1. 1 Time-scale Decomposition and Equivalent Rate Based Marking Yung Yi, Sanjay Shakkottai ECE Dept., UT Austin {yi,shakkott}@ece.utexas.edu Supratim Deb LIDS, MIT supratim@mit.edu

2. 2 Contents Introduction Marking Based Congestion Control, Motivation System Model and Problem Definition Source Update Model: Congestion Control Algorithm Intuition and Results Simulation Results Summary

3. 3 Marking Based Congestion Control Router reacts to the aggregate flow passing through it Marks packets during congestion control Explicit Congestion Notification (ECN) [Floyd 94] Active Queue Management (AQM) [Kelly 98, Kunniyur 01, Low 99, Towsley 00] Users adapt their transmission rate Congestion Control System Marking function at routers Rate Adaptation algorithm at sources marked packetunmarked packet Source decreases rate Source increases rate

4. 4 How Do Routers Mark Packets? 1 0 total arrival rate queue length Rate Based Marking Queue Based Marking Marking probability Adjust its transmission rate depending on volume of marks received marked packetunmarked packet Source decreases rate Source increases rate

5. 5 How Can We Simulate the Internet? Pure Packet Model: Discrete Event Simulation [ns2, pdns, parsec, ssfnet] Accurate transient behavior, but high complexity State changes at discrete events (message generated, packet arrival, packet departure, etc.) Computation: a sequence of event computations, processed in time stamp order Most of complexity: Queueing Dynamics Pure Fluid Model [Danzig 96, Towsley 00, 04, Hou 04, others] Fast and low complexity, but only steady state and approximate results Time-stepped evolution of system states Good in parallel processing slow, accurate, off-line fast, approximate, on-line

6. 6 Motivation In reality, A significant number of uncontrolled flows (e.g. multimedia and web mice) Queue based marking (e.g., REM and RED) is popular in the real implementation cf) REM: Random Exponential Marking, RED: Random Early Detection Question 1: Can queue dynamics be decoupled from user dynamics? Question 2: What is the implication on the marking function? Is there an equivalent marking function which depends only on “instantaneous” data transmission rate?

7. 7 Contents Introduction Marking Based Congestion Control, Motivation System Model and Problem Definition Source Update Model: Congestion Control Algorithm Intuition and Results Simulation Results Summary

8. 8 System Model n controlled flows, n uncontrolled flows Controlled flows Differential equation based controller with queue based marking Link Bandwidth: n c Capacity proportional to the number of flows Small Buffer Regime

9. 9 Modeling of Buffer Size: nB or B ? Queue buffer scale linearly with the # of flows or not ? Small Buffer Regime High Link speeds  need high-speed buffer with high cost Buffers need not scale with the link speed in order to achieve significant multiplexing gain [Cao & Ramanan 02] [Mandjes & Kim 01] [Mckeown 04]

10. 10 What Kind of Source Controller Model? [kelly 98, et el] Rate based Marking Queue Based Marking Uncontrolled rate Controlled rate : Utility function of i-th controlled flow : TCP controller : Proportional Fair Controller Problem in this research: Finding given

11. 11 Optimization Framework [Kelly et el.] c1c1 c2c2 x1x1 x2x2 x3x3 Differential Equation Based Distributed Congestion Control Algorithm Resource Constraints in Wired Networks

12. 12 Contents Introduction Marking Based Congestion Control, Motivation System Model and Problem Definition Source Update Model: Congestion Control Algorithm Intuition and Results Simulation Results Summary

13. 13 Intuition large number of uncontrolled flows (e.g., multimedia or web mice) large amount of randomness ……. Q length 1 round trip time Controlled flows (e.g., TCP flows) End-system controller influenced only through the (statistical) stationary queueing dynamics Queue Feedback (Ack) large number of cycles, where queue becomes empty Underlying Theory: Law of large numbers and Ergodic theorem

14. 14 Large System Limit Unscaled system: n flows Uncontrolled flows: Stationary point process aggregate arrival rate: not necessarily a Poisson process Limiting system M/D/1 queue with service rate: n uncontrolled flows (aggregate arrival rate = ) n  suitable scaling Poisson( ) n controlled flows (aggregate arrival rate = )

15. 15 Implications Low complexity model for large system dynamics No queueing dynamics in the model Simpler analysis and simulation Asynchronous event simulation  Synchronous time-stepped evolving simulation n  suitable scaling Queue Based Marking Function Rate Based Marking Function M/D/1 Queue Cf) Discrete Time Domain S. Deb and R. Srikant. Rate-based versus Queue-based models of congestion control. ACM Sigmetrics, June 2004.

16. 16 Equivalent Rate Based Marking Equivalent Rate Based Marking Function x: arrival rate of controlled flows Lambda: arrival rate of uncontrolled flows Depends only on the stationary distribution of an M/D/1 queue

17. 17 Sketch of Proof

18. 18 Example : REM [Low 99] REM’s queue based marking function Equivalent Marking Function (from P-K formula)

19. 19 Contents Introduction Marking Based Congestion Control, Motivation System Model and Problem Definition Source Update Model: Congestion Control Algorithm Intuition and Results Simulation Results Summary

20. 20 Simulation Results (1) Bottleneck bw: 100 x n pkts n = 100 ( n: # of controlled and uncontrolled flows ) TCP Sack, Proportional Fair Controller REM, RED Queue based marking scheme

21. 21 Simulation Results (2) Throughput Distribution of CWND

22. 22 Summary In the Internet Significant number of uncontrolled (short and unresponsive) flows Queue based marking is popular Randomness due to short and unresponsive flows in the Internet sufficient to decouple the dynamics of the router queues from those of end controllers  We can find an equivalent rate based marking function given the queue based marking function Easier analysis and simulation We can apply nice mathematical tools to the analysis Asynchronous event-driven simulation  Synchronous fluid model based time-stepped evolving simulation  leading to low simulation complexity

23. 23 References Y. Yi, S. Deb, and S. Shakkottai, “Short Queue Behavior and Rate Based Marking,” Proceedings of the 38th CISS, Princeton University, NJ, March, 2004. A longer version has been submitted to IEEE/ACM Transactions on Networking Cao and Ramanan, “A Poisson Limit for Buffer Overflow Probabilities,” Proc. IEEE Infocom, June, 2002. Daley and Vere-Jones, “An Introduction to the Theory of Point Processes,” Springer-Verlag, 1988. R. Srikant. "The Mathematics of Internet Congestion Control." Birkhauser, 2004.