Challenges in Packet Switched Setting

Transcription

1 Stochastic Models for Information Processing Bernard Fleury, Søren V. Andersen, Hans-Peter Schwefel 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) Note: slide-set will be complemented by formulas, mathematical derivations, and examples on the black-board! Page 1 Revision (SIPCOM9-3) Challenges in Packet Switched Setting Challenges in IP networks: HTTP Multiplexing of packets at nodes (L3) TCP Burstiness of IP traffic (L3-7) IP Impact of Dynamic Routing (L3) Link-Layer 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 L5-7 L4 L3 L2 Challenges in Wireless Networks: Wireless link models (channel models) MAC & LLC modeling RRM procedures Mobility models Cross layer optimization Analysis frequently with stochastic models Page 2

2 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(X n )= x n 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) Page 3 Revision (SIPCOM9-3) 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: E{X}=1/ λ; Var{X}=1/ λ 2, C 2 = Var{X} / [E{X}] 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) Page 4

3 Revision (SIPCOM9-3) Basic concepts III: Stochastic Processes Definition of process (X i ) (discrete) or (N t ) (continuous) Simplest type: X i independent and identically distributed (iid) Relevant Examples: Inter-arrival time process: X i Counting Process: n-1 N(t) = max{n Σ i=1 X i t}, alternatively N i ( ) = N(i ) N([i-1] ) Important Example: Poisson Process Assume i.i.d. exponential packet inter-arrival times (rate λ): X i :=T i -T i-1 Counting Process: Number of packets N t until time t Pr(N t =n)= (λt) n exp(- λt) / n! Properties: Merging: arrivals from two independent Poisson processes with rate λ 1 and λ 2 Poisson process with rate (λ 1 + λ 2 ) 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) Page 5 Revision (SIPCOM9-3) Continuous Time Markov Processes Defined by State-Space: finite or countable infinite, w/o.l.g. E={0,1,2,...,K} (K= also allowed) Transition rates: µ jk Holding time in state j: exponential with rate Σ k j µ jk =: µ j Transition probability from state j to k: µ jk / Σ l j µ jl = µ jk / µ j X t = RV indicating the current state at time t; π i (t):=pr(x t =i) Markov Property : transitions do not depend on history but only on current state t 0 <t 1 <...<t n, i 0,i 1,...i n,j E Computation of steady-state probabilities Chapman Kolmogorov Equations: dπ i (t)/dt = - µ i π i (t) + Σ j i µ ji π j (t) Flow-balance equations, steady-state (t ): µ i π i = Σ j i µ ji π j Here: restriction to irreducible, homogeneous processes Matrix Notation! Page 6

4 Revision (SIPCOM9-3) Queueing Models: Kendall Notation X/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. λ µ Finite buffer (size B) Scope here: Point-process models as opposed to fluidflow queues Page 7 Revision (SIPCOM9-3) M/M/1 queue Poisson arrival of packets (first M Markovian) with rate λ Exponentially distributed service times of rate µ (second M ) λ Single Server (1) FIFO service discipline Q t = 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) Infinite buffer µ 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) Page 8

5 Revision (SIPCOM9-3) M/M/1 queue: Performance λ λ λ µ µ µ Birth-Death Process Probability of i packets in queue [using flow-balance equations] π i := Pr(Q=i) = (1-ρ)* ρ i, where ρ= λ/ µ <1 Probability of idle server: π 0 = (1-ρ) Average Queue-length: E{Q}= ρ/(1-ρ) Average Delay (System Time): E{S}= E{Q}/ λ = 1/(µ-λ) Buffer Overflow Probabilities (PASTA principle) Pr(Q (a) B)= Pr(Q B) = ρ B Page 9 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?] Page 10

6 Content 1. Motivation 2. 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-Tails 6. Summary, Exercises Page 11 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 Methods Page 12

7 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 x 10 8 Traffic in OD Flow Eigenflow Mon Tue Wed Thu Fri Sat Sun Page Mon Tue Wed Thu Fri Sat Sun 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 Page 14

8 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. Page 15 Measuring Autocorrelation Coefficient of Autocorrelation (assumes stationarity): R(k) = Cov(X n,x n+k )/Var[X 0 ] = ( E[X n X n+k ] E 2 [X 0 ]) / Var[X 0 ] Page 16

9 Traffic Properties Poisson Arrival processes & M/M/1 type queues exponential distributions, memory-less Coefficient of variation: C 2 =Var(X)/[E{X}] 2 =1 Uncorrelated arrivals/services (Steady-state analysis) Actual Measurements (inter-cell times X i ): C 2 (X) between 13,,30 positive autocorrelation coefficient: ř(k) = (Σ i (X i -x)(x i+k -x)) / Var(X) > 0, slowly decaying with k Correlation Plot (Inter-cell times) Page 17 Bursty Models: ON/OFF Models Parameters: N sources, each average rate κ During ON periods: peak-rate λ p Mean duration of ON and OFF times κ = λ p ON/(ON+OFF) bursty traffic, when λ p >> κ Page 18

10 General hierarchical models Frequently used: Several levels with increasing granularity E.g. 3 levels: sessions, connections, packets Or: 5-level model: Page 19 Revision (SIPCOM9-3, MM4 Use Case) 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 Page 20

11 Revision (SIPCOM9-3, MM4 Use Case) Inter-departure time distribution Upstream: two modes, around 33 ms and 50 ms Downstream: single mode, around 60ms Page 21 Revision (SIPCOM9-3, MM4 Use Case) 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 Page 22

12 Revision (SIPCOM9-3, MM4 Use Case) Auto-correlation of inter-packet times Coefficient of auto-correlation Seemingly some periodic properties in game traffic Not achieved (of course) by two-state Markov model! Page 23 Content 1. Motivation 2. 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-Tails 6. Summary, Exercises Page 24

13 Markov Modulated Poisson Processes Example: Single-Source ON/OFF model Page 25 Multiplexed ON/OFF Models Parameters: N sources, each average rate κ During ON periods: peak-rate λ p Mean duration of ON and OFF times ON & OFF Times exponential MMPP representation with N+1 states (Exercises!) Page 26

14 Content 1. Motivation 2. 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-Tails 6. Summary, Exercises Page 27 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: Page 28

15 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) Page 29 Computational aspects Simple iteration method (*) may lead to long processing times Numerical properties Page 30

16 (optional) MMPP/M/1/B Queues Finite buffer-space, loss model Generator matrix Mixed matrix geometric solution Approaches to obtain matrix factor S Page 31 Content 1. Motivation 2. 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-Tails 6. Summary, Exercises Page 32

17 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 Page 33 Light Tails, Heavy Tails Light Exponential or faster decline f 1 (x) = 2 exp(-2(x-1)) Heavy Slower than any exponential f 2 (x) = x -2 Page 34

18 Examining Tails Best done using log-log complementary CDFs Plot log(1-f(x)) vs log(x) 1-F 2 (x) 1-F 1 (x) Page 35 Heavy Tails Arrive pre-1985: Scattered measurements note high variability in computer systems workloads : Detailed measurements note long distributional tails File sizes Process lifetimes : Attention focuses specifically on (approximately) polynomial tail shape: heavy tails post-1998: Heavy tails used in standard models Page 36

19 Power Tails, Mathematically We say that a random variable X is power tailed if: α P[ X > x ] ~ x 0 < α 2 a( x) where a(x) ~ b(x) means lim ( ) = 1. x b x Focusing on polynomial shape allows Parsimonious description Capture of variability in α parameter Page 37 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 Page 38

20 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 Page 39 Evidence: Web File Sizes Barford et al., World Wide Web, 1999 Page 40

21 Evidence: Process Lifetimes Harchol-Balter and Downey, ACM TOCS, 1997 Page 41 The Bad News Workload metrics following heavy tailed distributions are extremely variable 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 Page 42

22 Heavy Tails in Practice Power tails with α=0.8 Large observations dominate statistics (e.g., sample mean) Page 43 Self-Similar Properties: =10 s =1 s Measurement of intervall-based Counting Process =0.1 s =0.01 s Self-Similar Poisson Traditional Bursty Page 44

23 Long-Range Dependence in Measurements Correlation 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 Page 45 Mathematical Definitions Self-Similarity with Hurst Parameter H: [N i ( )] i = [s -H N j (s )] j d 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,... 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! k Page 46

24 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) Page 47 Performance Impact `Blow-up Regions i 0 =1,...,N for LRD traffic Radical delay increase at transitions Power-Tailed queuelength-distr....with changing exponents β(i 0 ) Page 48

25 Summary 1. Motivation 2. 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-Tails 6. Summary, Exercises Page 49 Non-exponential Service Times: M/G/1 Queues Poisson Arrivals (rate λ) service times general (ME) distributed (i.i.d. RVs Y i ) Utilization: ρ = λ E{Y} [ 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 Page 50

26 References Traffic Measurements H. Gogl, Measurement and Characterization of Traffic Streams in High-Speed Wide Area Networks, VDI Verlag, E. Matthiesen, J. Larsen, F. Olufsen: Quality of Service for Computer Gaming An Evaluation of DiffServ, Student Report, Aalborg University, Spring M. Neuts: Matrix geometric Solutions in Stochastic Models, John Hopkins University Press, M. Neuts: Structured stochastic matrices of M/G/1 type and their applications. Dekker, G. Latouche, V. Ramaswami: Introduction to matrix-analytic methods in stochastic modeling. ASA-SIAM Series on Statistics and Applied Probability H.-P. Schwefel: Performance Analysis of Intermediate Systems Serving Aggregated ON/OFF Traffic with Long-Range Dependent Properties, Dissertation, TU Munich, [Appendices B,C,D,F] K. Meier-Hellstern, W. Fischer: MMPP Cookbook, Performance Evaluation 18, p P. Fiorini et al.: Auto-correlation Lag-k for customers departing from Semi-Markov Processes, Technical Report TUM-I9506, TU München, July M. Crovella: Network Traffic Modeling, PhD lecture, Aalborg University, February Page 51 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 λ p =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 that Matlab 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. Page 52