phd course: performance & reliability analysis of ip-based communication networks
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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
Page 2Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 3Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 4Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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!
Page 5Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 6Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 7Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 8Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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>
Page 9Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Examples II: Erlangian Distributions• Sum of T exponentially distributed RVs• Probability density function:
• Moments:
• Limit (T, T~1/T): deterministic RV
• Matrix-Exponential Representation
Page 10Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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]
Page 11Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Truncated Power-Tail Distributions: cntd.
Page 12Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 13Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 14Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 15Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
M/ME/1//N Queueing Systems (cntd.)
• Normalization (r(n)=1)
• Probabilities at– Arrival instances
– Departure instances
Page 16Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
M/ME/1 Queue
Page 17Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 18Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
ME/M/1 Queueing Systems: Solution• Solution (Matrix-geometric solution reduces to geometric!)
• Performance Parameters
Page 19Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
ME/M/1 Queue: Geometric parameter s
Page 20Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Exercise 1:
1. ME distributions
2. M/ME/1 Queue
3. ME/M/1 Queue
Detailed tasks, see attached sheet.
Page 21Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 22Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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!
Page 23Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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]
Page 24Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 25Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
MMPP Representation: Multiplexed ON/OFF models
i=1 (needed for TCP
models, not done here)
Page 26Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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)
Page 27Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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)
Page 28Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 29Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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)
Page 30Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 31Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 32Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 33Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 34Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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)
Page 35Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 36Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Computation of mean First Passage Times: recursive equations
• Recursive Equations for Hu(n)
• Similarly for u(i)
Page 37Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 38Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 39Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 40Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
• 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
Page 41Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 42Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 43Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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))
Page 44Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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]
Page 45Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Page 46Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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]
Page 47Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Summary 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
Page 48Hans Peter Schwefel
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Exercises II:
Complete Exercise 1d (transient analysis)
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