Recent Progress on a Statistical Network Calculus

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Recent Progress on a Statistical Network Calculus

• Jorg Liebeherr
• Department of Computer Science
• University of Virginia

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Collaborators
Almut Burchard Robert Boorstyn Chaiwat
Oottamakorn Stephen Patek Chengzhi Li
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Contents

• 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 Endtoend Statistical Service
  Guarantees. (2nd revised version), Technical
  Report, 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, July 2002.

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Service Guarantees
Receiver
Switch
Sender

• A deterministic service gives worst-case
  guarantees
• Delay ? d
• A statistical service provides probabilistic
  guarantees
• Pr Delay ? d ? ? or Pr Loss ? l ? ?

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Multiplexing Gain

• Sources of multiplexing gain
• Traffic Conditioning (Policing, Shaping)
• Scheduling
• Statistical Multiplexing of Traffic

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Scheduling

• Scheduling algorithm determines the order in
  which traffic is transmitted

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Designing Networks for Multiplexing Gain
Scheduling
Traffic Characterization/ Conditioning
Service /Admission Control
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Designing Networks for Multiplexing Gain
Scheduling
Traffic Characterization/ Conditioning
Service /Admission Control
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Designing Networks for Multiplexing Gain
By now The design space for determi-nistic
guarantees is well understood.
Scheduling
Deterministic service
Multiple Buckets
Traffic Characterization/ Conditioning
Service /Admission Control
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Designing Networks for Multiplexing Gain
Still open Is there an elegant framework to
reason about statistical guarantees? ?
Statistical Network Calculus
Scheduling
?
Multiple Buckets
Traffic Characterization/ Conditioning
Service /Admission Control
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Related Work (small subset)
Deterministic network calculusCruz, 1991
Effective bandwidth in network calculusChang 94
Effective Bandwidth J. Hui 88Guerin et.al.
91Kelly 91Gibbens, Hunt 91
(min,) algebra for det. networks Agrawal
et.al. 99Chang 98LeBoudec 98
ServiceCurvesCruz 95

• Motivation for our work on statistical network
  calculus
• Maintain elegance of deterministic calculus
• (2) Exploit know-how of statistical multiplexing

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Source Assumptions

• Arrivals Aj(t,t?) are random processes
• Deterministic Calculus
• (A1) Additivity For any t1 lt t2 lt t3, we have
• (A2) Subadditive Bounds Traffic Aj is
  constrained by a subadditive deterministic
  envelope Aj as follows
• with

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Source Assumptions
Statistical Calculus (A1) (A2) (A3)
Stationarity The Aj are stationary random
variables (A4) Independence The Ai and Aj (i?j)
are stochastically independent (No assumptions
on arrival distribution!)
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Aggregating Arrivals
Flow 1
. . .
C
Flow N
Buffer with Scheduler
Regulated arrivals
Regulator
Arrivals from multiple flows Deterministic
Calculus Worst-case of multiple flows is sum of
the worst-case of each flow
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2000
Aggregating Arrivals

• Statistical Calculus
• To bound aggregate arrivals we define a function
  that is a bound on the sum of multiple flows with
  high probability ? Effective Envelope
• Effective envelopes are non-random functions

• effective envelope

• strong effective envelope

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Obtaining Effective Envelopes
with with
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Effective vs. Deterministic Envelope
Envelopes
Amin (Pt, srt) Type 1 flows P 1.5 Mbps r
.15 Mbps s 95400 bits Type 2 flows P 6
Mbps r .15 Mbps s 10345 bits
Type 1 flows
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Effective vs. Deterministic Envelope
Envelopes
Traffic rate at t 50 msType 1 flows
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Scheduling Algorithms

• Consider a work-conserving scheduler with rate R

• Consider class-q arrival at t with tdq

Class-p arrivals from class p which Are
transmitted before tagged arrival.
Arrivals from class p
Deadline of Tagged arrival
Tagged arrival
Limit (Scheduler Dependent)

• Tagged arrival has no delay bound violation if

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Scheduling Algorithms
Arrivals from class p
with FIFO SP EDF
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2000
Admission Control for Scheduling Algorithms
with Deterministic Envelopes
with Effective Envelopes
with Strong Effective Envelopes
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Effective vs. Deterministic Envelope
Envelope
C 45 Mbps, e 10-6Delay bounds Type 1
d1100 ms, Type 2 d210 ms,
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Effective Envelopes and Effective Bandwidth
2002
Effective Bandwidth (Kelly, Chang)
Given a(s,t), an effective envelope is given by
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Effective Envelopes and Effective Bandwidth
Now, we can calculate statistical service
guarantees for 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, e 10-6
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Statistical Network Calculus with Min-Plus Algebra
D(t)
A(t)
S(t)
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Convolution and Deconvolution operators

• Convolution operation
• Deconvolution operation
• Impulse function

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Service Curves (Cruz 1995)

• A (minimum) service curve for a flow is a
  function S such that
• Examples
• Constant rate service curve
• Service curve with delay guarantees

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Network Calculus Main Results (Cruz, Chang,
LeBoudec)

1. Output Envelope is an envelope for the
   departures
2. Backlog bound is an upper bound for the
   backlog B
3. Delay bound An upper bound for the delay is

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Network Service Curve (Cruz, Chang, LeBoudec)
Traffic Conditioning
S3
S1
Receiver
S2
Sender
Network Service Curve If S1, S2 and S3 are
service curves for a flow at nodes, then Snet
S1 S2 S3 is a service curve for the entire
network.
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2001
Statistical Network Calculus
A (minimum) service curve for a flow is a
function S such that A (minimum) effective
service curve for a flow is a function S? such
that
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2001
Statistical Network Calculus Theorems

1. Output Envelope is an envelope for the
   departures
2. Backlog bound is an upper bound for the
   backlog
3. Delay bound A probabilistic upper bound for the
   delay , i.e.,

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2002
Effective Network Service Curve
Network Service Curve If S1,?, S2 ,? SH ,? are
effective service curves for a flow at nodes,
then .
A good network service curve can be obtained by
working with a modified service curve definition
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What is the cause of the problem with the
network effective service curve?
D1 A2
A1
D2
S2, ?
S1, ?
Sender
Receiver
In the convolution the range 0,t where the
infimum is taken is a random variable that does
not have an a priori bound.
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2002
Statistical Per-Flow Service Bounds
Service available to aggregate SC

• Given
• Service guarantee to aggregate (SC ) is known
• Total Traffic is known
• What is a lower bound on the service seen by a
  single flow?

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2002
Statistical Per-Flow Service Bounds
Service available to aggregate SC
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

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Effective service curve of a single flow
Type 1 flows
Bandwidth needed by a per-flow allocation to meet
a delay bound of d10ms
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2002
Number of flows that can be admitted
Type 1 flows Goal probabilisticdelay bound
d10ms
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Conclusions

• Convergence of deterministic and statistical
  analysis with new constructs
• Effective envelopes
• Effective service curves
• Preserves much (but not all) of the deterministic
  calculus
• Open issues
• So far Often need bound on busy period or other
  bound on relevant time scale.
• Many problems still open for multi-node calculus

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Adaptive service curves

• Modified convolution operation

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Adaptive service curves

• adaptive service curve
• Many service curves are adaptive (? Cruz/Okino,
  LeBoudec)
• Obtain service curve with t00

l-adaptive service curve
l-adaptive effective service curve
strong (l-adaptive) effective service curve
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Effective Network Service Curve
Traffic Conditioning
A
T3,l,e
T1,l,e
Receiver
T2,l,e
Sender
Network Service Curve If T1,l,e, T2,l,e, and
T3,l,e, are strong effective service curves for
a flow at nodes, then Tnet,l, 3e T1,l,e
T2,l,e T3,l,e is a service curve for the
entire network.
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Recover original effective setwork curve
Given a strong effective service curve T l,e . If
the backlog clears in any time interval of length
l with probability e1 , i.e, Then is an
effective service curve