1. Hybrid Modeling of TCP Congestion Control João P. Hespanha, Stephan Bohacek, Katia Obraczka, Junsoo Lee University of Southern California

2. Background TCP/IP –Transmission Control Protocol/Internet Protocol –WWW, Telnet, FTP –UNIX, Windows 98, Windows 2000 all include TCP/IP –The evolution of TCP/IP is supported by Internet Engineering Task Force(IETF) –Window based congestion control –If congestion occurs reduce sending rate to half, otherwise increase window size by 1 for each round trip time

3. Congestion control in data networks Congestion control problem: How to adjust the sending rates of the data sources to make sure that the bandwidth B of the bottleneck link is not exceeded? B sources destinations B is unknown to the data sources and possibly time-varying

4. Congestion control in data networks q( t ) ´ queue size r 1 bps r 2 bps r 3 bps rate · B bps Congestion control problem: How to adjust the sending rates of the data sources to make sure that the bandwidth B of the bottleneck link is not exceeded? queue (temporary storage for data)

5. Congestion control in data networks When  i r i exceeds B the queue fills and data is lost (drops) rate · B bps ) drop (discrete event) Event-based control: The sources adjust their rates based on the detection of drops r 1 bps r 2 bps r 3 bps q( t ) ´ queue size queue (temporary storage for data)

6. Window-based rate adjustment w i (window size) ´ number of packets that can remain unacknowledged for by the destination 1 st packet sent e.g., w i = 3 t 2 nd packet sent 3 rd packet sent 1 st packet received & ack. sent 2 nd packet received & ack. sent 3 rd packet received & ack. sent 1 st ack. received ) 4 th packet can be sent t source idestination i w i effectively determines the sending rate r i : round-trip time t0t0 t1t1 t2t2 t3t3 00 11 22

7. Window-based rate adjustment w i (window size) ´ number of packets that can remain unacknowledged for by the destination ´ sending rate total round-trip time propagation delay per-packet transmission time time in queue until transmission This mechanism is still not sufficient to prevent a catastrophic collapse of the network if the sources set the w i too large queue gets full longer RTT rate decreases queue gets empty negative feedback

8. TCP Reno congestion control 1.While there are no drops, increase w i by 1 on each RTT 2.When a drop occurs, divide w i by 2 disclaimer: this is a simplified version of Reno that ignores some interesting phenomena… Network/queue dynamicsReno controllers  drop occurs drop detected (one RTT after occurred)  (congestion controller constantly probe the network for more bandwidth)

9. Switched system model for TCP queue-not-full queue-full (drop occurs) (drop detected) transition enabling condition state reset

10. Switched system model for TCP queue-not-full queue-full (drop occurs) (drop detected)  2 { 1, 2 } alternatively… continuous dynamics discrete dynamics reset dynamics  = 2  = 1

11. Linearization of the TCP model Time normalization ´ define a new “time” variable  by queue-not-full queue-full In normalized time, the continuous dynamics become linear 1 unit of  ´ 1 round-trip time

12. Switching-by-switching analysis queue-not-fullqueue full queue-not-fullqueue full queue-not-fullqueue full t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 ´ k th time the system enters the queue-not-full mode x1x1 x2x2 T state space x1x1 x2x2 impact map queue-not-full queue-full

13. Switching-by-switching analysis queue-not-fullqueue full queue-not-fullqueue full queue-not-fullqueue full t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 ´ k th time the system enters the queue-not-full mode x1x1 x2x2 T Theorem. The function T is a contraction. In particular, Therefore x k ! x 1 as k !1 x 1 ´ constant x ( t ) ! x 1 ( t ) as t ! 1 x 1 (t) ´ periodic limit cycle

14. NS-2 simulation results 0 100 200 300 400 500 01020304050 Window and Queue Size (packets) time (seconds) window size w 1 window size w 2 window size w 3 window size w 4 window size w 5 window size w 6 window size w 7 window size w 8 queue size q Router R1 Router R2 TCP Sources TCP Sinks Bottleneck link 20Mbps/20ms Flow 1 Flow 2 Flow 7 Flow 8 N1N1 N2N2 N7N7 N8N8 S1S1 S2S2 S7S7 S8S8

15. Results queue-not-fullqueue full queue-not-fullqueue full queue-not-fullqueue full t0t0 t1t1 t2t2 t3t3 t4t4 t5t5 t6t6 Window synchronization: convergence is exponential, as fast as.5 k Steady-state formulas: average drop rate average RTT average throughput

19. What next? queue-not-full queue-full One drop per flow is very specific to this network: all flows share the same queue similar propagation delays for all flows constant bit-rate cross traffic “drop-tail” queuing discipline r 1 bps r 2 bps r 3 bps B bps queue Other models for drops:

20. What next? Other models for drops: How many drops? t q( t )q( t ) queue-not-full queue full q max # of drops ´ s queue-full (inrate  outrate) Which flows suffer drops? number of packets that are out for flow i total number of packets that are out This probabilistic hybrid model seems to match well with packet-level simulations, e.g., with drop-head queuing disciplines. Analysis ???

21. What next? More general networks: qAqA qCqC qDqD qBqB flow 1 flow 2 flow 3

22. What next? More general networks: portion of the queue due to flow i outgoing rate of flow i total queue size  drop occurs qAqA qCqC qDqD qBqB flow 1 flow 2 flow 3

23. What next? More general networks: qCqC qDqD qAqA qBqB flow 1 flow 2 flow 3  drop occurs

24. What next? Even multicast (current work)… qCqC qDqD qAqA qBqB flow 1 flow 2  drop occurs

25. Conclusions Hybrid systems are promising to model network traffic in the context of congestion control: retain the low-dimensionality of continuous approximations to traffic flow are sufficiently expressive to represent event-based control mechanisms Hybrid models are interesting even as a simulation tool for large networks for which packet-by-packet simulations are not feasible Complex networks will almost certainly require probabilistic hybrid systems

26. END

27. Switching-by-switching analysis state space xkxk x k+1 impact map queue-not-full queue-full Impact maps are difficult to compute because their computation requires: Solving the differential equations on each mode (in general only possible for linear dynamics) Intersecting the continuous trajectories with a surface (often transcendental equations) It is often possible to prove that T is a contraction without an explicit formula for T…