Stochastic Models for Information Processingkom.aau.dk/project/sipcom/SIPCom05/sites/sipcom9/courses/SIPCom… · • Mm1 Channel Characterization and Measurements ... cumulative

Page 1 Hans Peter SchwefelSIPCom9-4: Stochastic Models Lecture 5, Fall05

Stochastic Models for Information Processing

• Mm1 Channel Characterization and Measurements (bfl)

• Mm2 Discrete-time Markov Chains, HMMs (hps)

• Mm3 Loss models and error concealment (sva)

• Mm4 Modeling of Medium Access Schemes (hps)

• Mm5 Traffic and Performance Models (hps)

[email protected]

http://www.kom.auc.dk/~hps

Bernard Fleury, Søren V. Andersen, Hans-Peter Schwefel

Note: slide-set will be complemented by formulas, mathematical derivations, and examples on the black-board!


Challenges in IP networks:• Multiplexing of packets at nodes (L3)• Burstiness of IP traffic (L3-7)• Impact of Dynamic Routing (L3)• Performance impact of transport layer, in particular TCP (L4)• Wide range of applications different traffic & QoS requirements (L5-7)• Feedback: performance traffic model, e.g. for TCP traffic, adaptive applications

Challenges in Wireless Networks:• Wireless link models (channel models)• MAC & LLC modeling• RRM procedures• Mobility models• Cross layer optimization

Analysis frequently with ‚stochastic‘ models

Challenges in Packet Switched SettingHTTP

TCP

IPLink-Layer

L5-7

L4

L3

L2

Revision (SIPCOM9-3)


Basic concepts• Probabilities

– ’Random experiment’ with set of possible results Ω– Axiomatic definition on event set V(Ω)

• 0≤Pr(A)≤1; Pr(∅)=0; Pr(A∪B)=Pr(A)+Pr(B) if A∩B=∅ [ A,B∈℘(Ω) ]– Conditional probabilities: Pr(A|B)=Pr(A∩B) / Pr(B)

• Random Variables (RV)– Definition: X: Ω→ú; Pr(X=x)=Pr(X-1(x))– Probability density function f(x), cumulative distribution function F(x)=Pr(X≤x),

reliability function (complementary distr. Function) R(x)=1-F(x)=Pr(X>x)

– Expected value, moments: E(Xn)=∫ xn f(x) dx– Relevant Examples, e.g.:

• number of packets that arrive at the access router in the next hour (discrete)• Buffer occupancy (#packets) in switch x at time y (discrete)• Number of downloads (’mouse clicks’) in the next web session (discrete)• Time until arrival of the next IP packet at a base station (continuous)



Basic concepts: Exponential Distributions

Important Case: Exponentially distributed RV• Single parameter: rate λ• Density function f(x)= λ exp(- λx), x>0• Cdf: F(x)=1-exp(- λx), Reliability function: R(x)=exp(- λx)• Moments: EX=1/ λ; VarX=1/ λ2, C2 = VarX / [EX]2 = 1

Important properties:• Memory-less: Pr(X>x+y | X>x) = exp(- λy) • Properties of two independent exponential RV: X with rate λ, Y with rate µ

– Distribution of min(X,Y): exponential with rate (λ+µ)– Pr(X<Y)= λ/(λ+µ)



Basic concepts III: Stochastic Processes• Definition of process (Xi) (discrete) or (Nt) (continuous)

– Simplest type: Xi independent and identically distributed (iid)• Relevant Examples:

– Inter-arrival time process: Xi– Counting Process:

N(t) = maxn | Σi=1 Xi ≤t , alternatively Ni(∆) = N(i∆) – N([i-1]∆)

Important Example: Poisson Process• Assume i.i.d. exponential packet inter-arrival times (rate λ): Xi:=Ti-Ti-1• Counting Process: Number of packets Nt until time t

– Pr(Nt=n)= (λt)n exp(- λt) / n!• Properties:

– Merging: arrivals from two independent Poisson processes with rate λ1 and λ2 Poisson process with rate (λ1+ λ2)

– Thinning: arrivals from a Poisson process of rate λ are discarded independently with probability p Poisson process with rate (1-p) λ

– Central Limit Theorem: superposition of n independent processes results in the limit n→∞in a Poisson process (under some conditions on the processes)

n-1



• Defined by– State-Space: finite or countable infinite, w/o.l.g. E=0,1,2,...,K (K=∞ also allowed)– Transition rates: µjk

• Holding time in state j: exponential with rate Σk≠j µjk =: µj

• Transition probability from state j to k: µjk / Σl≠j µjl = µjk / µj

• Xt = RV indicating the current state at time t; πi(t):=Pr(Xt=i)• ’Markov Property’: transitions do not depend on history but only on current state

t0<t1<...<tn , ∀i0,i1,...in ,j ∈E

• Computation of steady-state probabilities– Chapman Kolmogorov Equations: dπi(t)/dt = - µi πi(t) + Σj≠i µji πj(t) – Flow-balance equations, steady-state (t ∞): µi πi = Σj≠i µji πj

• Here: restriction to irreducible, homogeneous processes

Continuous Time Markov Processes


Matrix Notation!


Queueing Models: Kendall NotationX/Y/C[/B] Queues (example: M/M/1, GI/M/2/10, M/M/10/10, ...)• X: Specifies Arrival Process

– M=Markovian Poisson– GI=General Independent iid

• Y: Specifies Service Process (M,G(I),...)• C: Number of Servers• B: size of finite waiting room (buffer)

[also counting the packet in service]– If not specified: B=∞

• Often also specified: service discipline– FIFO: First-In-First-Out (default)– Processor Sharing: PS– Last-in-first-out LIFO (preemptive or non-preemptive)– Earliest Deadline First (EDF), etc.

Finite buffer (size B)

µλ

Scope here: Point-process models as opposed to fluid-flow queues



M/M/1 queue• Poisson arrival of packets (first ’M’ Markovian) with rate λ• Exponentially distributed service times of rate µ (second ’M’)• Single Server (1)• FIFO service discipline

Qt = Number of packets in system is continuous-time Markov Process

’Derived’ Parameter:• Utilization, ρ= λ/ µ : if ρ≥1, instable case (no steady-state q.l.d)

Performance Parameters• Queue-length distribution: π(t) , steady-state limit: π=lim π(t) (if ρ<1)• Queue-length that an arriving customer sees• Waiting/System time distribution• Buffer-Overflow Probability for level B = Pr(arriving customers sees buffer occupancy B or

higher)

Infinite buffer

µλ



M/M/1 queue: Performance

• Birth-Death Process– Probability of i packets in queue [using flow-balance equations]

πi := Pr(Q=i) = (1-ρ)* ρi , where ρ= λ/ µ <1– Probability of idle server: π0 = (1-ρ)– Average Queue-length: EQ= ρ/(1-ρ)– Average Delay (System Time): ES= EQ/ λ = 1/(µ-λ)– Buffer Overflow Probabilities (PASTA principle)

Pr(Q(a)≥B)= Pr(Q≥B) = ρB

λ λ λ

µµ µ



Discussion: Models for packet traffic• Poisson assumption for packet arrivals may be applicable for highly

aggregated traffic (core networks), but otherwise traffic tends to be bursty– High data rates in ftp download but less activity between downloads– http: activities after mouse-clicks– Video streaming: high data rates in frame transmissions– Interactive Voice: talk and silent periods

• Model Modifications:– Bulk Arrival processes– ON/OFF models– Hierarchical models

• [Assumption of exponential service times ?]


Content1. Motivation2. Traffic Characteristics

• Daily profiles, stationarity• Autocorrelation, ON/OFF models• Examples:

• ATM Traffic Measurements• Gaming Traffic

3. Markov Modulated Poisson Processes• Definition• Basic Properties

4. Performance Models: MMPP/M/1 Queues• Matrix formulation, Quasi-Birth Death Processes• Steady-State Queue-length probabilities• Solving the quadratic matrix equation

5. Long-Range Dependence and Heavy-Tails6. Summary, Exercises


Daily Profiles: Stationarity

• “A stationary process has the property that the mean, variance and autocorrelation structure do not change over time.”

• Informally: “we mean a flat looking series, without trend, constant variance over time, a constant autocorrelation structure over time and no periodic fluctuations (seasonality).”

NIST/SEMATECH e-Handbook of Statistical Methodshttp://www.itl.nist.gov/div898/handbook/

Source: M. Crovella


The 1-Hour / Stationarity Connection• Nonstationarity in traffic is primarily a result of varying human behavior over

time• The biggest trend is diurnal• This trend can usually be ignored up to timescales of about an hour, especially

in the “busy hour”

Mon Tue Wed Thu Fri Sat Sun

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

1.35

x 108

Traf

fic in

OD

Flo

w 8

4

Mon Tue Wed Thu Fri Sat Sun−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

Eig

enflo

w 2

9

Source: M. Crovella


Traffic Modeling: A Reasonable Approach

• Fully characterizing a stochastic process can be impossible – Potentially infinite set of properties to capture– Some properties can be very hard to estimate

• A reasonable approach is to concentrate on two particular properties:

marginal distribution and autocorrelation

Source: M. Crovella


Autocorrelation

• Once we have characterized the marginals, we know a lot about the process.

• In fact, if the process consisted of i.i.d. samples, we would be done.• However, most traffic has the property that its measurements are not

independent.• Lack of independence usually results in autocorrelation• Autocorrelation is the tendency for two measurements to both be greater

than, or less than, the mean at the same time.

Source: M. Crovella


Measuring AutocorrelationCoefficient of Autocorrelation (assumes stationarity):R(k) = Cov(Xn,Xn+k)/Var[X0] = ( E[Xn Xn+k] – E2[X0]) / Var[X0]


Traffic Properties

Correlation Plot (Inter-cell times)

Poisson Arrival processes & M/M/1 type queues•exponential distributions, memory-less

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

•Uncorrelated arrivals/services

•(Steady-state analysis)

Actual Measurements (inter-cell times Xi):• C2(X) between 13,…,30

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


Bursty Models: ON/OFF Models

Parameters:• N sources, each average rate κ

• During ON periods: peak-rate λp

• Mean duration of ON and OFF times

κ = λp ON/(ON+OFF)

’bursty’ traffic, when λp >> κ


General hierarchical models• Frequently used: Several levels with increasing granularity

– E.g. 3 levels: sessions, connections, packets– Or: 5-level model:


Measurement: Game Traffic• Traffic Measurements of Counter-Strike Session

– Architecture:• 2 clients, 1 server• Within switched 100Mb/s Ethernet

• Measurement of UDP traffic– Inter-packet time distribution

(inter-departure time)

– Packet Size distribution– Correlation properties (auto-, cross-)

• … for four different flows– Flow 1: Client 1 Server– Flow 2: Client 2 Server– Flow 3: Server Client 1– Flow 4: Server Client 2

Revision (SIPCOM9-3, MM4 Use Case)


Inter-departure time distribution• Upstream: two modes, around 33 ms and 50 ms• Downstream: single mode, around 60ms



Modeling of Gaming Traffic• Downstream (server client)

– i.i.d. normally distributed• Upstream (client server)

– Mix of two normal distributions– Normal distributions fitted with the mean and standard deviation of the

traffic sample– Selection of ‘short’ or ‘long’ inter-departure time determined by discrete

time Markov chain• Transition matrix derived from traffic samples



Auto-correlation of inter-packet times

Coefficient of auto-correlationSeemingly some periodic properties in game traffic

• Not achieved (of course) by two-state Markov model!










Markov Modulated Poisson Processes

Example: Single-Source ON/OFF model


Multiplexed ON/OFF Models

Parameters:• N sources, each average rate κ

• During ON periods: peak-rate λp

• Mean duration of ON and OFF times

ON & OFF Times exponential MMPP representation with N+1 states (Exercises!)









MMPP/M/1 Queues• State space: <n,i>, n=# packets in queue, i=state of modulating process• order states lexiographically

Generator matrix Block-tri-diagonal: Quasi-Birth-Death Process

• Steady-state probability vector: • Matrix Form of Balance Equations Matrix Geometric Solution

• From boundary conditions:


Matrix-Geometric Factor: R• From Balance equations. Sufficient condition for R:

with

• Performance Parameters:– Scalar queue-length probabilities– Average queue-length

• How to solve quadratic matrix equation:– Simple iteration

– Spectral decomposition (not discussed here)


Computational aspects• Simple iteration method (*) may lead

to long processing times • Numerical properties


(optional) MMPP/M/1/B Queues• Finite buffer-space, loss model• Generator matrix

• Mixed matrix geometric solution

• Approaches to obtain matrix factor S









Distributional Tails

• A particularly important part of a distribution is the (upper) tail

• P[X>x]• Large values dominate statistics and

performance• “Shape” of tail critically important

Source: M. Crovella


Light Tails, Heavy Tails

• Light – Exponential or faster decline

• Heavy – Slower than any exponential f2(x) = x-2

f1(x) = 2 exp(-2(x-1))

Source: M. Crovella


Examining Tails

• Best done using log-log complementary CDFs

• Plot log(1-F(x)) vs log(x)

1-F2(x)

1-F1(x)

Source: M. Crovella


Heavy Tails Arrivepre-1985: Scattered measurements note high variability in computer systems

workloads1985 – 1992: Detailed measurements note “long” distributional tails

– File sizes– Process lifetimes

1993 – 1998: Attention focuses specifically on (approximately) polynomial tail shape: “heavy tails”

post-1998: Heavy tails used in standard models

Source: M. Crovella


Power Tails, Mathematically

We say that a random variable X is power tailed if:

where a(x) ~ b(x) means .1lim )()( =

∞→ xbxa

x

Focusing on polynomial shape allowsParsimonious descriptionCapture of variability in α parameter

20 ~][ ≤<> − ααxxXP

Source: M. Crovella


A Fundamental Shift in Viewpoint

• Traditional modeling methods have focused on distributions with “light” tails– Tails that decline exponentially fast (or faster)– Arbitrarily large observations are vanishingly rare

• Heavy tailed models behave quite differently– Arbitrarily large observations have non-negligible probability– Large observations, although rare, can dominate a system’s

performance characteristics

Source: M. Crovella


Heavy Tails are Surprisingly Common

• Sizes of data objects in computer systems– Files stored on Web servers– Data objects/flow lengths traveling through the Internet– Files stored in general-purpose Unix filesystems– I/O traces of filesystem, disk, and tape activity

• Process/Job lifetimes• Node degree in certain graphs

– Inter-domain and router structure of the Internet– Connectivity of WWW pages

• Zipf’s Law

Source: M. Crovella


Evidence: Web File Sizes

Barford et al., World Wide Web, 1999 Source: M. Crovella


Harchol-Balter and Downey,

ACM TOCS, 1997

Source: M. Crovella

Evidence: Process Lifetimes


The Bad News

• Workload metrics following heavy tailed distributions are extremelyvariable

• For example, for power tails:– When α ≤ 2, distribution has infinite variance– When α ≤ 1, distribution has infinite mean

• In practice, empirical moments are slow to converge – or nonconvergent• To characterize system performance, either:

– Attention must shift to distribution itself, or – Attention must be paid to timescale of analysis

Source: M. Crovella


Heavy Tails in Practice

Power tailswith α=0.8

Large observations dominate statistics (e.g., sample mean)

Source: M. Crovella


Self-Similar Properties:

Measurementof intervall-basedCounting Process

Self-Similar Poisson Traditional Bursty

∆=0.01 s

∆=10 s

∆=1 s

∆=0.1 s


Long-Range Dependence in MeasurementsCorrelation Plot (Inter-cell times) Aggregated Variance Plot

r(k) ~ k -α+1 with α≈1.4 Var X(m) ~ m -α+1 Var X

Long-Range Dependence found in ATM measurements


Mathematical DefinitionsSelf-Similarity with Hurst Parameter H: [Ni(∆)]i = [s-H Nj(s∆)]jd

Second Order Self-Similarity: r(m)(k) ≡ r(k) for all m,k=1,2,....

where r(m)(k) is autocorrelation of averaged, m-aggregated process

Asymptotic Second Order Self-Similarity: lim r(m)(k) / r(k) = 1

for all m=1,2,...k→∞

Long-Range Dependence: Σ r(k) → ∞ (assuming r(k)≥0 )

with special case: r(k) ~ 1/k(α-1) , 1<α<2

The latter processes are asymptotic second order self-similar!


ON/OFF models with Heavy-Tails

• Power-Tail distribution Very long ON periods can occur

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

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


Performance Impact

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

• Radical delay increase at transitions

• Power-Tailed queuelength-distr.

• ...with changing exponents β(i0)


Summary1. Motivation2. Traffic Characteristics







Non-exponential Service Times: M/G/1 Queues

• Poisson Arrivals (rate λ)• service times general (ME) distributed (i.i.d. RVs Yi)

– Utilization: ρ = λ EY [ Pr(Q=0)=1- ρ for infinite buffer model, as we will see later]

• Solution Approaches: – Embedded Markov Chain (see e.g. Kleinrock)

Pollaczek-Khinchin Formula:

where C^2:=coefficient of variation of service times– Matrix-Exponential Distributions Quasi Birth-Death Processes

existence of Matrix-geometric solution


References• Traffic Measurements

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

– E. Matthiesen, J. Larsen, F. Olufsen: ‚Quality of Service for Computer Gaming – An Evaluation of DiffServ‘, Student Report, Aalborg University, Spring 04. www.control.auc.dk/~04gr832b/

• 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.

• M. Crovella: ’Network Traffic Modeling’, PhD lecture, Aalborg University, February 2004.


Exercises:A network operator asks you to assist him in dimensioning his access router. The operator expects that during the daily busy hours, the router is used by N=10 users, each of them independently generates traffic according to an ON/OFF process with exponential ON and OFF periods. On periods have mean ON=10s with a Poisson packet rate of λp=6pck/sec for a single user. OFF periods show mean Z=20 seconds. The aggregated traffic stream of N=10 users can be described by an MMPP with K=11 states.

a. Determine the Q and the L matrix of the MMPP (in MATLAB).b. Compute the stationary probability vector pi of the MMPP. What is the average packet rate generated by the

MMPP?c. Determine the queue-length distribution of an MMPP/M/1 queue with service rate nu=30pck/sec via the

following steps.i. Compute the coefficient matrices A0,A1,A2 for the quadratic matrix equation for the rate matrix R.

ii. Solve the quadratic matrix equation via the simple iterative method from the lecture. Measure the time thatMatlab needs to do so.

iii. Use R to compute and plot the queue-length probabilities and the average queue-length.d. Compare the average queue-length with an M/M/1 queue of same utilization.

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Stochastic Models for Information Processingkom.aau.dk/project/sipcom/SIPCom05/sites/sipcom9/courses/SIPCom… · • Mm1 Channel Characterization and Measurements ... cumulative