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### Nonlinear Dynamics in TCP/IP networks

Ljupco Kocarev

Institute for Nonlinear Science

University of California San Diego

Outline

- What is nonlinear time series analysis?
- Evidence of nonlinear behavior in TCP/IP networks
- Conclusions and open problems

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R. E. Kalman, 1956

“Nonlinear aspects of sampled-data control systems”

Markov process with transition probabilities:

Fact:Deterministic chaos as a fundamental concept is by now well established and described in literature. The mere fact that simple deterministic systems generically exhibit complicated temporal behavior in the presence of nonlinearity has influenced thinking and intuition in many fields.

What is nonlinear time series analysis?

Nonlinear time series analysis is a tool for study of compex and nonlienar dynamics from measurements

- H. D. I. Abarbanel, “Analysis of Observed Chaotic Data” Springer, New York (1996)
- H. Kantz and T. Schreiber, “Nonlinear Time Series Analysis” Cambridge University Press, Cambridge (1997)
- Software packageTISEAN (publicly available)

Phase space representation: Delay coordinates, Embedding parameter, Principal components, Poincaré sections, SVD filters

Visualization, non-stationarity: Recurrence plots, Space-time separation plot

Nonlinear prediction: Model validation, Nonlinear prediction, Finding unstable periodic orbits, Locally linear prediction, Global function fits

Nonlinear noise reduction: Simple nonlinear noise reduction, Locally projective nonlinear noise reduction, Nonlinear noise reduction

Lyapunov exponents: Maximal exponent, Lyapunov spectrum

Dimensions and entropies: Correlation dimension, Information dimension

Testing for nonlinearity: Surrogate data, Iterative Fourier transform method, General constrained randomization, Measuring weak nonlinearity

1982 first attempt to apply chaos theory to power grids

- 1997 connection between chaos and blackouts began to tighten when researchers started to work with actual blackout data
- 2004 The Unruly Power Grid – cover story of the August issue of Spectrum

Two opposite classes of systems:

Nonlinear and fully deterministic systemsStochastic systems

Assumption:The bulk of real world time series falls in neither of these limiting categories because they reflect nonlinear and deterministic responses and effectively stochastic components at the same time.

Evidence of nonlinear behavior in TCP/IP networks

Complex Dynamics in Communication Networks

(Edited by L. Kocarev and G. Vattay)

to be published by Springer 2005

Nonlinear Dynamics of TCP and its Implications to Network Performance

A. Veres and M. Boda

- The model consists of two end-hosts, both running Linux kernel version 2.4
- Two hosts can be far from each other: the propagation delay in the lab experiment is emulated by the NIST Net network emulator
- tcptrace utility: for calculating the window size as a function of time

TCP congestion window dynamics at increasing speeds.

Each figure shows both TCP window processes one on top of the other.

Propagation delay 100ms

Impact of a perturbing packet (which happens exactly at 60sec)

on TCP window dynamics at different service rates.

TCP rate processes at different buffer sizes

- (service rate 1200 kbps, delay 100 ms)
- When the buffer size is 10 packets, the traffic looks random and shows short timescale variations
- When the buffer size is 5 and 20 packets, we see long, alternating periods of high and low transmission rates

Spatio-tempral graph of 30 TCP window processes sharing a single bottleneck. Time flows from left to right, light shades represent large windows, dark shaded represent low windows. Spatio-temporal graph of the original system (top). Spatio-temporal graph of the perturbed system (middle). Difference between the two systems (bottom).

The difference between the two systems increase

at an average rate of every second

Dynamics of Congestion Control

A. C. Gilbert

Many experiments and the intuitive explanations of these experiments show that TCP sources competing for bandwidth on a congested link will synchronize through the weak coupling inherent in congestion control.

The graphs show the evolution of packet arrival rates and queue occupancies at a bottleneck link shared by 50 TCP sources sending an infinitely long file. On the top are results for a drop-tail policy; on the bottom are those for RED.

There is strong aggregate periodic behavior, made more clear by the strong component in the discrete Fourier transform of the arrival rate (below each figure).

The more pronounced periodic behavior caused by RED is counter to the commonly held intuition that a randomized drop-policy would prevent periodic behavior by ‘desynchronizing’ TCP sources.

Aggregate arrival rate shows periodic behavior with fixed RTTs with both drop-tail and RED

In this figure RTT is 140ms: aggregate rate still fluctuates with a period of about 2 seconds, and

the periodicity is more prominent with RED

Statistical Properties of Chaos in Communication Networks

G. Vattay et al.

On Dynamics of Transport Protocols Over Wide-Area Internet Connections

N. S. V. Rao, J. Gao and L. O. Chua

Number of traces using single and two competing TCP streams on two different connections from ORNL to Georgia Institute of Technology (GaTech) and to Louisiana State University (LSU) are collected

First connection: high-bandwidth (OC192 at 10Gbps) with relatively low backbone traffic and a round-trip time of about 10 milliseconds

Second connection: much lower bandwidth (10 Mbps) with higher levels of traffic and a round trip-time of about 26 milliseconds

Power spectral analysis of these data does not show any dominant peaks, and hence, the dynamics are not simply oscillatory

Data was measured on the Internet with ‘live’ background traffic, it is apparently more complicated and realistic than ns-2 traces

are vectors constructed from a scalar time series using the embedding theorem

Brackets denote the ensemble average of all possible (Vi,Vj) pairs

For low-dimensional chaotic systems, the curves for different shells form a common envelope, and the slope of the envelope is an estimate of the largest positive Lyapunov exponent.

The dynamics cannot be characterized as pure deterministic chaos, since in no case can we observe a well-defined linear envelope. Thus the random component of the dynamics due to competing network traffic is evident and can not simply be ignored.

- The data is not simply noisy, since otherwise we should have observed that is almost flat when k > (m-1)L. Thus, the deterministic component of dynamics which is due to the transport protocol plays an integral role and must be carefully studied.
- The features (ii) and (iii) indicate that the Internet transport dynamic contains both chaotic and stochastic components.

Conclusions and open problems

There exist plenty of theoretical and simulation evidences of nonlinear dynamics and chaos in TCP/IP networks

There exist only a few measurement evidences of nonlinear dynamics and chaos in TCP/IP networks

In terms of actual Internet traffic the question of the deterministic (chaotic) nature of transport dynamics is still open

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