Interacting Network Elements: Chaos and Congestion Propagation

Gábor Vattay

Department of Physics of Complex Systems

Eötvös University, Budapest, Hungary

Web servers

Webclient

Traffic

Convergence of technology

• Internet protocol (IP) takes over

• The Information has to be cut into packets

• Packets get a universal IP address and handled by heterogeneous network elements

From servers to users

User

Ser

vers

ACK-s

The flow

Internet

Basics of traffic modeling

Router (telephone exchange)

Incoming phone

calls

Outgoing phone

lines

N Q

Erlang’s formula (1917)

- Analyzed the phone calls in a small danish

village and came up with a robust model

Number of subscribers: N

Number of outgoing lines: Q

Call arrival rate [calls/sec]

Call holding times [sec]

What is the distribution of occupied lines ?

/1

dtn

dtnN

dtN

tn

tn

tn

dttn

)(1

1)(

)(

1)(

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0 1 2 Q

N N N

2 Q3

(t)Pn Prob. To have n occupied lines at time t

(t)Pn)((t)P)1(n(t)Pdt

dPn1n1n

n NN

Markovian model for line occupancy

n=

Q

k

k

k

N0n

n

0n

!P1,

,n!

PP

Poisson distribution

10

N

n

dtn

dtnN

dtN

tndttntn

)(1

1

0

1

)()()(

dtnNdtn 11

N

nn *0

NdtdtnNdtn 2)1()1()( 222

tDn 2)( On short time scales the process is Brownian

Typical internet traffic traces

W. E. Leland et al. SIGCOMM 93

1/f noise in ‘ping’ traces

I. Csabai, Journal of Physics A27, L417 (1994)

Modeling Internet traffic

It is harder to smooth out Internet traffic Paxson & Floyd 1995

Fractal traffic modeling)(tx traffic on a heavily used link [packets/sec]

t

t dttxtY ')'()( aggregated traffic

YYtY )( average+fluctuation

mY

2)( Y

average number of packets/sec m

mean variance of fluctuations

Y

Y 2)( relative variance or time variance

1~)( 2

Y

Yfor Poisson traffic

HY

Y1

2

1~)(

for Internet traffic

H=0.8

Hurst exponent on the internet …

…and the brain …

Mathematical tools

ationautocorrel theis where

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~ that assume We

,')'( where

,... :nfluctuatio Traffic

:Unit time

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Long range dependence (LRD)

Internet as a large dynamical system

TCP Congestion Control

• End-to-end principle

• Round trip time RTT

• Packet loss detection,

time-out, out of sequence

packet

• Packet loss probability

• Acknowledgement

• Congestion window: number

of unacknowledged packets

out in the network

Slow-start

• w=1

• each time an ACK arrives two new packets are sent w’ = w + 1

• In each round trip time the cwnd doubles

• Slow-start is terminated after the first packet loss, cwnd is halved w’=[w/2]

Congestion avoidance

• One new packet is sent out at each ACK

w’ = w+1/w

• If cwnd is an integer, then two packets are sent out

• At each packet loss the cwnd is halved w’=[w/2]

Chaos

Simplest network model

Periodicity

Veres & Boda INFOCOM 2000

Chaos

Veres & Boda INFOCOM 2000

Liapunov properties

Veres & Boda INFOCOM 2000

Veres & Boda INFOCOM 2000

3 TCPs with different round trip times

Vattay, Marodi, Steger 2002

Congestion window evolution

Poincarè surface of section

Symbols: 1,2,3

Fractal dimension of the attractor

Symbol sequence probability

Topological entropy

Basin of attraction

2 TCPs surface of section

Topological entropy

Interaction of flows

Interacting traffic flows

• Traffic flows crossing the same bottleneck can inherit scaling properties from each other

Kenesi, Molnár, Veres, Vattay SIGCOMM 2000

Mode locking structure of adaptation

Buchta & Vattay 2003

TCP

Background (UDP)

Bandwidth C

Congestion propagation

Fukuda &Takayasu 1999

Router-to-router congestion propagation

A congestion propagation model

Vattay, Steger, Vaderna 2003

Simulation results: 10 queues, 10 TCP

50 queues, 1 TCP/queuedeffect propagation

10 queues, 5 TCP/queue, web traffic

1TCP/que with initial delay t_dt_d [ms] timespan 0-4000 sec

0                                                0.01                                                0.1                                                0.2                                                0.3                                                0.4                                                0.5                                                0.6                                                

Measuring the speed of propagation: center of mass velocity

                 

                 

                 

                 

                 

td[ms] Site/sec

0 -0.154421

.1 -0.140778

.2 -0.141784

.3 -0.120683

.4 -0.059653

.5 -0.103517

.6 -0.032555

What causes congestion propagation?

1 TCP/queue (ns2) Our fluid model using Baccelli-Hong (2002)

Only one assumption is needed: packet loss is more likely for a joining TCP

flow at the router