PhD Course: Performance & Reliability Analysis of IP-based Communication Networks

Page 1Hans Peter Schwefel

PHD Course: Performance & Reliability Analysis, Day 3, Spring04

PhD Course: Performance & Reliability Analysis of IP-based Communication NetworksHenrik Schiøler, Hans-Peter Schwefel, Mark Crovella, Søren Asmussen

• Day 1 Basics & Simple Queueing Models(HPS)

• Day 2 Traffic Measurements and Traffic Models (MC)

• Day 3 Advanced Queueing Models & Stochastic Control (HPS)

• Day 4 Network Models and Application (HS)

• Day 5 Simulation Techniques (SA)

• Day 6 Reliability Aspects (HPS,HS)

Organized by HP Schwefel & H Schiøler



Reminder: Day 1 – Exponential models1. Intro & Review of basic stochastic concepts

• Random Variables, Exp. Distributions, Stochastic Processes

• Markov Processes: Discrete Time, Continuous Time, Chapman-Kolmogorov Eqs., steady-state solutions

2. Simple Qeueing Models• Kendall Notation• M/M/1 Queues, Birth-Death processes• Multiple servers, load-dependence, service disciplines• Transient analysis• Priority queues

3. Simple analytic models for bursty traffic• Bulk-arrival queues• Markov modulated Poisson Processes (MMPPs)

4. MMPP/M/1 Queues and Quasi-Birth Death Processes



Properties of Network Traffic

• Daily/weekly profiles

• Burstiness/Long-Range Dependence/Self-Similarity

• Application specific properties

• Impact of Transport Protocols (in particular TCP)

Actual Measurements (ATM-basedGerman University backbone, 1998)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 Correlation Plot (Inter-cell times)

Reminder: Day 2 – Traffic Properties



Content 1. Matrix Exponential (ME) Distributions• Definition & Properties: Moments, distribution function

• Examples• Truncated Power-Tail Distributions

2. Queueing Systems with ME Distributions• M/ME/1//N and M/ME/1 Queue• ME/M/1 Queue

3. Performance Impact of Long-Range-Dependent Traffic • N-Burst model, steady-state performance analysis

4. Transient Analysis• Parameters, First Passage Time Analysis (M/ME/1)

5. Stochastic Control: Markov Decision Processes• Definition• Solution Approaches• Example

Note: this slide-set contains only the outline of the lecture; many formulas and the mathematical derivations will be presented on the black-board!



Notation (Summary)

• 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



Matrix Exponential Distributions

• Replace single state with open system S containing m states, described by

• Entrance vector p=[p1,…, pm]

• State leaving Rates: M=diag(1,…,m)

• Transition Probabilities: P=(pij)

• Note: q:= ’ - P’ 0 (in contrast to discrete time Markov chains)

• Interested in Random VariableX:= time to leave S | entered according to p



Matrix Exponential Distributions: Properties

• Mean Time to leave S:where

– Trivial Example: Exponential distribution, E{X}=pV’=1/

• Distribution of X:

– Density: f(x)=- dR(x)/dx = p B exp(-xB) ’

• k-th Moments:

• Matrix Exponential Distributions Phase-Type Distributions



Examples I: Hyper-exponential Distributions

• Weighted Mixture of two (or m) exponentials:

• Moments:

frequently used to approximate distributions with high variance

• Matrix-Exponential Representation: <p,B>



Examples II: Erlangian Distributions• Sum of T exponentially distributed RVs• Probability density function:

• Moments:

• Limit (T, T~1/T): deterministic RV

• Matrix-Exponential Representation



Examples III: (Truncated) Power-Tail Distributions• Definitions:

– Sub-exponential distributions

– Power-Tail distributions

– Example: Pareto Distribution

• Properties:– Moments

– Residual Times

• Truncated Power-Tail Distributions– Representation as hyper-exponentials

– Systematic control of Power-tail Range

– [easy to generate in simulation models]



Truncated Power-Tail Distributions: cntd.











M/G/1 and M/ME/1 Queueing Models• Poisson Arrivals (rate )

• service times general (ME) distributed (i.i.d. RVs Xi, described by <p,B>)

– Utilization: = E{X} [ Pr(Q=0)=1- for infinite buffer model, as we will see later]

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

Pollaczek-Khinchin Formula:

– Quasi-Birth Death Processes existence of Matrix-geometric solution

– Limit of finite-customer systems: M/ME/1//N models



M/ME/1//N Queueing Systems: steady state queue-length distribution

• Define– r(n;N) := Pr(’n customers at S1’)

– ((n;N))i := Pr(’n customers at S1 & server S1 in state i’)

• Balance Equations

with

• Matrix-Geometric Solution



M/ME/1//N Queueing Systems (cntd.)

• Normalization (r(n)=1)

• Probabilities at– Arrival instances

– Departure instances



M/ME/1 Queue



ME/M/1 Queueing Systems• Definition:

– Service Times exponential (rate ),

– Inter-Arrival Times Xi iid, described by <p,B>)

• In principle same approach as for M/ME/1:– Existence of Matrix-geometric solution known from QBD theory– Consider solution of finite ME/M/1//N system– Take limit N

• Finite ME/M/1//N system identical to M/ME/1//N system (but consider S2 instead of S1) = 1 / [E{X} ]

• Obstacles– Lim UN does not exist (U has eigenvalues > 1)

consider lim sNUN where s is the smallest eigenvalue of A



ME/M/1 Queueing Systems: Solution• Solution (Matrix-geometric solution reduces to geometric!)

• Performance Parameters



ME/M/1 Queue: Geometric parameter s



Exercise 1:

1. ME distributions

2. M/ME/1 Queue

3. ME/M/1 Queue

Detailed tasks, see attached sheet.











Reminder: Self-Similarity/Long-Range Dependence

• Self-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: r(k) (assuming r(k)0 )

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

The latter processes are asymptotically second order self-similar!



Self-Similar/LRD Traffic Models

• Fractional Brownian Motion / Fractional Gaussian Noise

N(t) = m t + sqrt(a m) Z(t) where Z(t) continuous, normally distributed, E{Z2}= t2H , 0.5<H<1

... is self-similar with parameter H and long-range dependent with =3-2H

• ON/OFF Traffic with Power-Tailed ON periods (1< <2)... is long-range dependent

• ... [others, e.g. chaotic maps, F-ARIMA]



ON/OFF Models

Parameters:

• N sources, each average rate

• During ON periods: peak-rate p

MMPP representation(exponential case):

• Mean duration of ON and OFF times

• ON times general Matrix exponential: <p,B>: Pr(ON>t) = p exp(-Bt) ´

N



MMPP Representation: Multiplexed ON/OFF models

i=1 (needed for TCP

models, not done here)



N-Burst with 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)



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)



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



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)



Looking at Burstiness once more• Def. (see Day1): range b=0 (not bursty) to b=1 (very bursty):

b:=1-/p , Average Rate of Source , Peak Rate of Source p

• Investigating burstiness:– Vary p

– Keep #packets per ON period scale ON duration accordingly

– Extreme Cases:• b=0 (p= ): Poisson arrivals, no burstiness

• b=1 (p= ): Bulk arrivals

• Mean packet delay shown for – Single ON/OFF source (N=1)– Utilization =0.5– Tail Exponent =1.4 for ON periods



Fluid-Flow or Point-Process Models?

• Experiment: scale ’packet-size’ by factor k– Packet rate p k p , service rate k

– Limit:

• k : fluid-flow model

• Mean packet delay shown for – Single ON/OFF source (N=1)– Utilization =0.5– TPT-30 distribution with

Tail Exponent =1.4 for ON periods











Transient Behavior: Motivation

Example:

• ON/OFF traffic with TPT ON time

• Queueing Model with large buffer

Observations (Simulation):

• 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



Transient Analysis: Basics• Selected transient parameters

– Transient Queue-length Probabilities: Pr(Q(t)=n)– First Passage Time Analysis

FPT(n) = RV reflecting the time until the buffer occupancy reaches level n for the first time (given some well-defined initial state, normally empty queue)

– Busy Period Analysis• Duration of busy period• Pr(buffer occupancy reaches maximum QL n during busy period)

– Transient Overflow Probabilities: (t,n) = Pr(FPT(n)<t)

• Transient parameters depend on initial conditions– E.g. Empty queue (Q(0)=0), state of arrival process, state of service process (for correlated

service events)



Computation of mean First Passage Times: demonstrated for M/ME/1 Queues

• Poisson Arrivals (rate ), service times <p,B>

• Approach:

– [pu(i)]j:=Pr(server in state j when queue reaches level i for the first time)

– [u(i)]j:= Mean time it takes to have i+1 customers for the first time,

given started at queue-length i in server state j

• Introduce Matrix

– [Hu(n)]ij:=Pr(server in state j when queue goes from level nn+1 for the first time ,

given server started in state i at queue-length n)

• Rewrite



Computation of mean First Passage Times: recursive equations

• Recursive Equations for Hu(n)

• Similarly for u(i)



Example: Mean First Passage Times N-Burst/M/1

• mFPT grows by Power-Law B similar blow-up effects for changing as for steady-state analysis



Optional: Traffic/Performance Models for TCP

End-to-End flow/congestion control Feedback: network behavior (congestion) ingress traffic No separation traffic model/network model possible New traffic/performance models required

Approaches:1. Connection level models2. Sender/receiver behavior & fixpoint iteration3. Integrated models

Traffic Model

Network Model

Performance Values (Delay, Loss, etc.)

feedback



Example: Fixpoint Iteration

Network Model:

• E.g. M/M/1/K model or bulk-arrival queue

TCP model, e.g. [Padhye et al., Sigcomm 98]

b: # packets acknowledged by ACK

RTT: Average Round-Trip-Time (including queuing delay)

Wmax: Maximum size of congestion window

T0: Average Time-Out interval

p: Fraction of retransmitted packets

TCPModel

Network Model

Initial loss rate p, delay d Average

packet rate

Update: loss rate p, delay d

When stable: result



• Multiplexing of packets at nodes (L3)• Burstiness of IP traffic (L3-7)• Impact of Routing (L3)• Performance impact of transport layer,

in particular TCP (L4)• Wide range of applications different traffic & QoS requirements (L5-7)

Traffic Modelling Challenges

HTTP

TCP

IP

Link-Layer

L5-7

L4

L3

L2

Day1 & 3

[touched]

Traffic Model

Mobility Model

Network Model

Performance Values (Delay, Loss, etc.)

TCP / adaptive applics.

Wireless Link Properties











Markov Decision Processes: Motivation & Definition

• Markov Process definition– State-Space

– Transition Probabilities/Rates

Computation of System behavior (e.g. performance parameters)

Extension: Investigate ’optimal control’ of such a Markov model• Markov Decision Processes (MDPs)

– State-Space E

– Actions: a(s,t) A, sE

– Transition Probabilities depend on current state s(t) and on selected action a(s,t)

– Rewards [Costs/Utility]: r(s,a)

• Goal: Find ‘selection of actions’ that maximize some reward criterion

• Simple Example: extended Gilbert-Elliot Channel Model



MDPs: Policies• Policies

=(d1,d2,...), sequence of decision rules

– Decision rules: di: EA -- specification of action depending on state s

• Types of Policies– Randomized vs. deterministic

– Stationary vs. non-stationary

• For a given stationary, deterministic policy =(d,d,...), the MDP is a (homogeneous) Markov process, enriched by the reward function, r(s,d(s))



MDPs: Optimality CriterionsOptimality criterion: Maximize expected reward• Finite horizon (covering N steps)

vN, (s0)=E,s0{ r(Xi, d(Xi) } [sum runs from i=1 to N]

• Infinite horizon– Total expected reward

v (s0)=E,s0{ r(Xi, d(Xi) } [sum runs from i=1 to ]

– Expected average rewardv (s0)=lim 1/N E,s0{ r(Xi, d(Xi) } [sum runs from i=1 to N; lim

N]

– Total expected discounted reward

v (s0)=E,s0{ i r(Xi, d(Xi) } [sum runs from i=1 to ] , 0< <1

• for stationary policy =(d,d,...) and initial state X0=s0

[Assumptions: discrete time, finite action space]



MDPs: Solution Approaches and applications

Solution Approaches:• Value iteration• Policy iteration• Linear Programming• Dynamic Programming

Applications:• Call admission control & handoff control• Paging/mobility tracking• Radio Resource Management• Scheduling/Buffer Management• Routing

’Learning’ Approaches:• Neural Networks• Genetic Algorithms• Q-Learning



MDPs: References• Cassandras, Lafortune, ’Introduction to Discrete Event Systems’, Chapt. 9, 1999.

• Heyman, Sobel (ed.), ‘Stochastic Models’, Chapt. 8, 1990

• Bertsekas, ‘Dynamic Programming and Stochastic Control’, 1976

• Martin.L.Puterman, Markov Decision Processes: Discrete Stochastic Dynamic Programming, John Wiley &

Sons, Inc. 1994.• R.Ramjee, R.Nagarajan and D.Towsley, On Optimal Call Admission Control in Cellular Networks, IEEE

INFOCOM 1994

• R.Rezaiifar, A.Makowski and S.Kumar, Stochastic control of Handoffs in Cellular Networks, IEEE JSAC, vol.13., no.7. 1995

• El. El-Alfi, Y.Yao, H.Heffes, A Model-Based Q-Learning Scheme for Wireless Channel Allocation with Prioritized Handoff, IEEE Globecom 2001.

• U.Madhow, M.L.Honig, and K.Steiglitz, Optimization of Wireless Resources for Personal Communications Mobility Tracking, IEEE INFOCOM, 1994.

• V.Wong, V.Leung, An Adaptive Distance-Based Location Update Algorithm for Next-Generation PCS Networks, IEEE JSAC,vol.19,no.10,Oct.2001.

[Acknowledgment:List of References partially from Presentation of Bang Wang, NUS, Sept. 02]



Summary 1. Matrix Exponential (ME) Distributions• Definition & Properties: Moments, distribution function








Exercises II:

Complete Exercise 1d (transient analysis)

Documents

PhD Course: Performance & Reliability Analysis of IP-based Communication Networks