A Statistical Network Calculus for Computer Networks Jorg Liebeherr Department of Computer Science...

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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

CollaboratorsCollaborators

• Almut Burchard • Robert Boorstyn• Chaiwat Oottamakorn • Stephen Patek• Chengzhi Li • Florin Ciucu

• 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

““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

(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

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

0 200 400 600 800 1000

Frame number

Peak rate

Mean rate

effectivebandwidth

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

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.

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

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

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

Related Work (small subset)Related Work (small subset)

flow 1 for service

support to

needed Resources

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

0 200 400 600 800 1000

Frame number

Traffic Conditioning

• Traffic conditioning is typically done at the network edge• Reshaping traffic increases delays and/or losses

0 200 400 600 800 1000Frame number

Traffic ConditioningTraffic Conditioning

• 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

Multiplexing GainMultiplexing Gain

Flow 1Worstcasearrivals

Flow 2Flow 3

Without statistical multiplexingB

Worst-casebacklog

Flow 1Flow 2Flow 3

Arrivals

With statistical multiplexing

Backlog

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

The importance of Statistical The importance of Statistical MultiplexingMultiplexing

ngMultiplexi

lStatistica

Calculus

Network

ticDeterminis

Calculus

Network

Stochastic

• At high data rates, statistical multiplexing gain dominates the effects of scheduling and traffic characterization

Arrivals from a flow j are a random process

• Stationarity: The are stationary random processes

• Independence: The and are stochastically independent

Traffic CharacterizationTraffic Characterization

Leaky Buckets:

Regulatedarrivals

Each flow isregulated

Bufferwith Scheduler

Flow 1

Flow N )(* NA

),(1 ttA... ),( ttAN

A*=min (Pt,+t)

Traffic is constrained by a subadditive deterministic envelope such that

Regulated ArrivalsRegulated Arrivals

)(*1 A

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”

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

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:

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

),(1 ttA... ),( ttAN

Aggregating ArrivalsAggregating Arrivals

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

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

Traffic rate at t = 50 msType 1 flows

Effective vs. Effective vs. Deterministic Deterministic Envelope Envelope EnvelopesEnvelopes

Scheduling AlgorithmsScheduling Algorithms

• Work-conserving scheduler with unit rate that serves Q classes

• Class-q traffic has delay bound dq

• Scheduling algorithm:

Scheduler

Static Priority (SP):

Earliest Deadline First (EDF):

Deterministic ServiceNever a delay bound violation if:

Statistical ServiceDelay bound violation with if:

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

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

Effective Envelopes and Effective Bandwidth Effective Envelopes and Effective Bandwidth

Effective Bandwidth (Kelly 1996)

Given , an effective envelope is given by

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

........

............

..........

backlog=B(s)

delay=W(s)

A(t) D(t)

Statistical Network Calculus with Min-Plus Statistical Network Calculus with Min-Plus AlgebraAlgebra

•Convolution operation:

•Deconvolution operation

f*g(t)

Convolution and Deconvolution operatorsConvolution and Deconvolution operators

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

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

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

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

Type 1 flows:

Goal: probabilisticdelay bound d=10ms

Number of flows that can be admittedNumber of flows that can be admitted

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.

Network Service CurvesNetwork Service Curves

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

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

where are free parameters

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

ThroughFlows

Node HNode 2Node 1

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)

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