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

2Contents

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

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 packet unmarked packet

Source decreases rateSource increases rate

4

How Do Routers Mark Packets?

1

0total arrival rate

queue length

Rate Based Marking

Queue Based Marking

Markingprobability

Adjust its transmission rate depending on volume of marks received

marked packet unmarked packet

Source decreases rateSource increases rate

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

6Motivation

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 “instanta

neous” data transmission rate?

7Contents

Introduction Marking Based Congestion Control, Motivation

System Model and Problem Definition Source Update Model: Congestion Control Algorithm

Intuition and Results

Simulation Results

Summary

8System 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

9Small Buffer Regime

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

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

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Optimization Framework [Kelly et el.]

c1 c2

x1

x2x3

Differential Equation Based

Distributed Congestion Control Algorithm

Resource Constraints in Wired Networks

12Contents

Introduction Marking Based Congestion Control, Motivation

System Model and Problem Definition Source Update Model: Congestion Control Algorithm

Intuition and Results

Simulation Results

Summary

13Intuition

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

14Large 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 = )

15Implications

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 BasedMarking Function

Rate BasedMarking Function

M/D/1 Queue

Cf) Discrete Time DomainS. Deb and R. Srikant. Rate-based versus Queue-based models of congestion control. ACM Sigmetrics, June 2004.

16Equivalent 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

17Sketch of Proof

18Example : REM [Low 99]

REM’s queue based marking function

Equivalent Marking Function (from P-K formula)

19Contents

Introduction Marking Based Congestion Control, Motivation

System Model and Problem Definition Source Update Model: Congestion Control Algorithm

Intuition and Results

Simulation Results

Summary

20Simulation 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

21Simulation Results (2)

Throughput

Distributionof CWND

22Summary 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

23References

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.