The time scales of a stochastic network with failures Mathieu - - PowerPoint PPT Presentation

The time scales of a stochastic network with failures

Mathieu Feuillet

joint work with Philippe Robert

YEQT-V

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT For a specific file

An inspiring example: a DHT From a global perspective

An inspiring example: a DHT From a global perspective

An inspiring example: a DHT From a global perspective

An inspiring example: a DHT From a global perspective

An inspiring example: a DHT From a global perspective

An inspiring example: a DHT

T ypical questions: What is the maximum number of files that the system can sustain? What is the decay rate of the network? General problem: Study the evolution of a large distributed system with failures.

Background

• Reliability theory:

System Reliability Theory: Models, Statistical methods and Applications, Raudand, Hoyland,2003. Very few queueing analysis of large systems.

• Statistical studies:

fta.inria.fr

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

Model

• A set of FN files.
• At most d copies per file.
• Each copy is lost at rate μ.
• Capacity of duplication: λN.
• A file with 0 copies is lost.

Model

For 0 ≤ i ≤ d: Xi(t): number of files with i copies. Back-up policy: Recovery capacity λN allocated to the files with the minimum number of copies. (xi, xi+1) → (xi − 1, xi+1 + 1) at rate λN if x1 = x2 = · · · = xi−1 = 0 and x1 > 0.

Model

(Xi(t), 0 ≤ i ≤ d) is a transient Markov Process. X0(t) + X1(t) + · · · + Xd(t) = FN. Unique absorbing state (FN, 0 . . . , 0). Assume lim

N→∞

FN N = β > 0. General problem: Estimate the decay rate of the network.

Case d = 2

(X0(t)) (X1(t)) (X2(t)) μx1 λN1{x1>0} 2μx2

Case d = 2

State (x0, x1, FN − x0 − x1). (X0(t)) (X1(t)) (X2(t)) μx1 λN1{x1>0} 2μ(FN − x0 − x1)

Different behaviors

Three time scales:    t → t/N t → t t → Nt Three regimes: Overloaded network: 2β > ρ = λ/μ. Critical case: 2β = ρ. Underloaded case: 2β < ρ.

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

Time scale: t → t/N

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

Time scale: t → t/N

(L1(t)): an M/M/1 queue    ergodic if 2β < ρ, null recurrent if 2β = ρ, transient if 2β > ρ. (L1(t)) ∼ Nβ λ1{x1>0} 2μβ

Time scale: t → t/N

(L1(t)): an M/M/1 queue    ergodic if 2β < ρ, null recurrent if 2β = ρ, transient if 2β > ρ. (L1(t)) ∼ Nβ λ1{x1>0} 2μβ

No loss!

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

Overloaded network

If 2β > ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to    x0(t) = (β − ρ/2)(1 + e−2μt − 2e−μt), x1(t) = (2β − ρ)(e−μt − e−2μt), x2(t) = (β − ρ/2)e−2μt + ρ/2.

λ 2μ

β t

2 copies 1 copy 0 copies

T echnical point: Generalized Skorokhod problem.

Overloaded network

If 2β > ρ, (X0(t)/N, X1(t)/N, X2(t)/N) converges to    x0(t) = (β − ρ/2)(1 + e−2μt − 2e−μt), x1(t) = (2β − ρ)(e−μt − e−2μt), x2(t) = (β − ρ/2)e−2μt + ρ/2.

λ 2μ

β t

2 copies 1 copy 0 copies

Some losses!

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

2 copies

T echnical point: Generalized Skorokhod problem.

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

2 copies

No loss!

The critical case

If 2β = ρ,

• XN

0(t)

• N

, XN

1(t)

• N
• ⇒

t Y(u)du, Y(t)

• where

dY(t) =

• 2λ dB(t)+μ
• 2γ − 3Y(t) − 2μ

t Y(u)du

• dt

with the constraint Y(t) ≥ 0. t

1 copy 0 copies

Underloaded network

If 2β < ρ, X2(t)/N ⇒ β X1(t) ⇒ G Geometric r.v. w. param. 2β/ρ, (X0(t)) ⇒ (Nα(t)) with α = 2μβ(ρ − 2β). Nα(t) G ∼ βN μx1 λN1{x1>0} 2μNβ

Underloaded network

If 2β < ρ, X2(t)/N ⇒ β X1(t) ⇒ G Geometric r.v. w. param. 2β/ρ, (X0(t)) ⇒ (Nα(t)) with α = 2μβ(ρ − 2β). Nα(t) G ∼ βN μx1 λN1{x1>0} 2μNβ

No significant losses!

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

Time scale t → Nt

lim

N→+∞

X0(Nt) N

• = Ψ(t),

Gt Geometric r.v. with par. 2(β − Ψ(t))/ρ ∼ NΨ(t) Gt ∼ N(β − Ψ(t)) Stochastic averaging: At “time” Nt, X1 behaves as an M/M/1 process at equilibrium: +1 at rate 2μ(β − Ψ(t)) −1 at rate λ.

Time scale t → Nt

lim

N→+∞

X0(Nt) N

• = Ψ(t),

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

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. As t → ∞ then Ψ(t) ∼ β − e−2(β+t)/ρ. t → Nt “correct” time scale to describe decay.

Decay rate of the network

TN(δ) = inf{t ≥ 0 : XN

0(t) ≥ δN}

Theorem: lim

N→∞

TN(δ) N = − ρ 2 log(1 − δ) − δβ.

Contents

Introduction Model Time scale: t → t/N Time scale: t → t Time scale: t → Nt Case d ≥ 3

Case d ≥ 3

If ρ > dβ d time scales: t → Nkt, 0 ≤ k ≤ d − 1, 1 . . .

p − 1

p . . .

d − 1

d λN1{x1=x2=xp−1=0,xp−1>0} pμxp Time scale of decay: t → Nd−1t

At time scale t → Nd−k+1.

1 . . .

k − 1

k . . .

d − 1

d Empty Independent finite r.v. A cascade of time scales.

Conclusion

• A very simple but rich model.
• A rule of the thumb for dimensioning:

Trade-off between * the capacity dβ < ρ * the decay rate of order Nd−1. Future research:

• Distributed back-up mechanism
• Geometrical considerations

Thank you!

Annexes

T echnical corner: Skorokhod

Main difficulty: discontinuity of λN✶x1 > 0.

T echnical corner: Skorokhod

Main difficulty: discontinuity of λN✶x1 > 0. Usual solution: Skorokhod problem. X(t) = + R(t)

Constrained process Pushing process

Z(t)

Free process

+ ⇓

• Conv. of (Z(t)

Skorokhod Convergence of (X(t))

T echnical corner: Skorokhod

Main difficulty: discontinuity of λN✶x1 > 0. Usual solution: Skorokhod problem. X(t) = + R(t)

Constrained process Pushing process

Z(t)

Free process

Does not apply here!

T echnical corner: Skorokhod

Main difficulty: discontinuity of λN✶x1 > 0. Our approach: Generalized Skorokhod problem. X(t) = + R(t)

Constrained process Pushing process

G(X)(t)

Functional

+ ⇓

• Conv. of free equation

Skorokhod Convergence of (X(t))