Upload
others
View
6
Download
0
Embed Size (px)
Citation preview
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
Background rate (Kbps)
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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)
Coefficient of Variation
Base(400Kbps)Det(400Kbps)
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)
Coefficient of Variation
Base(430Kbps)Det(430Kbps)
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)
Coefficient of Variation
Base(450Kbps)Det(450Kbps)
Stoc_mvBC(450Kbps)
UDP delay versus Cv in mixed traffic-type queue.
HybridSimulations
R.G.Cole
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
)
Background rate (Kbps)
SimulationDeterministic
Stochastic
TCP goodput versus rate in mixed traffic-type queue.
Stochastic model matches well with simulation resultsfor TCP dynamics.
HybridSimulations
R.G.Cole
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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)
Coefficient of Variation
Base(300Kbps)Det(300Kbps)
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)
Coefficient of Variation
Base(400Kbps)Det(400Kbps)
Stoc_mvBC(400Kbps)Stoc(400Kbps)
0
2000
4000
6000
8000
10000
0.9 1 1.1 1.2 1.3 1.4 1.5
Goo
dput
(B
ps)
Coefficient of Variation
Base(450Kbps)Det(450Kbps)
Stoc_mvBC(450Kbps)Stoc(450Kbps)
0
2000
4000
6000
8000
10000
0.9 1 1.1 1.2 1.3 1.4 1.5
Goo
dput
(B
ps)
Coefficient of Variation
Base(500Kbps)Det(500Kbps)
Stoc_mvBC(500Kbps)Stoc(500Kbps)
TCP goodput versus Cv in mixed traffic-type queue.
HybridSimulations
R.G.Cole
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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
IntroductionMotivation
Hybrid SimulationModels
StochasticHybridSimulationsOur Approach
Brownian MotionModel Example
Initial ResultsSimulation Model
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.