Bandwidth Allocation in Large Stochastic Networks Mathieu Feuillet - - PowerPoint PPT Presentation

Bandwidth Allocation in Large Stochastic Networks

Mathieu Feuillet

Soutenance de thèse

12/07/2012

Introduction

Modeling

Network Traffic Performance Objectives:

• Modeling
• Design
• Dimensioning

3

What Are We T alking About?

• In a distributed storage system with failures,

what is the life expectancy of a file?

• Does the Internet collapse if users are selfish

and don’t use congestion control?

• Does CSMA/CA, as used in WiFi, ensure

efficient use of bandwidth?

4

Contents

Mathematical tools Modeling Scaling methods Stochastic averaging Examples Unreliable File System The Law of the Jungle Flow-Aware CSMA

5

Modeling

Modeling

Network Traffic Performance Objectives:

• Modeling
• Design
• Dimensioning

T

• ols:
• Markov processes
• Queueing models
• Scaling methods

7

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.

8

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.

8

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 (for

d ≥ 2).

9

Scaling Methods

Scaling Methods

Principle: N a scaling parameter Analyze 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 on some specific part of sample paths.

11

Scaling Methods

Principle: N a scaling parameter Analyze 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 on some specific part of sample paths. There may be more than one time scale of interest!

11

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 a deterministic dynamical system:

˙ x(t) = F(x(t))

12

Classical Example: Fluid Limit

¯

X(t)

• =

X(Nt)

N

• ,

with N = X(0).

Scaling parameter: initial state Time scale: t → Nt

13

Classical Example: Fluid Limit

¯

X(t)

• =

X(Nt)

N

• ,

with N = X(0).

Scaling parameter: initial state Time scale: t → Nt

Fluid limit reaches 0 ⇓ Process is stable

13

Example: Fluid Limit of M/M/1 Queue

¯ X(t)

• =

X(Nt) N

• ,

with N = X(0). 1 ·102 0.5 1 ·101 X(0) = 10 t X(t)

14

Example: Fluid Limit of M/M/1 Queue

¯ X(t)

• =

X(Nt) N

• ,

with N = X(0). 1 ·103 0.5 1 ·102 X(0) = 102 λ = 0.8 μ = 1 t X(t)

14

Example: Fluid Limit of M/M/1 Queue

¯ X(t)

• =

X(Nt) N

• ,

with N = X(0). 1 ·104 0.5 1 ·103 X(0) = 103 λ = 0.8 μ = 1 t X(t)

14

Example: Fluid Limit of M/M/1 Queue

¯ X(t)

• =

X(Nt) N

• ,

with N = X(0). 1 ·105 0.5 1 ·104 X(0) = 104 λ = 0.8 μ = 1 t X(t)

14

Example: Fluid Limit of M/M/1 Queue

¯ X(t)

• =

X(Nt) N

• ,

with N = X(0). 1 ·106 0.5 1 ·105 X(0) = 105 λ = 0.8 μ = 1 t X(t)

14

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]

15

T echnical Corner

Proof of the tightness of the scaled process XN (ΨN(t)) ΦN

• Stochastic Differential Equation representation
• f (XN(t)) with martingales
• Standard tightness criteria

16

T echnical Corner

Proof of the tightness of the scaled process XN (ΨN(t)) ΦN

• Stochastic Differential Equation representation
• f (XN(t)) with martingales
• Standard tightness criteria

Difficulties:

• Discontinuities: Skorokhod Problem T

echniques

• Stochastic averaging

16

T echnical Corner

Proof of the tightness of the scaled process XN (ΨN(t)) ΦN

• Stochastic Differential Equation representation
• f (XN(t)) with martingales
• Standard tightness criteria

Difficulties:

• Discontinuities: Skorokhod Problem T

echniques

• Stochastic averaging

16

T echnical Corner

Proof of the tightness of the scaled process XN (ΨN(t)) ΦN

• Stochastic Differential Equation representation
• f (XN(t)) with martingales
• Standard tightness criteria

Difficulties:

• Discontinuities: Skorokhod Problem T

echniques

• Stochastic averaging

Each example has its specific difficulties

16

Stochastic Averaging

A Deterministic Example

Deterministic sequences (xN(t)) and (yN(t)) with: ˙ xN(t) = NF(xN(t)), ˙ yN(t) = G(xN(t), yN(t))

18

A Deterministic Example

Deterministic 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

18

A Deterministic Example

Deterministic 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)).

18

A Deterministic Example

Deterministic 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))

18

A Deterministic Example

Deterministic sequences (xN(t)) and (yN(t)) with: ˙ xN(t) = NF(xN(t), yN(t)), Fast time-scale ˙ yN(t) = G(xN(t), yN(t)) Slow time-scale

19

A Deterministic Example

Deterministic sequences (xN(t)) and (yN(t)) with: ˙ xN(t) = NF(xN(t), yN(t)), Fast time-scale ˙ yN(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)

19

A Deterministic Example

Deterministic sequences (xN(t)) and (yN(t)) with: ˙ xN(t) = NF(xN(t), yN(t)), Fast time-scale ˙ yN(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

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)) Equilibrium Fixed point Stationary x⋆

y

distribution πy Convergence Regularity of Regularity of y → x⋆

y

y → πy

20

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)) Equilibrium Fixed point Stationary x⋆

y

distribution πy Convergence Regularity of Regularity of y → x⋆

y

y → πy

20

References

Statistical mechanics: [Bogolyubov 62] Stochastic calculus: [Khasminskii 68], [Papanicolaou et al. 77], [Freidlin-Wenzell 79]. Loss networks: [Kurtz 92], [Hunt-Kurtz 94]

21

Contributions

The 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

22

Contributions

The 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

22

Example 1: An Unreliable File System

Model

1 1 2 2 3 3 Back-up: λN βN files 2 copies/file

24

Model

1 2 2 3 3 Back-up: λN βN files 2 copies/file μ 1

Each copy is lost at rate μ

24

Model

1 2 2 3 3 Back-up: λN βN files 2 copies/file

24

Model

1 2 2 3 3 Back-up: λN βN files 2 copies/file

A file with 1 copy can be backed up

24

Model

1 1 2 2 3 3 Back-up: λN βN files 2 copies/file

A file with 1 copy can be backed up

24

Model

1 1 2 2 3 3 Back-up: λN βN files 2 copies/file

24

Model

1 1 2 2 3 3 Back-up: λN βN files 2 copies/file 2

A file with 0 copies is lost

24

Model

1 1 2 3 3 Back-up: λN βN files 2 copies/file 2

A file with 0 copies is lost

24

Model

1 1 3 3 Back-up: λN βN files 2 copies/file

A file with 0 copies is lost

24

Model

1 1 3 3 Back-up: λN βN files 2 copies/file 1

24

Model

1 3 3 Back-up: λN βN files 2 copies/file 3

24

Model

1 3 Back-up: λN βN files 2 copies/file 3

24

Model

1 Back-up: λN βN files 2 copies/file 1

24

Model

∅

Back-up: λN βN files 2 copies/file

What is the decay rate of the network?

24

Model

Xi(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 λN1{x1>0} 2μx2

25

Model

Xi(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 λN1{x1>0} 2μ(βN − x0 − x1)

25

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.

26

Time scale: t → t/N

(X0(t/N)) (X1(t/N)) (X2(t/N)) μ x1 N λ1{x1>0} 2μ N (βN − x1 − x0)

27

Time scale: t → t/N

(L1(t)): an M/M/1 queue

• ergodic if 2β < ρ,

transient if 2β > ρ. (L1(t)) ∼ Nβ λ1{x1>0} 2μβ

28

Time scale: t → t/N

(L1(t)): an M/M/1 queue

• ergodic if 2β < ρ,

transient if 2β > ρ. (L1(t)) ∼ Nβ λ1{x1>0} 2μβ

No loss!

28

Time scale: t → t

Overloaded network

If 2β > ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to a deterministic process (x0(t), x1(t), x2(t)).

λ 2μ

β t

x2(t) : 2 copies x1(t) : 1 copy x0(t) : 0 copies

29

Time scale: t → t

Overloaded network

If 2β > ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to a deterministic process (x0(t), x1(t), x2(t)).

λ 2μ

β t

x2(t) : 2 copies x1(t) : 1 copy x0(t) : 0 copies

A fraction N(β − ρ/2) is lost!

29

Time scale: t → t

Underloaded network

If 2β < ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to    x0(t) = 0, x1(t) = 0, x2(t) = β.

λ 2μ

β t

x2(t) : 2 copies

30

Time scale: t → t

Underloaded network

If 2β < ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to    x0(t) = 0, x1(t) = 0, x2(t) = β.

λ 2μ

β t

x2(t) : 2 copies

No significant loss!

30

Time Scale t → Nt

lim

N→+∞

X0(Nt) N

• = Ψ(t),

where Ψ(t) is the unique solution of Ψ(t) = μ t 2μ(β − Ψ(s)) λ − 2μ(β − Ψ(s)) ds.

31

Time Scale t → Nt

lim

N→+∞

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 t

0 copies

t → Nt is the “correct” time scale to describe decay.

31

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.

32

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 E(X1,s(∞))ds = t 2μ(β − Ψ(s)) λ − 2μ(β − Ψ(s)) ds.

32

T echnical Corner

Step 1 Radon measures: tightness of (μN) with 〈μN, g〉 = 1 N Nt g

• XN

1(s), s

• ds

33

T echnical Corner

Step 1 Radon measures: tightness of (μN) with 〈μN, g〉 = 1 N Nt g

• XN

1(s), s

• ds

Step 2 Control of limits of (μN): lim

N→∞

1 N Nt XN

1(s)ds = Ψ(t) =

t 〈πs, I〉 ds t πs(N)ds = t

33

T echnical Corner

Step 1 Radon measures: tightness of (μN) with 〈μN, g〉 = 1 N Nt g

• XN

1(s), s

• ds

Step 2 Control of limits of (μN): lim

N→∞

1 N Nt XN

1(s)ds = Ψ(t) =

t 〈πs, I〉 ds t πs(N)ds = t Here: Proof by stochastic domination

33

T echnical Corner

Step 1 Radon measures: tightness of (μN) with 〈μN, g〉 = 1 N Nt g

• XN

1(s), s

• ds

Step 2 Control of limits of (μN): lim

N→∞

1 N Nt XN

1(s)ds = Ψ(t) =

t 〈πs, I〉 ds t πs(N)ds = t Here: Proof by stochastic domination Step 3 Identification of πs with martingale techniques and balance equations.

33

Decay Rate of the Network

TN(δ) = inf

• t ≥ 0 : XN

0(t) ≥ δβN

• Theorem:

lim

N→∞

TN(δ) N = − ρ 2 log(1 − δ) − δβ. δ TN(δ)/N

34

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

35

Example 2: The Law of the Jungle

Context

Congestion control:

• Rate adjustment to limit packet loss
• Retransmission of lost packets

37

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

37

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

Bandwidth Sharing Networks

[Massoulié Roberts 00]

1 2

• 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 congestion

policy

38

Bandwidth Sharing Networks

[Massoulié Roberts 00]

λ1 λ0 λ2 μ1ϕ1 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 congestion

policy

38

Resource Allocation

Usually, α-fair policies are considered [MW00]. Here:

• Sources send at their maximum rate (1 or a)
• T

ail dropping: At each link, output rates are proportional to input rates

39

Resource Allocation

Usually, α-fair policies are considered [MW00]. Here:

• Sources send at their maximum rate (1 or a)
• T

ail dropping: At each link, output rates are proportional to input rates x1 x0 x2

39

Resource Allocation

Usually, α-fair policies are considered [MW00]. Here:

• Sources send at their maximum rate (1 or a)
• T

ail dropping: At each link, output rates are proportional to input rates x1 x0 x2 ϕ1 =

x1 x0+x1

α =

x0 x0+x1

39

Resource Allocation

Usually, α-fair policies are considered [MW00]. Here:

• Sources send at their maximum rate (1 or a)
• T

ail dropping: At each link, output rates are proportional to input rates x1 x0 x2 ϕ1 =

x1 x0+x1

α =

x0 x0+x1

ϕ0 = min

• α,

α x2a+α

• ϕ2 = min
• x2a,

x2a α+x2a

• 39

Ergodicity Condition

1 2 Optimal ergodicity condition: ρ0 + ρ1 < 1, ρ0 + ρ2 < 1 where ρi = λi/μi. We know α-fair policies are optimal [BM02].

40

Ergodicity Condition

1 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?

40

Fluid Limits

x2 ϕ0 = min

• α,

α x2a+α

• ϕ2 = min
• x2a,

x2a α+x2a

• ϕ1 =

x1 x0+x1

α =

x0 x0+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 − μ1

z1(t) z0(t)+z1(t),

˙ z2(t) = λ2 − μ2.

41

Fluid Limits

x2 ϕ0 = min

• α,

α x2a+α

• ϕ2 = min
• x2a,

x2a α+x2a

• ϕ1 =

x1 x0+x1

α =

x0 x0+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 − μ1

z1(t) z0(t)+z1(t),

˙ z2(t) = λ2 − μ2. If ρ2 < 1, (z2(t)) reaches 0 in finite time.

41

Fluid Limits

x2 ϕ0 = min

• α,

α x2a+α

• ϕ2 = min
• x2a,

x2a α+x2a

• x1

x0 ϕ1 =

x1 x0+x1

α Classes 0 and 1 are frozen: πα

2 is the stationary distribution of class 2

¯ Φ0(α) = Eπα

2

• Φ0
• α,

α x2a + α

• .

42

Fluid Limits

When z2(t) = 0:            ˙ z0(t) = λ0 − μ0 ¯ ϕ0

• z0(t)

z0(t) + z1(t)

• ,

˙ z1(t) = λ1 − μ1 z1(t) z0(t) + z1(t) , ˙ z2(t) = 0. with ¯ Φ0(α) = Eπα

2

• Φ0
• α,

α x2a + α

• .

43

Fluid Limits

When z2(t) = 0:            ˙ z0(t) = λ0 − μ0 ¯ ϕ0

• z0(t)

z0(t) + z1(t)

• ,

˙ z1(t) = λ1 − μ1 z1(t) z0(t) + z1(t) , ˙ z2(t) = 0. with ¯ Φ0(α) = Eπα

2

• Φ0
• α,

α x2a + α

• .

Stochastic averaging

43

Ergodicity Conditions

Ergodicity conditions: ρ1 < 1, ρ2 < 1, ρ0 < ¯ ϕ0(1 − ρ1) Optimal conditions: ρ1 < 1, ρ2 < 1, ρ0 < min(1 − ρ1, 1 − ρ2)

44

Ergodicity Conditions

Ergodicity 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)

44

Ergodicity Conditions

Ergodicity 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!

44

Impact of Maximum Rate a

Class 1: ρ1 Class 0: ρ0

Optimal a = 1 a = 0.1 a = 0.01

45

Impact of Maximum Rate a

Class 1: ρ1 Class 0: ρ0

Optimal a = 1 a = 0.1 a = 0.01

What happens when a → 0 ?

45

Scaling the Maximum Rate a

We freeze α and consider the process (XS

2(t)) with

Q-matrix: q(x2, x2 + 1) = λ2, q(x2, x2 − 1) = μ2 min

• x2a,

x2a α + x2a

• 46

Scaling the Maximum Rate a

We freeze α and consider the process (XS

2(t)) with

Q-matrix: q(x2, x2 + 1) = λ2, q(x2, x2 − 1) = μ2 min

• x2

a S , x2a/S α + x2a/S

• Time-scale: t → St

46

Scaling the Maximum Rate a

We freeze α and consider the process (XS

2(t)) with

Q-matrix: q(x2, x2 + 1) = λ2, q(x2, x2 − 1) = μ2 min

• x2

a S , x2a/S α + x2a/S

• Time-scale: t → St

(XS

2(St)/S) ⇒ (x2(t)) with

˙ x2(t) = λ2 − μ2 min

• ax2(t),

x2(t)a α + x2(t)a

• 46

Scaling the Maximum Rate a

We freeze α and consider the process (XS

2(t)) with

Q-matrix: q(x2, x2 + 1) = λ2, q(x2, x2 − 1) = μ2 min

• x2

a S , x2a/S α + x2a/S

• Time-scale: t → St

(XS

2(St)/S) ⇒ (x2(t)) with

˙ x2(t) = λ2 − μ2 min

• ax2(t),

x2(t)a α + x2(t)a

• Fixed point:

x2 = ρ2 a max

• 1,

α 1 − ρ2

• 46

Scaling the Maximum Rate a

(XS

2(St)/S) t→∞

− − → XS

2(∞)/S S→∞

 

• 

 S→∞

(x2(t)) − − →

t→∞

x2(∞)

47

Scaling the Maximum Rate a

(XS

2(St)/S) t→∞

− − → XS

2(∞)/S S→∞

 

• 

 S→∞

(x2(t)) − − →

t→∞

x2(∞)

Convergence of processes ⇓ Convergence of stationary distribution

47

Scaling the Maximum Rate a

(XS

2(St)/S) t→∞

− − → XS

2(∞)/S S→∞

 

• 

 S→∞

(x2(t)) − − →

t→∞

x2(∞)

Convergence of processes ⇓ Convergence of stationary distribution

lim

a→0

¯ Φ0(1 − ρ1) = min(1 − ρ1, 1 − ρ2) The policy is asymptotically optimal

47

Conclusion

• Analysis of equilibrium,
• Inversion of limits: scaling on stationary

distributions

• Impact of access rates

Extensions:

• Linear networks with L links
• Second order scaling: speed of convergence.
• Upstream trees

Open problem:

• General acyclic networks

48

Example 3: Flow-Aware CSMA

Model

The network is represented by a conflict graph 1 2 3 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.

50

Model

The network is represented by a conflict graph 1 2 3

wireless link

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.

50

Model

The 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.

50

Conflict Graph

1 2 3 λ1 λ2 λ3 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 μ1 1 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 μ2 2 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 μ3 3 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 μ1 μ3 1 3 Schedules: ∅, {1}, {2}, {3}, {1, 3}.

51

Conflict Graph

1 2 3 λ1 λ2 λ3 Schedules: ∅, {1}, {2}, {3}, {1, 3}. Optimal stability region: convex hull of schedules

51

Conflict Graph

1 2 3 λ1 λ2 λ3 Schedules: ∅, {1}, {2}, {3}, {1, 3}. Optimal stability region: convex hull of schedules In this example: {ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1} with ρi = λi/μi

51

Conflict Graph

1 2 3 λ1 λ2 λ3 Schedules: ∅, {1}, {2}, {3}, {1, 3}. Optimal stability region: convex hull of schedules In this example: {ρ1 + ρ2 ≤ 1, ρ2 + ρ3 ≤ 1} with ρi = λi/μi

Stability region?

51

Standard CSMA

∼ exp(α) Back-off ∼ exp(1) Transmission ∼ exp(α) Back-off

52

Standard CSMA

∼ exp(α) Back-off ∼ exp(1) Transmission ∼ exp(α) Back-off

Optimal?

52

Standard CSMA

∼ exp(α) Back-off ∼ exp(1) Transmission ∼ exp(α) Back-off

Optimal?

1 2 3 ρ1 ρ2 ρ3

0.5 1 0.5 1

ρ1 = ρ3 ρ2

Optimal Actual

52

Flow-Aware CSMA

Proposed modification of CSMA: Exponential backoff time for each flow ∼ exp(αx1) Back-off ∼ exp(1) Transmission ∼ exp(αx1) Back-off

53

Flow-Aware CSMA

Proposed 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:

53

Flow-Aware CSMA

Proposed 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 flow dynamics. When N → ∞, (YN(t)): classical loss network.

53

Flow-Aware CSMA

Proposed 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 flow dynamics. When N → ∞, (YN(t)): classical loss network.

Stochastic averaging

53

Optimality of Flow-Aware CSMA

Theorem: Flow-aware CSMA algorithm is optimal for any network. Sketch of proof:

• Asymptotically behaves as Max-Weight.
• Deduce a Lyapunov function and apply

Foster’s criterion.

54

Conclusion

• An optimal and fully distributed channel

access mechanism

• Limiting process: jump process
• Simplification of the problem

Extension:

• Multi-channel

Open problem:

• Initial problem still open

55

General Conclusion

Three examples:

• Capacity of an unreliable file system
• Law of the Jungle
• Flow-Aware CSMA

56

General Conclusion

Three 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

56

General Conclusion

Three 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 rare

phenomenon

56

General Conclusion

Three 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 rare

phenomenon

Many interesting open questions. . .

56

Thank you!