Stochastic Network Calculus

Stochastic Network Calculus - Assessing the Performance of the Future InternetStochastic Network Calculus Assessing the Performance of the Future Internet
Markus Fidler joint work with Amr Rizk
Institute of Communications Technology Leibniz Universitat Hannover
April 22, 2010
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Motivation Traffic characteristics Related results Research goal
FBm sample path envelope Definition of envelopes FBm envelopes Backlog bound
End-to-end analysis Leftover service curves Network service curves Scaling properties
c© Markus Fidler | IKT LUH | 2/27
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• delay
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Queuing networks
Queuing theory has been used since the 60’s to understand the performance of computer networks, most prominently to prove the efficiency of packet switching over circuit switching.
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I product forms for tandem queues
I simple results for multiplexing and de-multiplexing of traffic
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Queuing networks
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Queuing networks
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zHX(t) d= X(zt)
I long-range dependence: Hurst parameter H ∈ (0.5, 1) auto-covariance v(t) at lag t:
∑∞ t=0 v(t) =∞
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zHX(t) d= X(zt)
∑∞ t=0 v(t) =∞
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zHX(t) d= X(zt)
∑∞ t=0 v(t) =∞
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zHX(t) d= X(zt)
∑∞ t=0 v(t) =∞
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Fractional Brownian motion Fractional Brownian motion (fBm) is a model for self-similar traffic with long memory. FBm arrivals A(t) are composed of a mean rate λ and a correlated Gaussian increment process Z(t) with zero mean
A(t) = λt+ Z(t).
I variance σ2
α(θ, t) = 1 θt
2 t2H−1
α lies between the mean rate and peak rate and characterizes the resource requirements of a flow. Note the continuous growth with t.
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Fractional Brownian motion Fractional Brownian motion (fBm) is a model for self-similar traffic with long memory. FBm arrivals A(t) are composed of a mean rate λ and a correlated Gaussian increment process Z(t) with zero mean
A(t) = λt+ Z(t).
I variance σ2
α(θ, t) = 1 θt
2 t2H−1
α lies between the mean rate and peak rate and characterizes the resource requirements of a flow. Note the continuous growth with t.
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Lindley’s recursion Model of a buffered work-conserving link with capacity C
I A(t), D(t) cumulative arrivals, resp., departures in [0, t) I B(t) = A(t)−D(t) backlog, i.e. unfinished work, at time t
A(t) D(t)C
By induction with B(0) = 0
B(t) = max τ∈[0,t]
{A(τ, t)− C(t− τ)}
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Largest term approximation Lindley’s recursion for the backlog B(t) at a system with capacity C
B(t) = sup τ∈[0,t]
{A(τ, t)− C(t− τ)}.
P[B > b] = P
] .
For fBm arrivals known backlog bounds are derived from an approximation by the largest term
P[B > b] ≈ sup τ∈[0,t]
{ P [A(τ, t)− C(t− τ) > b]
} that strictly provides, however, only a lower bound.
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Largest term approximation Lindley’s recursion for the backlog B(t) at a system with capacity C
B(t) = sup τ∈[0,t]
{A(τ, t)− C(t− τ)}.
P[B > b] = P
] .
For fBm arrivals known backlog bounds are derived from an approximation by the largest term
P[B > b] ≈ sup τ∈[0,t]
{ P [A(τ, t)− C(t− τ) > b]
} that strictly provides, however, only a lower bound.
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Performance bounds for fBm traffic The fundamental finding is the asymptotic resp. approximate result
lim b→∞
= εa
that has been derived from I the Gaussian increments [Norros, ’94]
I large deviations theory [Duffield, O’Connell, ’95]
I statistical envelopes [Fonseca, Mayor, Neto, ’00].
It provides clear rules for dimensioning, e.g. regarding packet loss
I is it better to double the buffer size b or
I is it better to double the unused capacity C − λ?
The result is, however, limited to through traffic at a single system.
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Performance bounds for fBm traffic The fundamental finding is the asymptotic resp. approximate result
lim b→∞
= εa
that has been derived from I the Gaussian increments [Norros, ’94]
I large deviations theory [Duffield, O’Connell, ’95]
I statistical envelopes [Fonseca, Mayor, Neto, ’00].
It provides clear rules for dimensioning, e.g. regarding packet loss
I is it better to double the buffer size b or
I is it better to double the unused capacity C − λ?
The result is, however, limited to through traffic at a single system.
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Networks under fBm cross-traffic The goal of this work is to provide end-to-end performance bounds for a through flow in a network under fBm cross-traffic.
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Acr,1 Acr,2 Acr,n
We seek to answer: Given the delay of a through flow at single system is w. What is the end-to-end delay for n tandem systems?
To this end, we use the framework of stochastic network calculus. Essential intermediate steps are
I an envelope for fBm sample paths I a leftover service curve for systems under fBm cross-traffic I a convolution form for service curves of tandem systems
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Networks under fBm cross-traffic The goal of this work is to provide end-to-end performance bounds for a through flow in a network under fBm cross-traffic.
1 2 n
Acr,1 Acr,2 Acr,n
We seek to answer: Given the delay of a through flow at single system is w. What is the end-to-end delay for n tandem systems?
To this end, we use the framework of stochastic network calculus. Essential intermediate steps are
I an envelope for fBm sample paths I a leftover service curve for systems under fBm cross-traffic I a convolution form for service curves of tandem systems
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time
[from Liebeherr, statcalc talk, ’05]
I the point-wise envelope is violated at most with probability 1/4 at any point in time
I the sample path envelope is violated at most by a share of 1/4 of the sample paths
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time
[from Liebeherr, statcalc talk, ’05]
I the point-wise envelope is violated at most with probability 1/4 at any point in time
I the sample path envelope is violated at most by a share of 1/4 of the sample paths
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Construction of sample path envelopes A point-wise definition of stochastic envelope E(t) is
P[A(τ, t)− E(t− τ) > 0] ≤ εp for any τ ∈ [0, t] as opposed to the sample path definition
P[∃τ ∈ [0, t] : A(τ, t)− E(t− τ) > 0] ≤ εs
that is equivalent to the backlog-like form (E(t) = b+ Ct)
P
For a set of supporting points the union bound yields
P
t∑ τ=0
εp.
For t→∞ the difficulty is to select εp such that its sum is bounded.
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P[∃τ ∈ [0, t] : A(τ, t)− E(t− τ) > 0] ≤ εs that is equivalent to the backlog-like form (E(t) = b+ Ct)
P
P
t∑ τ=0
εp.
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P[∃τ ∈ [0, t] : A(τ, t)− E(t− τ) > 0] ≤ εs that is equivalent to the backlog-like form (E(t) = b+ Ct)
P
P
t∑ τ=0
εp.
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Construction of fBm envelopes With Chernoff’s bound an envelope with point-wise violation probability εp(t) follows from the effective bandwidth of fBm traffic. The optimal solution is
E(t) = λt+ √ −2 log εp(t)σtH .
To derive a sample path envelope we have to select εp(t) such that it has a finite sum. We choose
εp(t) = ηt 2β
resulting in E(t) = λt+
√ −2 log ησtH+β.
Selecting β ∈ (0, 1−H) ensures a linear growth of E(t) in t.
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Construction of fBm envelopes With Chernoff’s bound an envelope with point-wise violation probability εp(t) follows from the effective bandwidth of fBm traffic. The optimal solution is
E(t) = λt+ √ −2 log εp(t)σtH .
To derive a sample path envelope we have to select εp(t) such that it has a finite sum. We choose
εp(t) = ηt 2β
resulting in E(t) = λt+
√ −2 log ησtH+β.
Selecting β ∈ (0, 1−H) ensures a linear growth of E(t) in t.
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I the point-wise violation probability decays with t
I the decay is, however, slower than exponential
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2β for all
t ∈ [0,∞) the envelope is violated by sample paths at most with
εs = Γ( 1
.
We optimize the parameters η and β to find a closed form backlog bound from the envelope:
I the optimal η follows by fitting E(t) below b+ Ct when deriving the backlog bound b at a server with capacity C
I a near optimal β is derived using Stirling’s approximation of the Gamma function Γ(x) ≈
√ 2π/x (x/e)x for x 1
After some algebra we express the sample path violation probability εs as a multiple of the known approximation εa. This supports earlier conclusions from εa by a rigorous sample path argument.
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2β for all
t ∈ [0,∞) the envelope is violated by sample paths at most with
εs = Γ( 1
.
We optimize the parameters η and β to find a closed form backlog bound from the envelope:
I the optimal η follows by fitting E(t) below b+ Ct when deriving the backlog bound b at a server with capacity C
I a near optimal β is derived using Stirling’s approximation of the Gamma function Γ(x) ≈
√ 2π/x (x/e)x for x 1
After some algebra we express the sample path violation probability εs as a multiple of the known approximation εa. This supports earlier conclusions from εa by a rigorous sample path argument.
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0 100 200 300 400 10
−30
T=0.6T=0.4T=0.2 ms
I the EBB model includes all traffic types that have exponentially bounded burstiness, e.g. Markov on-off traffic
I the decay of tail probabilities for fBm traffic log ε ∼ −b2−2H
is much slower Weibull-type making buffering less effective
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P
] ≤ ε.
infτ{A(τ) + S(t− τ)} =: A⊗ S(t) is known as min-plus convolution.
Ath
Acr
Dcr
Given cross-traffic with sample path envelope Ecr(t) at a server with capacity C. A leftover service curve for the through traffic is
Sth(t) = max{0, Ct− Ecr(t)}.
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P
] ≤ ε.
infτ{A(τ) + S(t− τ)} =: A⊗ S(t) is known as min-plus convolution.
Ath
Acr
Dcr
Given cross-traffic with sample path envelope Ecr(t) at a server with capacity C. A leftover service curve for the through traffic is
Sth(t) = max{0, Ct− Ecr(t)}.
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...Ath
Acr,1 Acr,2 Acr,n
A network service curve can be derived by recursive insertion of the definition of service curve. In case of deterministic service curves
Dn = An⊗Sn = Dn−1⊗Sn = An−1⊗Sn−1⊗Sn = · · · = A1⊗ni=1Si
A simple recursive insertion of stochastic service curves
P
] ≤ ε(b).
is, however, not possible since the definition I requires sample path arguments for the arrivals A(t) I makes only point-wise statements for the departures D(t)
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...Ath
Acr,1 Acr,2 Acr,n
A network service curve can be derived by recursive insertion of the definition of service curve. In case of deterministic service curves
Dn = An⊗Sn = Dn−1⊗Sn = An−1⊗Sn−1⊗Sn = · · · = A1⊗ni=1Si
A simple recursive insertion of stochastic service curves
P
] ≤ ε(b).
is, however, not possible since the definition I requires sample path arguments for the arrivals A(t) I makes only point-wise statements for the departures D(t)
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The construction of stochastic service curves that provide sample path guarantees for the departures has been open for a while.
The method [Ciucu, Burchard, Liebeherr, ’05]
I resorts to the union bound
I introduces a slack rate and thus
I achieves integrable violation probabilities
We provide a solution for end-to-end leftover service curves under fBm cross-traffic and optimize its parameters as before.
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n
O ( n(log n)
1 2−2H
result O(n log n) [Ciucu, Burchard, Liebeherr, ’05] is recovered.
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−1
m s]
Hcr = 0.5
Hcr = 0.7
Hcr = 0.75
Hcr = 0.8
Under LRD spare capacity is essential for network performance, e.g. for Hcr = 0.75 (0.5) halving the spare capacity increases the delay bound tenfold (twofold).
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We contributed
I a sample path envelope for fBm traffic that complements and agrees with previous approximate results
I a sample path leftover service curve for systems under fBm cross-traffic
to derive performance bounds for networks under fBm cross-traffic.
We showed that end-to-end bounds under fBm cross-traffic grow
as O ( n(log n)
I O(n) for deterministic traffic [LeBoudec, Thiran, ’01]
I Θ(n log n) for EBB traffic [Ciucu, Burchard, Liebeherr, ’05]
I O(n) for statistically independent EBB traffic [Fidler, ’06]
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References
I Amr Rizk, Markus Fidler: End-to-end Performance Bounds for Networks under Long-memory fBm Cross-traffic, http://arxiv.org/abs/0909.0633, September 2009.
I Amr Rizk, Markus Fidler: Sample Path Bounds for Long-memory fBm Traffic, Proc. of IEEE INFOCOM MC, March 2010.
I Amr Rizk, Markus Fidler: End-to-end Performance Bounds for Networks under Long-memory fBm Cross-traffic, Proc. of IEEE IWQoS, June 2010.

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Stochastic Network Calculus