Interacting Network Elements: Chaos and Congestion Propagation Gábor Vattay Department of Physics of Complex Systems Eötvös University, Budapest, Hungary
Web servers Web client
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 Servers ACK-s
The flow
Internet
Basics of traffic modeling
Router (telephone exchange) Incoming phone calls Outgoing phone lines NQ
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 ?
012Q Prob. To have n occupied lines at time t Markovian model for line occupancy n= Poisson distribution
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 traffic on a heavily used link [packets/sec] aggregated traffic average+fluctuation average number of packets/sec m mean variance of fluctuations relative variance or time variance
for Poisson traffic for Internet traffic H=0.8
Hurst exponent on the internet …
…and the brain …
Mathematical tools
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
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/queue deffect propagation
10 queues, 5 TCP/queue, web traffic
1TCP/que with initial delay t_d t_d [ms]timespan sec
Measuring the speed of propagation: center of mass velocity
td[ms] Site/sec
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