Worst-caseAnalysisofNetworks UsingNetwork Calculusafsec.asr.cnrs.fr/wp-content/uploads/2011/01/ENS Lyon NC.pdf · ENS Lyon, December 7, 2010 14 Dipartimento di Ingegneria della Informazione

Worst-case Analysis of NetworksUsing Network Calculus

Giovanni SteaComputer Networking Group

Department of Information EngineeringUniversity of Pisa, Italy

ENS Lyon, December 7, 2010

ENS Lyon, December 7, 2010 2

Dipartimento di Ingegneria dellaInformazione

OutlineOutline

• Motivation• Performance analysis of real-time traffic

• Network Calculus• Basic modeling: arrival and service curves

• Concatenation, bounds• “Pay burst only once” principle / IntServ

• Advanced modeling: aggregate scheduling



RealReal--time traffictime traffic

• Expected to represent the bulk of the traffic in the Internet soon• Skype users: 107-108

• Cisco white paper: video traffic volume to surpass P2P in 2010

• Revenue-generating only if reliable• Reliability boils down to “packets meeting

deadlines”• End-to-end delay bounds are required



Performance analysisPerformance analysis

• Tagged flow (of packets) traversing a path

• Cross traffic

• Many queueing points (routers)

• How to compute a bound on the e2e delay?



Performance analysis (2)Performance analysis (2)

• Service Level Agreement with upstream neighbor

• I will carry • up to X Mbps of your traffic • from A to B

• within up to Y ms (!!)• for Z$

A

B



Modeling a network with NC (1)Modeling a network with NC (1)

Fixed delays: propagation, …

Variable delay: queueing

Per-flow scheduling Aggregate scheduling



Modeling a Modeling a queueingqueueing point with NCpoint with NC

• A scheduler serves its queues on a time sharing basis:

• Most guarantee that a queue is visited:• For a minimum amount of time• After a computable maximum vacation

Round robin

Strict priority



Modeling a Modeling a queueingqueueing point with NC (2)point with NC (2)

• Minimum service over a maximum interval• A minimum guaranteed rate• With a latency (when the server is away)• The latency is upper bounded

• Round robin schedulers (DRR, PDRR, …)• Fair Queueing schedulers (PGPS, WFQ,

WF2Q, STFQ, SCFQ, …)• Strict priority (for the queue at highest prio)



Example: a Round Robin schedulerExample: a Round Robin scheduler

• Fixed length packets, 2 packets per queueservice

Slope C

latency

Slope R<C



TF

P2

TF

P3

TF

P4

TF

P5

TF

P6

TF

P7

t

F2

P1

TF

P1

t

service

Example: a priority schedulerExample: a priority scheduler

• Strict non preemptive priority, queuescheduled at top priority

Slope C

latency

Service curve: summarizes the service received in a worst case by a backlogged tagged flow

Models the presence of other queues

Rate-latency service curves most common in practice




• Worst-case behavior for my queue:• served at minimum rate

• with maximum latency




• Nodes transform functions of time( )A t ( )D t

TF

P1

TF

P2

TF

P3

TF

P4

Server with

latency

capacity

R<C

time

bits ( )A t ( )D t

( )tβ

0t

( )0A t

( )0D t

( ) ( )D t A tβ≥ ⊗

( )( ) ( )

0inf

s t

A t

A s t s

β

β≤ ≤

⊗ =

+ −

( ) ( )0

tA s t s dsβ⋅ −∫



Concatenation propertyConcatenation property

• Assume a flow traverses a tandem of nodes

1 2

( )( )

1,2

1 2

t

t

ββ β

=

⊗

( )A t ( )D t ( ) ( )1,2D t A tβ≥ ⊗

It works for any # of nodes



Concatenation property (2)Concatenation property (2)

• At each of the N nodes:• Cross traffic could be anything• Schedulers need not be the same

• mid ’90s: analyzing a tandem of identical schedulers • Took a whole PhD

• Was enough for a top-level journal paper (Parekh ’93, Saha ’98,…, all on IEEE/ACM TNet)

1 2



An endAn end--toto--end delay bound (1)end delay bound (1)

• A bound on the delay for a given arrival process can easily be computed

time

bits ( )A t

Dmax

( )2e eA tβ⊗

( )2e e tβ

( )A t ( )D t




• Traffic shaping/policing• Check conformance with a pre-specified profile

• Drop/delay out-of-profile traffic• Employed to verify SLA conformance

packets shaper network

A

B




• Arrival curve: a constraint on the arrival process

time

bits ( )A t

( )tα

( ) ( ) ( )A t A s t sα− ≤ −

( ) ( )A t A tα≤ ⊗

example:

a token bucketburst B, rate r

B

r



Backlog and delay bounds in NCBacklog and delay bounds in NC

• service curve for a path + arrival curve for a flow

time

bits

( )tα

D

D is a bound on the e2e delay

B B is a bound on the backlog

Bounds are tight



Convolution of rateConvolution of rate--latency curveslatency curves

( ) ( )( ) ( )

1,2 1 2

1 20inf

s t

t t

s t s

β β β

β β≤ ≤

= ⊗

= + −( ) ( )i i it R t Lβ += ⋅ −

bits

bits

L1

L2

L1+L2

min(R1,R2)R1

R2

• Sum of the latencies, min of the rates



““Pay bursts only oncePay bursts only once”” principleprinciple

• When traversing a tandem, your DB consists of• N latencies• One burst delay

bits

( )tα

1min i

i N

B

R≤ ≤

1 1

1min

N N

i ii ii i

i N

B BD L L

R R= =

≤ ≤

= + < +

∑ ∑

B

Pay burst only once(at the min rate)

1

N

iiL

=∑ D(1…N) < D(1)+…+D(N)



IntServIntServ architecture (1)architecture (1)

• RSVP protocol

• FlowSpec: burst and rate of the flow• i.e., its token bucket arrival curve

• Path message• Includes FlowSpec and required delay bound• Collects node latencies at each hop

PATHL1

PATHL1+L2

PATHL1+…+LN




• At the destination• compute what rate you need to reserve

• Based on your required delay D

bits

( )tα

B R

B1

N

iiL

=∑

D

1

max ,N

ii

BR r

D L=

= − ∑

r




• RESV message travels back and reservesa rate R at each node• The delay bound is guaranteed from now on

RESVR



TrafficTraffic aggregationaggregation

• Aggregation as "the" solution for scalableprovisioning of QoS in core networks

• Internet:• Differentiated Services• MPLS

Per-flow resource management

(e.g., packet scheduling) just doesn’t scale



PerPer--aggregate schedulingaggregate scheduling

• Packets of an aggregate normally queued FIFO

• Arbitration (scheduling) among aggregates• Forwarding guarantees for the aggregate



NC with aggregate schedulingNC with aggregate scheduling

• Computations get more involved…

Multi-queue scheduler

FIFO queues



Traffic aggregation (cont.)Traffic aggregation (cont.)

• Aggregates change along a path• Per-node guarantees for different sets of flows

at each node• Cannot use concatenation of SCs

?

bits ( )A t



Performance evaluation problemPerformance evaluation problem

• Users care about their flows, not aggregates• Users want e2e delay bounds, not per-node

forwarding guarantees

How to compute

per-flow end-to-end delay bounds

from per-aggregate per-node guarantees?



Performance AnalysisPerformance Analysis

• Tandem network of FIFO rate-latency elements• All nodes have a rate-latency SC for the aggregate• All flows have a leaky-bucket AC

( )1,3

( )1,1 ( )2,2

( )2,3

( )3,3

1 2 3Tagged flow

Cross flows



Leftover service curvesLeftover service curves

• Th. FIFO Mux - Minimum Service Curves• [Cruz et al., ‘98]

( ) ( ) ( ) 2 1 , 0' , tttt τα ττβ βτ +

> ⋅ −= − ≥

( )tβ

( )0' ,tβ τ

( )1' ,tβ τ?



The LUDB methodologyThe LUDB methodology

LUDB: Least Upper Delay Bound

Step 1:Apply the FIFO Mux theorem iteratively so as to

“remove” all cross-flows

Ref: L. Lenzini, E. Mingozzi, G. Stea, "A Methodology for Computing End-to-end Delay Bounds in FIFO-multiplexing Tandems" Elsevier Performance Evaluation, 65 (2008), pp. 922-943



The LUDB methodology (2)The LUDB methodology (2)

( )1,3

( )1,1 ( )2,2

( )2,3

( )3,3

1 2 3

( )1,3

( )2,3

1' 2 ' 3'

( )1,3

( )2,3

1' 2' 3'⊗

( )1,3 1' ( )2' 3' '⊗

( )1,3

( )1,1 ( )2,2

( )2,3

( )3,3

1 2 3

( )1,3 ( )1' 2 ' 3' '⊗ ⊗

( )tα



The LUDB methodology (3)The LUDB methodology (3)

• An n-dimensional infinity of e2e SCs for the tagged flow• n = # of cross-flows

• Delay bound = fn. of n parameters

• Step 2• Solve an optimization problem

• The minimum is the best, i.e. tightest, delay bound

( ) 10min ,...,

inLUDB D

ττ τ

≥=



Nested vs. nonNested vs. non--nested tandemsnested tandems

• Nested iff• path(f1)∩∩∩∩path(f2) =∅∅∅∅ (disjoint)• path(f1)⊆⊆⊆⊆path(f2) (nested)

• You can only compute an e2e SC for the tagged flow in a nested tandem

( )1,3

( )1,2

( )2,3

( )3,3

1 2 3

( )1,3

( )2,3

( )3,3

1 2 3Nested

Non-nested



NonNon--nested tandems: MSA and STAnested tandems: MSA and STA

( )1,3α

( )1,2α

( )2,3α

1 2 3

( )1,3α

( )1,2α

( )2,3α

1 2 3

( )( )32,3α

( )( )31,3α

( )1,3α

( )1,2α

( )2,3α

1 2 3

( )( )32,3α

Ref: L. Bisti, L. Lenzini, E. Mingozzi, G. Stea, "Numerical analysis of worst-case end-to-end delay bounds in FIFO tandem networks “, submitted (Oct 2010)



Two important pointsTwo important points

• The LUDB method:1. Is scalable enough

• You can use it in paths of 30+ nodes

2. Yields accurate bounds• Close to a flow’s Worst-Case Delay• (Sometimes)

Ref: L. Bisti, L. Lenzini, E. Mingozzi, G. Stea, "Estimating the Worst-case Delay in FIFO Tandems Using Network Calculus", Proc. VALUETOOLS'08, Athens, Greece, October 21-23, 2008



Nested tandems: Piecewise linear problemNested tandems: Piecewise linear problem

• Piecewise linear objective, linear constraints• Explode it recursively into Ω simplexes

• Problem: Ω grows exponentially• 30-node tandem: Ω≈109

• Most simplexes are infeasible: Ω→106

• Use a heuristic sampling: Ω→102, very reliable

( ) 10min ,...,

inLUDB D

ττ τ

≥=

( )( )

1

11... 0,..., 0

min min ,...,i

j n

j nj

f

LUDB Dτ

τ τ

τ τ= Ω ≥

≥

=



Nested tandems: nesting treeNested tandems: nesting tree

• Build the nesting tree

• Every leaf is a SC• Up: FIFO mux• Siblings: ⊗⊗⊗⊗

1 65432

( )1, 6

( )2, 3 ( )4, 6

( )3, 3 ( )4, 4 ( )5, 6

( )6, 6

( )1,6

( )2,3

( )3,3

( )5,6

( )6,6( )4,6

( )4,4

1

2

3 4 5

6

Early identification of infeasible simplexes

9 6: 10 10Ω →



NestedNested tandem: tandem: heuristicheuristic LUDBLUDB

• Solve only very few “promising” simplexes• Very fast (indeed)

• High chance to attain the optimum (80%+)• Maximum error negligible (1%-2%)

0

0.02

0.04

0.06

0.08

0.1

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10

k=2, l=3k=2, l=4k=2, l=5k=3, l=3

ma

xim

um r

elat

ive

erro

r

% o

f exa

ct m

atch

es

n



NonNon--nested tandems: where to cut?nested tandems: where to cut?

• Rule of thumb: the less, the better• Sever all the dependencies• Primary sets of cuts PSCs

1 2 3 4

1

2

3

4

F

× × ××

×

( )( )

( )( )

( )( )

1,2 1,3 2,3

2,3 3,4 3,4

1

2

3

4

DM

×× × ×

× ×

321 4

( )1,2

( )2,3

( )3,4( )1,3

( )1,4

Repeat the algorithm (MSA/STA) for each

PSC (costly!)



NonNon--nested tandemsnested tandems

• Complexity is exp. with tandem length• Feasible for up to 20-30 nodes

0.01

0.1

1

10

100

1000

104

10 15 20 25 30

MSA, 20%MSA, 50%MSA, 100%

STA, 20%STA, 50%STA, 100%

h-STA, 20%h-STA, 50%h-STA, 100%

com

puta

tion

time

(s)

N

MSA

h-STASTA



What you get for complexityWhat you get for complexity

• LUDB bounds exponentially better than (naïve) per-node bounds

1

10

100

1000

10 15 20 25 30

20%50%100%

per

-nod

e de

lay

/ LU

DB

ra

tio

N

500-1000 times smaller



AccuracyAccuracy

• Delay bounds are as useful as they are tight, i.e. close to the Worst-Case Delay• WCD unknown (to date)

• End-to-end analysis is fundamental

( )1,3

( )1,2

( )2,3

( )3,3

1 2 3

( )1,3

( )2,3

( )3,3

1 2 3



SinkSink--tree networkstree networks

1

N iii

i

DB

LCR=

= +∑Interfering flow

at node 10

Interfering flow

at node 6

Ref: L. Lenzini, L. Martorini, E. Mingozzi, G. Stea, "Tight End-to-end Per-flow Delay Bounds in FIFO Multiplexing Sink-tree Networks",

Perf. Evaluation, 63/9-10, Oct. ’06, pp. 956-987

• Closed-form delay bound

• = Worst-Case Delay• Proof by construction



Accuracy (2)Accuracy (2)

• e2e analysis is however not sufficient• Sink trees: nested tandems, all paths are (j,N)

WCD = LUDB• Tune flows so that a bit experiences a delay = LUDB

• However, in other nested tandems

WCD < LUDB

N1N −1 2( )1,N

( ),N N( )1,N N−( )2,N



A conclusive negative answerA conclusive negative answer

• “extend” flow (j, N-1) to (j, N)• New WCD larger or equal (Theorem)

• Compute the LUDB for both tandems• New LUDB strictly smaller than the old one• the LUDB may be larger than the WCD

( )1,2

( )1,1

1 2

T

( ) ( ) 1,2 , 1,11 2

T

W W LUDBLUDB≤≤≤≤ ≤≤≤≤ <



Flow extensionFlow extension

• Other applications• Turn a non-nested tandem into a nested one

• Nodes must be overprovisioned, otherwise FE is useless

( )1,3

( )*1,3

( )2,3

1 2 3

( )1,3

( )1,2

( )2,3

1 2 3



Assessing the accuracyAssessing the accuracy

• Compute a (heuristic) lower bound• Tune flows carefully so as to “maximize” delay

• Compare the lower bound and the LUDB

• The smaller the interval, the tighter the bound

• Relative Overrating Bound (ROB):

1 - LowerBound/LUDB

[ ],WCD LowerBound LUDB∈



1

Lower bounds on the WCTTLower bounds on the WCTT

• Benchmark the accuracy of the LUDB• Works with nested and non-nested tandems

140

150

160

170

180

190

200

210

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

LUDB lower bound

dela

y (m

s)

Rf



Case Studies Case Studies –– nested tandemsnested tandems

N1N −1 2( )1,N

( )1,1( )1,2( )1, 1N −

0

0.05

0.1

0.15

0.2

0.25

2 4 6 8 10 12 14 16

20%50%100%

RO

B

N

Node utilization



Case studies: nonCase studies: non--nested tandemsnested tandems

N1N −1 5432

( )1,N

( )1,1 ( )2,3

( )1,2 ( )3,4

( )4,5 ( ),N N

( )1,N N−

1c 2c 2 1Nc − 2Nc

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

2 4 6 8 10 12 14 16

20%50%100%

RO

B

N

Node utilization



The DEBORAH toolThe DEBORAH tool

• Compute LUDBs and lower bounds• In all kind of tandems (nested, non nested)• Using both MSA and STA

• Highly optimized C++ code• Makes a lot of difference

• Try it out, it’s free!• Available with source code at http://info.iet.unipi.it/~cng/• Straightforward textual syntax • Works under Linux, Windows, MacOS

• Drop us an email and let us know what you think of it



ConclusionsConclusions

• Network Calculus allows one to compute e2e delay bounds • Easy and tight in a per-flow scheduling

environment• Complex and not always tight in an

aggregate-scheduling environment



ReferencesReferences

1. J.-Y. Le Boudec, P. Thiran, Network Calculus, Springer LNCS vol. 2050, 2001 (available online).

2. C. S. Chang, Performance Guarantees in Communication Networks, Springer, New York, 2000.

3. Y. Jiang, Y. Liu, “Stochastic Network Calculus”, Springer 2008

4. R.L. Cruz. “A calculus for network delay, part i: Network elements in isolation”. IEEE Transactions on Information Theory, Vol. 37, No. 1, March 1991, pp. 114-131.

5. R.L. Cruz. “A calculus for network delay, part ii: Network analysis”. IEEE Transactions on Information Theory, Vol. 37, No. 1, March 1991, pp. 132–141.



More referencesMore references

1. L. Lenzini, E. Mingozzi, G. Stea, "A Methodology for Computing End-to-end Delay Bounds in FIFO-multiplexing Tandems" Elsevier Performance Evaluation, 65 (2008) 922-943

2. L. Lenzini, L. Martorini, E. Mingozzi, G. Stea, "Tight End-to-end Per-flowDelay Bounds in FIFO Multiplexing Sink-tree Networks", ElsevierPerformance Evaluation, Vol. 63, Issues 9-10, October 2006, pp. 956-987

3. L. Lenzini, E. Mingozzi, G. Stea, "Delay Bounds for FIFO Aggregates: a Case Study", Elsevier Computer Communications 28/3, February 2005, pp. 287-299

4. L. Bisti, L. Lenzini, E. Mingozzi, G. Stea, "Estimating the Worst-case Delayin FIFO Tandems Using Network Calculus", Proceedings ofVALUETOOLS'08, Athens, Greece, October 21-23, 2008

5. L. Lenzini, E. Mingozzi, G. Stea, "End-to-end Delay Bounds in FIFO-multiplexing Tandems", Proc. of VALUETOOLS'07, Nantes (FR), October 23-25, 2007.

6. L. Lenzini, E. Mingozzi, G. Stea, "Delay Bounds for FIFO Aggregates: a Case Study", Proceedings of the 4th COST263 International Workshop on Quality of Future Internet Services (QoFIS '03), Stockholm, Sweden, October 1-3, 2003 LNCS 2811/2003, pp. 31-40



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Worst-caseAnalysisofNetworks UsingNetwork Calculusafsec.asr.cnrs.fr/wp-content/uploads/2011/01/ENS Lyon NC.pdf · ENS Lyon, December 7, 2010 14 Dipartimento di Ingegneria della Informazione