Nonlinear Dynamics in TCP/IP networks. Ljupco Kocarev Institute for Nonlinear Science University of California San Diego. Outline. What is n onlinear time series analysis? Evidence of nonlinear behavior in TCP/IP networks Conclusions and open problems. x n+1. 1. 1/2. 1/3. 1/2.
Institute for Nonlinear Science
University of California San Diego
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.
Nonlinear time series analysis is a tool for study of compex and nonlienar dynamics from measurements
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
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.
Complex Dynamics in Communication Networks
(Edited by L. Kocarev and G. Vattay)
to be published by Springer 2005
A. Veres and M. Boda
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.
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
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
G. Vattay et al.
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.
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