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A Statistical Network Calculus for ComA Statistical Network Calculus for Computer Networksputer Networks
Jorg Liebeherr
Department of Computer Science
University of Virginia
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CollaboratorsCollaborators
• Almut Burchard • Robert Boorstyn• Chaiwat Oottamakorn • Stephen Patek• Chengzhi Li • Florin Ciucu
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• R. Boorstyn, A. Burchard, J. Liebeherr, C. Oottamakorn. “Statistical Service Assurances for Packet Scheduling Algorithms”, IEEE Journal on Selected Areas in Communications. Special Issue on Internet QoS, Vol. 18, No. 12, pp. 2651-2664, December 2000.
• A. Burchard, J. Liebeherr, and S. D. Patek. “A Calculus for End–to–end Statistical Service Guarantees.” (2nd revised version), Technical Report CS-2001-19, May 2002.
• J. Liebeherr, A. Burchard, and S. D. Patek , “Statistical Per-Flow Service Bounds in a Network with Aggregate Provisioning”, Infocom 2003.
• C. Li, A. Burchard, J. Liebeherr, “Calculus with Effective Bandwidth”, Technical Report CS-2003-20, November 2003.
• F. Ciucu, A. Burchard, J. Liebeherr, ",A Network Service Curve Approach for the Stochastic Analysis of Networks”, ACM Sigmetrics 2005, to appear.
PapersPapers
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““Toy Models” in Computer Networking Toy Models” in Computer Networking
• Learn from Physics: Wide use of toy models
… that capture key characteristics of studied system
… that permit back-of-the-envelope calculations
… that are usable by non-theorists
• Simple models have played a major role in the evolution and development of data networks• Queueing Networks• Effective Bandwidth• (Deterministic) Network Calculus
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(Product Form) Queueing Networks(Product Form) Queueing Networks
• Jackson (50’s), Kelly, BCMP (70’s)
• Flow of “jobs” in system of queues and servers
• Applications: Provided motivation for packet-switching (Kleinrock’s PhD thesis)
Main result: Steady state probability of queue occupancey n = (n1, n2, … , nk) :
P(n ) = P(n1) P(n2) … P(nk)
Limitations: Limited to Poisson traffic Limited scheduling algorithms
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Effective BandwidthEffective Bandwidth
Hui, Mitra, Kelly (90s)• Describes bandwidth needs o
f complex traffic by a number
• Application: admission control in ATM networks
Can consider: service guarantees wide variety of traffic (incl. LRD)
statistical multiplexingLimitations:
not well suited for scheduling
MPEG-Compressed Video Trace
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Network CalculusNetwork Calculus
• Cruz, Chang, LeBoudec (90’s)
• Worst case delay and backlog bounds for fluid flow traffic
• Application: design of new schedulers (WFQ) new services (IntServ).
SenderReceiver
S3S1
S2
Snet
Limitations: No random losses No statistical multiplexing, therefore pessimistic
• Main result: If S1, S2 and S3 describes the service at each node, then Snet = S1 * S2 * S3 describes the service given by the network as a whole.
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State-of-the-artState-of-the-art
• No analysis methodology is widely used today.• Today, a lot of networking research relies on simulation and
measurements to validate new designs • Simulation and measurement are generally not suitable for
evaluation of radically new designs
Requirements Queueing networks
Effective bandwidth
Network calculus
Traffic classes (incl. self-similar, heavy-tailed)
Limited Broad Broad(but loose)
Scheduling Limited No Yes
QoS (bounds on loss, throughput, delay)
Very limited
Loss, throughput
Deterministic
Statistical Multiplexing
Some Yes No
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Motivation: Motivation: Develop network calculus into newDevelop network calculus into new “Toy Model”“Toy Model”
Today, fundamental progress in networking is hampered by the lack of methods to evaluate how radically new designs will perform.
• Opportunity: Simple (`toy') models that permit fast (`back-of-the-envelope') evaluations can become an enabling factor for breakthrough changes in networking research
• Approach: Probabilistic version of network calculus (stochastic network calculus) is a candidate for a new class of toy models for networking
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RateVarianceEnvelopeKnightly `97
Effective Bandwidth:
J. Hui ’88Guerin et.al. ’91Kelly `91Gibbens, Hunt `91
Deterministic network calculusCruz `91
Effective bandwidth in network calculusChang `94
(min,+) algebra for det. networks:
Agrawal et.al. `99Chang `98LeBoudec `98
ServiceCurvesCruz `95
Cruz calculus with probabilistic trafficKurose `92
Exponentially/stochasti-cally. bounded burstinessYaron/Sidi `93Starobinski/Sidi `99
Stochastically bounded service curveQiu et.al.`99
1985 1990 1995 2000
Our goals:
(1) Maintain elegance of deterministic calculus
(2) Exploit statistical multiplexing
(3) Try to express other models
2005
Related Work (small subset)Related Work (small subset)
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flow 1 for service
support to
needed Resources
N
flows N for service
support to
needed Resources
Multiplexing gain is the raison d’être for packet networks.
Sources of multiplexing gain:
• Traffic characterization and conditioning
• Scheduling
• Statistical Multiplexing
Multiplexing GainMultiplexing Gain
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MPEG-Compressed Video Trace
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Tra
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Traffic Conditioning
• Traffic conditioning is typically done at the network edge• Reshaping traffic increases delays and/or losses
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Traffic ConditioningTraffic Conditioning
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• Scheduling algorithm determines the order in which traffic is transmitted
• Examples:• Different loss priorities priority scheduling• Traffic with rate guarantees rate-based scheduling (WFQ, WRR)• Delay constraints deadline-based scheduling (EDF)
SchedulingScheduling
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Multiplexing GainMultiplexing Gain
Flow 1Worstcasearrivals
Flow 2Flow 3
Time
Without statistical multiplexingB
ack
log
Worst-casebacklog
Flow 1Flow 2Flow 3
Time
Bac
klo
g
Arrivals
With statistical multiplexing
Backlog
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Example of Statistical Multiplexing: Example of Statistical Multiplexing: Retirement SavingsRetirement Savings
Life expectancy: Normal(=75, =10) years
Retiring Age: 65 years
Interest: 0%
Withdrawal: $50,000 per year
How much money does a person need to save (with confidence of 95% or 99%)?
Life expectancy in a group of N people is Normal(, N).
N=1 person (Individual Savings): 95% confidence: 10 + 2= 30 years $1.5 Mio.99% confidence: 10 + 2= 40 years $2 Mio.
N=100 people (Pooled Savings): 95% confidence: 10 + 2= 12 years $600,00099% confidence: 10 + 2= 13 years $650,000
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The importance of Statistical The importance of Statistical MultiplexingMultiplexing
Gain
ngMultiplexi
lStatistica
Calculus
Network
ticDeterminis
Calculus
Network
Stochastic
• At high data rates, statistical multiplexing gain dominates the effects of scheduling and traffic characterization
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Arrivals from a flow j are a random process
• Stationarity: The are stationary random processes
• Independence: The and are stochastically independent
Traffic CharacterizationTraffic Characterization
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Leaky Buckets:
Regulatedarrivals
Each flow isregulated
Bufferwith Scheduler
Flow 1
Flow N )(* NA
C
),(1 ttA... ),( ttAN
A*=min (Pt,+t)
P
Traffic is constrained by a subadditive deterministic envelope such that
Regulated ArrivalsRegulated Arrivals
)(*1 A
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Definition:Definition: Effective envelope for is a function such that
Note: Effective envelope is not a sample path bound. Often, we need a stronger version of the effective envelope!
Effective envelopeEffective envelope
Define a function that bounds traffic with high probability “Effective Envelope”
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Sample Paths and EnvelopesSample Paths and Envelopes
Samplepaths
Note: All envelopes are non-random functions
Effective envelopeAt any time, at most one sample path is violated
Stronger effective envelopeAt most one sample path is violated
Deterministic envelopeNever violated
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Probabilistic Sample Path BoundProbabilistic Sample Path Bound
A strong effective envelope for an interval of length is a function which satisfies
Relationship between the envelopes is established as follows:
with
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Arrivals from multiple flows:
Deterministic Network Calculus: Deterministic Network Calculus: Worst-case of multiple flows is sum of the worst-case of each flow
Regulatedarrivals
Traffic Conditioning
Bufferwith Scheduler
Flow 1
Flow N
)(A*1
)(A*N
C
),(1 ttA... ),( ttAN
Aggregating ArrivalsAggregating Arrivals
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Effective Envelopes for aggregated flowsEffective Envelopes for aggregated flows
Stochastic Calculus: Stochastic Calculus: Exploit independence and extract statistical multiplexing gain by calculating
• For example, using the Chernoff Bound, we can obtain
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Type 1 flows:P =1.5 Mbps = .15 Mbps=95400 bits
Type 2 flows:P = 6 Mbps = .15 Mbps= 10345 bits
Type 1 flows
strong effectiveenvelopes
Effective vs. Effective vs. Deterministic Deterministic Envelope Envelope EnvelopesEnvelopes
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Traffic rate at t = 50 msType 1 flows
Effective vs. Effective vs. Deterministic Deterministic Envelope Envelope EnvelopesEnvelopes
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Scheduling AlgorithmsScheduling Algorithms
• Work-conserving scheduler with unit rate that serves Q classes
• Class-q traffic has delay bound dq
• Scheduling algorithm:
Scheduler
)(A*1
)(A*N
.
.
.
Static Priority (SP):
Earliest Deadline First (EDF):
Deterministic ServiceNever a delay bound violation if:
Statistical ServiceDelay bound violation with if:
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Statistical Multiplexing vs. SchedulingStatistical Multiplexing vs. Scheduling
Statistical multiplexing makes a big difference
Scheduling has small impact
Example: MPEG videos with delay constraints at C= 622 Mbps Deterministic service vs. statistical service (= 10-6)
Thick lines: EDF SchedulingDashed lines: SP scheduling
dterminator=100 ms dlamb=10 ms
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More interesting traffic typesMore interesting traffic types
• So far: Traffic of each flow was regulated
• Next: Consider different traffic types:
• On-Off traffic
• Fraction Brownian Motion (FBM) traffic
• Approach: Exploit literature on Effective Bandwidth
• Describes traffic in terms of a function
• Expressions have been derived for many traffic types
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Effective Envelopes and Effective Bandwidth Effective Envelopes and Effective Bandwidth
Effective Bandwidth (Kelly 1996)
Given , an effective envelope is given by
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Comparisons of statistical service guarantees for different schedulers and traffic types
Schedulers:
SP- Static PriorityEDF – Earliest Deadline FirstGPS – Generalized Processor Sharing
Traffic:
Regulated – leaky bucketOn-Off – On-off sourceFBM – Fractional Brownian Motion
C= 100 Mbps, = 10-6
Effective Envelopes and Effective Bandwidth Effective Envelopes and Effective Bandwidth
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.....
........
............
.
D(t)
A(t)
s
..........
..
backlog=B(s)
delay=W(s)
S(t)
A(t) D(t)
Statistical Network Calculus with Min-Plus Statistical Network Calculus with Min-Plus AlgebraAlgebra
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•Convolution operation:
•Deconvolution operation
t
f(t)
g(t)
f*g(t)
Convolution and Deconvolution operatorsConvolution and Deconvolution operators
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1. Output Envelope: is an envelope for the departures
2. Backlog bound: is an upper bound for the backlog
3. Delay bound: An upper bound for the delay is
Cruz `95: A service curve for a flow is a function S such that:
(min,+) results(min,+) results (Cruz, Chang, LeBoudec(Cruz, Chang, LeBoudec))
Deterministic (min,+)Deterministic (min,+) Network CalculusNetwork Calculus
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1. Output Envelope: is an envelope for the departures with probability
2. Backlog bound: is an upper bound for the backlog with probability
3. Delay bound: An upper bound for the delay with probability is
An effective service curve for a flow is a function such that:
(min,+) results(min,+) results
Stochast Network CalculusStochast Network Calculus
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Given:• Service guarantee to aggregate (C ) is known• Total Traffic is known
What is a lower bound on the service seen by a single flow?
Allocated capacity C
Sender Receiver
Statistical Per-Flow Service BoundsStatistical Per-Flow Service Bounds
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Can show:
is an effective service curve for a flow where is a strong effective envelope and is a probabilistic bound on the busy
period
Allocated capacity C
Sender Receiver
Statistical Per-Flow Service BoundsStatistical Per-Flow Service Bounds
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Type 1 flows:
Goal: probabilisticdelay bound d=10ms
Number of flows that can be admittedNumber of flows that can be admitted
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SenderReceiver
S3S1S2
Deterministic Network Service Curve (Cruz, Chang, LeBoudec)(Cruz, Chang, LeBoudec) :
If are service curves for a flow at nodes, then
Snet = S1 * S2 * S3
is a service curve for the entire network.
Snet
Network Service CurvesNetwork Service Curves
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Network Service Curve:
If S1,, S2 , … SH , are effective service curves for a flow, then for all
.
Unfortunately, this network service is not very useful!
Finding a suitable network service curve has been a longstanding open problem. A solution is presented in an upcoming ACM Sigmetrics 05 paper.
Network Service Curve in a Stochastic CalcNetwork Service Curve in a Stochastic Calculusulus
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Effective Network Service CurveEffective Network Service Curve
• Revise the definition of the effective service curve to
• Define
Theorem: A network service curve is given by
with
where are free parameters
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Application of Network Service CurveApplication of Network Service Curve
• Analyze end-to-end delay of through flows for Markov Modulated On-Off Traffic
• Compare delay with network service curve to a summation of per-node bounds
...
CrossFlows
CrossFlows
CrossFlows
CrossFlows
CrossFlows
CrossFlows
ThroughFlows
ThroughFlows
Node HNode 2Node 1
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ExampleExample
• Peak rate: P = 1.5 MbpsAverage rate: = 0.15 Mbps
• T= 1/ + 1/ = 10 msec
• C = 100 Mbos• Cross traffic = through traffic • = 10-9
• Addition of per-node bounds grows O(H3)
• Network service curve bounds grow O(H log H)
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ConclusionsConclusions
• Presented aspects of stochastic network calculus• Preserves much (but not all) of the deterministic ca
lculus• Can express many existing results on:
• Deterministic calculus • Effective bandwidth• Other models (EBB, not shown)
• Many open issues
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ConclusionsConclusions
Requirements
Queueing networks
Effective bandwidth
Network calculus
Traffic classes (incl. self-similar, heavy-tailed)
Limited Broad Broad(but loose)
Scheduling Limited No Yes
QoS (bounds on loss, throughput delay)
Very limited
Loss, throughput
Deterministic
Statistical Multiplexing
Some Yes No
Stochastic network calculus
Broad
Yes
Yes
Yes