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Time-scale Decomposition and Equivalent Rate Based Marking

Time-scale Decomposition and Equivalent Rate Based Marking. Supratim Deb LIDS, MIT supratim@mit.edu. Yung Yi, Sanjay Shakkottai ECE Dept., UT Austin {yi,shakkott}@ece.utexas.edu. Contents. Introduction Marking Based Congestion Control, Motivation System Model and Problem Definition

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Time-scale Decomposition and Equivalent Rate Based Marking

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

  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. Marking Based Congestion Control marked packet unmarked packet • 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 Source decreases rate Source increases rate

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

  5. slow, accurate, off-line fast, approximate, on-line 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

  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. 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. 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 System Model

  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] Small Buffer Regime

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

  11. Optimization Framework [Kelly et el.] Resource Constraints in Wired Networks x1 c1 c2 x2 x3 Differential Equation Based Distributed Congestion Control Algorithm

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

  14. Large System Limit n controlled flows (aggregate arrival rate = ) Poisson( ) n  suitable scaling n uncontrolled flows (aggregate arrival rate = ) • 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:

  15. 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 Implications M/D/1 Queue n  suitable scaling Rate BasedMarking Function Queue BasedMarking Function Cf) Discrete Time DomainS. Deb and R. Srikant. Rate-based versus Queue-based models of congestion control. ACM Sigmetrics, June 2004.

  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 thestationary distribution of an M/D/1 queue

  17. Sketch of Proof

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

  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. 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. Simulation Results (2) Throughput Distributionof CWND

  22. 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 Summary

  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.

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