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
Department of Computer Science
University of Virginia
… that capture key characteristics of studied system
… that permit back-of-the-envelope calculations
… that are usable by non-theorists
Main result: Steady state probability of queue occupanceyn = (n1, n2, … , nk) :
P(n ) = P(n1) P(n2) … P(nk)
Hui, Mitra, Kelly (90s)
not well suited for scheduling
Today, fundamental progress in networking is hampered by the lack of methods to evaluate how radically new designs will perform.
Deterministic network calculusCruz `91
Effective bandwidth in network calculusChang `94
J. Hui ’88Guerin et.al. ’91Kelly `91Gibbens, Hunt `91
(min,+) algebra for det. networks:
Agrawal et.al. `99Chang `98LeBoudec `98
Cruz calculus with probabilistic trafficKurose `92
Exponentially/stochasti-cally. bounded burstinessYaron/Sidi `93Starobinski/Sidi `99
Stochastically bounded service curveQiu et.al.`99
Multiplexing gain is the raison d’être for packet networks.
Sources of multiplexing gain:
Without statistical multiplexing
With statistical multiplexing
Life expectancy: Normal(m=75, s=10) years
Retiring Age:65 years
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
Each flow isregulated
Traffic is constrained by a subadditive deterministic envelope such that
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!
Stronger effective envelope
At most one sample path is violated
At any time, at most one sample path is violated
Note: All envelopes are non-random functions
A strong effective envelope for an interval of length is a function which satisfies
Relationship between the envelopes is established as follows:
Arrivals from multiple flows:
Deterministic Network Calculus: Worst-case of multiple flows is sum of the worst-case of each flow
Stochastic Calculus: Exploit independence and extract statistical multiplexing gain by calculating
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
Traffic rate at t = 50 msType 1 flows
Never a delay bound violation if:
Delay bound violation with if:
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
Effective Bandwidth (Kelly 1996)
Given , an effective envelope is given by
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
Cruz `95:A service curve for a flow is a function S such that:
(min,+) results(Cruz, Chang, LeBoudec)
An effective service curve for a flow is a function such that:
Allocated capacity C
Allocated capacity C
is an effective service curve for a flow where is a strong effective envelope and is a probabilistic bound on the busy period
Type 1 flows:
Goal: probabilisticdelay bound
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.
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:
If S1,, S2 , … SH , are effective service curves for a flow, then for all
Theorem: A network service curve is given by
where are free parameters