page 1 hans peter schwefel traffic analysis ii: mm3/4, laqt, fall03 traffic theory and queueing...

Hans Peter SchwefelTraffic Analysis II: MM3/4, LAQT, Fall03

Traffic Theory and Queueing Systems II

www.control.auc.dk/~henrik/undervisning/trafik2/oversigt.html

by Henrik Schiøler & Hans-Peter Schwefel

• Mm1 M/G/1 queues

• Mm2 Matrix Analytic Methods I

• Mm3 Matrix Analytic Methods II

• Mm4 Network Calculus I

• Mm5 Network Calculus II

www.kom.auc.dk/~hps/teaching


MotivationScenario: Packet-Based Network

Goals: QoS Provision & Network Planning

Necessary: Performance Model/ Traffic Model

German University Backbone 1999 (DFN)

• Burstiness

• Daily Profile

• Application Mixes

• Protocol Impact (e.g. TCP)

Peculiarities of Network Traffic:


Traffic Properties

Correlation Plot (Inter-cell times)

In Part I of this course:Markov chains, M/M/1 queues, Birth-Death processes

•exponential distributions, memory-less

•Coefficient of variation: C2=Var(X)/[E{X}]2=1

•Uncorrelated arrivals/services

•Steady-state analysis

Actual Measurements (inter-cell times Xi):

• C2(X) between 13,…,30

• positive autocorrelation coefficient: ř(k) = (i (Xi-x)(Xi+k-x)) / Var(X) > 0, slowly decaying with k


Content of MM2 (and MM3): Matrix Analytic Methods

1. Matrix-Exponential (ME) Distributions1.1 Mean Time to Leave S (E{X})1.2 Distribution of X1.3 Examples (HYP-2, Erl-T)1.4 Power-Tail Distributions

2. M/ME/1 Queues2.1 M/ME/1//N Systems2.2 Open M/ME/1 System

3. ME/M/1 Queues4. Transient Analysis5. Markov Modulated Poisson Processes

(MMPPs)

6. Outlook

References: L=[Lipsky], S=[Schwefel], N=[Neuts]

L3.1.1, SB.1

L3.1.2, SB.1

L3.2, SB.2

L3.3.4, SB.2

L4.1, SD.2

L4.2.1, L4.2.3, SD.2

L5.1, SD.4

L4.5.1, (SF.3)

SC.2, SD.5, SF.1, SD.6

SC.1, SC.3


Content of (MM2 and) MM3: Matrix Analytic Methods

1. Matrix-Exponential (ME) Distributions2. M/ME/1 Queues3. ME/M/1 Queues4. Transient Analysis5. Markov Modulated Poisson Processes (MMPPs)

5.1 MMPP Definition5.2 MMPP/M/1 Queues, Quasi-Birth-Death

Processes5.3 MMPP/M/1/B loss systems5.4 Applications (ON/OFF Processes)

6. Outlook6.1 Semi Markov Processes6.2 M/G/1 Type and G/M/1 Type Queues

References: L=[Lipsky], S=[Schwefel], N=[Neuts]

L3.1.1, L3.1.2, L3.2, L3.3.4,SB.1, SB.2

L4.1, L4.2.1, L4.2.3, SD.2

L5.1, SD.4

L4.5.1, (SF.3)

SC.2

SD.5, SF.1

SD.6

SC.1, SC.3


Notation• Matrices: underlined capitals: B , V, ...

– Unit Matrix: I = diag([1,1,…,1])

• Row vectors: underlined, lower-case: p, u, …

• Column vectors: primed ’ = [1,1,…,1]’

• Random Variables: Capitals: X, U, …

• Expected Values: E{X}, E{X2},…

• Coefficient of correlation: r(X,Y)=E{(X-E{X})*(Y-E{Y})} / (std(X)*std(Y))auto-correlation coefficient (process (Xi) ): r(k) = r(Xi, Xi+k)

• Queueing Systems:– Infinite buffer: G/G/1

– Finite loss systems: G/G/1/B

– Finite number of customers: G/G/1//K


References• Traffic Measurements

– H. Gogl, ’Measurement and Characterization of Traffic Streams in High-Speed Wide Area Networks’, VDI Verlag, 2001.

• L. Lipsky: ’Queueing Theory, a linear algebraic approach’, Mac Millan, 1992; Extended version in preparation. [Chpts. 3,4,5]

• M. Neuts: ’Matrix geometric Solutions in Stochastic Models’, John Hopkins University Press, 1981.

• M. Neuts: ’Structured stochastic matrices of M/G/1 type and their applications.’ Dekker, 1989.• G. Latouche, V. Ramaswami: ’Introduction to matrix-analytic methods in stochastic modeling’.

ASA-SIAM Series on Statistics and Applied Probability 5. 1999.• H.-P. Schwefel: ’Performance Analysis of Intermediate Systems Serving Aggregated

ON/OFF Traffic with Long-Range Dependent Properties’, Dissertation, TU Munich, 2000. [Appendices B,C,D,F]

• K. Meier-Hellstern, W. Fischer: ’MMPP Cookbook’, Performance Evaluation 18, p.149-171. 1992.

• P. Fiorini et al.: ’Auto-correlation Lag-k for customers departing from Semi-Markov Processes’, Technical Report TUM-I9506, TU München, July 1995.


1. Matrix Exponential Distributions


1.3 Erlangian Distributions


1.4 Truncated Power-Tail Distributions


2. M/ME/1//N Systems


3. ME/M/1 Queue: Geometric parameter s


4. Transient Behavior: Motivation

• Highly correlated overflow events

• Large fluctuations between different observation intervals

Steady-state Overflow-Prob.not always meaningful

Transient analysis necessary

Simulation: Fraction of overflowed packets


4. Transient Behavior: Computation of mean First Passage Times (M/ME/1)


5. Markov Modulated Poisson Processes


5.4 Multiplexed ON/OFF Sources: N-Burst Model

• N sources, each average rate (=16.3 cells/ms)

• During ON periods: peak-rate p (p =140 cells/ms)

• Mean # cells per ON period, np (np=9.1)

• ON duration Matrix-Exponential, <p,B>: Pr(ON>t) = p exp(-Bt) ´ MMPP representation


5.4 TPT-T ON-time distribution

• Power-Tail distribution Very long ON periods can occur

• Exponent : Heavier Tails for lower ; Impact on exponent in AKF

• Truncated Tail: Power-Tail Range, Maximum Burst Size (MBS)


5.4 N-Burst /M/1 Performance: Blow-up Points

• `Blow-up Regions´ i0=1,...,N for LRD traffic

• Radical delay increase at transitions

• Power-Tailed queuelength-distr.

• ...with changing exponents (i0)

• Tail truncated at qN(MBS)


5.4 Cause of Blow-up Regions

• Oversaturation periods caused by a mimimum of i0 long-term active sources

• Duration of oversaturation periods: PT with exponent =i0(-1)+1 matters for performance, not or H


5.4 Consequences: Power-Laws for BOP & CLP

• Buffer Inefficiency: BOP ~ B1-

CLP ~ B1-

• Drop-off for B>qN

• Power-Law growth of mCD(MBS) when

<2

BUT: Steady-State reached in 4-8 daily Busy Hours ?

BOP = Buffer Overflow Probability (N-Burst/M/1) , CLP = Cell Loss Probability (N-Burst/M/1/B)


5.4 Transient Behavior

• Highly correlated overflow events

• Large fluctuations between different observation intervals

Steady-state Overflow-Prob.not always meaningful

Transient analysis necessary

Simulation: Fraction of overflowed packets


5.4 Transient Overflow Events

Mean First Passage Times Transient Overflow Probabilities &conditional Overflow Ratio• mFPT grows by Power-Law B

Blow-up effects for changing (or N or p)

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