geometric distribution (“failure” prob. ≈ λdt)

• Value set: SX = (0,∞)• Probability density function (pdf):


• Mean value: E[X] = 0∞ λx exp(−λx) dx = 1/λ

• Second moment: E[X2] = 0∞ λx2 exp(−λx) dx = 2/λ2

• Variance: D2[X] = E[X2] − E[X]2 = 1/λ2

0),(Exp >∼ λλX

0,)( >= − xexf xX

λλ

0,1)( >−=≤= − xexXPxF xX

λ

43


Memoryless property of exponential distribution

• Exponential distribution has so called memoryless property:for all x,y ∈ (0,∞)

– Prove!

• Tip: Prove first that PX > x = e−λx

• Application:– Assume that the call holding time is exponentially distributed with mean h

(min).– Consider a call that has already lasted for x minutes.

Due to memoryless property, this gives no information about the length of the remaining holding time:it is distributed as the original holding time and, on average, lasts still hminutes!

| yXPxXyxXP >=>+>

44


Minimum of exponential random variables

• Let X1 ∼ Exp(λ1) and X2 ∼ Exp(λ2) be independent. Then

and

• Prove! – Tip: See slide 15

)(Exp,min: 2121min λλ +∼= XXX

2,1,21

min ∈== + iXXP ii λλ

λ

45


Standard normal (Gaussian) distribution

– limit of the “normalized” sum of IID r.v.s with mean 0 and variance 1 (cf. slide 48)

• Value set: SX = (−∞,∞)• Probability density function (pdf):


• Mean value: E[X] = 0 (symmetric pdf)

• Variance: D2[X] = 1

)1,0(N∼X

221

21:)()( x

X exxf −==π

ϕ

∞−=Φ=≤= xX dyyxxXPxF )(:)(:)( ϕ

46


Normal (Gaussian) distribution

– if (X − µ)/σ ∼ N(0,1)

• Value set: SX = (−∞,∞)• Probability density function (pdf):


• Mean value: E[X] = µ + σE[(X − µ)/σ] = µ (symmetric pdf around µ)

• Variance: D2[X] = σ2D2[(X − µ)/σ] = σ2

0,),,(N 2 >ℜ∈∼ σµσµX

( )σµ

σ ϕ −== xXX xFxf 1)(')(

( )σµ

σµ

σµ −−− Φ=≤=≤= xxX

X PxXPxF :)(

47


Properties of the normal distribution

• (i) Linear transformation: Let X ∼ N(µ,σ2) and α,β ∈ ℜ. Then

• (ii) Sum: Let X1 ∼ N(µ1,σ12) and X2 ∼ N(µ2,σ2

2) be independent.Then

• (iii) Sample mean: Let Xi ∼ N(µ,σ2), i = 1,…n, be independent and identically distributed (IID). Then

),(N: 22σαβαµβα +∼+= XY

),(N 22

212121 σσµµ ++∼+ XX

),(N: 21

1

1 σµn

n

iinn XX ∼=

=

48


Central limit theorem (CLT)

• Let X1,…, Xn be independent and identically distributed (IID)with mean µ and variance σ2 (and the third moment exists)

• Central limit theorem:

• It follows that

)1,0(N)(i.d.

/1 →− µ

σ nnX

),(N 21σµnnX ≈

49


Contents

• Basic concepts





50


Other random variables

• In addition to discrete and continuous random variables, there are so called mixed random variables

– containing some discrete as well as continuous portions

• Example:– The customer waiting time W in an M/M/1 queue has an atom at zero

(PW = 0 = 1 − ρ > 0) but otherwise the distribution is continuous

FW(x)

x0

1

0

1 − ρ


6. Introduction to stochastic processes

2


Contents

• Basic concepts

• Poisson process

• Markov processes• Birth-death processes

3


Stochastic processes (1)

• Consider some quantity in a teletraffic (or any) system

• It typically evolves in time randomly– Example 1: the number of occupied channels in a telephone link

at time t or at the arrival time of the nth customer

– Example 2: the number of packets in the buffer of a statistical multiplexer at time t or at the arrival time of the nth customer

• This kind of evolution is described by a stochastic process– At any individual time t (or n) the system can be described by a random

variable– Thus, the stochastic process is a collection of random variables

4



• Definition: A (real-valued) stochastic process X = (Xt | t ∈ I) is a collection of random variables Xt

– taking values in some (real-valued) set S, Xt(ω) ∈ S, and

– indexed by a real-valued (time) parameter t ∈ I.

• Stochastic processes are also called random processes(or just processes)

• The index set I ⊂ ℜ is called the parameter space of the process

• The value set S ⊂ ℜ is called the state space of the process

• Note: Sometimes notation Xt is used to refer to the whole stochastic process (instead of a single random variable related to the time t)

5



• Each (individual) random variable Xt is a mapping from the sample space Ω into the real values ℜ:

• Thus, a stochastic process X can be seen as a mapping from the sample space Ω into the set of real-valued functions ℜI (with t ∈ I asan argument):

• Each sample point ω ∈ Ω is associated with a real-valued function X(ω). Function X(ω) is called a realization (or a path or a trajectory)of the process.

)(: ωω tt X,X ℜ→Ω

)(: ωω X,X Iℜ→Ω

6


Summary

• Given the sample point ω ∈ Ω– X(ω) = (Xt(ω) | t ∈ I) is a real-valued function (of t ∈ I)

• Given the time index t ∈ I,– Xt = (Xt(ω) | ω ∈ Ω) is a random variable (as ω ∈ Ω)

• Given the sample point ω ∈ Ω and the time index t ∈ I,– Xt(ω) is a real value

7


Example

• Consider traffic process X = (Xt | t ∈ [0,T]) in a link between two telephone exchanges during some time interval [0,T]

– Xt denotes the number of occupied channels at time t

• Sample point ω ∈ Ω tells us

– what is the number X0 of occupied channels at time 0,

– what are the remaining holding times of the calls going on at time 0,

– at what times new calls arrive, and – what are the holding times of these new calls.

• From this information, it is possible to construct the realization X(ω) ofthe traffic process X

– Note that all the randomness in the process is included in the sample point ω– Given the sample point, the realization of the process is just a (deterministic)

function of time

8


Traffic process

654321

6543210

time

time




traffic volume

call holdingtime

chan

nels

nr o

f cha

nnel

s

9


Categories of stochastic processes

• Reminder:– Parameter space: set I of indices t ∈ I– State space: set S of values Xt(ω) ∈ S

• Categories:– Based on the parameter space:

• Discrete-time processes: parameter space discrete• Continuous-time processes: parameter space continuous

– Based on the state space:• Discrete-state processes: state space discrete• Continuous-state processes: state space continuous

• In this course we will concentrate on the discrete-state processes (with either a discrete or a continuous parameter space (time))

– Typical processes describe the number of customers in a queueing system (the state space being thus S = 0,1,2,...)

10


Examples

• Discrete-time, discrete-state processes– Example 1: the number of occupied channels in a telephone link

at the arrival time of the nth customer, n = 1,2,...– Example 2: the number of packets in the buffer of a statistical multiplexer

at the arrival time of the nth customer, n = 1,2,...

• Continuous-time, discrete-state processes– Example 3: the number of occupied channels in a telephone link

at time t > 0– Example 4: the number of packets in the buffer of a statistical multiplexer

at time t > 0

11


Notation

• For a discrete-time process,

– the parameter space is typically the set of positive integers, I = 1,2,…

– Index t is then (often) replaced by n: Xn, Xn(ω)

• For a continuous-time process,

– the parameter space is typically either a finite interval, I = [0, T], or all non-negative real values, I = [0, ∞)

– In this case, index t is (often) written not as a subscript but in parentheses: X(t), X (t;ω)

12


Distribution

• The stochastic characterization of a stochastic process X is made by giving all possible finite-dimensional distributions

where t1,…, tn ∈ I, x1,…, xn ∈ S and n = 1,2,...• In general, this is not an easy task because of dependencies between

the random variables Xt (with different values of time t)• For discrete-state processes it is sufficient to consider probabilities of

the form

– cf. discrete distributions

,, 11 ntt xXxXPn

≤≤

,, 11 ntt xXxXPn

==

13


Dependence

• The most simple (but not so interesting) example of a stochasticprocess is such that all the random variables Xt are independent ofeach other. In this case

• The most simple non-trivial example is a discrete state Markov process. In this case

• This is related to the so called Markov property:– Given the current state (of the process),

the future (of the process) does not depend on the past (of the process), i.e. how the process has arrived to the current state

,..., 11 11 nttntt xXPxXPxXxXPnn

≤≤=≤≤

=== ,..., 11 ntt xXxXPn

|| 1121 1121 −====⋅=− ntntttt xXxXPxXxXPxXP

nn

14


Stationarity

• Definition: Stochastic process X is stationary if all finite-dimensional distributions are invariant to time shifts, that is:

for all ∆, n, t1,…, tn and x1,…, xn

• Consequence: By choosing n = 1, we see that all (individual) random variables Xt of a stationary process are identically distributed:

for all t ∈ I. This is called the stationary distribution of the process.

,,,, 11 11 nttntt xXxXPxXxXPnn

≤≤=≤≤ ∆+∆+

)( xFxXP t =≤

15


Stochastic processes in teletraffic theory

• In this course (and, more generally, in teletraffic theory) various stochastic processes are needed to describe

– the arrivals of customers to the system (arrival process)

– the state of the system (state process)

• Note that the latter is also often called as traffic process

16


Arrival process

• An arrival process can be described as

– a point process (τn | n = 1,2,...) where τn tells the arrival time of the nth

customer (discrete-time, continuous-state)

• non-decreasing: τn+1 ≥ τn kaikilla n– thus non-stationary!

• typically it is assumed that the interarrival times τn − τn-1 areindependent and identically distributed (IID) renewal process

• then it is sufficient to specify the interarrival time distribution

• exponential IID interarrival times Poisson process

– a counter process (A(t) | t ≥ 0) where A(t) tells the number of arrivals up to time t (continuous-time, discrete-state)

• non-decreasing: A(t+∆) ≥ A(t) for all t,∆ ≥ 0– thus non-stationary!

• independent and identically distributed (IID) increments A(t+∆) − A(t)with Poisson distribution Poisson process

17


State process

• In simple cases – the state of the system is described just by an integer

• e.g. the number X(t) of calls or packets at time t– This yields a state process that is continuous-time and discrete-state

• In more complicated cases, – the state process is e.g. a vector of integers (cf. loss and queueing network

models)

• Typically we are interested in – whether the state process has a stationary distribution– if so, what it is?

• Although the state of the system did not follow the stationary distribution at time 0, in many cases state distribution approaches the stationary distribution as t tends to ∞

18


Contents

• Basic concepts

• Poisson process


19


Bernoulli process

• Definition: Bernoulli process with success probability p is an infinite series (Xn | n = 1,2,...) of independent and identical random experiments of Bernoulli type with success probability p

• Bernoulli process is clearly discrete-time and discrete-state

– Parameter space: I = 1,2,…

– State space: S = 0,1

• Finite dimensional distributions (note: Xn’s are IID):

• Bernoulli process is stationary (stationary distribution: Bernoulli(p))

−=−=

=====

−

=

−∏ i ii iii xnxn

i

xx

nnnn

pppp

xXPxXPxXxXP

)1()1(

,...,

1

1

1111

20


Definition of a Poisson process

• Poisson process is the continuous-time counterpart of a Bernoulli process

– It is a point process (τn | n = 1,2,...) where τn tells tells the occurrence time of the nth event, (e.g. arrival of a client)

– “failure” in Bernoulli process is now an arrival of a client

• Definition 1: A point process (τn | n = 1,2,...) is a Poisson process with intensity λ if the probability that there is an event during a short time interval (t, t+h] is λh + o(h) independently of the other time intervals– o(h) refers to any function such that o(h)/h → 0 as h → 0– new events happen with a constant intensity λ: (λh + o(h))/h → λ– probability that there are no arrivals in (t, t+h] is 1 − λh + o(h)

• Defined as a point process, Poisson process is discrete-time and continuous-state

– Parameter space: I = 1,2,…– State space: S = (0, ∞)

21


Poisson process, another definition

• Consider the interarrival time τn − τn-1 between two events (τ0 = 0)

– Since the intensity that something happens remains constant λ, the ending of the interarrival time within a short period of time (t, t+h], after it has lasted already the time t, does not depend on t (or on other previous arrivals)

– Thus, the interarrival times are independent and, additionally, they have the memoryless property. This property can be only the one of exponential distribution (of continuous-time distributions)

• Definition 2: A point process (τn | n = 1,2,...) is a Poisson processwith intensity λ if the interarrival times τn − τn−1 are independent and identically distributed (IID) with joint distribution Exp(λ)

22


Poisson process, yet another definition (1)

• Consider finally the number of events A(t) during time interval [0,t]– In a Bernoulli process, the number of successes in a fixed interval would

follow a binomial distribution. As the “time slice” tends to 0, this approaches a Poisson distribution.

– Note that A(0)=0

• Definition 3: A counter process (A(t) | t ≥ 0) is a Poisson process with intensity λ if its increments in disjoint intervals are independent and follow a Poisson distribution as follows:

• Defined as a counter process, Poisson process is continuous-time and discrete-state

– Parameter space: I = [0, ∞)

– State space: S = 0,1,2,…

)(Poisson)()( ∆∼−∆+ λtAtA

23


Poisson process, yet another definition (2)

• One dimensional distribution: A(t) ∼ Poisson(λt)– E[A(t)] = λt, D2[A(t)] = λt

• Finite dimensional distributions (due to independence of disjoint intervals):

• Poisson process, defined as a counter process is not stationary, but it has stationary increments

– thus, it doesn’t have a stationary distribution, but independent and identically distributed increments

)()(

)()()(

)(,...,)(

11

121211

11

−− −=−−=−=

===

nnnn

nn

xxtAtAP

xxtAtAPxtAP

xtAxtAP

24


Three ways to characterize the Poisson process

• It is possible to show that all three definitions for a Poisson process are, indeed, equivalent

A(t)

τ1 τ2 τ3 τ4

τ4−τ3

event with prob. λh+o(h)no event with prob. 1−λh+o(h)

25


Properties (1)

• Property 1 (Sum): Let A1(t) and A2(t) be two independent Poisson processes with intensities λ1 and λ2. Then the sum (superposition) process A1(t) + A2(t) is a Poisson process with intensity λ1 + λ2.

• Proof: Consider a short time interval (t, t+h]– Probability that there are no events in the superposition is

– On the other hand, the probability that there is exactly one event is

)()(1))(1))((1( 2121 hohhohhoh ++−=+−+− λλλλ

))())((1())(1))((( 2121 hohhohhohhoh ++−++−+ λλλλ

λ1λ2

λ1+λ2

)()( 21 hoh ++= λλ

26


Properties (2)

• Property 2 (Random sampling): Let τn be a Poisson process with intensity λ. Denote by σn the point process resulting from a random and independent sampling (with probability p) of the points of τn. Then σn is a Poisson process with intensity pλ.

• Proof: Consider a short time interval (t, t+h]– Probability that there are no events after the random sampling is

– On the other hand, the probability that there is exactly one event is

)(1))()(1())(1( hohphohphoh +−=+−++− λλλ

)())(( hohphohp +=+ λλ

λpλ

27


Properties (3)

• Property 3 (Random sorting): Let τn be a Poisson process with intensity λ. Denote by σn

(1) the point process resulting from a random and independent sampling (with probability p) of the points of τn.Denote by σn

(2) the point process resulting from the remaining points. Then σn

(1) and σn(2) are independent Poisson processes with

intensities λp and λ(1 − p).• Proof: Due to property 2, it is enough to prove that the resulting two

processes are independent. Proof will be ignored on this course.

λλp

λ(1-p)

28


Properties (4)

• Property 4 (PASTA): Consider any simple (and stable) teletrafficmodel with Poisson arrivals. Let X(t) denote the state of system at time t (continuous-time process) and Yn denote the state of the system seen by the nth arriving customer (discrete-time process). Then the stationary distribution of X(t) is the same as the stationary distribution of Yn.

• Thus, we can say that – arriving customers see the system in the stationary state– PASTA= “Poisson Arrivals See Time Avarages”

• PASTA property is only valid for Poisson arrivals– and it is not valid for other arrival processes– consider e.g. your own PC. Whenever you start a new session, the system

is idle. In the continuous time, however, the system is not only idle but also busy (when you use it).

29


Contents

• Basic concepts

• Poisson process


30


Markov process

• Consider a continuous-time and discrete-state stochastic process X(t)– with state space S = 0,1,…,N or S = 0,1,...

• Definition: The process X(t) is a Markov process if

•

for all n, t1< … < tn+1 and x1,…, xn +1• This is called the Markov property

– Given the current state, the future of the process does not depend on its past (that is, how the process has evolved to the current state)

– As regards the future of the process, the current state contains all the required information

==== ++ )(,,)(|)( 1111 nnnn xtXxtXxtXP

)(|)( 11 nnnn xtXxtXP == ++

31


Example

• Process X(t) with independent increments is always a Markov process:

• Consequence: Poisson process A(t) is a Markov process:– according to Definition 3, the increments of a Poisson process are

independent

))()(()()( 11 −− −+= nnnn tXtXtXtX

32


Time-homogeneity

• Definition: Markov process X(t) is time-homogeneous if

for all t, ∆ ≥ 0 and x, y ∈ S– In other words,

probabilities PX(t + ∆) = y | X(t) = x are independent of t

)0(|)()(|)( xXyXPxtXytXP ==∆===∆+

33


State transition rates

• Consider a time-homogeneous Markov process X(t)

• The state transition rates qij, where i, j ∈ S, are defined as follows:

• The initial distribution PX(0) = i, i ∈ S, and the state transition ratesqij together determine the state probabilities PX(t) = i, i ∈ S, by theKolmogorov (backwards/forwards) equations

• Note that on this course we will consider only time-homogeneous Markov processes

)0(|)(lim: 1

0iXjhXPq

hhij ===

↓

34


Exponential holding times

• Assume that a Markov process is in state i• During a short time interval (t, t+h] , the conditional probability that there

is a transition from state i to state j is qijh + o(h) (independently of the other time intervals)

• Let qi denote the total transition rate out of state i, that is:

• Then, during a short time interval (t, t+h] , the conditional probability that there is a transition from state i to any other state is qih + o(h)(independently of the other time intervals)

• This is clearly a memoryless property• Thus, the holding time in (any) state i is exponentially distributed with

intensity qi

≠=

ijiji qq :

35


State transition probabilities

• Let Ti denote the holding time in state i and Tij denote the (potential) holding time in state i that ends to a transition to state j

• Ti can be seen as the minimum of independent and exponentially distributed holding times Tij (see lecture 5, slide 44)

• Let then pij denote the conditional probability that, when in state i, there is a transition from state i to state j (the state transition probabilities);

ijij

i TT≠

= min

i

ijijiij q

qTTPp ===

)(Exp,)(Exp ijijii qTqT ∼∼

36


State transition diagram

• A time-homogeneous Markov process can be represented by a state transition diagram, which is a directed graph where

– nodes correspond to states and

– one-way links correspond to potential state transitions

• Example: Markov process with three states, S = 0,1,2

0statetostatefromlink >⇔ ijqji

2

0

1

q20 q01

q21

q12

−+++−

+−= 0

0

Q

37


Irreducibility

• Definition: There is a path from state i to state j (i → j) if there is a directed path from state i to state j in the state transition diagram.

– In this case, starting from state i, the process visits state j with positive probability (sometimes in the future)

• Definition: States i and j communicate (i ↔ j) if i → j and j → i.

• Definition: Markov process is irreducible if all states i ∈ Scommunicate with each other

– Example: The Markov process presented in the previous slide is irreducible

38


Global balance equations and equilibrium distributions

• Consider an irreducible Markov process X(t), with state transition rates qij

• Definition: Let π = (πi | πi ≥ 0, i ∈ S) be a distribution defined on the state space S, that is:

It is the equilibrium distribution of the process if the following globalbalance equations (GBE) are satisfied for each i ∈ S:

– It is possible that no equilibrium distribution exists, but if the state space is finite, a unique equilibrium distribution does exist

– By choosing the equilibrium distribution (if it exists) as the initial distribution, the Markov process X(t) becomes stationary (with stationary distribution π)

(N)1= ∈Si iπ

(GBE) ≠≠ = ij jijij iji qq ππ

39


Example

2

0

1

1 1

µ

1

(N)1210 =++ πππ

1)1(

(GBE)11

11

12

201

20

⋅=+⋅⋅+⋅=⋅

⋅=⋅

πµπµπππ

ππ

µµµ

µ πππ +++

+ ===

31

231

131

0 ,,

−−

−=

µ1

10

01

Q

40


Local balance equations

• Consider still an irreducible Markov process X(t).with state transition rates qij

• Proposition: Let π = (πi | πi ≥ 0, i ∈ S) be a distribution defined on the state space S, that is:

•

If the following local balance equations (LBE) are satisfied for each i,j ∈ S:

•then π is the equilibrium distribution of the process.

• Proof: (GBE) follows from (LBE) by summing over all j ≠ i

• In this case the Markov process X(t) is called reversible (looking stochastically the same in either direction of time)

(N)1= ∈Si iπ

(LBE)jijiji qq ππ =

41


Contents

• Basic concepts

• Poisson process


42


Birth-death process

• Consider a continuous-time and discrete-state Markov process X(t)– with state space S = 0,1,…,N or S = 0,1,...

• Definition: The process X(t) is a birth-death process (BD) if state transitions are possible only between neighbouring states, that is:

• In this case, we denote

– In particular, we define µ0 = 0 and λN = 0 (if N < ∞)

01|| =>− ijqji

0: 1, ≥= −iii qµ0: 1, ≥= +iii qλ

43


Irreducibility

• Proposition: A birth-death process is irreducible if and only if λi > 0 for all i ∈ S\N and µi > 0 for all i ∈ S\0

• State transition diagram of an infinite-state irreducible BD process:

• State transition diagram of a finite-state irreducible BD process:

1 2λ1

µ2

0λ0

µ1

λ2

µ3

1 N-1λ1

µ2

0λ0

µ1

λN−2

µN−1

NλN−1

µN

44


Equilibrium distribution (1)

• Consider an irreducible birth-death process X(t)

• We aim is to derive the equilibrium distribution π = (πi | i ∈ S) (if it exists)

• Local balance equations (LBE):

• Thus we get the following recursive formula:

• Normalizing condition (N):

(LBE)11 ++= iiii µπλπ

∏=

+−

+==

i

jiii

j

j

i

i

101

1

1 µλ

µλ ππππ

(N)11

01 == ∏

∈ =∈

−

Si

i

jSii

j

jµ

λππ

44

45


1

1 10

10

11 1,

−∞

= ==

+== ∏∏ −−

i

i

j

i

ji

j

j

j

jµ

λµ

λπππ

Equilibrium distribution (2)

• Thus, the equilibrium distribution exists if and only if

• Finite state space:The sum above is always finite, and the equilibrium distribution is

• Infinite state space:If the sum above is finite, the equilibrium distribution is

∞< ∏∈ =

−

Si

i

j j

j

1

1µ

λ

1

1 10

10

11 1,

−

= ==

+== ∏∏ −− N

i

i

j

i

ji

j

j

j

jµ

λµ

λπππ

45 46


Example

1 2λ

µ

(N)1)1( 20210 =++=++ ρρππππ

ii

ii

ii

ρππ

µλρρππµπλπ

0

1

1

(LBE))/:(

=

==

=

+

+

21 ρρρπ++

=

i

i

0λ

µ

46

−−

−=

µλµ

λ

0

0

Q

47


Pure birth process

• Definition: A birth-death process is a pure birth process ifµi = 0 for all i ∈ S

• State transition diagram of an infinite-state pure birth process:

• State transition diagram of a finite-state pure birth process:

• Example: Poisson process is a pure birth process (with constant birth rate λi = λ for all i ∈ S = 0,1,…)

• Note: Pure birth process is never irreducible (nor stationary)!

1 2λ10

λ0 λ2

1 N-1λ10

λ0 λN−2N

λN−1

TKK · 2004-01-12 · Lect01.ppt S-38.145 - Introduction to Teletraffic Theory – Spring 2004 1 1....

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Transcript of TKK · 2004-01-12 · Lect01.ppt S-38.145 - Introduction to Teletraffic Theory – Spring 2004 1 1....