outline - unipi.it nc.pdf · wf2q, stfq, scfq, …) • strict priority (for the queue at highest...
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Network Network CalculusCalculus: : A A worstworst--casecase theorytheory
forfor QoS QoS guaranteesguarantees in in packetpacket networksnetworks
Giovanni SteaComputer Networking Group
Department of Information EngineeringUniversity of Pisa, Italy
IMT Lucca, Oct 28, 2009
IMT Lucca, October 28, 2009 2
Dipartimento di Ingegneria dellaInformazione
OutlineOutline
• Motivation• Performance analysis of real-time traffic
• Why classical queueing theory is unfit
• Network Calculus• Basic modeling: arrival and service curves• Concatenation, bounds
• “Pay burst only once” principle / IntServ• Advanced modeling: aggregate scheduling
• Stochastic Network Calculus
IMT Lucca, October 28, 2009 3
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 4
Dipartimento di Ingegneria dellaInformazione
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?
IMT Lucca, October 28, 2009 5
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 6
Dipartimento di Ingegneria dellaInformazione
Network Calculus and Network Calculus and QueueingQueueing TheoryTheory
• 100 years of Queueing Theory• 1909. A. K. Erlang “The Theory of Probabilities and
Telephone Conversations”.• Originated in the area of telecommunications• Developed and applied in a variety of areas• Erlang Centennial held in Denmark, April 2009.
• ~20 years of Network Calculus• 1991. R. L. Cruz, “A Calculus for Network Delay”.• Recent development of queueing theory for
computer networks
IMT Lucca, October 28, 2009 7
Dipartimento di Ingegneria dellaInformazione
NC and NC and QueueingQueueing Theory (2)Theory (2)
• Queueing theory requires models for traffic• Simplistic models required for tractability• What if the traffic mix changes?
• New applications (social networks, etc.)• Flash crowds (e.g., a football match)• Topology modifications (routing, link upgrades)
• Queueing theory mainly concerned with average performance metrics• Real-time traffic needs bounds
IMT Lucca, October 28, 2009 8
Dipartimento di Ingegneria dellaInformazione
Modeling a network with NC (1)Modeling a network with NC (1)
Fixed delays: propagation, …
Variable delay: queueing
Per-flow scheduling Aggregate scheduling
IMT Lucca, October 28, 2009 9
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 10
Dipartimento di Ingegneria dellaInformazione
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)
IMT Lucca, October 28, 2009 11
Dipartimento di Ingegneria dellaInformazione
Example: a Round Robin schedulerExample: a Round Robin scheduler
• Fixed length packets, 2 packets per queueservice
Slope C
latency
Slope R<C
IMT Lucca, October 28, 2009 12
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 13
Dipartimento di Ingegneria dellaInformazione
Modeling a Modeling a queueingqueueing point with NC (3)point with NC (3)
• Worst-case behavior for my queue:• served at minimum rate
• with maximum latency
IMT Lucca, October 28, 2009 14
Dipartimento di Ingegneria dellaInformazione
Modeling a Modeling a queueingqueueing point with NC (4)point with NC (4)
• 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⋅ −∫
IMT Lucca, October 28, 2009 15
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 16
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 17
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 18
Dipartimento di Ingegneria dellaInformazione
An endAn end--toto--end delay bound (2)end delay bound (2)
• 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
IMT Lucca, October 28, 2009 19
Dipartimento di Ingegneria dellaInformazione
An endAn end--toto--end delay bound (3)end delay bound (3)
• 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
IMT Lucca, October 28, 2009 20
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 21
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 22
Dipartimento di Ingegneria dellaInformazione
““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)
IMT Lucca, October 28, 2009 23
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 24
Dipartimento di Ingegneria dellaInformazione
IntServIntServ architecture (2)architecture (2)
• 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
IMT Lucca, October 28, 2009 25
Dipartimento di Ingegneria dellaInformazione
IntServIntServ architecture (3)architecture (3)
• RESV message travels back and reservesa rate R at each node• The delay bound is guaranteed from now on
RESVR
IMT Lucca, October 28, 2009 26
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 27
Dipartimento di Ingegneria dellaInformazione
PerPer--aggregate schedulingaggregate scheduling
• Packets of an aggregate normally queued FIFO
• Arbitration (scheduling) among aggregates• Forwarding guarantees for the aggregate
IMT Lucca, October 28, 2009 28
Dipartimento di Ingegneria dellaInformazione
NC with aggregate schedulingNC with aggregate scheduling
• Computations get more involved…
Multi-queue scheduler
FIFO queues
IMT Lucca, October 28, 2009 29
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 30
Dipartimento di Ingegneria dellaInformazione
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?
IMT Lucca, October 28, 2009 31
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 32
Dipartimento di Ingegneria dellaInformazione
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?
IMT Lucca, October 28, 2009 33
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 34
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 35
Dipartimento di Ingegneria dellaInformazione
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
≥=
IMT Lucca, October 28, 2009 36
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 37
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 38
Dipartimento di Ingegneria dellaInformazione
ScalabilityScalability
• Computation times for very nasty non-nested tandems
Ref: L. Bisti, L. Lenzini, E. Mingozzi, G. Stea, “Computation and TightnessAssessment of End-to-end Delay Bounds in FIFO Tandems UsingNetwork Calculus", submitted to journal, 2008
0,001
0,01
0,1
1
10
100
1000
104
10 15 20 25 30
20%50%100%
Tlu
db (
s)
N
30 nodes, 465 flows, 20 mins
IMT Lucca, October 28, 2009 39
Dipartimento di Ingegneria dellaInformazione
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☺
�
IMT Lucca, October 28, 2009 40
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 41
Dipartimento di Ingegneria dellaInformazione
Accuracy (2)Accuracy (2)
• Compared to (naïve) per-node analysis
1
10
100
1000
10 15 20 25 30
20%50%100%pe
r-n
ode
dela
y / L
UD
B r
atio
N
500-1000 times smaller
IMT Lucca, October 28, 2009 42
Dipartimento di Ingegneria dellaInformazione
Stochastic Network CalculusStochastic Network Calculus
• Brings in a probabilistic framework• Better captures statistical multiplexing
• Concatenation results still hold
• Currently active field of research• SIGMETRICS, VALUETOOLS
( ) ( )D t A tβ≥ ⊗ ( ) ( ){ } ( )P A t D t x g xβ⊗ − > ≤
deterministic stochastic
IMT Lucca, October 28, 2009 43
Dipartimento di Ingegneria dellaInformazione
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
• Only method known so far• Stochastic extensions: promising research
area• Better account for statistical multiplexing
IMT Lucca, October 28, 2009 44
Dipartimento di Ingegneria dellaInformazione
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.
IMT Lucca, October 28, 2009 45
Dipartimento di Ingegneria dellaInformazione
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
IMT Lucca, October 28, 2009 46
Dipartimento di Ingegneria dellaInformazione
Thank you for listeningThank you for listening
• Questions? • Comments?