How to model a TCP/IP network i l 20 tusing only 20 parameters

K. Mills (NIST), E. Schwartz (CMU) & J. Yuan (Tsinghau U)

visual hash of MesoNet source code from http://www.wordle.net/

( ), ( ) ( g )Winter Simulation Conference – Dec 8, 2010

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Outline

• Goal – Problem – Solution• Scale Reduction: Theory and Practicey• Overview of the 20 MesoNet Parameters• Parameter Explanations in 5 Categories

– Network (4 parameters)Network (4 parameters)

– Sources & Receivers (4 parameters)

– User Behavior (6 parameters)

– Protocols (3 parameters)( p )

– Simulation & Measurement Control (3 parameters)

• Describe Sample Use of Model• Discuss Simulation Resource RequirementsDiscuss Simulation Resource Requirements• Conclusions

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Goal – Problem – Solution• Goal – compare proposed Internet congestion control

algorithms under a wide range of controlled, repeatable conditionsrepeatable conditions

• Problem – real network not controllable and bl b d l ll repeatable; test beds currently too small; most

network simulation models have large search space and require infeasible memory and processing

f l f k bl resources for large, fast networks; more tractable fluid-flow simulators are currently inaccurate

• Solution – design a reduced scale network simulation model (MesoNet) that is easy to configure and tractable to computetractable to compute

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Simulating large, fast networks across many conditions and congestion control algorithms requires scale reduction in both model parameters & responses

Scale Reduction: Theory & Practice

algorithms requires scale reduction in both model parameters & responses

y1, …, yz = f(x1|[1,…,l] …, xp|[1,…,l])

Response State‐Space Stimulus State‐Space

(232)1000 O(109633) [ 1080 = atoms in visible universe]

Parameter ReductionMultidimensional ResponseReduction (2 )

Discard parameters not germane to study – reduce by 944 parameters

O(10 ) [ ]

(232)56 O(10539)

20Group related remaining parameters– reduce by 36 parameters

22 Responses

CorrelationAnalysis &Clustering

PrincipalComponentsAnalysis

This Talk

Use experiment design theory to reducebi i 256

220

(232)20 O(10192)

O(106)

Model ReductionSelect only 2 values for each parameter

Level Reduction

g

7 Responses 4 ResponsesDomainAnalysis

parameter combinations to 256

Use sensitivity analysisto identity six mostsignificant parameters

220‐12 256ExperimentDesign Theory

26‐1 32

SensitivityAnalysis7 Responses

Talk given Dec. 6 @ 2:00 PMTalk given Dec. 6 @ 1:30 PM

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Use experiment design theory again to reduceparameter combinations to 32

g

Model Reduction for MesoNet Simulator

• Need to identify and retain only parameters germane to topic being studied (we identified 56 such parameters)

• Need to examine retained parameters to identify groups ofrelated parameters defining aspects of same macro-parameter (after grouping we identified 20 parametersparameter (after grouping we identified 20 parametersrelevant to a study of Internet congestion control)

For a full explanation of our reasoning and our entire study report see NIST Special Publication 500-282:y p pStudy of Proposed Internet Congestion Control MechanismsAvailable online at http://www.nist.gov/itl/antd/Congestion_Control_Study.cfm

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MesoNet – a TCP/IP network model using only 20 parametersusing only 20 parameters

Category Identifier Name

NetworkX1 Topology X2 Propagation DelayNetwork

Configuration X2 Propagation Delay X3 Network Speed X4 Buffer Provisioning

Sources & X5 Number of Sources & Receivers X6 Distribution of Sources

Receivers X7 Distribution of Receivers X8 Source & Receiver Interface Speeds X9 Think Time

X10 Patience X11 Web Object Size for BrowsingUser

Behavior

X11 Web Object Size for Browsing

X12 Proportion & Size of Larger File Downloads

X13 Selected Spatiotemporal Congestion X14 Long-lived Flows g

Protocols X15 Congestion Control Algorithms X16 Initial Congestion Window Size X17 Initial Slow Start Threshold

Simulation & M t

X18 Measurement Interval Size X19 Si l i D iMeasurement

Control X19 Simulation Duration X20 Startup Pattern

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Parameter X1 is the Topology = Routers + Links + Routes + Propagation Delaysp g y

AK

A1

A1b

A1c

A1a

A2b A2cA2a

C1

C1b

C1c

C1a H1bH1cH1a

H1d

H1e

H1f

K1

K2K0a

K1a

K1bK1c K1d

K2a K2b

K2c

K2d

J1b J1c

I2

I2a

I1

II0a

I1a

I1b I1c I1d

I2gI2f

I2e

I2dI2cI2b

B2d

A2

C

J

K

L

C1

C2

C2bC2c

C2a

H2b

H2c

H2a

H1H1f

H2fH2e

H2d

L2

L1

L1a L1bL1c

L1d

L2a

L2b

L0aL0b

J1

J2

J1aJ1b J1c

J1d

J1e

J1f

J2a

J2b

J2c

J2dJ2e J2f

B2

B2a

B2b

B2cB2d

B2eB2f

B2g

H

H2

D

M

B

L

N

L2 L2b

L2c

L2dN1aN1b

N1c

N1d

N1e

N1fN2

M2b

M2cM2d M2e

M2f

M2

M2g

M2aM1a

M1M1b

M1cM1dM1e M1f M1g

F2aF1

F1a

F1b F1c F1d

F2b

D2

D2a

D2gD2f

D2dD2cD2b

D2eB0aB1B1a

B1b

B1c B1dN1

E

G

FO

E1

E1b

E1a

G1a

N1f

N2aN2b

N2c N2dN2e

N2fM1f g

O0a

O1aO1b

O1cO1c

O1

O2a

O2

O2b O2c O2d

O2e

O2f

O2g

F2

F2g F2f

F2d

F2c

F0a

F2e

D1D1a

D1b

D1c D1d

D0a

E2

PE1b

E1c

E2bE2cE2a

G1b

G1c

G1eG1d

G2b G2cG2a

G2e

G2dG1f

G2fP1 P2

P1a

P1b

P1c

P1dP2a

P2bP2c P2d

P2e

P2f

P2gG1 G2

3 Router Tiers: Backbone Point of Presence (PoP) Access3 Router Tiers: Backbone – Point of Presence (PoP) – Access3 Access Router Classes: Typical – Fast – Directly Connected1 ingress/egress path from access routers to backbone routers 7

Topology Link Characteristics and Scaling Propagation Delay with Parameter X2

A

C

D

E

G

F

J

K

M

B

L

N

O

A1

A1b

A1c

A1a

A2b A2cA2a

C1

C1b

C1c

C1a

C2

C2bC2c

C2a

H2b

H2c

H2a

H1

H1bH1cH1a

E1E1a

H1d

H1e

H1f

H2fH2e

H2d

G1a

K1

K2K0a

K1a

K1bK1c K1d

K2a K2b

K2c

K2d

L2

L1

L1a L1bL1c

L1d

L2a

L2b

L2c

L2d

L0aL0b

J1

J2

J1aJ1b J1c

J1d

J1e

J1f

J2a

J2b

J2c

J2dJ2e J2f

N1aN1b

N1c

N1d

N1eN1f

N2

N2aN2b

N2c N2dN2e

N2f

M2b

M2cM2d M2e

M2f

M2

M2g

M2aM1a

M1M1bM1c

M1dM1e M1f M1g

O0a

O1aO1b

O1cO1c

O1

O2a

O2

O2b O2c O2d

O2e

O2f

O2g

I2

I2a

I1

II0a

I1a

I1b I1c I1d

I2gI2f

I2e

I2dI2cI2b

F2

F2aF1

F1a

F1b F1c F1d

F2g F2f

F2d

F2c

F2b

F0a

F2e

D2

D2a

D1D1a

D1b

D1c D1d

D2gD2f

D2dD2cD2b

D2e

D0a

B0aB1B1a

B1b

B1c B1d

B2

B2a

B2b

B2cB2d

B2eB2f

B2g

H

E2

H2

A2

N1

p g yP

E1bE1c

E2bE2cE2a

G1b

G1c

G1eG1d

G2b G2cG2a

G2e

G2dG1f

G2fP1 P2

P1a

P1b

P1c

P1dP2a

P2bP2c P2d

P2e

P2f

P2gG1 G2

Link# Endpoints Cost Metric Prop. Delay (ms) X2 = 0.5 X2 = 2

• Packets incur propagation delay when transiting a link

p p y ( )1 A-B 50 21 10.5 42 2 B-C 10 25 12.5 50 3 B-D 50 8 4 16 4 B-L 223 75 37.5 150 5 C-H 100 12 6 24 6 D-E 10 10 5 20 7 D F 108 33 16 5 66

• Cost metric used to compute routes from source backbone router to destination backbone

7 D-F 108 33 16.5 668 E-G 100 33 16.5 66 9 F-G 10 7 3.5 1410 F-H 50 12 6 24 11 F-I 55 22 11 4412 G-O 104 23 11.5 46 13 G-P 110 19 9 5 38 router to destination backbone

router13 G P 110 19 9.5 3814 I-H 10 14 7 28 15 I-J 50 8 4 16 16 I-K 147 22 11 4417 J-L 60 20 10 40 18 K-L 50 7 3.5 1419 L-M 50 12 6 24 20 L-N 39 6 3 1221 L-O 10 14 7 28 22 M-O 10 6 3 12 23 N-O 10 8 4 16 24 O-P 10 14 7 28

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Defined Speed Relationships among Router Classes used to Scale Router Speeds with Parameter X3used to Scale Router Speeds with Parameter X3

(MesoNet simplification – only routers have speeds)

Parameter Value Speed Relationships Speed Scaling with X3 s1 X3 Router Class Speed X3 = 800 X3 = 1600 s2 4 Backbone s1 x BBspeedup 1600 3200s2 4 Backbone s1 x BBspeedup 1600 3200s3 10 PoP s1/ s2 400 800 BBspeedup 2 N-Class s1/ s2/ s3 40 80 Bfast 2 F-Class s1/ s2/ s3 x Bfast 80 160 Bdirect 10 D-Class s1/ s2/ s3 x Bdirect 400 800Bdirect 10 D-Class s1/ s2/ s3 x Bdirect 400 800

P t X4 l t th B ff P i i iParameter X4 selects the Buffer ProvisioningAlgorithm, which generally interacts with networkspeed and propagation delayspeed and propagation delay

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Three Parameters Determine Number (X5) and Distribution of Sources (X6) and Receivers (X7)Distribution of Sources (X6) and Receivers (X7)

Combination of parameters X5, X6, and X7 determine distribution of fl i th t l d i th i l tiflows in the topology during the simulation

Sample Computation of Number and Distribution of Sources and Receivers (given Topology on Slide 7 and base # Sources = 100, X5 = 3, probNs = 0.1, probNsf = 0.6, probNr = 0.8, probNrf = 0.1 )

Class #routers srcs/router #srcs %srcs rcvrs/router #rcvrs %rcvrs Flow class %flows

N-class 122 90 10,980 31.6 960 117,120 95.3 NN-flows 30.1 FN-flows 60.5

F-class 40 540 21,600 62.2 120 4,800 3.9 FF-flows 2.4 DN-flows 6 1DN-flows 6.1

D-class 8 270 2,160 6.2 120 960 0.8 DF-flows 0.74 DD-flows 0.05

Parameter X8 defines the probability that sources and receivers connect to the topology at 1 Gbps or 100 Mbps

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User Behavior Represented via Sources^

Too Slow (reactive sources only)

Select Think Time(Exponential Distribution)

Select Think Time(Exponential Distribution)

Too Long(reactive sources only)

Select Think Time(E ti l

Think Time Expired

Select Receiver &File Size (Pareto Distribution)

Select Receiver &File Size (Pareto Distribution)

(Exponential Distribution)

Finished

Select Think Time

File Size (Pareto Distribution)

Parameter x9 specifies averageThink Time

Parameter x10 specifies User Patience(probability source is reactive)

Select Think Time(Exponential Distribution)

^Note: this simplified diagram omits a flow connection phase that occurs before sending and also the potential for the connection phase to fail – after which source enters Thinking11

User Traffic CharacterizationParameter X11 characterizes Web Objects ( on, )

Size of Web Objects

on average size (packets)

shape of Pareto distributionProbability of Web Object

(1 – Fp – Sp – Mp)

charact r z s j ( , )

shape of Pareto distribution

Larger File Size Multipliers Larger File Probabilities

( p p p)

Parameter x12 characterizes Larger Files [(Fx, Fp), (Sx, Sp), (Mx, Mp)]

Fx documents

Sx software downloads

Mx movies

Fp documents

Sp software downloads

Mp movies

Jumbo File Characteristics

Jx size multiplier for jumbo files

Parameter X13 characterizes Jumbo Files (Jx, Jon, Joff)

p j

Jon fraction of simulated time after which jumbo file transfers begin

Joff fraction of simulated time after which jumbo file transfers end

Parameter X14 characterizes number, location and start and stop times for Long-Lived Flows 12

Assignment of Three Protocol ParametersP t 15 ifiParameter x15 specifies (prTCP, prHSTCP, prCTCP, prSCALABLE, prFAST, prHTCP, prBICTCP)

Congestion Control Algorithm Identifier Probability of Source Implementation Transmission Control Protocol (TCP) 1 prTCP High Speed TCP (HSTCP) 2 prHSTCPHigh Speed TCP (HSTCP) 2 prHSTCPCompound TCP (CTCP) 3 prTCP Scalable TCP (STCP) 4 prSTCP FAST AQM Scalable TCP (FAST) 5 prFAST Hamilton TCP (HTCP) 6 prHTCP Bi I C ti (BIC) 7 BICBinary Increase Congestion (BIC) 7 prBIC

 

160

180

Parameter x16 specifies initial

80

100

120

140

160

cwnd

initial sst linear increase

Parameter x16 specifies initial congestion window (cwnd)

0

20

40

60

80c

initial cwnd

exponential increase Parameter x17 specifies initial

slow start threshold (sst)0

0 10 20 30 40time

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Simulation Measurement & Control

Parameter x18 specifies measurementinterval size5000

6000

s

2000

3000

4000

endi

ng F

low

s

Parameter x19 specifies number ofmeasurement intervals to simulate

0

1000

2000Se

Parameter x20 specifies startup patternfor sources

0 50 100 150 200 250Time

Count of Flows in the Sending State Measured every M = 200 ms for MI = 250 intervals – Simulation Duration (.2 s x 250 =) 50 s

0 25 % f t ti t 0 0 08 % f t ti– 0.25 % of sources starting a t=0, 0.08 % of sources startingafter an average delay 33 % of think time, 0.17 % of sources starting after an average delay 66 % of think time and remainingsources starting after average delay of think time

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Combining MesoNet Parameters with2-Level Orthogonal Fractional Factorial (OFF) Experiment Design

Global Local

X5

+X2X3

+

_

_

+

X5

+

X1

X2X3

+

__

_

+

X5

+X2X3

+

__

_

+

X5

+X2X3

+

__

_

+

X7

Example ComparingOFF Design

vs

Global Local

X4X1 +

_

_ + X4X1 +

_ +

X5

+

+

X5

+

+

X4X1 +

_ + X4X1 +

_ +

X5

+

+

X5

+

+

OFF Design 1-FAT Design

X5

X7 vs.Factor-at-a-Time (FAT)Design for 7 parameters

X4

+

X1

X2X3

+__

_

_

+ X4

+

X1

X2X3

+__

_

_

+ X4

+

X1

X2X3

+___

_

_

+ X4

+

X1

X2X3

+___

_

_

+

X4

X6

Comparing 7 Congestion Control Algorithms with 2-Level design for, 9 MesoNetParameters requires (29 x 7 =) 3584 runs

At 28 processor hours per run and with 48 available processors, theseAt 28 processor hours per run and with 48 available processors, theseruns would require about 2090 hours (87 days)

Adopting a 29-4 OFF experimental design would reduce the resource requirement to only (32 x7) = 224 runs, which could be completed in about 130 hours (1 week)to only (3 x7) runs, wh ch could be completed n about 30 hours ( week)

Cost: misses 29 - 25 parameter combinations 15

Two Sample Experiments using 29-4 Orthogonal Fractional Factorial Design

Definition of the 32 Parameter Configurations used to Simulate a Modest Size, Moderate Speed Network in Experiment #1. F E i #2 d l f X3 d # S l i li d b 10 Si l

One Experiment Design – Two Experiments

  Factor-> X2 X3 X4 X5 X7 X9 X11 X12 X15Condition -- -- -- -- -- -- -- -- --

1 1 800 0.5 3 0.7 5000 100 0.04/0.004/0.0004 0.72 1 1600 0.5 2 0.3 5000 100 0.04/0.004/0.0004 0.3

Values of 11 Fixed Parameters

For Experiment #2 red values for X3 and # Sources were multiplied by 10 to Simulate a Larger, Faster Network.

3 2 800 0.5 2 0.7 5000 100 0.02/0.002/0.0002 0.34 2 1600 0.5 3 0.3 5000 100 0.02/0.002/0.0002 0.75 1 800 1 2 0.3 5000 100 0.02/0.002/0.0002 0.76 1 1600 1 3 0.7 5000 100 0.02/0.002/0.0002 0.37 2 800 1 3 0.3 5000 100 0.04/0.004/0.0004 0.38 2 1600 1 2 0.7 5000 100 0.04/0.004/0.0004 0.79 1 800 0.5 3 0.3 7500 100 0.02/0.002/0.0002 0.310 1 1600 0.5 2 0.7 7500 100 0.02/0.002/0.0002 0.711 2 800 0 5 2 0 3 7500 100 0 04/0 004/0 0004 0 7

Parameter Assigned Value X1 Abilene Topology (Backbone: 11 routers and 14 links; 22 PoP routers; 139 Access routers) X6 probNs = 0.1, probNsf = 0.6 X7 probNr = 0.6, probNrf = 0.2

X10 0 (all users have infinite patience) X13 Jon = 1; Joff = 1; Jx = 1 (no explicit spatiotemporal congestion)11 2 800 0.5 2 0.3 7500 100 0.04/0.004/0.0004 0.7

12 2 1600 0.5 3 0.7 7500 100 0.04/0.004/0.0004 0.313 1 800 1 2 0.7 7500 100 0.04/0.004/0.0004 0.314 1 1600 1 3 0.3 7500 100 0.04/0.004/0.0004 0.715 2 800 1 3 0.7 7500 100 0.02/0.002/0.0002 0.716 2 1600 1 2 0.3 7500 100 0.02/0.002/0.0002 0.317 1 800 0.5 2 0.3 5000 150 0.02/0.002/0.0002 0.318 1 1600 0.5 3 0.7 5000 150 0.02/0.002/0.0002 0.719 2 800 0.5 3 0.3 5000 150 0.04/0.004/0.0004 0.7

X13 Jon 1; Joff 1; Jx 1 (no explicit spatiotemporal congestion)X14 no long-lived flows X16 initial cwnd = 2 (default Microsoft WindowsTM value) X17 initial sst = 231/2 (arbitrary large value) X18 M = 200 ms X19 MI = 18,000 (x .2 M =) 3600 s X20 prON = 0.25, prONsecond = 0.08, prONthird = 0.1719 2 800 0.5 3 0.3 5000 150 0.04/0.004/0.0004 0.7

20 2 1600 0.5 2 0.7 5000 150 0.04/0.004/0.0004 0.321 1 800 1 3 0.7 5000 150 0.04/0.004/0.0004 0.322 1 1600 1 2 0.3 5000 150 0.04/0.004/0.0004 0.723 2 800 1 2 0.7 5000 150 0.02/0.002/0.0002 0.724 2 1600 1 3 0.3 5000 150 0.02/0.002/0.0002 0.325 1 800 0.5 2 0.7 7500 150 0.04/0.004/0.0004 0.726 1 1600 0.5 3 0.3 7500 150 0.04/0.004/0.0004 0.327 2 800 0.5 3 0.7 7500 150 0.02/0.002/0.0002 0.3

X20 prON 0.25, prONsecond 0.08, prONthird 0.17

Each of the 32 parameter combinationswere run against 7 congestion control protocols

28 2 1600 0.5 2 0.3 7500 150 0.02/0.002/0.0002 0.729 1 800 1 3 0.3 7500 150 0.02/0.002/0.0002 0.730 1 1600 1 2 0.7 7500 150 0.02/0.002/0.0002 0.331 2 800 1 2 0.3 7500 150 0.04/0.004/0.0004 0.332 2 1600 1 3 0.7 7500 150 0.04/0.004/0.0004 0.7

– requiring 7 x 32 = 224 simulations

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Is MesoNet Computationally Tractable?32 bit SLX 64 bit SLX

Experiment #1 Experiment #2 CPU hours (224 runs) 5,857.18 94,355.28 Avg. CPU hours/Run 26.15 421.23

32-bit SLX 64-bit SLX

Avg. CPU hours/Run 26.15 421.23Min. CPU hours/Run 12.58 203.04 Max. CPU hours/Run 43.97 739.04 Avg. Memory Usage (Mbytes) 196.56 2,392.41

Required 35 Required 11 Required 35 processor weeks

Required 11 processor years

Parallel simulation of configurations reduced this to:

E i t #1 Sl S ll N t k E i t #2 L F t N t k

1 week using48 processors

31 days using48 processors

Parallel simulation of configurations reduced this to

Experiment #1 – Slow, Small Network Experiment #2 – Large, Fast NetworkStatistic Flows Completed Data Packets Sent Flows Completed Data Packets Sent Avg./Run 11,466,429 3,414,017,482 116,317,093 33,351,040,358 Min./Run 7,258,056 2,138,998,764 72,944,797 21,069,357,409 Max./Run 17,390,781 5,048,119,166 175,947,632 50,932,067,100Max./Run 17,390,781 5,048,119,166 175,947,632 50,932,067,100Total All Runs 2,568,480,122 764,739,915,978 26,055,028,851 7,470,633,040,199

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Comparing MesoNet with RossNet* Parallel Network Simulator(Throughput/Latency Tradeoff)

MesoNet limited to 1 processor per simulation, using sequential SLX simulator

Simulation Experiment Event Rate (events/second)

RossNet can use 2-4 processors per simulationSimulation Experiment Event Rate (events/second)MesoNet Experiment #1 – 32-bit SLX – 1 processor per simulation 725,359 MesoNet Experiment #2 – 64-bit SLX – 1 processor per simulation 439,864 RossNet Simulation of small topologies – 2 to 4 processors per simulation 256,244 RossNet Simulation of AT&T topology – 2 to 4 processors per simulation 150,720

RossNet speedups from parallel simulation averaged just under 1.7 (max. 3.2) when using 4 processors

Given 48 processors, MesoNet can run 48 simulations in parallel, while RossNet simulations using 4 processors can run only 12 simulations in parallel

RossNet requires a speedup of 4 to equal the throughput of MesoNetRossNet requires a speedup of 4 to equal the throughput of MesoNet

If sufficient processors exist to run all RossNet simulations in parallel, thenRossNet might provide superior latency to MesoNet*Yaun, G., D. Bauer, H. Bhutada, C. Carothers, M. Yukel and S. Kalyanaraman. 2003. Large-Scale Network Simulation Techniques: Examples of TCP and OSFP Models. In SIGCOM Computer Communications Review, 33:3, 27-41.

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Conclusions• Defined a concise TCP/IP model using only 20

parameters

• Showed how the model can be combined with 2-level orthogonal fraction factorial techniques to design efficient experimentsefficient experiments

• Demonstrated how to carefully explore a parameter y p pspace using parallel instances of a sequential simulator

• Found our model and approach competitive with a • Found our model and approach competitive with a parallel TCP/IP simulator, which required additional processors to achieve the same throughput

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Related Work

• More Parallel Simulators – (throughput/latency tradeoff)

Riley, G., M. Ammar, F. Fujimoto, A. Park, K. Perumalla and D. Xu. 2004. A Federated Approach to Distributed N t k Si l ti I ACM T ti M d li d C t Si l ti 14 2 116 148Network Simulation. In ACM Transactions on Modeling and Computer Simulation, 14:2, 116-148.

Zeng. X., R. Bagrodia and M. Gerla. 1998. GloMoSim: a Library for Parallel Simulation of Large-scale Wireless Networks. In Proceedings of the 12th Workshop on Parallel and Distributed Simulations, 154-161.

• Fluid-Flow Simulators – (inaccurate)

Towsley, D., V. Misra and W. Gong. 2000. Fluid-based analysis of a network of AQM routers supporting TCP flows with an application to RED. In Proceedings of SIGCOMM, 30:4, 151,160.pp g

Yi, Y. and S. Shakkottai. 2007. FluNet: A hybrid internet simulator for fast queue regimes, In Computer Networks: The International Journal of Computer and Telecommunications Networking, 51:18, 4919-4937.

• Hybrid Continuous-Time/Discrete-Event Simulators– (promising)Hy r ont nuous m /D scr t E nt S mu ators (prom s ng)

Lee, J., S. Bohacek, J. Hespanha and K. Obraczka. 2007. Modeling Communication Networks with Hybrid Systems. In IEEE/ACM Transactions on Networking, 15:3, 630-643.

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