A Statistical Network Calculus for Computer Networks

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A Statistical Network Calculus for Computer
Networks

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

2
Collaborators

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

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Papers

• 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 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.

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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

• Jackson (50s), Kelly, BCMP (70s)
• Flow of jobs in system of queues and servers
• Applications Provided motivation for
  packet-switching (Kleinrocks 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 Bandwidth

• Hui, Mitra, Kelly (90s)
• Describes bandwidth needs of complex traffic by a
  number
• Application admission control in ATM networks

Peak rate
effectivebandwidth
Mean rate

• Can consider
• service guarantees
• wide variety of traffic (incl. LRD)
• ? statistical multiplexing
• Limitations
• ? not well suited for scheduling

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Network Calculus

• Cruz, Chang, LeBoudec (90s)
• Worst case delay and backlog bounds for fluid
  flow traffic
• Application design of new schedulers (WFQ) new
  services (IntServ).

• 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.

• Limitations
• No random losses
• No statistical multiplexing, therefore pessimistic

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State-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 Develop network calculus into new
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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Related Work (small subset)
2005
1985
1990
1995
2000
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Multiplexing Gain

• Multiplexing gain is the raison dêtre for packet
  networks.
• Sources of multiplexing gain
• Traffic characterization and conditioning
• Scheduling
• Statistical Multiplexing

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Traffic Conditioning
Traffic Conditioning

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

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Scheduling

• 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)

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Multiplexing Gain
Without statistical multiplexing
Worstcasearrivals
Flow 1
Flow 2
Flow 3
Time
With statistical multiplexing
Arrivals
Flow 1
Flow 2
Flow 3
Time
Backlog
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Example of Statistical Multiplexing Retirement
Savings

• Life expectancy Normal(m75, s10) 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(m, s / ?N).
• N1 person (Individual Savings) 95 confidence
  10 2s 30 years ? 1.5 Mio.99 confidence
  10 2s 40 years ? 2 Mio.
• N100 people (Pooled Savings) 95 confidence
  10 2s 12 years ? 600,00099 confidence 10
  2s 13 years ? 650,000

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The importance of Statistical Multiplexing

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

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Traffic Characterization

• Arrivals from a flow j are
  a random process
• Stationarity The are stationary random
  processes
• Independence The and are
  stochastically independent

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Regulated Arrivals
Flow 1
. . .
C
Flow N
Each flow isregulated
Buffer with Scheduler
Regulated arrivals
Traffic is constrained by a subadditive
deterministic envelope such that
Leaky Buckets
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Effective envelope
Define a function that bounds traffic with high
probability ? Effective Envelope
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!
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Sample Paths and Envelopes
Note All envelopes are non-random functions
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Probabilistic 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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Aggregating Arrivals
Flow 1
. . .
C
Flow N
Traffic Conditioning
Buffer with Scheduler
Regulated arrivals
Arrivals from multiple flows Deterministic
Network Calculus Worst-case of multiple flows is
sum of the worst-case of each flow
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Effective Envelopes for aggregated flows

• Stochastic Calculus Exploit independence and
  extract statistical multiplexing gain by
  calculating
• For example, using the Chernoff Bound, we can
  obtain

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Effective vs. Deterministic Envelope
Envelopes
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
strong effective envelopes
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

• 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)
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Statistical Multiplexing vs. Scheduling
Example MPEG videos with delay constraints at C
622 Mbps Deterministic service vs.
statistical service (e 10-6)
dterminator100 ms dlamb10 ms
Thick lines EDF SchedulingDashed lines SP
scheduling
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Scheduling vs. Statistical Multiplexing
C 45 Mbps, e 10-6Delay bounds Type 1
d1100 ms, Type 2 d210 ms,
Thick lines EDF SchedulingThin lines SP
scheduling
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More 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 Bandwidth (Kelly 1996)
Given , an effective envelope is given by
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Effective Envelopes and Effective Bandwidth
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, 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

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Deterministic (min,) Network Calculus
Cruz 95 A service curve for a flow is a
function S such that
(min,) results (Cruz, Chang, LeBoudec)

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

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Stochast Network Calculus
An effective service curve for a flow is a
function such that
(min,) results

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

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Statistical Per-Flow Service Bounds
Allocated capacity C

• 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?

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Statistical Per-Flow Service Bounds
Allocated capacity C
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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Number of flows that can be admitted
Type 1 flows Goal probabilisticdelay bound
d10ms
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Network Service Curves
S3
S1
Receiver
S2
Sender
Deterministic Network Service Curve (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.
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Network Service Curve in a Stochastic Calculus
Network Service Curve If S1,?, S2 ,? SH ,?
are effective service curves for a flow, then
for all .
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Effective 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 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

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Example

• Peak rate P 1.5 MbpsAverage rate r 0.15
  Mbps
• T 1/m 1/l 10 msec

• C 100 Mbos
• Cross traffic through traffic
• e 10-9

• Addition of per-node bounds grows O(H3)
• Network service curve bounds grow O(H log H)

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Conclusions

• Presented aspects of stochastic network calculus
• Preserves much (but not all) of the deterministic
  calculus
• Can express many existing results on
• Deterministic calculus
• Effective bandwidth
• Other models (EBB, not shown)
• Many open issues

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Conclusions
Stochastic network calculus
Broad
Yes
Yes
Yes
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