hybrid simulations mixed stochastic and event flowsrgcole/presentations/kobayashi_ws_2008.pdf ·...

HybridSimulations

R.G.Cole

IntroductionMotivation

Hybrid SimulationModels

StochasticHybridSimulationsOur Approach

Brownian MotionModel Example

Initial ResultsSimulation Model

UDP Results

TCP Results

Future Work

Summary

Mixed Stochastic and Event FlowsBrownian Motion Modeling for Simulation Dynamics

Robert G. Cole1, George Riley2, Derya Cansever3 andWilliam Yurcick4

1Johns Hopkins University2Georgia Institue of Technology

3SI International, Inc.4University of Texas - Dallas

09 May 2008 / Kobayashi Workshop 2008

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Outline

1 IntroductionMotivationHybrid Simulation Models

2 Stochastic Hybrid SimulationsOur ApproachBrownian Motion Model Example

3 Initial ResultsSimulation ModelUDP ResultsTCP Results

4 Future Work

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

MotivationScalable Network Simulation Models

Future network design and modeling requires largescale, high fidelity simulations capability.Training requires real-time speedup of networksimulations.Parallelization of network simulations not always usefuldue to lack of topological communities of interest.Hybrid analytic/event simulations appear to be anattractive alternative.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Hybrid Simulation Types

Network models:Partitioned models, e.g., packet edge and analytic coreMixed node models, i.e., packet and analytic trafficmixed at each network queue.

Analytic models:Deterministic models, e.g., dynamics described by fluidand differential equations.Stochastic models, e.g., Brownian motion models ofqueue dynamics.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Challenges to Stochastic, Mixed Node Models

Simulation of network of queues.

Mixing the two traffic types at a single, finite queue.Splitting and merging traffic flows within networksimulation.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Approach to Hybrid Stochastic Simulations

Comparison of deterministic versus stochastic fluid mixing at queue.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Brownian Motion Models

GI/G/1/K queueing system

Cumulative Distribution Function, F (w , t |w0, t = 0) satisfiesthe Fokker-Planck Equation [Kobayashi, 1974a],[Kobayashi, 1974b] and [Heyman, 1975]

∂

∂tF = −m

∂F∂w

+12σ2 ∂

2F∂w2

(1)

where m = λs − µ, and σ2 = λs × C2v + µ× C2

v ,µ.

The Fokker-Planck equation has an analytic solution

F (w , t |w0, t = 0) = α× Φ(w − w0 −mt

σ√

t)

+ β × e2mw/σ2Φ(−w − w0 −mt

σ√

t) (2)

Φ(±w − w0 −mt

σ√

t) =

1√

2π

Z ±w−w0−mtσ√

t

−∞e−x2/2dx (3)

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Boundary Conditions

Initial Conditions:

F (w , t = 0) = H(w − w0); H(x) is the Heavy-side Function.

Upper and lower limits on work in system:Common BCs are

limw→0

F (w , t |w0, t = 0) = 0 (4)

limw→wmax

F (w , t |w0, t = 0) = 1 (5)

An alternative set of BCs is

limw→max [0,w0−µt]

F (w , t |w0, t = 0) = 0 (6)

limw→wmax

F (w , t |w0, t = 0) = 1 (7)

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Rationale for Boundary Conditions

Diagram of rationale for the boundary conditions.

Buffer size sets maximum and minimum limits to workin queue.Maximum drain rate (assuming no arrivals) sets shortterm limit on the minimum work in queue (2nd set of BCon previous slide).

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Simple Simulation Model

Simple simulation model in GTNets.

Background is hyper-exponential arrival, deterministicservice modeling UDP packetsForeground is exponential arrival, deterministic servicemodeling UDP packetsForeground is later modeled as TCP streamInvestigate foreground packet delay (UDP and TCP),loss (UDP and TCP), and goodput (TCP)

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

UDP Delay and Loss Results

0

100

200

300

400

500

250 300 350 400 450 500

Del

ay (

mse

c)

Background rate (Kbps)

SimulationDeterministicStoc_mvBC

0

0.2

0.4

0.6

0.8

1

250 300 350 400 450 500

Loss

pro

babi

lity


SimulationDeterministicStoc_mvBC

UDP delay and loss versus rate in mixed traffic-type queue.

Deterministic fluid model has no mechanism to allowforeground traffic buffer access at high utilization.Stochastic model allows foreground traffic access evenin overload situations.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

UDP Delay Results versus Cv

0

20

40

60

80

100

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Del

ay (

mse

c)

Coefficient of Variation

Base(300Kbps)Det(300Kbps)

Stoc_mvBC(300Kbps)

0

50

100

150

200

250

300

350

400

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Del

ay (

mse

c)



Stoc_mvBC(400Kbps)

100

150

200

250

300

350

400

450

500

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Del

ay (

mse

c)



Stoc_mvBC(430Kbps) 100

150

200

250

300

350

400

450

500

0.9 1.0 1.1 1.2 1.3 1.4

Del

ay (

mse

c)



Stoc_mvBC(450Kbps)

UDP delay versus Cv in mixed traffic-type queue.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

TCP Goodput Results

1

10

100

1000

10000

100000

250 300 350 400 450 500 550

Thr

ough

put (

Bps

)


SimulationDeterministic

Stochastic

TCP goodput versus rate in mixed traffic-type queue.

Stochastic model matches well with simulation resultsfor TCP dynamics.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

TCP Goodput versus Cv

5000

10000

15000

20000

25000

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6

Goo

dput

(B

ps)



Stoc_mvBC(300Kbps)Stoc(300Kbps)

0

5000

10000

15000

20000

0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Goo

dput

(B

ps)




0

2000

4000

6000

8000

10000

0.9 1 1.1 1.2 1.3 1.4 1.5

Goo

dput

(B

ps)




0

2000

4000

6000

8000

10000

0.9 1 1.1 1.2 1.3 1.4 1.5

Goo

dput

(B

ps)




TCP goodput versus Cv in mixed traffic-type queue.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Future Investigations and Efforts

Improve Mixed Queue Models:Not happy with Cv 6= 1 results.Possible solution: discretize distribution[Kobayashi, 1974b].Develop the background loss models.Possible approach: Bayes Theorem (see below).

Develop Network Flow Models:Only investigated mixing at single node models to date.Leverage literature of network queueing, e.g.,[Whitt, 1995], [Kobayashi and Mark, 2002], others.

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Loss Models of FluidApplication of Bayes Theorem

Estimating background packet loss from posterior density function.

Bayes Equation : P(A|B) = P(B|A)P(A)/P(B) (8)

p(wA, t ′|wB = w (−)i+1 , ti+1; w (+)

i , ti ) =p(wB = w (−)

i+1 , ti+1|wA, t ′)p(wA, t ′|w(+)i , ti )

p(wB = w (−)i+1 , ti+1|w

(+)i , ti )

(9)

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

Summary

Initial investigations into mixed, hybrid stochasticsimulation modelsMuch work to be done:

Discretize distributionsDevelop fluid loss modelsDevelop network flow models

OutlookInitial simulation results are encouragingNeed much more development and simulation studies

HybridSimulations

R.G.Cole






UDP Results

TCP Results

Future Work

Summary

References

Cole, R.G., Riley, G., Cansever, D. and W. Yurcik,Stochastic Process Models for Packet/Analytic-Based Network Simulations, IEEE/ACM Principles ofDistributed Simulation (PADS), Rome, Italy, June 2008.

Heyman, D.,A Diffusion Model Approximation for the GI/G/1 Queue in Heavy Traffic, Bell Labs Technical Journal,Vol. 54, No. 9, November 1975.

Kobayashi, H.,Applications of the Diffusion Approximation to Queueing Networks, I. Equilibrium Queue Distributions,Journal of the Association for Computing Machinery, Vol. 21, No. 2, 1974.

Kobayashi, H.,Applications of the Diffusion Approximation to Queueing Networks, I. Nonequilibrium Distributions andComputer Modeling, Journal of the Association for Computing Machinery, Vol. 21, No. 3, 1974.

H. Kobayashi and B. L. Mark,A Unified Theory for Queuing and Loss-Queueing Networks, IFORS’99 Conference, Beijing, August16-20, 1999.

H. Kobayashi and B. L. Mark,Generalized Loss Models and Queueing-Loss Networks, Int. Trans. in Operational Research, Vol. 9,No. 1, pp. 97-112, January 2002.

Whitt, W.,Variability Functions for Parametric-Decomposition Approximations of Queueing Networks.Management Science, vol. 41, No. 10, pp. 1704-1715, 1995.

hybrid simulations mixed stochastic and event flowsrgcole/presentations/kobayashi_ws_2008.pdf ·...

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