1. Control Theory in TCP Congestion Control and new “FAST” designs. Fernando Paganini and Zhikui Wang UCLA Electrical Engineering July 2002. Collaborators: Steven Low, John Doyle, Jiantao Wang (Caltech). Earlier versions: Sachin Adlakha, Sanjeewa Athuraliya.

2. Basic fluid-flow models Routing matrix : 1 2 3 Feedback mechanism: L communication links shared by S source-destination pairs.

3. Congestion Control Loop LINKS SOURCES ROUTING : link prices : aggregate link flows : source rates : aggregate prices per source Decentralized control at links and sources. Routing assuming fixed, or varying at much slower time-scale.

4. Link and Source Controls determine: Steady-state properties: –Utilization –Queues –Fairness Dynamic properties: –Speed of response –Stability, oscillations. –Sensitivity to noise.

5. Issues with currently deployed TCP: Steady-state properties: –Utilization: (can be affected by dynamics) –Queues (~full with DropTail). –Fairness (dependence on RTT) Dynamic properties: –Speed of response: Additive increase is too slow in large cwnd sizes (high capacity networks). –Stability, oscillations: Multiplicative decrease is too aggressive in large cwnd. –Sensitivity to noise.

6. Stability of TCP: ns-2 simulations 50 identical FTP sources, single link 9 pkts/ms, RED AQM Stable case, RTT= 40ms Queue Unstable case, RTT= 200ms Window

7. Fluid Modeling of TCP-Reno/RED (Misra, Gong, Towsley, Hollot ’00/’01, Low-Paganini-Doyle ’02) Additive Increase Multiplicative Decrease All quantities are real-valued.

8. Stability Analysis of TCP-Reno/RED Packet simulations validate both the region and the oscillation frequency at the onset of instability. Linearizing around equilibrium we find a stability region: Unstable for large equilibrium windows, which arise with high delay or, strikingly, high capacity! Unstable for Large delay Large capacity Small load Link capacity Stability region for the case of N identical sources.

9. Improving these limitations: Steady-state properties: –ECN allows us to decouple feedback from queuing, so in principle we can get high utilization, low delay. –Resource allocation issue can be addressed if we allow sources to pick a utility function. –Study this by optimization theory. Dynamic properties: –We need to negotiate the appropriate tradeoff between responding fast to track available bandwidth, but not so fast that everything oscillates: working close to the boundary of linear stability is the best compromise. –To study this tradeoff: control theory.

10. Optimization-based approaches (Kelly, Low, …) Steady-state properties: –Utilization –Queues –Fairness Dynamic properties: –Speed of response –Stability, oscillations. –Sensitivity to noise. Start from the equilibrium side: Source utility functions Characterize social optimum. Develop decentralized algorithms Worry about dynamics later: Stability without delay. Noise variances Stability margins to delay. Speed of response. Difficulty: hard to arrange for the dynamic tradeoffs, Specially for the wide variety of network scenarios.

11. Control Theory-Based Approach Steady-state: –Utilization –Queues –Fairness Dynamic properties: –Speed of response –Stability, oscillations. –Sensitivity to noise. Look for a scalable (network and delay independent) solution to the dynamic tradeoff (stability vs. speed or response) To get the other equilibrium property, both approaches can adapt things at a slower time-scale assuming a bound on the RTT:“primal-dual” solutions, similar but with minor differences. Can also impose one steady- state property: “Primal” solution (Vinnicombe) gives freedom on utility functions Our “dual” solution ensures utilization.

12. Dynamics and the role of delay Without delay, nothing would stop us from adapting the sources’ rates arbitrarily fast. In the presence of delay, there is a stability problem: e.g., controlling temperature of your shower. Special case of general principle in feedback systems: what limits the performance (e.g. speed of response) are characteristics of the open loop (bandwidth, delay). In this case, the only impediment is delay. In particular, this sets the time-scale of our response.

13. Stability/performance tradeoff in presence of delay Feedback design is about how fast you can respond while remaining stable: the limiting factor comes from the “plant” dynamics. In this problem, from delay. A simple example:

14. Congestion control loop with delays LINKS SOURCES p : link congestion measures or “prices” RTT: Routing/ Delay matrix:

15. Ingredients for scalable stability.

16. “Dual” solution with integrator at links. SINGLE LINK SOURCES

17. Nyquist argument for stability Note: if all delays are scaled by some constant, the plot does not change. In the time domain, only effect is a change in time-scale of response.

18. Extension to arbitrary networks LINKS SOURCES p : link prices : number of bottlenecks in source i’s path Local analysis around equilibrium. Routing matrices refer here only to bottleneck links.

19. Stability result

20. Global, nonlinear implementation Remark: Athuraliya and Low ’00 considered adding another integrator to clear the queue. However, scalable stability for arbitrary delays does not extend to that case.

21. Global, nonlinear implementation Static control law for sources : linearization requirement is “Elasticity” of demand decreases with delay, number of bottlenecks.

22. Properties of the nonlinear laws Global stability? Validate by –Flow simulation of differential equations using Matlab. So far, cases of local stability have been global. –Mathematical proof. Tools which combine delay and nonlinearity are very limited! We have partial results for single link, but with further parameter constraints. Equilibrium structure, fairness: Determined by the fixed utility, and possibly very unfair, since exponentials distinguish rates very sharply. Objective: allow freedom of choice in utility functions. This calls for source dynamics, which clashes with scalable stability; we can allow it if we only require scalability to “practical” RTTs, and adapt source dynamics slowly.

23. New solution with fairness tracking Can prove local stability under adequate parameter choices (next slide)

24. Linearized laws and requirements for stability. LINKS SOURCES

25. Recap, so far Our first flow control had scalable stability to arbitrary delays, plus exact tracking of link utilization. We can further assign the equilibrium resource allocation between sources at a slower (universally agreed-on) time-scale (slower than max RTT). Packet-implementations: –One approach based on ECN marking to communicate the price, to be described by Zhikui later. –Alternative (Choe and Low): use queuing delay as price, modifying TCP Vegas.