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 F N 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 : X i ( t ) : number of files with i copies. Back-up policy: Recovery capacity λN allocated to the files with the minimum number of copies. ( x i , x i + 1 ) → ( x i − 1 , x i + 1 + 1 ) at rate λN if x 1 = x 2 = · · · = x i − 1 = 0 and x 1 > 0.

Model ( X i ( t ) , 0 ≤ i ≤ d ) is a transient Markov Process. X 0 ( t ) + X 1 ( t ) + · · · + X d ( t ) = F N . Unique absorbing state ( F N , 0 . . . , 0 ) . Assume F N lim = β > 0 . N N → ∞ General problem: Estimate the decay rate of the network.

Case d = 2 2 μ x 2 ( X 0 ( t )) ( X 1 ( t )) ( X 2 ( t )) μ x 1 λ N 1 { x 1 > 0}

Case d = 2 State ( x 0 , x 1 , F N − x 0 − x 1 ) . 2 μ ( F N − x 0 − x 1 ) ( X 0 ( t )) ( X 1 ( t )) ( X 2 ( t )) μ x 1 λ N 1 { x 1 > 0}

Different behaviors Three time scales:  t → t / N  t → t  t → Nt Three regimes: Overloaded network: 2 β > ρ = λ/μ . Critical case: 2 β = ρ . Underloaded case: 2 β < 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 2 μ ( F N − x 1 − x 0 ) N ( X 0 ( t / N )) ( X 1 ( t / N )) ( X 2 ( t / N )) x 1 λ 1 { x 1 > 0} μ N

Time scale: t → t / N  ergodic if 2 β < ρ,  ( L 1 ( t )) : an M / M / 1 queue null recurrent if 2 β = ρ,  transient if 2 β > ρ. 2 μβ ( L 1 ( t )) ∼ N β 0 λ 1 { x 1 > 0}

Time scale: t → t / N  ergodic if 2 β < ρ,  ( L 1 ( t )) : an M / M / 1 queue null recurrent if 2 β = ρ,  transient if 2 β > ρ. 2 μβ ( L 1 ( t )) ∼ N β 0 λ 1 { x 1 > 0} No loss!

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

Overloaded network If 2 β > ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  = ( β − ρ/ 2 )( 1 + e − 2 μt − 2 e − μt ) , x 0 ( t )  = ( 2 β − ρ )( e − μt − e − 2 μt ) , x 1 ( t ) = ( β − ρ/ 2 ) e − 2 μt + ρ/ 2 .  x 2 ( t ) 2 copies 1 copy β 0 copies λ 2 μ t T echnical point: Generalized Skorokhod problem.

Overloaded network If 2 β > ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  = ( β − ρ/ 2 )( 1 + e − 2 μt − 2 e − μt ) , x 0 ( t )  = ( 2 β − ρ )( e − μt − e − 2 μt ) , x 1 ( t ) = ( β − ρ/ 2 ) e − 2 μt + ρ/ 2 .  x 2 ( t ) 2 copies 1 copy β 0 copies λ 2 μ t Some losses!

Underloaded network If 2 β ≤ ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  x 0 ( t ) = 0 ,  x 1 ( t ) = 0 ,  x 2 ( t ) = β. λ 2 μ 2 copies β t T echnical point: Generalized Skorokhod problem.

Underloaded network If 2 β ≤ ρ , ( X 0 ( t ) /N, X 1 ( t ) /N, X 2 ( t ) /N ) converges to  x 0 ( t ) = 0 ,  x 1 ( t ) = 0 ,  x 2 ( t ) = β. λ 2 μ 2 copies β t No loss!

The critical case If 2 β = ρ , �� t � X N X N � � 0 ( t ) 1 ( t ) , ⇒ Y ( u ) d u, Y ( t ) � � N N 0 where � t � � � d Y ( t ) = 2 λ d B ( t )+ μ 2 γ − 3 Y ( t ) − 2 μ Y ( u ) d u d t 0 with the constraint Y ( t ) ≥ 0. 1 copy 0 copies t

Underloaded network If 2 β < ρ, X 2 ( t ) /N ⇒ β X 1 ( t ) ⇒ G Geometric r.v. w. param. 2 β/ρ, ( X 0 ( t )) ⇒ ( N α ( t )) with α = 2 μβ ( ρ − 2 β ) . 2 μNβ N α ( t ) ∼ βN G μx 1 λN 1 { x 1 > 0}

Underloaded network If 2 β < ρ, X 2 ( t ) /N ⇒ β X 1 ( t ) ⇒ G Geometric r.v. w. param. 2 β/ρ, ( X 0 ( t )) ⇒ ( N α ( t )) with α = 2 μβ ( ρ − 2 β ) . 2 μNβ N α ( t ) ∼ βN G μx 1 λN 1 { x 1 > 0} 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 � X 0 ( Nt ) � lim = Ψ( t ) , N N → + ∞ G t Geometric r.v. with par. 2 ( β − Ψ( t )) /ρ ∼ N Ψ( t ) G t ∼ N ( β − Ψ( t )) Stochastic averaging: At “time” Nt, X 1 behaves as an M/M/ 1 process at equilibrium: + 1 at rate 2 μ ( β − Ψ( t )) − 1 at rate λ.

Time scale t → Nt � X 0 ( Nt ) � lim = Ψ( t ) , N N → + ∞ where Ψ( t ) unique solution of � t 2 μ ( β − Ψ( s ) Ψ( t ) = μ d s. λ − 2 μ ( β − Ψ( s )) 0

Time scale t → Nt � X 0 ( Nt ) � lim = Ψ( t ) , N N → + ∞ where Ψ( t ) unique solution in ( 0 , β ) of ( 1 − Ψ( t ) /β ) ρ/ 2 e Ψ( 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 T N ( δ ) = inf{ t ≥ 0 : X N 0 ( t ) ≥ δN } Theorem: T N ( δ ) ρ lim = − log ( 1 − δ ) − δβ. N 2 N → ∞

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 → N k t, 0 ≤ k ≤ d − 1, pμx p . . . . . . p 0 1 d p − 1 d − 1 λN 1 { x 1 = x 2 = x p − 1 = 0 ,x p − 1 > 0} Time scale of decay: t → N d − 1 t

At time scale t → N d − k + 1 . Empty . . . . . . 0 1 d k k − 1 d − 1 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 N d − 1 . Future research: - Distributed back-up mechanism - Geometrical considerations

Thank you!

Annexes

T echnical corner: Skorokhod Main difficulty: discontinuity of λN ✶ x 1 > 0.

T echnical corner: Skorokhod Main difficulty: discontinuity of λN ✶ x 1 > 0. Usual solution: Skorokhod problem. X ( t ) = Z ( t ) + R ( t ) Constrained process Pushing process Free process Conv. of ( Z ( t ) + Skorokhod ⇓ Convergence of ( X ( t ))

T echnical corner: Skorokhod Main difficulty: discontinuity of λN ✶ x 1 > 0. Usual solution: Skorokhod problem. X ( t ) = Z ( t ) + R ( t ) Constrained process Pushing process Free process Does not apply here!

T echnical corner: Skorokhod Main difficulty: discontinuity of λN ✶ x 1 > 0. Our approach: Generalized Skorokhod problem. X ( t ) = G ( X )( t ) + R ( t ) Constrained process Pushing process Functional Conv. of free equation + Skorokhod ⇓ Convergence of ( X ( t ))