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Bandwidth Allocation in LargeStochastic Networks
Mathieu Feuillet
Soutenance de thèse
12/07/2012
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Introduction
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Modeling
NetworkTraffic Performance
Objectives:- Modeling- Design- Dimensioning
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What Are We Talking About?
- In a distributed storage system with failures,what is the life expectancy of a file?
- Does the Internet collapse if users are selfishand don’t use congestion control?
- Does CSMA/CA, as used in WiFi, ensureefficient use of bandwidth?
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Contents
Mathematical toolsModeling
Scaling methods
Stochastic averaging
ExamplesUnreliable File System
The Law of the Jungle
Flow-Aware CSMA
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Modeling
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Modeling
NetworkTraffic Performance
Objectives:- Modeling- Design- Dimensioning
Tools:- Markov processes- Queueing models- Scaling methods
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Stochastic Models
State: (X(t)) a Markov jump process in Nd:
- Number of files,
- Number of active flows in the Internet,
- Number of messages to be transmitted.
Markov assumptions:- Poisson arrivals
- Exponentially distributed sizes/durations.
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Stochastic Models
State: (X(t)) a Markov jump process in Nd:
- Number of files,
- Number of active flows in the Internet,
- Number of messages to be transmitted.
Markov assumptions:- Poisson arrivals
- Exponentially distributed sizes/durations.
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Stochastic Models
State: (X(t)) a Markov jump process in Nd:
- generally, non-reversible,
- when ergodic, invariant distribution not known,
- results on transient properties are rare (ford ≥ 2).
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Scaling Methods
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Scaling Methods
Principle: N a scaling parameterAnalyze the evolution of the sample path of
XN(ΨN(t))
ΦN
as N→∞, for some convenient (ΨN(t)) and (ΦN).
Time scale t→ ΨN(t) is used as a tool to focus onsome specific part of sample paths.
There may be more than one time scale ofinterest!
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Scaling Methods
Principle: N a scaling parameterAnalyze the evolution of the sample path of
XN(ΨN(t))
ΦN
as N→∞, for some convenient (ΨN(t)) and (ΦN).
Time scale t→ ΨN(t) is used as a tool to focus onsome specific part of sample paths.
There may be more than one time scale ofinterest!
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Scaling Methods: Goals
Give a First order description of (XN(t)):
XN(ΨN(t)) ≈ ΦN.x(t)
where,(x(t)) is a simpler stochastic process or even adeterministic dynamical system:
x(t) = F(x(t))
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Classical Example: Fluid Limit
X(t)
=
X(Nt)
N
, with N = ‖X(0)‖.
Scaling parameter: initial state
Time scale: t 7→ Nt
Fluid limit reaches 0⇓
Process is stable
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Classical Example: Fluid Limit
X(t)
=
X(Nt)
N
, with N = ‖X(0)‖.
Scaling parameter: initial state
Time scale: t 7→ Nt
Fluid limit reaches 0⇓
Process is stable13
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Example: Fluid Limit of M/M/1 Queue
X(t)
=
X(Nt)
N
, with N = X(0).
0 1·102
0
0.5
1
·101
X(0) = 10
t
X(t
)
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Example: Fluid Limit of M/M/1 Queue
X(t)
=
X(Nt)
N
, with N = X(0).
0 1·103
0
0.5
1
·102
X(0) = 102
λ = 0.8 μ = 1
t
X(t
)
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Example: Fluid Limit of M/M/1 Queue
X(t)
=
X(Nt)
N
, with N = X(0).
0 1·104
0
0.5
1
·103
X(0) = 103
λ = 0.8 μ = 1
t
X(t
)
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Example: Fluid Limit of M/M/1 Queue
X(t)
=
X(Nt)
N
, with N = X(0).
0 1·105
0
0.5
1
·104
X(0) = 104
λ = 0.8 μ = 1
t
X(t
)
14
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Example: Fluid Limit of M/M/1 Queue
X(t)
=
X(Nt)
N
, with N = X(0).
0 1·106
0
0.5
1
·105
X(0) = 105
λ = 0.8 μ = 1
t
X(t
)
14
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References
Fluid limits for queueing systems:[Malyshev 93][Rybko-Stolyar 92][Dai 95]
Scaling methods:[Khasminskii 56][Freidlin-Wentzell 79][Ethier-Kurtz 86][Robert 03]
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Technical CornerProof of the tightness of the scaled process
XN (ΨN(t))
ΦN
- Stochastic Differential Equation representationof (XN(t)) with martingales
- Standard tightness criteria
Difficulties:- Discontinuities: Skorokhod Problem Techniques- Stochastic averaging
Each example has its specific difficulties
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Technical CornerProof of the tightness of the scaled process
XN (ΨN(t))
ΦN
- Stochastic Differential Equation representationof (XN(t)) with martingales
- Standard tightness criteria
Difficulties:- Discontinuities: Skorokhod Problem Techniques- Stochastic averaging
Each example has its specific difficulties
16
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Technical CornerProof of the tightness of the scaled process
XN (ΨN(t))
ΦN
- Stochastic Differential Equation representationof (XN(t)) with martingales
- Standard tightness criteria
Difficulties:- Discontinuities: Skorokhod Problem Techniques- Stochastic averaging
Each example has its specific difficulties
16
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Technical CornerProof of the tightness of the scaled process
XN (ΨN(t))
ΦN
- Stochastic Differential Equation representationof (XN(t)) with martingales
- Standard tightness criteria
Difficulties:- Discontinuities: Skorokhod Problem Techniques- Stochastic averaging
Each example has its specific difficulties16
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Stochastic Averaging
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t)),
Fast time-scale
yN(t) = G(xN(t),yN(t))
Slow time-scale
Fast time-scale:
xN(t/N) = F(xN(t/N)).
Slow time-scale: If x(t) tends to a fixed point x?:(yN(t)) converges to (y(t)) with
y(t) = G(x?,y(t))
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale:
xN(t/N) = F(xN(t/N)).
Slow time-scale: If x(t) tends to a fixed point x?:(yN(t)) converges to (y(t)) with
y(t) = G(x?,y(t))
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale:
xN(t/N) = F(xN(t/N)).
Slow time-scale: If x(t) tends to a fixed point x?:(yN(t)) converges to (y(t)) with
y(t) = G(x?,y(t))
18
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale:
xN(t/N) = F(xN(t/N)).
Slow time-scale: If x(t) tends to a fixed point x?:(yN(t)) converges to (y(t)) with
y(t) = G(x?,y(t))
18
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t),yN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale: When N→∞, yN(t/N) ≈ z
xN(t/N) ≈ F(x(t/N), z)
Slow time-scale: If (xN(t)) tends to a fixed point x?z:
(yN(t)) converges to (y(t)) with
y(t) = G
x?y(t)
,y(t)
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t),yN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale: When N→∞, yN(t/N) ≈ z
xN(t/N) ≈ F(x(t/N), z)
Slow time-scale: If (xN(t)) tends to a fixed point x?z:
(yN(t)) converges to (y(t)) with
y(t) = G
x?y(t)
,y(t)
19
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A Deterministic ExampleDeterministic sequences (xN(t)) and (yN(t)) with:
xN(t) = NF(xN(t),yN(t)), Fast time-scaleyN(t) = G(xN(t),yN(t)) Slow time-scale
Fast time-scale: When N→∞, yN(t/N) ≈ z
xN(t/N) ≈ F(x(t/N), z)
Slow time-scale: If (xN(t)) tends to a fixed point x?z:
(yN(t)) converges to (y(t)) with
y(t) = G
x?y(t)
,y(t)
19
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Stochastic vs Deterministic
Deterministic Stochastic
Fast process ODE Markov process(x(t)) (X(t))
x = F(x(t),y) Ω(y)
Slow process ODE Markov process(y(t)) (Y(t))
EquilibriumFixed point Stationary
x?y
distributionπy
Convergence Regularity of Regularity ofy 7→ x?
yy 7→ πy
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Stochastic vs Deterministic
Deterministic Stochastic
Fast process ODE Markov process(x(t)) (X(t))
x = F(x(t),y) Ω(y)
Slow process ODE Markov process(y(t)) (Y(t))
EquilibriumFixed point Stationary
x?y
distributionπy
Convergence Regularity of Regularity ofy 7→ x?
yy 7→ πy
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References
Statistical mechanics:[Bogolyubov 62]
Stochastic calculus:[Khasminskii 68],[Papanicolaou et al. 77],[Freidlin-Wenzell 79].
Loss networks:[Kurtz 92],[Hunt-Kurtz 94]
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ContributionsThe Law of the Jungle:
- Stochastic averaging
- Scaling over the stationary distributions
Flow-Aware CSMA:- Suboptimality of CSMA (mono/multi-channel)- Optimality of Flow-Aware CSMA (mono/multi)- Time-scale separation
An unreliable file system:- Three time-scales- Stochastic averaging (simpler proof)
Transient properties of Engset and Ehrenfest:- Positive martingales- Asymptotics on hitting times
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ContributionsThe Law of the Jungle:
- Stochastic averaging
- Scaling over the stationary distributions
Flow-Aware CSMA:- Suboptimality of CSMA (mono/multi-channel)- Optimality of Flow-Aware CSMA (mono/multi)- Time-scale separation
An unreliable file system:- Three time-scales- Stochastic averaging (simpler proof)
Transient properties of Engset and Ehrenfest:- Positive martingales- Asymptotics on hitting times
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Example 1:
An Unreliable File System
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Model
∅
1
1
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
24
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Model
∅1
1
1
2
2
3
3
Back-up: λN
βN files2 copies/file μ 1
2
2
1
3
31
Each copy is lost at rate μ24
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Model
∅1
1
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
24
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Model
∅1
1
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
A file with 1 copy can be backed up24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
A file with 1 copy can be backed up24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1
2
2
1
3
31
A file with 0 copies is lost24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
A file with 0 copies is lost24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
A file with 0 copies is lost24
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Model
∅1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
24
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Model
∅11
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
24
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Model
∅11
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
3
1
24
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Model
∅11
1
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
3
1
24
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Model
∅
1
11
2
2
3
3
Back-up: λN
βN files2 copies/file
μ 1 2
2
1
3
31
What is the decay rate of the network?24
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ModelXi(t) : number of files with i copies at time t.
(X0(t),X1(t),X2(t)): a transient Markov Process.
X0(t) +X1(t) +X2(t) = βN.
A unique absorbing state (βN,0,0).
(X0(t)) (X1(t)) (X2(t))
μx1 λN1x1>0
2μx2
2μ(βN− x0 − x1)
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ModelXi(t) : number of files with i copies at time t.
(X0(t),X1(t),X2(t)): a transient Markov Process.
X0(t) +X1(t) +X2(t) = βN.
A unique absorbing state (βN,0,0).
(X0(t)) (X1(t)) (X2(t))
μx1 λN1x1>0
2μx2
2μ(βN− x0 − x1)
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Different Behaviors
Three time scales:
t → t/N
t → t
t → Nt
Three regimes:
Overload: 2β > ρ = λ/μ,Critical load: 2β = ρ,Underload: 2β < ρ.
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Time scale: t→ t/N
(X0(t/N)) (X1(t/N)) (X2(t/N))
μx1
N
λ1x1>0
2μ
N(βN− x1 − x0)
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Time scale: t→ t/N
(L1(t)): an M/M/1 queue
¨
ergodic if 2β < ρ,
transient if 2β > ρ.
0 (L1(t)) ∼ Nβ
λ1x1>0
2μβ
No loss!
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Time scale: t→ t/N
(L1(t)): an M/M/1 queue
¨
ergodic if 2β < ρ,
transient if 2β > ρ.
0 (L1(t)) ∼ Nβ
λ1x1>0
2μβ
No loss!
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Time scale: t→ tOverloaded network
If 2β > ρ, (X0(t)/N,X1(t)/N,X2(t)/N) converges to adeterministic process (x0(t),x1(t),x2(t)).
λ2μ
β
t
x2(t) : 2 copiesx1(t) : 1 copyx0(t) : 0 copies
A fraction N(β− ρ/2) is lost!
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Time scale: t→ tOverloaded network
If 2β > ρ, (X0(t)/N,X1(t)/N,X2(t)/N) converges to adeterministic process (x0(t),x1(t),x2(t)).
λ2μ
β
t
x2(t) : 2 copiesx1(t) : 1 copyx0(t) : 0 copies
A fraction N(β− ρ/2) is lost!
29
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Time scale: t→ tUnderloaded network
If 2β < ρ, (X0(t)/N,X1(t)/N,X2(t)/N) converges to
x0(t) = 0,x1(t) = 0,x2(t) = β.
λ2μ
β
t
x2(t) : 2 copies
No significant loss!
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Time scale: t→ tUnderloaded network
If 2β < ρ, (X0(t)/N,X1(t)/N,X2(t)/N) converges to
x0(t) = 0,x1(t) = 0,x2(t) = β.
λ2μ
β
t
x2(t) : 2 copies
No significant loss!30
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Time Scale t→ Nt
limN→+∞
X0(Nt)
N
= Ψ(t),
where Ψ(t) is the unique solution of
Ψ(t) = μ
∫ t
0
2μ(β−Ψ(s))
λ− 2μ(β−Ψ(s))ds.
β
t0 copies
t→ Nt is the “correct” time scale to describedecay.
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Time Scale t→ Nt
limN→+∞
X0(Nt)
N
= Ψ(t),
where Ψ(t) unique solution in (0, β) of
(1−Ψ(t)/β)ρ/2 eΨ(t)+t = 1.
β
t0 copies
t→ Nt is the “correct” time scale to describedecay.
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A Stochastic Averaging Phenomenon
∼ NΨ(t) X1,t(∞)
∼ N(β−Ψ(t))
Fast time scale: At “time” Nt,(X1(Nt+u/N),u ≥ 0): an M/M/1 with transition rates:
+1 at rate 2μ(β−Ψ(t))
−1 at rate λ.
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A Stochastic Averaging Phenomenon
∼ NΨ(t) X1,t(∞)
∼ N(β−Ψ(t))
Slow time scale: (X0(Nt)/N) “sees” only X1 at equi-librium:
Ψ(t)” = ”μ
∫ t
0E(X1,s(∞)) ds =
∫ t
0
2μ(β−Ψ(s))
λ− 2μ(β−Ψ(s))ds.
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Technical CornerStep 1 Radon measures: tightness of (μN) with
⟨μN,g⟩ =1
N
∫ Nt
0g
XN1 (s), s
ds
Step 2 Control of limits of (μN):
limN→∞
1
N
∫ Nt
0XN1 (s) ds = Ψ(t) =
∫ t
0⟨πs, I⟩ds
∫ t
0πs(N) ds = t
Here: Proof by stochastic dominationStep 3 Identification of πs with martingaletechniques and balance equations.
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Technical CornerStep 1 Radon measures: tightness of (μN) with
⟨μN,g⟩ =1
N
∫ Nt
0g
XN1 (s), s
ds
Step 2 Control of limits of (μN):
limN→∞
1
N
∫ Nt
0XN1 (s) ds = Ψ(t) =
∫ t
0⟨πs, I⟩ds
∫ t
0πs(N) ds = t
Here: Proof by stochastic dominationStep 3 Identification of πs with martingaletechniques and balance equations.
33
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Technical CornerStep 1 Radon measures: tightness of (μN) with
⟨μN,g⟩ =1
N
∫ Nt
0g
XN1 (s), s
ds
Step 2 Control of limits of (μN):
limN→∞
1
N
∫ Nt
0XN1 (s) ds = Ψ(t) =
∫ t
0⟨πs, I⟩ds
∫ t
0πs(N) ds = t
Here: Proof by stochastic domination
Step 3 Identification of πs with martingaletechniques and balance equations.
33
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Technical CornerStep 1 Radon measures: tightness of (μN) with
⟨μN,g⟩ =1
N
∫ Nt
0g
XN1 (s), s
ds
Step 2 Control of limits of (μN):
limN→∞
1
N
∫ Nt
0XN1 (s) ds = Ψ(t) =
∫ t
0⟨πs, I⟩ds
∫ t
0πs(N) ds = t
Here: Proof by stochastic dominationStep 3 Identification of πs with martingaletechniques and balance equations.
33
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Decay Rate of the Network
TN(δ) = inf¦
t ≥ 0 : XN0 (t) ≥ δβN©
Theorem:
limN→∞
TN(δ)
N= −
ρ
2log(1− δ)− δβ.
δ
TN
(δ)/N
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Conclusion
- Three different time scales
- A first example of stochastic averaging
- Asymptotics on a transitory property.
Extensions:- Number of copies: d > 2⇒ d− 1 times scales
- Decentralized back-up (mean-field)
Open problem:- Modeling a DHT: geometrical considerations
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Example 2:
The Law of the Jungle
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Context
Congestion control:- Rate adjustment to limit packet loss- Retransmission of lost packets
No congestion control:- No rate adjustment- Sources send at their maximum rate- Coding to recover from packet loss
Does this bring congestion collapse?
37
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Context
Congestion control:- Rate adjustment to limit packet loss- Retransmission of lost packets
No congestion control:- No rate adjustment- Sources send at their maximum rate- Coding to recover from packet loss
Does this bring congestion collapse?
37
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Context
Congestion control:- Rate adjustment to limit packet loss- Retransmission of lost packets
No congestion control:- No rate adjustment- Sources send at their maximum rate- Coding to recover from packet loss
Does this bring congestion collapse?
37
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Bandwidth Sharing Networks[Massoulié Roberts 00]
1
0
2
λ1
λ0
λ2μ1ϕ1
μ0ϕ0
μ2ϕ2
- A flow: a stream of packets
- Flows are considered as a fluid
- Users divided in classes/routes
- Poisson arrivals/Exponential sizes
- Resource allocation determined by congestionpolicy
38
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Bandwidth Sharing Networks[Massoulié Roberts 00]
1
0
2
λ1
λ0
λ2μ1ϕ1
μ0ϕ0
μ2ϕ2
- A flow: a stream of packets
- Flows are considered as a fluid
- Users divided in classes/routes
- Poisson arrivals/Exponential sizes
- Resource allocation determined by congestionpolicy
38
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Resource Allocation
Usually, α-fair policies are considered [MW00].
Here:- Sources send at their maximum rate (1 or a)- Tail dropping: At each link, output rates are
proportional to input rates
x1
x0
x2
ϕ1 = x1x0+x1
α = x0x0+x1
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
39
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Resource Allocation
Usually, α-fair policies are considered [MW00].
Here:- Sources send at their maximum rate (1 or a)- Tail dropping: At each link, output rates are
proportional to input rates
x1
x0
x2
ϕ1 = x1x0+x1
α = x0x0+x1
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
39
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Resource Allocation
Usually, α-fair policies are considered [MW00].
Here:- Sources send at their maximum rate (1 or a)- Tail dropping: At each link, output rates are
proportional to input rates
x1
x0
x2
ϕ1 = x1x0+x1
α = x0x0+x1
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
39
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Resource Allocation
Usually, α-fair policies are considered [MW00].
Here:- Sources send at their maximum rate (1 or a)- Tail dropping: At each link, output rates are
proportional to input rates
x1
x0
x2
ϕ1 = x1x0+x1
α = x0x0+x1
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
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Ergodicity Condition
1
0
2
Optimal ergodicity condition:
ρ0 + ρ1 < 1, ρ0 + ρ2 < 1
where ρi = λi/μi.
We know α-fair policies are optimal [BM02].
What about our policy?
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Ergodicity Condition
1
0
2
Optimal ergodicity condition:
ρ0 + ρ1 < 1, ρ0 + ρ2 < 1
where ρi = λi/μi.
We know α-fair policies are optimal [BM02].
What about our policy?
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Fluid Limits
x2
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
ϕ1 = x1x0+x1
α = x0x0+x1
x1
x0
If x2 0, class 2 uses virtually all the second link.If (z0(t), z1(t), z2(t)) is a fluid limit with z2(0) > 0,
z0(t) = λ0,
z1(t) = λ1 − μ1z1(t)
z0(t)+z1(t) ,
z2(t) = λ2 − μ2.
If ρ2 < 1, (z2(t)) reaches 0 in finite time.
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Fluid Limits
x2
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
ϕ1 = x1x0+x1
α = x0x0+x1
x1
x0
If x2 0, class 2 uses virtually all the second link.If (z0(t), z1(t), z2(t)) is a fluid limit with z2(0) > 0,
z0(t) = λ0,
z1(t) = λ1 − μ1z1(t)
z0(t)+z1(t) ,
z2(t) = λ2 − μ2.
If ρ2 < 1, (z2(t)) reaches 0 in finite time.41
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Fluid Limits
x2
ϕ0 = min
α, αx2a+α
ϕ2 = min
x2a,x2a
α+x2a
x1
x0
ϕ1 = x1x0+x1
α
Classes 0 and 1 are frozen:πα2 is the stationary distribution of class 2
Φ0(α) = Eπα2
Φ0
α,α
x2a+ α
.
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Fluid LimitsWhen z2(t) = 0:
z0(t) = λ0 − μ0ϕ0
z0(t)
z0(t) + z1(t)
,
z1(t) = λ1 − μ1z1(t)
z0(t) + z1(t),
z2(t) = 0.
with
Φ0(α) = Eπα2
Φ0
α,α
x2a+ α
.
Stochastic averaging
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Fluid LimitsWhen z2(t) = 0:
z0(t) = λ0 − μ0ϕ0
z0(t)
z0(t) + z1(t)
,
z1(t) = λ1 − μ1z1(t)
z0(t) + z1(t),
z2(t) = 0.
with
Φ0(α) = Eπα2
Φ0
α,α
x2a+ α
.
Stochastic averaging
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Ergodicity ConditionsErgodicity conditions:
ρ1 < 1, ρ2 < 1,
ρ0 < ϕ0(1− ρ1)
Optimal conditions:
ρ1 < 1, ρ2 < 1,ρ0 < min(1− ρ1,1− ρ2)
But:ϕ0(1− ρ1) < min(1− ρ2,1− ρ1)
Not optimal!
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Ergodicity ConditionsErgodicity conditions:
ρ1 < 1, ρ2 < 1,
ρ0 < ϕ0(1− ρ1)
Optimal conditions:
ρ1 < 1, ρ2 < 1,ρ0 < min(1− ρ1,1− ρ2)
But:ϕ0(1− ρ1) < min(1− ρ2,1− ρ1)
Not optimal!
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Ergodicity ConditionsErgodicity conditions:
ρ1 < 1, ρ2 < 1,
ρ0 < ϕ0(1− ρ1)
Optimal conditions:
ρ1 < 1, ρ2 < 1,ρ0 < min(1− ρ1,1− ρ2)
But:ϕ0(1− ρ1) < min(1− ρ2,1− ρ1)
Not optimal!
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Impact of Maximum Rate a
Class 1: ρ1
Cla
ss0
:ρ
0
Optimala = 1a = 0.1a = 0.01
What happens when a→ 0 ?
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Impact of Maximum Rate a
Class 1: ρ1
Cla
ss0
:ρ
0
Optimala = 1a = 0.1a = 0.01
What happens when a→ 0 ?
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Scaling the Maximum Rate aWe freeze α and consider the process (XS2(t)) withQ-matrix:
q(x2,x2 + 1) = λ2,
q(x2,x2 − 1) = μ2 min
x2a,x2a
/S
α + x2a
/S
Time-scale: t 7→ St
(XS2(St)/S)⇒ (x2(t)) with
x2(t) = λ2 − μ2 min
ax2(t),x2(t)a
α + x2(t)a
Fixed point:
x2 =ρ2
amax
1,α
1− ρ2
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Scaling the Maximum Rate aWe freeze α and consider the process (XS2(t)) withQ-matrix:
q(x2,x2 + 1) = λ2,
q(x2,x2 − 1) = μ2 min
x2a
S,
x2a/S
α + x2a/S
Time-scale: t 7→ St
(XS2(St)/S)⇒ (x2(t)) with
x2(t) = λ2 − μ2 min
ax2(t),x2(t)a
α + x2(t)a
Fixed point:
x2 =ρ2
amax
1,α
1− ρ2
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Scaling the Maximum Rate aWe freeze α and consider the process (XS2(t)) withQ-matrix:
q(x2,x2 + 1) = λ2,
q(x2,x2 − 1) = μ2 min
x2a
S,
x2a/S
α + x2a/S
Time-scale: t 7→ St
(XS2(St)/S)⇒ (x2(t)) with
x2(t) = λ2 − μ2 min
ax2(t),x2(t)a
α + x2(t)a
Fixed point:
x2 =ρ2
amax
1,α
1− ρ2
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Scaling the Maximum Rate aWe freeze α and consider the process (XS2(t)) withQ-matrix:
q(x2,x2 + 1) = λ2,
q(x2,x2 − 1) = μ2 min
x2a
S,
x2a/S
α + x2a/S
Time-scale: t 7→ St
(XS2(St)/S)⇒ (x2(t)) with
x2(t) = λ2 − μ2 min
ax2(t),x2(t)a
α + x2(t)a
Fixed point:
x2 =ρ2
amax
1,α
1− ρ2
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Scaling the Maximum Rate a
(XS2
(St)/S)t→∞−−→ XS
2(∞)/S
S→∞
y
yS→∞
(x2(t)) −−→t→∞
x2(∞)
Convergence of processes⇓
Convergence of stationary distribution
lima→0
Φ0(1− ρ1) = min(1− ρ1,1− ρ2)
The policy is asymptotically optimal
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Scaling the Maximum Rate a
(XS2
(St)/S)t→∞−−→ XS
2(∞)/S
S→∞
y
yS→∞
(x2(t)) −−→t→∞
x2(∞)
Convergence of processes⇓
Convergence of stationary distribution
lima→0
Φ0(1− ρ1) = min(1− ρ1,1− ρ2)
The policy is asymptotically optimal
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Scaling the Maximum Rate a
(XS2
(St)/S)t→∞−−→ XS
2(∞)/S
S→∞
y
yS→∞
(x2(t)) −−→t→∞
x2(∞)
Convergence of processes⇓
Convergence of stationary distribution
lima→0
Φ0(1− ρ1) = min(1− ρ1,1− ρ2)
The policy is asymptotically optimal47
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Conclusion- Analysis of equilibrium,
- Inversion of limits: scaling on stationarydistributions
- Impact of access rates
Extensions:- Linear networks with L links
- Second order scaling: speed of convergence.
- Upstream trees
Open problem:- General acyclic networks
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Example 3:
Flow-Aware CSMA
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ModelThe network is represented by a conflict graph
1 2 3
wireless link potential interference
For each node i:- Xi(t) ∈ N: number of flows at time t- Yi(t) = 1 if node is active at time t, 0 otherwise.
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ModelThe network is represented by a conflict graph
1 2 3
wireless link
potential interference
For each node i:- Xi(t) ∈ N: number of flows at time t- Yi(t) = 1 if node is active at time t, 0 otherwise.
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ModelThe network is represented by a conflict graph
1 2 3
wireless link potential interference
For each node i:- Xi(t) ∈ N: number of flows at time t- Yi(t) = 1 if node is active at time t, 0 otherwise.
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2 μ3
1 2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2 μ3
1 2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1
μ2 μ3
1
2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1
μ2
μ3
1
2
3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2
μ3
1 2
3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1
μ2
μ3
1
2
3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2 μ3
1 2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedules
In this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2 μ3
1 2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Conflict Graph
1 2 3
λ1 λ2 λ3
μ1 μ2 μ3
1 2 3
Schedules: ∅, 1, 2, 3, 1,3.
Optimal stability region: convex hull of schedulesIn this example: ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1with ρi = λi/μi
Stability region?
51
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Standard CSMA
∼ exp(α)
Back-off∼ exp(1)
Transmission∼ exp(α)
Back-off
Optimal?
1 2 3
ρ1 ρ2 ρ3
0 0.5 10
0.5
1
ρ1 = ρ3
ρ2
OptimalActual
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Standard CSMA
∼ exp(α)
Back-off∼ exp(1)
Transmission∼ exp(α)
Back-off
Optimal?
1 2 3
ρ1 ρ2 ρ3
0 0.5 10
0.5
1
ρ1 = ρ3
ρ2
OptimalActual
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Standard CSMA
∼ exp(α)
Back-off∼ exp(1)
Transmission∼ exp(α)
Back-off
Optimal?
1 2 3
ρ1 ρ2 ρ3
0 0.5 10
0.5
1
ρ1 = ρ3
ρ2
OptimalActual
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Flow-Aware CSMAProposed modification of CSMA:
Exponential backoff time for each flow
∼ exp(αx1)
Back-off∼ exp(1)
Transmission∼ exp(αx1)
Back-off
The process (X(t),Y(t)) is difficult to analyze:
Idea: Separate network dynamics and flowdynamics.When N→∞, (YN(t)): classical loss network.
Stochastic averaging
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Flow-Aware CSMAProposed modification of CSMA:
Exponential backoff time for each flow
∼ exp(αx1)
Back-off∼ exp(1)
Transmission∼ exp(αx1)
Back-off
The process (X(t),Y(t)) is difficult to analyze:
Idea: Separate network dynamics and flowdynamics.When N→∞, (YN(t)): classical loss network.
Stochastic averaging
53
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Flow-Aware CSMAProposed modification of CSMA:
Exponential backoff time for each flow
∼ exp(αNx1)
Back-off∼ exp(N)
Transmission∼ exp(αNx1)
Back-off
The process (XN(t),YN(t)) is difficult to analyze:
Idea: Separate network dynamics and flowdynamics.When N→∞, (YN(t)): classical loss network.
Stochastic averaging
53
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Flow-Aware CSMAProposed modification of CSMA:
Exponential backoff time for each flow
∼ exp(αNx1)
Back-off∼ exp(N)
Transmission∼ exp(αNx1)
Back-off
The process (XN(t),YN(t)) is difficult to analyze:
Idea: Separate network dynamics and flowdynamics.When N→∞, (YN(t)): classical loss network.
Stochastic averaging
53
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Optimality of Flow-Aware CSMA
Theorem:Flow-aware CSMA algorithm is optimal for anynetwork.
Sketch of proof:- Asymptotically behaves as Max-Weight.
- Deduce a Lyapunov function and applyFoster’s criterion.
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Conclusion
- An optimal and fully distributed channelaccess mechanism
- Limiting process: jump process
- Simplification of the problem
Extension:- Multi-channel
Open problem:- Initial problem still open
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General ConclusionThree examples:
- Capacity of an unreliable file system
- Law of the Jungle
- Flow-Aware CSMA
Mathematical tools:- Several examples of scalings
- A simpler proof for stochastic averaging
and. . .- Scalings: A set of powerful tools
- Stochastic averaging: a not so rarephenomenon
Many interesting open questions. . .
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General ConclusionThree examples:
- Capacity of an unreliable file system
- Law of the Jungle
- Flow-Aware CSMA
Mathematical tools:- Several examples of scalings
- A simpler proof for stochastic averaging
and. . .- Scalings: A set of powerful tools
- Stochastic averaging: a not so rarephenomenon
Many interesting open questions. . .
56
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General ConclusionThree examples:
- Capacity of an unreliable file system
- Law of the Jungle
- Flow-Aware CSMA
Mathematical tools:- Several examples of scalings
- A simpler proof for stochastic averaging
and. . .- Scalings: A set of powerful tools
- Stochastic averaging: a not so rarephenomenon
Many interesting open questions. . .
56
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General ConclusionThree examples:
- Capacity of an unreliable file system
- Law of the Jungle
- Flow-Aware CSMA
Mathematical tools:- Several examples of scalings
- A simpler proof for stochastic averaging
and. . .- Scalings: A set of powerful tools
- Stochastic averaging: a not so rarephenomenon
Many interesting open questions. . . 56
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Thank you!