Modeles Trafic

7/28/2019 Modeles Trafic

1/96

Traffic Models

S. Vaton

Traffic Models

Sandrine VATON {[email protected]}Telecom Bretagne

TELECOM Sud Paris, may 5th 2008


2/96

Traffic Models

S. VatonTraffic Models (1/2)

Goals :

to analyze traffic measurements (file sizes, trafficvolumes/min., packet sizes (in bytes), packet interarrival

times, packet headers, etc...), to identify somecharacteristic properties (correlation, laws ofdistribution, etc...)to propose some traffic models, most of the timestatistical models,that take into account the main properties discovered in

the measured traffic,knowledge : statistics, statistical processes, signalprocessing, time series ...


3/96

Traffic Models

S. VatonTraffic Models (2/2)

Why do we need some traffic models ?

to design some traffic generators for lab-experiments,to be used as an input in some analytical performancemodels (queuing systems, large deviations theory, ...)

to detect some anomalies in the traffic (attacks,equipment failures, etc...) (as some significant deviationfrom some usual distribution of the traffic)knowledge : computer-based simulation, queuing

systems, large deviations theory, decision theory (e.g.detection of abrupt changes)


4/96

Traffic Models

S. VatonGoals of This Class

Taxonomy of traffic models :markovian models : from the exponential distribution toBMAP processesnon-markovian models : heavy-tailed distributions,

self-similarity, long-range dependence

Desired skills :to clarify some definitions (long-range dependence,self-similarity, etc...)to detect some properties in the traffic (heavy tails,

long-range dependence, etc...)parameter estimation : the value of the parameters of amodel are estimated from traffic measurementsto produce synthetic traffic from theoretical trafficmodelsto understand the impact of some characteristicproperties (e.g. : large buffer asymptotics for a queue

when the input traffic is self-similar)


5/96

Traffic Models

S. Vaton

Introduction

Exponential

Distribution,PoissonProcess

ContinuousTime MarkovChains

MarkovQueues

Phase-typedistributions

MAPProcesses

BMAPProcesses

Deuxieme partie II

Markovian Models


6/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Markovian Models

1 Introduction

2 Exponential Distribution, Poisson Process

3 Continuous Time Markov Chains

4 Markov Queues

5 Phase-type distributions

6 MAP Processes

7 BMAP Processes


7/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

A little bit of history... (1/3)

markovian models : from the Poisson model (Erlang,

1907) to BMAP processes (years 1980s)

markovian modelsvery usual traffic modelsrelatively simple usage (closed-form performanceformulas)

justify Markov models of queues

solved problems : parameter estimation, performance ofqueues when the input traffic is Markov


8/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses


but...Paxson and Floyd, The failure of Poissonmodelling, 1995

interarrival times are not exponential,sporadicity over several time scales,slow decay of correlations (long-range dependence),heavy-tailed distributions (duration/volume of TCPconnections ; file sizes, transfer time, read time inWWW, etc...),self-similarity of the cumulated workload, etc...


9/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses


Consequences :

standard markovian models are inaccurateQoS predictions (delay, loss, etc...) that are based onmarkovian models are dramatically optimistic

it becomes necessary to :design new traffic models (heavy tails, self-similarity,

long-range dependence, etc...)

Pareto distribution, Weibull distribution, -stabledistribution, fractional Brownian motion (fBm), fractional

Gaussian noise (fGn), fractional ARIMA process

(fARIMA), etc...parameter estimation for these new models

performance evaluation for these new models (queues),

traffic generators, etc...


10/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Markovian models : taxonomy (1/2)

exponential distribution :service time (time that is necessary in order to serve aclient), interarrival time (time between the arrivals of twoconsecutive clients)the oldest and simplest model (Erlang, beginning of theXXth century)

exponential distribution of the interarrival times = thecount process is a Poisson processcoefficient of variation cv = 1, 1 parameter only (mean =1/)

Phase-type distributions :

they generalize the exponential distribution,coefficient of variation cv > 1 or cv < 1, but severalparametersmore general, but more complicatedassociation of several exponential distributions in seriesor in parallel (several phases)

Erlang, Hypo-Exp, Hyper-Exp, Cox, etc...


11/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Markovian models : taxonomy (2/2)

MAP (Markovian Arrival Processes)make it possible to take into account time correlation intraffic (correlation in the arrival of successive packets,bursts, etc...)generalize the Poisson process and Phase-typedistributions

short-term correlation (contrary to long-rangedependent models)

BMAP (Batch Markovian Arrival Processes)as MAP, they take into account time correlation in traffic(correlation in the series of interarrival times),

but moreover, they take into account another feature(packet size, class of service,some field in the packetheader, etc...)2 traffic descriptors = {timestamp, size} (batch pointprocess)a batch (size of the packet, class of service, etc...) isassociated to the arrival time


12/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Taxonomy of Markovian Models

MAP BMAP

MMPP PH

MMPP2

IPP

COX HypoEXP

Erlang

HyperEXP

EXP


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Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Characterization of Continuous R.V.

Characterization of a continuous random variable X :

probability density function (pdf) p(x)

P(x1 X x2) =x2x1

p(u) du

cumulative distribution function (cdf) F(x)

F(x) = P(X x) =x

p(u) du

or complementary cdf Fc(x)

Fc(x) = P(X > x) = 1 F(x) =

+

xp(u) du

moments E(Xn), n = 1, 2, 3, . . .,

moment generating function M() = E(exp(X)),

cumulant generating function () = log M()


14/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Exponential Distribution : Definition

Definition : exponential distribution with parameter :probability density function (pdf) :

x < 0 p(x) = 0x

0 p(x) = exp(

x)

cumulative distribution function :

x 0, F(x) = 0x 0, F(x) =

x0 p(u) du = 1 exp(x)

complementary cumulative distribution function :

x 0, Fc(x) = 1x 0, Fc(x) =

+x p(u) du = exp(x)

ffi


15/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

1st and 2nd-order cumulants

Exponential distribution : 1st and 2nd order cumulants

let us consider a random variable X Exp() :

mean [1st order cumulant] :E

(X) = 1/, (unit of :sec1),

variance [2nd order cumulant] :

var(X) = E(X2) (E(X))2 = (1/)2,standard deviation : std(X) = var(X) = 1/,coefficient of variation : cv(X) = std(X)/E(X) = 1,

T ffi M d l


16/96

Traffic Models

S. Vaton

Introduction

Exponential



MarkovQueues


MAPProcesses

BMAPProcesses

Memoryless Property

Memoryless property :

let X be a continuous R.V. with exponential distribution,X Exp() ; then the memoryless property holds :

P(X x + u | X x) = P(X u),x 0,u 0

it means that if X represents a duration for example, thetime spent since start time does not change anything

on the distribution of the time left before the end ;

typically in the framework of queuing theory, X could

be the service time or the interarrival time (timebetween arrivals of two consecutive clients)

this property is fundamental since it justifies the

solution of some queues as Markov chains (e.g. queues

with Poisson input and exponential service times)

Traffic Models


17/96

Traffic Models

S. Vaton

Introduction

ExponentialDistribution,PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Poisson Process (1/3)

Count Process

Temps

Nombre de Clients

0

1

2

3

4

5

6

dj arrivs N(t)

Temps t

client 2 client 3 client 4 client 5 client 6client 1

E1 E2 E3 E4 E5

T1 T2 T3 T4 T5 T6arrivee arriveearrivee arrivee arrivee arrivee

temps entre arrivees

The count process N(t) counts, for example, the number of

arrivals of clients on [0, t].

Traffic Models


18/96

Traffic Models

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses


Poisson Process : definition

A count process N(t) is a Poisson process (with parameter) if the following conditions are satisfied :

the increments of N(t) are independent, which meansthat if t1 < t2 < t3 < t4 then N(t2) N(t1) andN(t4) N(t3) are independent,stationary increments : the distribution of

N(t + )N(t) (0 < t, ) is the same as the distributionof N() N(0),N(t + )

N(t) is a Poisson R.V. with mean

P(N(t + ) N(t) = k) = exp() ()k

k!

is the average number of events (arrival of clients, etc...)

per time unit (unit of : sec1).

Traffic Models


19/96

Traffic Models

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses


Poisson process : exponential interarrival times

Let {N(t), t R} be a count process, let T1 T2 T3 . . .be the arrival times, and let

E1 = T2 T1, E2 = T3 T2, E3 = T4 T3, . . . be thecorresponding interarrival times.

The following proposals are equivalent :

{N(t), t R} is a Poisson process with rate ,

E1, E2, E3, . . . are independent and identicallydistributed R.V. with distribution Exp().

Traffic Models

C i Ti M k Ch i (1/2)


20/96

Traffic Models

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses

Continuous Time Markov Chains (1/2)

Continuous Time Markov Chains : definition

A continuous time discrete state Markov chain (CTMC) is acontinuous time process with discrete state space which

satisfies to the weak Markov property :

P(Xtn = in | Xtn1 = in1, Xtn2 = in2, . . . , X t0 = i0)= P(Xtn = in | Xtn1 = in1),t0 < t1 < . . . < tn, i0, i1, . . . , in E

E

t

1

2

3

4

5

6

0

tn1 tnt0 t1 t2 t3

Traffic Models

C ti Ti M k Ch i (2/2)


21/96

Traffic Models

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses

Continuous Time Markov Chains (2/2)

Interpretation of the weak Markov property

future values of the process Xt depend on the presentvalue only

the only memory is the current value of the process

past and future are independent conditionnally to the

present value of the process

Traffic Models

Ch t i ti f CTMC (1/2)


22/96

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses

Characterization of a CTMC (1/2)

Characterization of a continuous time discrete state Markov

chain

A continuous time discrete state Markov chain can becharacterized by :

its state transition diagram

or, equivalently by its infinitesimal generator

Traffic Models

Ch t i ti f CTMC (2/2)


23/96

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses

Characterization of a CTMC (2/2)

Example : continuous time discrete state Markov chain,

state space E = {1, 2, 3, 4} :state transition diagram

0.1

1

2

0.80.4

0.3

0.14 1.9

3

infinitesimal generator

Q =

0.5 0.1 0.3 0.10.8 1.2 0 0.40 0 0 0

0 0 1.9 1.9

Traffic Models

State Transition Diagram


24/96

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses



0.1

1

2

0.80.4

0.3

0.14 1.9

3

For example, if the process is in state 1 then

it remains in state 1 for a time distributed as an exponentialR.V. with parameter 0.5 = 0.1 + 0.3 + 0.1 (average value1/0.5 = 2 seconds)

and after that time the process changes its state for state 2with proba. 0.1/0.5 = 1/5 or state 3 with proba. 0.3/0.5 = 3/5

or state 4 with proba. 0.1/0.5 = 1/5.

Traffic Models

Infinitesimal Generator


25/96

S. Vaton

Introduction



MarkovQueues


MAPProcesses

BMAPProcesses

Infinitesimal Generator

The infinitesimal generator of this example is the followingmatrix :

Q =

0.5 0.1 0.3 0.10.8 1.2 0 0.40 0 0 0

0 0 1.9 1.9

non-diagonal elements are transition rates : qij is the rate oftransition from state i to state j (i = j)

non-diagonal elements are always positive or nulldiagonal elements are always negative or null

the sum of the elements over each row is always 0 ;qii =

j=i qij

Traffic Models

Markov Queues


26/96

S. Vaton

Introduction

ExponentialDistribution,

PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Markov Queues

Markov Queues

if the following conditions are satisfied :1 the arrival of clients is a Poisson process (i.e. the

interarrival times are independent and identicallydistributed as Exp())

2 the service times are distributed as Exp()

then the number of clients in the system is a continuoustime discrete states Markov chain

the weak Markov property holds for the number of

clients in the system because the memoryless property

is satisfied by the interarrival times and the servicetimes

the theory of Markov chains is a powerful framework for

solving some performance evaluation problems

(typically queues with Poisson arrivals and exponential

service times)

Traffic Models

Example : M/M/1


27/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Example : M/M/1

Example of the M/M/1 :

M/M/1

arrival of clients : Poisson process with rate (averagenumber of clients per second)

service times : Exponential distribution with parameter (average service time : 1/ seconds)

1 server

no limitation on the size of the queue (infinite buffer)

FIFO : First In First Out

Traffic Models

M/M/1 : State Transition Diagram


28/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/1 : State Transition Diagram

State Transition Diagram : the number of clients in the

system (service or buffer) is a CTMC with the following state

transition diagram

... ... ...0 1 2 3

arrival rate : (rate at which clients arrive to thesystem, transitions n n + 1, n 0)departure rate : (rate at which clients leave thesystem, transitions n

n

1, n

1)

Traffic Models

M/M/1 : Steady-State Distribution


29/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/1 : Steady-State Distribution

Stability :

the system is stable if the condition < is satisfied

the clients should arrive at a rate lower than the rate at which the server is able to serve them

= is the load factor (unit : Erlangs) : condition of

stability : < 1Steady-state distribution :

If < 1 then the number of clients (service or buffer)converges in distribution to the following stationary

distribution

(n) = (1 )n, n 0

(n) is the probability that n clients are in the system

(in steady state)

Traffic Models

M/M/1 : Performance Measures


30/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/1 : Performance Measures

Performance Measures :

performance measures (input/output rate, server

utilization, average delay, etc...) are easily obtainedfrom the steady state distribution (n)

input/output rate :

server utilisation :

average delay :

1

1

1 =1

+1

1

Traffic Models

M/M/C/C


31/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/C/C

M/M/C/C

1

2

C

.

.

.

.

.

.

.

.

....

lost

M/M/C/C :arrival of clients : Poisson process with rate

service times : Exponential distribution with rate

C servers, no buffer

if a client arrives and if a server is free then the client startsservice immediatel ; otherwise the client is lost

Traffic Models

M/M/C/C : State Transition Diagram


32/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/C/C : State Transition Diagram

State Transition Diagram : the number of clients in thesystem (service or buffer) is a CTMC with the following state

transition diagram

... ... ...1 2 30 CC1

32 4 C(C 1)

arrival rate : (transitions n n + 1, n 0)departure rate : n (number of active servers ,transitions n n 1, 1 n C)

Traffic Models

M/M/C/C : Steady-State Distribution


33/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

M/M/C/C : Steady State Distribution

Steady-state distribution :

the system is always stable (systems with finite buffers

are always stable)

steady-state distribution

(n) = (0)n

n! 0 n CPerformance Measures :

the probability that a client is lost is given by the

Erlang-B formula

Erlang-B formula :

EB(, C) =C/C!

Cn=0

n/n!

Traffic Models

Phase-type distributions


34/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Phase type distributions

Phase-type distributionsPH-type distributions generalize the exponential

distribution

PH-type distribution = association of n exponentialdistribution in series or in parallel (n phases)

more general than the Exponential distribution (in

particular, they make it possible to take into account

some cases when Cv = 1)but more parameters than the Exponential distribution

Neuts, Matrix Geometric Solutions in Stochastic

Models, Johns Hopkins University Press, 1981

popular cases of PH-type distributions : Erlang,

Hypo-Exponential, Hyper-Exponential, Cox, etc...

Traffic Models

Popular Particular Cases (1/4)


35/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

opu a a t cu a Cases ( )

Erlang-distribution

... ... ... ...2

k1

FIG.: Erlang-k Distribution

X = X1 + X2 + . . . + Xk with Xi Exp()

Traffic Models

S VPopular Particular Cases (2/4)


36/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

p ( )

Hypo-Exponential Distribution

... ... ... ...

1 2 k

FIG.: Hypo-Exponential Distribution with k Phases, Ho(k)

X = X1 + X2 + . . . + Xk with Xi Exp(i)

Ho(k) distribution = k Exponential Phases in series

Erlang-k distribution is a particular case of Ho(k) distribution(when 1 = 2 = . . . = k)

in the case of the Ho(k) distribution it holds that Cv < 1,

hence the name hypo-exponential

Traffic Models

S V tPopular Particular Cases (3/4)


37/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

p ( )

Hyper-Exponential Distribution

.

.

.

.

.

.

.

.

.

.

...

.

.

2

k

1

pk

p2

p1

FIG.: Hyper-Exponential Distribution with k phases, Hr(k)

Hr(k) distribution = k Exponential phases in parallel

Cv > 1 hence the name Hyper-exponential

Traffic Models

S VatonPopular Particular Cases (4/4)


38/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

p ( )

Coxian distribution

1 2 3 k... ... ... ... k1

1 a1 1 a2 1 ak1

1 2 3 ka1 a2 ak1

FIG.: Coxian Distribution with k phases, Co(k)

Co(k) distribution = association of k Exponential phases inseries and parallel

Traffic Models

S VatonPH-type Distributions : General Case


39/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

yp

Er(k), Ho(k), Cox(k), Hr(k) are particular cases of

PH-type distributions

PH-type distribution : association of several exponential

phases (in series or in parallel)

PH-type distributions have been introduced by Neuts,

Matrix Geometric Solutions in Stochastic Models,Johns Hopkins University Press, 1981

in the general case, a PH-type distribution

is characterized by a continuous time Markov chain withone absorbing state

absorbing means that once the Markov chain hasreached that state the chain cannot leave the state (no

transitions from that state)the PH-type distribution represents the time spent in thedifferent transient states before the absorption occurs

k = number of phases = number of transient states

Traffic Models

S VatonExamples (1/2)


40/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

2/3

1/3

3

1 2

FIG.: PH-type distribution with k = 3 phases

Characterization by the following CTMC :

state space {e, 1, 2, 3} (e, absorbing state)

initial distribution : = (1/3, 0, 2/3) for the transient states {1, 2, 3}and 0 for the absorbing state e

infinitesimal generator

Q =

0

BB@

0 0 0 00 1 1 02 0 2 0

3 0 0 3

1

CCA

Traffic Models

S VatonExamples (2/2)


41/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Exercice :

In the case of Ho(k), Hr(k), and Co(k) distributions please

find out the CTMC that characterizes the distribution :

state space,

state transition diagram,

initial distribution,

infinitesimal generator.

Traffic Models

S. VatonCharacterization


42/96

S. Vaton

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

infinitesimal generator of the corresponding CTMC

Q =

0 0 0 . . . . . . 0 0t1t2t3 T......tk

where

T is a matrix of size k k which characterizes thetransition rates between transient statest is a vector of size k 1 such that t = T e wheree = [1 . . . 1]

initial distribution : , vector of size 1 k which characterizesthe initial distribution (the absorbing state is omitted)

Traffic Models

S. VatonMarkovian Arrival Processes (MAP)


43/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Markovian Arrival Processes (MAP)

modelisation of the arrivals (of packets, etc...) in thecase of correlated arrivals (bursts, etc...)

Temps

FIG.: Correlated Arrivals

MAP processes = generalization of PH-type

distributions

contrary to PH-type distributions, MAP processes take

into account the correlations between the successive

interarrival times

a popular particular case of MAP = the Markov

Modulated Poisson Process (MMPP)

Traffic Models

S. VatonMMPP Process (1/2)


44/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Markov Modulated Poisson Process (MMPP)

MMPP (Markov Modulated Poisson Process)

MMPP = Poisson process in which the rate dependson the state of a CTMC with finite state space

MMPP(2)

CMTC

FIG.: MMPP with 2 states

particular case of MMPP = Interrupted Poisson Process

(IPP)(2 states, 1 = 0,2 = 0)

Traffic Models

S. VatonMMPP Process (2/2)


45/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Markov Modulated Poisson Process

MMPP = particular case of Doubly Stochastic PoissonProcess (DSPP)

DSPP = Poisson process in which the rate (t) is itself

a stochastic process

Poisson P(N(t) = k) = exp(t) (t)kk!DSPP P(N(t) = k) = exp( t0 (u) du) (

Rt

0(u) du)k

k!

MMPP = Poisson process for which the rate (t) is aCTMC with states {1, 2, . . . , k}MMPP = particular case of MAP process

Traffic Models

S. VatonMarkovian Arrival Process (MAP)


46/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Markovian Arrival Processes (MAP)

MAP = generalization of PH-type distributions

correlation between successive interarrivals (contraryto PH-type distributions)

once the absorption has occurred the Markov chainstarts again from one of the transient states, taking intoaccount the last visited transient statethe distribution of the state from which the Markov chainstarts again depends on the last visited transient state

this distribution was noted in the case of PH-type

distributions;in the case of MAP processes there is one distribution

(i), i = 1, 2, . . . , k for each transient statei {1, 2, . . . , k})

Traffic Models

S. VatonExample of the MMPP(2) (1/2)


47/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Example : the case of the MMPP(2)

CTMC with 2 states

when the Markov chain is in state 1, arrivals as a

Poisson process with rate 1 ; when the Markov chain isin state 2, arrivals as a Poisson process with rate 2

1 2

Poisson(1) Poisson(2)

FIG.: MMPP a 2 etats

Traffic Models

S. VatonExample of the MMPP(2) (2/2)


48/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

MMPP(2) as a generalization of PH-type distributions

CTMC with one absorbing state estate space : {e, 1, 2}state transition diagram and infinitesimal generator

1 2

e

21

0@ 0 0 01 (1 + ) 2 (2 + )

t =

12

T = (1 + )

(2 + )

2 initial distributions : 1 = (1, 0) if the absorption hasoccurred from state 1 ; 2 = (0, 1) if the absorption has

occurred from state 2

Traffic Models

S. VatonBMAP Processes (1/2)


49/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Batch Markovian Arrival Processes (BMAP)

BMAP = generalization of MAP processesBMAP processes make it possible to take into account

the interarrival times but also the values of a given

batch (packet size, type of packet, Class of Service,

etc...)

Examples of batches :packet size (in bytes)TCP flag : SYN, SYN+ACK, FIN, etc...more generally, any information with finite state space{1, 2, . . . , m}

CMTC(2)

BMAP

FIG.: BMAP Process, k = 2, m = 3 (red/green/blue)

Traffic Models

S. VatonBMAP Processes (2/2)


50/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

Example : batch MMPP(2), (1=red ; 2=green ; 3=blue)

CTMC with 2 states

when the CTMC is in state 1, the packets are producedas a Poisson(1) process, packets are red with proba. =1/3, green with proba.=1/3, blue with proba.=1/3

when the CTMC is in state 2, the packets are producedas Poisson(2), packets are red with proba. = 1/2,

green with proba. = 0, blue with proba. = 1/2

Traffic Models

S. VatonTaxonomy of Markovian Models


51/96

Introduction


PoissonProcess


MarkovQueues


MAPProcesses

BMAPProcesses

MAP BMAP

MMPP PH

MMPP2

IPP

COX HypoEXP

Erlang

HyperEXP

EXP

Traffic Models

S. Vaton


52/96

Heavy-TailedDistributions

Long-Range

Dependence

Self Similarity

AgregationModels

Impact onPerformance

Troisieme partie III

Non Markovian Models

Traffic Models

S. VatonCriticism of Markovian Models (1/2)


53/96


Long-Range

Dependence

Self Similarity

AgregationModels


Paxson and Floyd, The failure of Poisson modelling, 1995

interrarival times are not exponential,

bursts over several time scales,slowly-decaying correlations (long-range dependence),

heavy tails (in duration/volume of TCP connections ; file

sizes, transfer times, reading times in WWW, etc...),

self-similarity of the cumulated workload, etc...

Traffic Models

S. VatonCriticism of Markovian Models (2/2)


54/96


Long-Range

Dependence

Self Similarity

AgregationModels


Goals of this class

clarify the concepts of heavy tails, long-range

dependence, self-similarity

introduce some tests to detect long-range dependence,

etc... in trafficintroduce some traffic models with heavy tails,

long-range dependence, self similarity, etc...

temptative explanation of the source of long-range

dependence in trafficstudy of the impact of long-range dependence on QoS

(delays, buffer occupation, etc...)

Traffic Models

S. VatonNon Markovian Models


55/96


Long-Range

Dependence

Self Similarity

AgregationModels


8 Heavy-Tailed Distributions

9 Long-Range Dependence

10 Self Similarity

11 Agregation Models

12 Impact on Performance

Traffic Models

S. VatonHeavy-Tailed Distributions (1/3)


56/96


Long-Range

Dependence

Self Similarity

AgregationModels


Heavy-Tailed Distributions :

Definition : a distribution is heavy-tailed if the

complementary cumulative distribution function decays

more slowly than the exponential :

> 0, limt exp(t)Fc

(t) = where Fc(t) = P(X t) is the complementarycumulative distribution function.

Examples of heavy-tailed distributions :

Pareto distribution : Fc(t) = ( ttmin ) with tmin, > 0(power-law tailed) ; infinite variance if < 2 ; infinitemean, infinite variance if 1Weibull distribution : Fc(t) exp((t/a)c) with c < 1and a > 0 ; heavy-tailed but not power-law tailed

Traffic Models



57/96


Long-Range

DependenceSelf Similarity

AgregationModels


Heavy-Tailed Distributions and Trafficused in order to take into account phenomenons ofextreme variability in trafficexample : duration and volume of TCP connections, offile sizes, of Web pages, etc...

Indications of heavy tails :mice/elephant phenomenon : a small number of flowsrepresent a large proportion of traffic in volume (Paretodistribution, also called 80/20 distribution)very large variability

extremely large flows (in volume) are not an exceptionhigh variance (of flow sizes, etc...)

power-law tails : the complementary cumulative

distribution function is linear in log-log scale

Traffic Models



58/96


Long-Range


AgregationModels


Example : traffic volume on one interface of a router on a

Local Area Network (5 days of traffic measurements)

0 20 40 60 80 100 1200

2

4

6

8

10

12

14x 10

6

Time (unit=hour)

TrafficVolume

Traffic Models

S. Vaton-stable Distributions


59/96


Long-Range


AgregationModels


-stable distributions :

often used in order to model very impulsive

phenomenons (bursts)

in what follows we restrict ourselves to -stablesymetric distributions

the distribution is approximately gaussian around the

mean value

but, when < 2, the -stable distribution isheavy-tailed (more precisely, power-law tailed)

Traffic Models

S. VatonInfluence of the parameter


60/96


Long-Range


AgregationModels


0 50 100 150 200 250 300 350 400 450 5005

4

3

2

1

0

1

2

3

4

5

0 50 100 150 200 250 300 350 400 450 50015

10

5

0

5

10

15

20

25

0 50 100 150 200 250 300 350 400 450 500100

50

0

50

100

150

0 50 100 150 200 250 300 350 400 450 5008

6

4

2

0

2

4

6

8

10

12 x 10

4

FIG.: -stable distributions with = 2.0, = 1.5, = 1.0, = 0.5

The phenomenon is more and more impulsive for small

values of the arameter . Traffic Models

S. VatonCharacterization of -stable distribution


61/96


Long-RangeDependence

Self Similarity

AgregationModels


-stable distributions : characterization

no analytical expression of the probability densityfunction in the general case (particular cases : = 2,Gaussian distribution ; = 1, Cauchy distribution)

characterization by the moment generating function :

M() =E

(exp(X)) = exp(

)parameters of the distribution :

, characteristic exponent (Levy index), values0 < 2, location parameter, (mean if 1 <

2, median if

0 < 1), dispersion parameter (dispersion of the valuesaround , similar to the variance of the Gaussiandistribution)other parameter : , skewness parameter, 1 1 ;symetric -stable distributions : = 0

Traffic Models

S. Vaton

H T il d

2 Important Particular Cases


62/96



Self Similarity

AgregationModels


2 important particular cases of -stable distributions

Gaussian distribution if = 2 :

f=2,,(x) =

1

4 exp((x

)2

4 )

Cauchy distribution if = 1 :

f=1,,(x) =1

2 + (x )2

Traffic Models

S. Vaton

H T il d

Tails of -Stable Distributions


63/96



Self Similarity

AgregationModels


Tails of -stable distributions

for < 2 the -stable distribution is heavy-tailed (moreprecisely, power law tailed) ;

Pareto distribution : Fc(t) = P(X t) = ( txmin ) ; thePareto distribution behaves as a power-law on all its

range of values [xmin, +[-stable distributions with < 2 behave as a power-lawin the tails of the distribution : limt t

Fc(t) = cst.

influence of the Levy index :

the smallest the value of (0 < < 2) the strongest isthe phenomenon of heavy tailsfor small values of extremely large values of the R.V.are not the exception ; modelisation of very impulsivephenomenons

Traffic Models

S. Vaton

Heavy Tailed

Moments of -Stable Distributions


64/96



Self Similarity

AgregationModels

Impact onPerformance Moments of -stable distributions

if < 2 then the variance of X is infinite

if moreover 1 then the mean of X is also infinite

Traffic Models

S. Vaton

Heavy Tailed

Generation of -Stable Distributions


65/96



Self Similarity

AgregationModels


-stable distributions : generation methods

we restrict ourselves to the case of symetric -stabledistributions even if similar expressions exist in the

general case

generation of a symetric -stable distribution

W, exponential R.V. with parameter = 1 ; , R.V. withuniform distribution over ] /2, +/2[

X =sin()

(cos())1/(

cos((1 ))W

)1

then X is -stable with characteristic exponent (standard symetric -stable distribution : = 0, = 0, = 1)

Traffic Models

S. Vaton

Heavy-Tailed

Generation of -Stable Distributions


66/96



Self Similarity

AgregationModels


2 important particular cases : = 2, Gaussian distribution

X =sin(2)

(cos())1/2(

cos()

W)1/2 = 2

W sin()

Box-Muller algorithm for the generation of the N(0, 1)distribution.

= 1, Cauchy distribution

X = tan()

This is a classical method for the generation of the

Cauchy distribution.

Traffic Models

S. Vaton

Heavy-Tailed

Long Range Dependence (LRD)


67/96

Heavy TailedDistributions


Self Similarity

AgregationModels


Paxson et Floyd, The failure of Poisson modellinglong-term correlations in traffic

this long-term correlation is contradictory with classical

(Markovian) traffic models (Poisson, MMPP, MAP, etc...)

Long-Range Dependence (LRD)definition

signature (autocovariance function, power density

spectrum, variance-time analysis)

some examples of LRD processes (fractional Gaussiannoise, fARIMA)

generation of LRD processes

Traffic Models

S. Vaton

Heavy-Tailed

Spectral Representation


68/96

Heavy TailedDistributions


Self Similarity

AgregationModels


2nd order stationarity (stationarity of the covariances)Let Xt be a discrete time stochastic process (time series).This process is 2nd order stationary if :

E(Xt) = m does not depend on t

E(Xt+k

Xt)

m2 depends on k only (but does notdepend on t)

Representation of a Second Order Stationary Process

A 2nd order stationary process can be represented :

in the time domain by its autocorrelation function

in the frequency domain by its power density spectrum

= repartition of the energy of the signal on the different

frequencies

Traffic Models

S. Vaton

Heavy-Tailed

Spectral Representation


69/96

yDistributions


Self Similarity

AgregationModels


Autocorrelation Function

The autocorrelation function of a 2nd order stationaryprocess is defined as follows :

(k) = E(Xt+kXt) m2

SpectrumThe power density spectrum of the process is the discrete

Fourier transform of the autocorrelation function :

S() =

+

k=

(k)exp(ik)

The discrete time Fourier transform can be obtained as the

output of the Fast Fourier Transform (FFT) algorithm.

Traffic Models

S. Vaton

Heavy-Tailed

LRD : definition


70/96

yDistributions


Self Similarity

AgregationModels


Long-Range Dependence (LRD) can be defined as

follows :slow decay of the autocorrelation function (k) of a 2ndorder stationary signal

divergence in = 0 of the power density spectrum S()both definitions are equivalent

(k)k

= O(k2H2

), 0.5 < H < 1.0S()

= O(12H), 0.5 < H < 1.0.

Hurst parameter H

H is the Hurst parameter of the process

the process is LRD if 0.5 < H 1the closest to 1 is H, the strongest the LRDphenomenon isin the case of Markovian models, autoregressivemodels, etc... the Hurst parameter is H = 0.5 (no LRD)traffic analyses find out most of the time H

0.80

Traffic Models

S. Vaton

Heavy-Tailed

LRD in Traffic (1/2)


71/96

Distributions


Self Similarity

AgregationModels


Autocorrelation Function

measured traffic trace (measurements from a Local AreaNetwork)

synthetic traffic : traffic generated according to an AR(2)model : X(t) =

2i=1 aiX(t i) + (t), (t) WGN(0, 2)

-0.2

0

0.2

0.4

0.6

0.8

1

"AR2"

"LAN Traffic"

k

Traffic Models

S. Vaton

Heavy-TailedDi ib i

LRD in Traffic (2/2)


72/96

Distributions


Self Similarity

AgregationModels


Power Density Spectrum

measured traffic trace (Local Area Network)

synthetic traffic (AR(2) model)

1000

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

1e+11

0.0001 0.001 0.01 0.1

1000

10000

100000

1e+06

1e+07

1e+08

1e+09

1e+10

0.0001 0.001 0.01 0.1

AR2LAN Traffic

S()

S()

Traffic Models

S. Vaton

Heavy-TailedDi t ib ti

LRD : signatures

Si f LRD


73/96

Distributions


Self Similarity

AgregationModels


Signatures of LRD :

slow decay of the autocorrelation functiondivergence of the power density spectrum in = 0

bad averaging properties :

when the signal is averaged, strong variations remain inthe averaged signal

variance-time analysis is based on these bad averagingproperties

Variance-Time Analysis

if Xt is a LRD process with Hurst parameter H then

var( 1m

i=1,m

Xi)m

= O(m2H2)

on the contrary, in the case of classical models (MAP, etc...)it holds that var( 1mi=1,m Xi) = O(m

1)

Traffic Models

S. Vaton


Variance-Time Plot (1/2)


74/96

Distributions


Self Similarity

AgregationModels


Variance-Time Plot :

average the signal on windows of m samples

compute the variance sm of the averaged signalrepeat for different values of m

plot sm against m in log-log scale : straight line withslope (2H 2)

Traffic Models

S. Vaton


Variance-Time Plot (2/2)


75/96

Distributions


Self Similarity

AgregationModels


measured traffic trace (measurements over a LAN)

synthetic traffic (AR(2) model)

1

10

100

1000

10000

1 10 100 1000

"AR2""LAN Traffic"

m

sm

Traffic Models

S. Vaton


Fractional Gaussian Noise (1/4)


76/96

Distributions


Self Similarity

AgregationModels


Fractional Gaussian Noise (fGn)popular particular case of LRD process

defined by the following equation

(1

B)dXt = t t

W GN(0, 2)

where

d is a fractional coefficient : 0 < d < 0.5(1 B)d = k=0 Ckd (1)kBk where B is the shiftoperator i.e. BXt = Xt1

Ckd = (d+1)(k+1)(dk+1) where (x) =0 ettx1 dt

the fGn is a LRD process with Hurst parameter

H = d + 0.5

Traffic Models

S. Vaton




77/96

Distributions


Self Similarity

AgregationModels


Influence of the parameter H :

-4

-3

-2

-1

0

1

2

3

4

0 200 400 600 800 1000

-4

-3

-2

-1

0

1

2

3

4

0 200 400 600 800 1000-4

-3

-2

-1

0

1

2

3

4

0 200 400 600 800 1000

-4

-3

-2

-1

0

1

2

3

4

0 200 400 600 800 1000

H=0.5 H=0.65

H=0.95H=0.8

t

tt

t

XH

(t)

XH

(t)

XH

(t)

XH

(t)

Traffic Models

S. Vaton




78/96

Distributions


Self Similarity

AgregationModels


Autocorrelation function

0.001

0.01

0.1

1

0 20 40 60 80 100

H=0.65

H=0.8

H=0.95-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

H=0.65

H=0.8

H=0.95

H=0.3

H=0.45

LRD

LRD

slope 2H2

kk

H H

Traffic Models

S. Vaton




79/96


Self Similarity

AgregationModels


Autocorrelation Function and Power Density Spectrum

autocorrelation function

(k) = corr(Xt, Xt+k) =12 (|k + 1|2H 2|k|2H + |k 1|2H)

(k)k

= O(H(2H 1)k2H2)

power density spectrum

S() =22|1 ei|2d =

2

2|2sin(/2)|2d

and when 0 it holds thatS()

0 2

2||2d =

2

2||12H

Traffic Models

S. Vaton


Generation of a fGN

Method based on the FFT


80/96


Self Similarity

AgregationModels


Method based on the FFT

let Xt be a fGn, and S() its power density spectrumXt is a 2nd order stationary process and thus Xt isharmonizable

Xt =20 e

itS1/2()dB()

where Bt

is the standard Brownian motiondiscrete approximation of eitS1/2()

IN =N1

n=0 eit 2

N S1/2( 2N )(B(2(n+1)

N ) B( 2nN ))=

N1n=0 e

it 2N S1/2( 2N )

2N n

IN converges to Xt (in the sense of the L2 norm)Fast Fourier Transform algorithm (FFT)

S1/2( 2nN )

2N n, 0 n N 1

FFT

N1n=0 e

it 2nN S1/2( 2nN )

2N n, 1 t N

Traffic Models

S. Vaton


Self-similarity : Principle

Intuitive idea : deterministic self similarity


81/96


Self Similarity

AgregationModels


Intuitive idea : deterministic self-similarity

deterministic self-similarity : the same pattern is

repeated on all scales

statistical self-similarity : same characteristics (bursts,

etc...) at all time scales

Traffic Models

S. Vaton


Self-Similarity in Traffic


82/96


Self Similarity

AgregationModels


Paxson and Floyd, The failure of Poisson modellingself-similarity of the cumulated workload

variations are present at all time scales

no typical time scale for bursts, etc...

the cumulated workload is highly variable

Self-Similarity

definition of self-similarity

signatures of self-similarity

relation with LRD

example : fractional Brownian motion

Traffic Models

S. Vaton


Definition of Self-Similarity


83/96


Self Similarity

AgregationModels


Statistical self-similarity : definitionLet Yt be a continuous-time process. Yt is self-similar, withself-similarity coefficient H, if

{Yat, t 0}(d)

= {aH

Yt, t 0}

equality between the laws of the processes

a change of time scale does not change the shape of

the self-similar process

the interesting case is the case when 0.5 < H < 1.0

Traffic Models

S. Vaton


Signatures of Self-Similarity (1/3)

Indexes of self-similarity


84/96


Self Similarity

AgregationModels


Indexes of self similarity

mean/variance index (IDC, Index of Dispersion for Counts)

R/S index (Rescaled Adjusted Range, Pox diagram)

Index of Dispersion for Counts (IDC)

historically used in order to test the validity of the Poissonassumption ; gives an estimation method for the Hurst

exponent H

let Nt be a Poisson process with rate , thenvar(Nt) = moy(Nt) = t and IDC(t) = 1

let Yt be a self-similar process with coefficient H

IDC(t) = var(Y(t))moy(Y(t))

= var(A(t))t

t2H1

plot IDC(t) against t in log-log scale ; straight line with slope(2H 1)the case of the Poisson rocess corres onds to H = 0 5

Traffic Models

S. Vaton



R/S i d (P di )


85/96


Self Similarity

AgregationModels


R/S index (Pox diagram)

principle : the range of values taken by Yt (around alinear trend t) is very large when H is close to 1

R/S index : definition

R (Range) measures the maximal range of values takenby Yt around a linear trend

S (Standard Deviation) measures the standarddeviation of the process of increments Xt = Yt Yt1more precisely,

R, R(t, k) = max0ik[Yt+i Yt ik (Yt+k Yt)]min0ik[Yt+i Yt

i

k (Yt+k Yt)]S, S(t, k) =

k1

i=1,k(Xt+i Xt,k)2

avec Xt,k = k1

i=1,k Xt+i

Traffic Models

S. Vaton




86/96


Self Similarity

AgregationModels


R/S index (Pox diagram)

R/S index : results

If Yt is self-similar with Hurst parameter H then

E(R(t, k)/S(t, k)) = O(kH)

Pox diagram : plot the values of R/S against k inlog-log scale ; straight line with slope Hif H is in the range 0.5 < H < 1.0 then the process isself-similar ; otherwise straight line with slope 0.5

Traffic Models

S. Vaton


Self-Similarity and LRD


87/96


Self Similarity

AgregationModels


a self-similar process is never 2nd order stationary

(contrary to LRD processes)

Which model should we choose ?

if the time series looks stationary then a LRD processcan be usedcumulated workload : self-similar process ; example :fractional Brownian motion, stable Levy motion, etc...workload (traffic per second, per minute, etc...) : LRDprocess; example : fractional Gaussian noise, fARIMA,

etc...

Traffic Models

S. Vaton


Fractional Brownian Motion (1/2)


88/96


Self Similarity

AgregationModels


Refresher : standard Brownian motion Bt

continuous time Gaussian process

stationary and independent increments

distribution of the increments Bt

Bs

N(0, 2

|t

s|)

Fractional Brownian motion B(H)t (fBm)

continuous time Gaussian process

stationary increments

distribution of the incrementsB

(H)t B(H)s N(0, 2|t s|2H)

Traffic Models

S. Vaton


Fractional Brownian Motion (2/2)

Fractional Brownian motion : properties


89/96


Self Similarity

AgregationModels


p p

for H in the range 0.5 < H < 1.0 the increments are notindependent (contrary to the standard Brownian

motion)

the fractional Brownian motion is a self-similar process

a, {B(H)

at , t 0}(d)

= {aH

B

(H)

t , t 0}the case of the standard Brownian motion is the case

H = 0.5

Fractional Brownian motion and fractional Gaussian noise

let Xt be the increment process Xt associated to thefractional Brownian motion B

(H)t

Xt = B(H)t B(H)t1

Xt is a fractional Gaussian noise with coefficient H.

Traffic Models

S. Vaton


L R

Origins of Long Memory in Traffic


90/96


Self Similarity

AgregationModels


Agregation Models :

some temptative explanations of some phenomenons

observed in traffic (self-similarity of the cumulated

workload, LRD in the series of traffic/minute, interarrival

times, etc...)

principle : the superposition of short-term memoryeffects at the microscopic level can result in some

long-memory effects at the macroscopic level

most popular model :

superposition of a large number of ON/OFF sourceswith Pareto distributed ON and/or OFF periods...Willinger, Taqqu et al., Self-Similarity Through HighVariability : Statistical Analysis of LAN Traffic Data, 1997

Traffic Models

S. Vaton


L R

Superposition of ON/OFF Sources (1/2)


91/96


Self Similarity

AgregationModels


Agregation of ON/OFF sources

ON/OFF source :

alternance of On and Off periodsOn periods : some traffic is emitted, Off periods : the

source is silentExample : Web navigation = downlad times / reading

timesOn/Off model characterized by the distribution of Onand Off periods

ONOFF OFF OFFON

Agregation of On/Off sources :

superposition of a very high number of On/Off sourcesPareto-type distribution of the On and/or Off periods

Traffic Models

S. Vaton


Long Range

Superposition of ON/OFF Sources (2/2)

Origins of LRD in Traffic


92/96


Self Similarity

AgregationModels


Willinger, Taqqu et al., 1997

superposition of a high number of ON/OFF sources

Power-law tailed distribution of ON and OFF periods

(Pareto-type distribution) :

Fci (x)x

= O(xi ) i = 1

ON,i = 2

OF F

with 1 < i < 2 (finite mean, infinite variance)

2 limits : the number of microscopic sources goes to infinity,and the duration of their action goes to infinity

the cumulated workload converges to

either a fractional Brownian motion (Gaussianself-similar process)or a stable Levy motion (-stable self-similar process)

(the convergence depends on the order in which the limits

are taken).

Traffic Models

S. Vaton


Long Range

LRD and QoS


93/96


Self Similarity

AgregationModels


Impact of LRDimpact of LRD in traffic on some parameters of QoS

(delay, losses, . . . )

Markovian models of traffic

are dramatically optimistic

since they undervalue the probability of some rareevents (buffer overflows, very long delays, etc...)if the traffic is in fact LRD

case of LRD offered traffic

queues with LRD offered traffic

asymptotic performance results (large buffers, very longdelays, . . . )these results are based on large deviations theory

Traffic Models

S. Vaton


Long-Range

Example : LRD and Delay

Delay in a G/G/1 queue


94/96


Self Similarity

AgregationModels


2 cases : LRD offered traffic, offered traffic with short

memory

we are interested in the tails of the distribution of delays

(very long delays,. . . )

well-known particular case : M/M/1, the distribution of

the delay (service time + waiting time) is exponentialwith mean value (1/) 11distribution of the waiting-time W :

case of short-memory offered traffic

P(W > x) exp(x) exponentialcase of long-memory offered traffic

P(W > x) exp(x22H) Weibull

Traffic Models

S. Vaton


95/96

Quatrieme partie IV

Conclusions

Traffic Models

S. Vaton Summary


96/96

Summary of this Class

1 Markovian Models

Exponential distribution and Poisson process,PH-type distributions,MAP processes (special case of the MMPP)BMAP processes

2 Other Models

heavy-tailed distributions,Long Range Dependence,self-similarity

signatures in trafficexplanations of LRD in trafficimpact on performance (delays)

Questions?

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