PhD Course: Performance & Reliability Analysis
of IP-based Communication Networks
Henrik 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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Page 1
Hans Peter Schwefel
Reminder: Day 1 – Exponential models
1. Intro & Review of basic stochastic concepts
•
•
Random Variables, Exp. Distributions, Stochastic
Processes
Markov Processes: Discrete Time, Continuous Time, ChapmanKolmogorov 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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Reminder: Day 2 – Traffic Properties
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-based
German 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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Correlation Plot (Inter-cell times)
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Hans Peter Schwefel
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!
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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 Variable
X:= time to leave S | entered
according to p
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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>
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Examples II: Erlangian
Distributions
• Sum of T exponentially distributed RVs
• Probability density function:
• Moments:
•
Limit (T, T~1/T): deterministic RV
• Matrix-Exponential Representation
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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]
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Truncated Power-Tail Distributions: cntd.
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
M/ME/1//N Queueing Systems (cntd.)
• Normalization (r(n)=1)

• Probabilities at
– Arrival instances
– Departure instances
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
M/ME/1 Queue
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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
Hans Peter Schwefel
ME/M/1 Queueing Systems: Solution
• Solution (Matrix-geometric solution reduces to geometric!)
• Performance Parameters
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
ME/M/1 Queue: Geometric parameter s
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Exercise 1:
1. ME distributions
2. M/ME/1 Queue
3. ME/M/1 Queue
Detailed tasks, see attached sheet.
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Reminder: Self-Similarity/Long-Range Dependence
[Ni()]i =d [s-H Nj(s)]j
• Self-Similarity with Hurst Parameter H:
r(m)(k)  r(k)
• Second Order Self-Similarity:
for all m,k=1,2,....
where r(m)(k) is autocorrelation of averaged, m-aggregated process
• Asymptotic Second Order Self-Similarity:
(m)(k) / r(k) = 1
lim
r
k
for all m=1,2,...
• Long-Range Dependence:
with special case:
 r(k)  
(assuming r(k)0 )
r(k) ~ 1/k(-1) , 1<<2
The latter processes are asymptotically second order self-similar!
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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]
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
ON/OFF Models
Parameters:
•
N sources, each average rate 
•
Mean duration of ON and OFF times
•
During ON periods: peak-rate p
•
ON times general Matrix exponential:
<p,B>: Pr(ON>t) = p exp(-Bt) ´
MMPP representation
(exponential case):
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
N
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Hans Peter Schwefel
MMPP Representation: Multiplexed ON/OFF models
i=1
(needed for TCP
models, not done here)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
N-Burst /M/1 Performance: Blow-up Points
•
•
`Blow-up Regions´ i0=1,...,N for LRD
traffic
•
Power-Tailed queuelength-distr.
•
...with changing exponents (i0)
Radical delay increase at transitions
•
Tail truncated at qN(MBS)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Transient Behavior: Motivation
Example:
Simulation: Fraction of overflowed packets
• 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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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 nn+1 for the first time ,
given server started in state i at queue-length n)
•
Rewrite
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Computation of mean First Passage Times:
recursive equations
• Recursive Equations for Hu(n)
• Similarly for u(i)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Example: Mean First Passage Times N-Burst/M/1
• mFPT grows by PowerLaw B
 similar blow-up effects
for changing  as for
steady-state analysis
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
Traffic
Model
Network
Model
feedback
Approaches:
1. Connection level models
2. Sender/receiver behavior & fixpoint iteration
3. Integrated models
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Page 38
Performance
Values
(Delay, Loss,
etc.)
Hans Peter Schwefel
Example: Fixpoint Iteration
Update: loss rate p, delay d
Initial loss
rate p,
delay d
TCP
Model
Average
packet
rate
Network
Model
When stable: result
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Traffic Modelling Challenges
HTTP
Multiplexing of packets at nodes (L3)
Day1 & 3
TCP
Burstiness of IP traffic (L3-7)
IP
Impact of Routing (L3)
Link-Layer
Performance impact of transport layer, [touched]
in particular TCP (L4)
• Wide range of applications  different traffic & QoS requirements (L5-7)
•
•
•
•
Traffic
Model
TCP / adaptive
applics.
Mobility
Model
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
Network
Model
L5-7
L4
L3
L2
Performance
Values
(Delay, Loss,
etc.)
Wireless Link
Properties
Page 40
Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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, sE
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
MDPs: Policies
• Policies 
– =(d1,d2,...), sequence of decision rules
– Decision rules: di: EA -- 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))
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
MDPs: Optimality Criterions
[Assumptions: discrete time,
finite action space]
Optimality 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 reward
v (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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
MDPs: Solution Approaches and applications
Solution Approaches:
• Value iteration
• Policy iteration
• Linear Programming
• Dynamic Programming
’Learning’ Approaches:
• Neural Networks
• Genetic Algorithms
• Q-Learning
Applications:
• Call admission control & handoff control
• Paging/mobility tracking
• Radio Resource Management
• Scheduling/Buffer Management
• Routing
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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]
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
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
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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Hans Peter Schwefel
Exercises II:
Complete Exercise 1d (transient analysis)
PHD Course: Performance & Reliability Analysis, Day 3, Spring04
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