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A Statistical Network Calculus for Computer Networks. Jorg Liebeherr Department of Computer Science University of Virginia. Collaborators. Almut Burchard Robert Boorstyn Chaiwat Oottamakorn Stephen Patek Chengzhi Li Florin Ciucu. Papers.

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a statistical network calculus for computer networks

A Statistical Network Calculus for Computer Networks

Jorg Liebeherr

Department of Computer Science

University of Virginia

  • 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.
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 occupanceyn = (n1, n2, … , nk) :

P(n ) = P(n1) P(n2) … P(nk)


  • Limited to Poisson traffic
  • Limited scheduling algorithms
effective bandwidth
Effective Bandwidth

Hui, Mitra, Kelly (90s)

  • Describes bandwidth needs of complex traffic by a number
  • Application: admission control in ATM networks

Peak rate


Mean rate

Can consider:

  • service guarantees
  • wide variety of traffic (incl. LRD)

 statistical multiplexing


 not well suited for scheduling

network calculus







Network 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).
  • Main result: If S1, S2 and S3 describes the service at each node, then Snet = S1 * S2 * S3describes the service given by the network as a whole.


  • No random losses
  • No statistical multiplexing, therefore pessimistic
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
motivation develop network calculus into new toy model
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
related work small subset

Deterministic network calculusCruz `91

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

  • Our goals:
  • Maintain elegance of deterministic calculus
  • Exploit statistical multiplexing
  • Try to express other models

Cruz calculus with probabilistic trafficKurose `92

Exponentially/stochasti-cally. bounded burstinessYaron/Sidi `93Starobinski/Sidi `99

RateVarianceEnvelopeKnightly `97

Stochastically bounded service curveQiu et.al.`99

Related Work (small subset)






multiplexing gain
Multiplexing Gain

Multiplexing gain is the raison d’être for packet networks.

Sources of multiplexing gain:

  • Traffic characterization and conditioning
  • Scheduling
  • Statistical Multiplexing
traffic conditioning
Traffic Conditioning

Traffic Conditioning

  • Traffic conditioning is typically done at the network edge
  • Reshaping traffic increases delays and/or losses
  • 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)
multiplexing gain1




Multiplexing Gain

Without statistical multiplexing


Flow 1

Flow 2

Flow 3


With statistical multiplexing


Flow 1

Flow 2

Flow 3



example of statistical multiplexing retirement savings
Example of Statistical Multiplexing: Retirement Savings

Life expectancy: Normal(m=75, s=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(m, s / N).

N=1 person (Individual Savings): 95% confidence: 10 + 2s = 30 years  $1.5 Mio.99% confidence: 10 + 2s = 40 years  $2 Mio.

N=100 people (Pooled Savings): 95% confidence: 10 + 2s = 12 years  $600,00099% confidence: 10 + 2s = 13 years  $650,000

the importance of statistical multiplexing
The importance of Statistical Multiplexing
  • At high data rates, statistical multiplexing gain dominates the effects of scheduling and traffic characterization
traffic characterization
Traffic Characterization
  • Arrivalsfrom a flow j are a random process
  • Stationarity: The are stationary random processes
  • Independence: The and are stochastically independent
regulated arrivals
Regulated Arrivals

Flow 1





Flow N

Each flow isregulated


with Scheduler



Traffic is constrained by a subadditive deterministic envelope such that

Leaky Buckets:

effective envelope
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!

sample paths and envelopes

Stronger effective envelope

At most one sample path is violated

Deterministic envelope

Never violated


Effective envelope

At any time, at most one sample path is violated

Sample Paths and Envelopes

Note: All envelopes are non-random functions

probabilistic sample path bound
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:


aggregating arrivals
Aggregating Arrivals

Flow 1





Flow N

Traffic Conditioning


with Scheduler



Arrivals from multiple flows:

Deterministic Network Calculus: Worst-case of multiple flows is sum of the worst-case of each flow

effective envelopes for aggregated flows
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
effective vs deterministic envelope envelopes
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


Type 1 flows

effective vs deterministic envelope envelopes1
Effective vs. Deterministic Envelope Envelopes

Traffic rate at t = 50 msType 1 flows

scheduling algorithms

Deterministic Service

Never a delay bound violation if:

Statistical Service

Delay bound violation with if:

Scheduling Algorithms
  • Work-conserving scheduler with unit rate that serves Q classes
  • Class-q traffic has delay bound dq
  • Scheduling algorithm:





  • Static Priority (SP):
  • Earliest Deadline First (EDF):

Statistical multiplexing makes a big difference

Scheduling has small impact

Statistical Multiplexing vs. Scheduling

Example: MPEG videos with delay constraints at C= 622 Mbps

Deterministic service vs. statistical service (e = 10-6)

dterminator=100 ms dlamb=10 ms

Thick lines: EDF SchedulingDashed lines: SP scheduling

more interesting traffic types
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
effective envelopes and effective bandwidth
Effective Envelopes and Effective Bandwidth

Effective Bandwidth (Kelly 1996)

Given , an effective envelope is given by

effective envelopes and effective bandwidth1
Effective Envelopes and Effective Bandwidth

Comparisons of statistical service guarantees for different schedulers and traffic types


SP- Static PriorityEDF – Earliest Deadline FirstGPS – Generalized Processor Sharing


Regulated – leaky bucketOn-Off – On-off sourceFBM – Fractional Brownian Motion

C= 100 Mbps, e = 10-6

convolution and deconvolution operators
Convolution and Deconvolution operators
  • Convolution operation:
  • Deconvolution operation
deterministic min network calculus
Deterministic (min,+)Network Calculus

Cruz `95:A service curve for a flow is a function S such that:

(min,+) results(Cruz, Chang, LeBoudec)

  • Output Envelope: is an envelope for the departures
  • Backlog bound: is an upper bound for the backlog
  • Delay bound: An upper bound for the delay is
stochast network calculus
Stochast Network Calculus

An effective service curve for a flow is a function such that:

(min,+) results

  • Output Envelope: is an envelope for the departures with probability e
  • Backlog bound: is an upper bound for the backlog with probability e
  • Delay bound: An upper bound for the delay with probability eis
statistical per flow service bounds



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?
statistical per flow service bounds1



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

number of flows that can be admitted
Number of flows that can be admitted

Type 1 flows:

Goal: probabilisticdelay bound


network service curves


Network Service Curves






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.

network service curve in a stochastic calculus

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 Calculus

Network Service Curve:

If S1,, S2 , … SH , are effective service curves for a flow, then for all


effective network service curve
Effective 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 curve
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
  • C = 100 Mbos
  • Cross traffic = through traffic
  • e = 10-9
  • Peak rate: P = 1.5 MbpsAverage rate: r = 0.15 Mbps
  • T= 1/m + 1/l = 10 msec
  • Addition of per-node bounds grows O(H3)
  • Network service curve bounds grow O(H log H)
  • 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